Doctorat De L'universite

D E Lyon

Devant Le

M Laquerriere

Rapporteur Mme Vincent

M Piere-Simon

Jouk

Examinateur M Zhu

Ecole Sigle

Doctorale

Renée El

Melhem Bât

Pascal Blaise

R Gourdon

M Stéphane

ÉLECTROTECHNIQUE E E A Électronique

M Gérard Scorletti
email: gerard.scorletti@ec-lyon.fr

M Philippe
email: philippe.normand@univ-lyon1.fr

Mme Sylvie

Jean-Yves Buffière
email: jean-yves.buffiere@insa-lyon.fr

Stéphanie Cauvin

M Jocelyn Bonjour

Véronique Guichard

J Y Toussaint

M Christian Montes

N°d'ordre NNT : 2020LYSEI021

Keywords: Computed tomography, X-ray, Spectral CT, Photon-counting detector, Material decomposition, Machine learning, Sparse representation Tomographie, Rayons X, CT spectrale, Détecteur à comptage des photons, Décomposition de matériaux, Apprentissage machine, Représentation éparse domain material decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Unreliability in image-domain material decomposition . . . . . . 2.2.2 Optimization method of material decomposition . . . . . . .

Material decomposition is a fundamental and primordial problem in spectral photoncounting X-ray CT (sCT). The present thesis focuses on the development of material decomposition methods using spectral and morphological information embedded in multienergy sCT images. In this framework, three methods were developed. For the first method, by using bounded mass density, local joint sparsity and structural low-rank (DSR) in image domain, we achieve highly accurate decomposition of materials such as gadolinium, iodine and iron. The results on both numerical phantom and physical data demonstrated that the proposed DSR method leads to more accurate decomposition than usual pseudo-inverse method with singular value decomposition (SVD) and current popular sparse regularization method with L1-norm constraint. The second method works in a region-wise manner. It consists in optimizing basis materials based on spatio-energy segmentation of regions-of-interests (ROIs) in sCT images, reducing noise by averaging multi-energy spatial images, and performing a fine material decomposition involving optimized decomposition matrix, denoising regularization and sparsity regularization. The results on both digital and physical data showed that the proposed ROI-wise material decomposition method presents clearly higher reliability and accuracy compared to common decomposition methods based on total variation (TV) or L1-norm (lasso) regularization. In the third method, we propose the notion of super-energy-resolution (SER) sCT imaging, which is realized through establishing the relationship between simulation and physical phantoms by means of coupled dictionary learning in a pixel-wise way. The effectiveness of the proposed methods was validated on digital phantom, physical phantoms and in vivo data. The results showed that for the same decomposition method using lasso regularization, the proposed super-energy-resolution imaging presents much higher decomposition accuracy and detection ability compared to what can be provided by current sCT machine.

Résumé

La décomposition de matériaux est un problème fondamental et primordial dans la tomographie spectrale (sCT-spectral computed tomography) par rayons X basée sur des détecteurs à comptage de photons (PCD-photon-counting detector). La présente thèse porte sur le développement de méthodes de décomposition de matériaux en utilisant des informations spectrale et morphologique encodées dans des images sCT multi-énergie. Dans ce cadre, trois méthodes ont été développées. Pour la première méthode, en utilisant la densité de masse limitée, la parcimonie conjointe locale, et le faible rang structurel (DSR) dans le domaine de l'image, nous obtenons une décomposition très précise de matériaux tels que le gadolinium, l'iode et le fer. Les résultats sur les données numériques et physiques du fantôme ont démontré que la méthode DSR proposée conduit à une décomposition plus précise que la méthode pseudo-inverse habituelle avec décomposition en valeur singulière (SVD-singular value decomposition) et la méthode de régularisation parcimonieuse courante avec contrainte de norme L1 (lasso). La deuxième méthode opère par région. Elle consiste à optimiser les matériaux de base en se basant sur la segmentation spatio-énergétique des régions d'intérêt (ROI-regions-of-interests) dans les images sCT, à réduire le bruit en faisant le moyennage des images spatiales multi-énergie, et à effectuer une décomposition fine des matériaux impliquant une matrice de décomposition optimisée, une régularisation du débruitage et une régularisation parcimonieuse. Les résultats sur des données numériques et physiques ont montré que la méthode proposée de décomposition des matériaux ROI par ROI (ROI-wise-region-of-interests-wise) présente une fiabilité et une précision nettement supérieures à celles des méthodes de décomposition courantes fondées sur la régularisation de la variation totale (TV) ou de la norme L1. Dans la troisième méthode, nous proposons la notion d'imagerie sCT à super-résolution énergétique (SERsuper-energy-resolution), qui est réalisée en établissant la relation entre la simulation et les fantômes physiques au moyen d'un apprentissage par dictionnaire couplé, de manière pixel par pixel. L'efficacité de ces méthodes proposées a été validée sur des données de fantômes numériques, de fantômes physiques et in vivo. Les résultats montrent que, pour la même méthode de décomposition de matériaux utilisant la régularisation par lasso, l'imagerie à super-résolution énergétique proposée présente une précision de décomposition et un pouvoir de détection beaucoup plus élevé que ce que peut fournir la machine sCT actuelle. straints in Image Domain [Si-Mohamed et al., 2017a]) . . . . . . 1.3 X-ray tube spectrum for the tube acceleration voltage U =120 kV. The spectrum is mainly composed of a broad quantum distribution caused by Bremsstrahlung in the x-ray tube anode material. The characteristic lines are due to inner electron shell transitions during a photoelectric effect event in the anode material atoms. Measured on Philips spectral CT prototype [START_REF] Cormode | Multicolor spectral photon-counting computed tomography: in vivo dual contrast imaging with a high count rate scanner[END_REF], Si-Mohamed et al., 2018] Le scanner CT à bi-énergie (DECT-dual-energy CT) a ensuite été étudié par Alvarez et Macovski en 1976[START_REF] Alvarez | Energy-selective reconstructions in X-ray computerised tomography[END_REF], Macovski et al., 1976]. Les scanners DECT ont permis d'obtenir des informations sur les propriétés d'atténuation dépendantes de l'énergie (c'est-à-dire spectrales mais seulement deux) de l'objet. Différentes réalisations techniques ont été utilisées. La solution la plus simple est la technique dite de double kVp. Deux scans CT avec différentes tensions d'accélération du tube par rayons X sont effectués. Les deux ensembles de données qui en résultent contiennent des informations sur les caractéristiques spectrales d'atténuation des rayons X de l'objet. Ces données spectrales peuvent être utilisées pour obtenir des informations supplémentaires sur l'objet. Néanmoins, les scanners DECT n'effectuent qu'une analyse en deux points de l'atténuation des rayons X, ce qui améliore la caractérisation des tissus et permet une quantification assez précise de l'iode, mais est insuffisant pour discriminer si- Figure 1: Mesure de l'atténuation des rayons X en CT. Les photons des rayons X émis par la source passent à travers l'objet qui est généralement un patient pour des applications cliniques. Le détecteur mesure une intensité des rayons X quantiques, résolue spatialement sur le faisceau en éventail des rayons X. Il est à noter qu'un filtre en forme de noeud papillon placé devant le tube à rayons X avant d'irradier un objet quelconque est habituellement appliqué pour moduler le flux des rayons X.

et le mouvement peut créer des problèmes pour aligner les deux ensembles de données.

La tomographie spectrale (sCT-spectral CT) ou multi-énergie a récemment fait l'objet d'une attention de plus en plus intense, en raison des nouvelles avancées de la technologie des détecteurs à comptage de photons (PCD-photon-counting detector). Par rapport à l'énergie conventionnelle intégrant le CT ou DECT, le sCT peut compter le nombre de photons dans des bandes d'énergie séparées avec une seule exposition [Taguchi andIwanczyk, 2013, Faby et al., 2015]. De cette façon, plusieurs matériaux peuvent être identifiés simultanément. Un certain nombre de méthodes de décomposition de matériaux ont été reportées et différents systèmes expérimentaux sont en cours de développement.

Cependant, la plupart des recherches se concentrent sur l'imagerie des agents de contraste à numéro atomique élevé (numéro Z), tels que l'iode, le gadolinium et l'or, qui ont des bords K (i.e. k-edge) uniques dans la gamme d'énergie de détection. Il est beaucoup plus difficile de distinguer ceux qui ont un faible numéro atomique, surtout lorsque leur numéro Z est proche.

La figure 1 [START_REF] Kalender | An algorithm for noise suppression in dual energy CT material density images[END_REF], Wu et al., 2009, Matsumoto et al., 2011, Yu et al., 2012, Mendonca et al., 2014, McCollough et al., 2015, Zhao et al., 2016, Iyama et al., 2018]. Bien que le sCT ne soit pas encore utilisé en pratique clinique, elle offre des perspectives prometteuses pour de nombreuses Selon le type et le nombre de matériaux de base, la décomposition de matériaux dans le sCT peut être un problème inverse mal posé, voire mal conditionné. Un tel problème peut survenir avant [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography[END_REF], Zimmerman and Schmidt, 2015, Ducros et al., 2017], pendant [Foygel Barber et al., 2016, Mechlem et al., 2018, Hou et al., 2018], ou après [START_REF] Taguchi | Image-domain material decomposition using photon-counting CT[END_REF], Le Huy and Molloi, 2011, Faby et al., 2014, Li et al., 2015, Li et al., 2017] le processus de reconstruction de l'image. Un certain nombre de méthodes de décomposition de matériaux ont été rapportées dans la littérature. Elles peuvent être classées en deux types principaux: l'optimisation sans et avec régularisation.

La méthode pseudo-inverse avec SVD est une méthode d'optimisation habituelle sans régularisation [START_REF] Faby | CT calibration and dose minimization in image-based material decomposition with energy-selective detectors[END_REF]. Dans la méthode d'optimisation avec régularisation, une représentation parcimonieuse a souvent été utilisée car les données multi-énergie ont souvent une corrélation élevée. La principale motivation de la représentation parcimonieuse est d'utiliser la nature parcimonieuse des données multidimensionnelles. Une représentation parcimonieuse appropriée peut donner une décomposition précise des matériaux avec peu de bandes d'énergie. En général, la représentation parcimonieuse par minimisation de la norme L1 est considérée comme efficace dans de nombreux domaines [START_REF] Candes | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information[END_REF], Donoho, 2006] pour éviter le surlissage. D'autres méthodes basées sur la représentation parcimonieuse ont également été étudiées pour la décomposition de matériaux, y compris le gradient parcimonieuse via la variation totale bilatérale (TVB) [START_REF] Clark | Spectral diffusion: an algorithm for robust material decomposition of spectral CT data[END_REF] et la transformation parcimonieuse via le cadre serré [START_REF] Gao | Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM)[END_REF]. habituelle qui tend à donner une solution parcimonieuse pixel par pixel. En d'autres termes, la norme L1 limite le nombre de matériaux existant en un pixel, alors que la norme L2,1 impose aux pixels de partager les mêmes atomes de matériaux de base dans chaque petit patch local. Pour la présente application des images de sCT, la norme L2,1 nous permet de combiner les informations spatiales (lignes en X) et spectrales (colonnes en X).

En même temps, en plus des similitudes spectrales et spatiales, les images multi-énergie présentent également la cohérence structurelle, qui reflète le fait que les matériaux décomposés devraient partager les mêmes bords dans la même petite régions en raison de leur propriété locale dans l'image. De telles informations de bord dans une image de matériau sont également rares et peuvent être saisies par l'opérateur de gradient. Pour traiter les similitudes de bord, nous utilisons une représentation de bas rang sur les matrices de gradient de tous les matériaux décomposés. La contrainte de rang bas est réalisée via la norme nucléaire qui est l'enveloppe convexe de la matrice de rang [START_REF] Candès | Robust principal component analysis[END_REF].

La performance de la méthode proposée de décomposition du matériau DSR a été évaluée sur des données de fantômes numériques et physiques. Deux méthodes ont été comparées à la méthode de DSR proposée: (a) méthode pseudo-inverse avec SVD, (b) méthode de régularisation avec contrainte de la norme L1 (L1).

Les données de projection du sCT ont été simulées à l'aide du logiciel Virtual X-ray Imaging (VXI) de l'INSA [START_REF] Duvauchelle | Computer code to simulate X-ray imaging techniques[END_REF], puis reconstruites par la méthode pour résoudre le modèle proposé. Pour accélérer la convergence, l'initialisation des variables dans l'ADMM a été fixée comme les résultats de la pseudo-inverse avec la méthode DSR. Bien qu'il y ait quatre régularisations dans la fonction de coût, seuls le terme conjoint de sparsité et le terme structurel de rang inférieur ont les paramètres de régularisation λ 1 et λ 2 . Les résultats de la décomposition de matériaux sont clairement influencés par les paramètres de régularisation. Il s'agit en fait d'un problème persistant, puisque la résolution de la fonction de coût avec plus d'un paramètre de régularisation est encore un problème difficile [START_REF] Clark | Spectral diffusion: an algorithm for robust material decomposition of spectral CT data[END_REF],Gao et al., 2011,Zhang et al., 2016,Ding et al., 2018]. Les paramètres de régularisation scalaire λ 1 et λ 2 influencent respectivement leniveau de sparsité et de rang faible. Nous avons choisi les deux paramètres scalaires par validation croisée [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF]. Le paramètre Lagrangien augmenté τ qui est égal à la taille du pas de l'itération a aussi un impact évident sur la vitesse de convergence. En pratique, pour rendre les performances moins dépendantes du choix initial, nous avons mis à jour τ à chaque itération de la manière discutée dans [Boyd et al., 2010, WANG and[START_REF] Wang | [END_REF] (essayez de garder les normes primaires et doubles résiduelles dans l'ADMM à l'intérieur d'un facteur car elles convergent toutes les deux vers zéro). Enfin, en raison de l'utilisation de la propriété locale, le choix des paramètres de régularisation λ 1 , λ 2 et τ est également influencé par la taille du patch. En général, une taille de patch plus petite devrait avoir des paramètres de régularisation plus petits, sinon des erreurs évidentes se produiront près des bords des patchs dans les images de matériau décom- et al., 2011], dans lequel les images à différents bandes d'énergie sont mélangées mais le bruit est réduit. Une autre caractéristique du bruit dans le sCT est qu'il est dépendant du matériau, ce qui signifie que le bruit montre différents impacts sur les matériaux selon leurs catégories et leurs densités massiques. Par exemple, les matériaux semblables à l'eau (eau, poly(méthacrylate de méthyle) (PMMA) et tissu mou, etc.) avec des coefficients d'atténuation massique plus faibles et les matériaux avec des densités plus faibles sont beaucoup plus susceptibles à l'influence du bruit par rapport aux matériaux denses d'intérêt. Il a été montré dans [START_REF] Ducros | Regularization of nonlinear decomposition of spectral x-ray projection images[END_REF], Si-Mohamed et al., 2018] que les dilutions d'agents de contraste (p. ex., iode, gadolinium et or, etc.) à plus faible concentration ont un rapport contraste/bruit plus faible. L'image composite à bruit réduit qui en résulte est alors utilisée dans la décomposition fine du matériau. Le diagramme global de la décomposition de matériaux par ROIs est illustré à la figure 9.

(1) Basis material optimization: L'optimisation des matériaux de base, qui est au coeur de notre décomposition de matériaux ROI par ROI, comprend deux étapes: la segmentation spatio-énergétique et la sélection des matériaux de base. La première étape vise à séparer les images spatiales reconstruites en différents ROI. La deuxième étape consiste à supprimer les matériaux de base non pertinents en utilisant une décomposition grossière des matériaux et un seuillage de la population relative. (2) Noise-reduced composite image construction: Les bruits aux différentes bandes d'énergie sont indépendants lorsque l'effet de partage de charge du PCD peut être ignoré.

Ainsi, en faisant la moyenne des images spatiales multi-énergie dans la dimension de l'énergie, on obtient une image composite où le bruit est réduit et l'information est mélangée.

(3) Fine material decomposition: La décomposition fine des matériaux est la dernière étape de la méthode proposée de décomposition de matériaux par ROI, qui fait intervenir trois termes: le terme de fidélité des données basé sur la matrice de décomposition optimisée M, le terme de régularisation du débruitage impliquant l'image composite à bruit réduit Y c , et le terme de régularisation de la sparsité.

En particulier, le terme de régularisation du débruitage est consacré à l'exploitation Nous avons proposé une méthode de décomposition de matériaux ROI par ROI. La méthode proposée permet de décomposer les matériaux de manière plus fiable et plus précise. Ceci est principalement dû à l'introduction de la segmentation spatio-spectrale qui permet d'extraire les caractéristiques pertinentes codées dans les images sCT multiénergie pour l'optimisation des matériaux de base.

Les résultats montrent que la méthode de décomposition de matériaux par ROI favorise une plus grande fiabilité, alors que la sensibilité de la détection des matériaux est quelque peu sacrifiée en contrepartie. Cela peut expliquer en partie pourquoi seuls les matériaux ayant une concentration supérieure à une certaine limite ont pu être décomposés avec précision. C'est le cas de l'iode avec une concentration de 1 mg/cc par rapport à des concentrations plus élevées (Fig. 13).

En fait, le compromis entre la sensibilité et la fiabilité de la méthode proposée en termes de ROI est régularisé par le seuil T dans le seuillage de la population relative.

Un T trop petit ou trop grand a induit l'augmentation des erreurs de décomposition, car un T plus petit entraîne une fiabilité médiocre (mais une sensibilité élevée). En d'autres

DÉCOMPOSITION DE MATÉRIAUX DANS LE DOMAIN DE L'IMAGE DANS LA TOMOGRAPHIE SPECTRALE AUX RAYON X POUR DES APPLICATIONS MÉDICALES

termes, lorsque T est trop petit, le bruit et les erreurs de reconstruction auront un fort impact sur la précision de la décomposition. A l'inverse, un T excessif dégradera la capacité de décomposition pour les matériaux de faible concentration. Heureusement, les résultats montrent une fourchette relativement large pour le choix de T autour de la valeur optimale, ce qui implique que la méthode proposée est relativement peu sensible à T. (1) SER image synthesis based on coupled dictionary learning: Nous apprenons la correspondance entre les caractéristiques spectrales LER et SER en utilisant la méthode d'apprentissage par dictionnaire couplé (coupled dictionary learning-CDL) au niveau du pixel.

INTRODUCTION GÉNÉRALE

Il est inefficace d'apprendre directement la correspondance à partir de l'ensemble des images ou des patchs LER et SER en raison d'informations supplémentaires sur la morphologie et le spectre. En général, l'apprentissage des caractéristiques morphologiques de certains organes est un autre défi qui exige des images abondantes avec diverses informations spatiales. Nous le résolvons en apprenant la correspondance des paires de pixels LER et SER, plutôt que les images entières ou les patchs.

Avec cette stratégie, les échantillons d'entraînement ne contiennent que des informations spectrales. En conséquence, les fantômes physiques utilisés dans notre travail peuvent avoir des formes simples, mais des composants matériels abondants.

La CDL est généralement utilisée dans le domaine de la super résolution spatiale [START_REF] Yang | Image superresolution via sparse representation[END_REF],Wang et al., 2012,Hu et al., 2018]. Dans la théorie de l'apprentissage par dictionnaire, les caractéristiques d'un pixel y peuvent être représentées comme Currently, conventional CT is a widely used x-ray scanning technique in both clinical and industrial areas. It provides answers to urgent diagnostic tasks such as oncology tumor staging, acute stroke analysis, or radiation therapy planning and industrial missions such as quality analysis and luggage scanning.

Dual-energy methods for CT (DECT) were subsequently investigated by [START_REF] Alvarez | Energy-selective reconstructions in X-ray computerised tomography[END_REF][START_REF] Alvarez | Energy-selective reconstructions in X-ray computerised tomography[END_REF], Macovski et al., 1976]. 2013, Faby et al., 2015]]. In this way, multiple materials can be simultaneously identified.

A number of material decomposition methods have been reported and different experimental systems are under development for spectral CT. However, most researches focus on the imaging of contrast agents with high atomic numbers (Z number), such as iodine, gadolinium and gold, which have unique K-edges within detection energy range. It is much more difficult to distinguish those having low atomic numbers, especially when their Z numbers are close.

Targeting on medical application of spectral CT, our objective is to develop material decomposition methods to improve the performance of material decomposition, e.g. accuracy and reliability, etc.

Main contributions

The main contributions of this thesis include: the proposal of new material decomposition methods, the validation and assessment of the proposed methods through both simulated and physical sCT data. In the following, we detail the contributions in the three aspects:

• Material Decomposition in X-ray Spectral CT Using Multiple Constraints in Image Domain (Chapter 3).

X-ray spectral CT appears as a new promising imaging modality for the quantitative measurement of materials in an object, compared to conventional energyintegrating CT or dual energy CT. We consider material decomposition in spectral CT as an overcomplete ill-conditioned inverse problem. To solve the problem, we make full use of multi-dimensional nature and high correlation of multi-energy data and spatially neighboring pixels in spectral CT. Meanwhile, we also exploit the fact that material mass density has limited value. The material decomposition is then achieved by using bounded mass density, local joint sparsity and structural low-rank (DSR) in image domain. The results on numerical phantom demonstrate that the proposed DSR method leads to more accurate decomposition than usual pseudo-inverse method with singular value decomposition (SVD) and current popular sparse regularization method with L1-norm constraint.

• ROI-Wise Material Decomposition in Spectral Photon-counting CT (Chapter 4).

Spectral photon-counting X-ray CT (sCT) opens up new possibilities for the quantitative measurement of materials in an object, compared to conventional energyintegrating CT or dual energy CT. However, achieving reliable and accurate mate-

GENERAL INTRODUCTION

rial decomposition in sCT is extremely challenging, due to photon-counting detector limitations and interference between different basis materials. To cope with that, we propose a novel material decomposition method that works in a region-wise manner. The method consists in optimizing basis materials based on spatio-energy segmentation of regions-of-interests (ROIs) in sCT images, reducing noise by averaging multi-energy spatial images, and performing a fine material decomposition involving optimized decomposition matrix, denoising regularization and sparsity regularization. The effectiveness of the proposed method was validated on both digital and physical data. The results showed that the proposed ROI-wise material decomposition method presents clearly higher reliability and accuracy compared to common decomposition methods based on total variation (TV) or L1-norm (lasso) regularization.

• Super energy resolution based material decomposition in spectral CT (Chapter 5).

Spectral photon-counting X-ray CT (sCT) offers new possibilities for the quantitative decomposition of materials in an object, compared to conventional energyintegrating CT or dual-energy CT. However, achieving accurate material decomposition, especially for low-concentration materials, is still extremely challenging for current sCT, due to restricted energy resolution stemming from the trade-off between the number of energy bins and undesired factors such as random noise.

We propose to improve material decomposition by introducing the notion of superenergy-resolution in sCT. The super-energy-resolution material decomposition consists in learning the relationship between simulation and physical phantoms. To this end, a coupled dictionary learning method is utilized to learn such relationship in a pixel-wise way. The results on both physical phantoms and in vivo data showed that for the same decomposition method using lasso regularization, the proposed super-energy-resolution method achieves much higher decomposition accuracy and detection ability compared to physical sCT imaging.

Organization of thesis

The thesis manuscript is organized as follows:

In Chapter 1, entitled "Principles of Spectral CT", the principles of common X-ray CT are introduced, including X-ray source and interacting of X-ray with materials; the .1: X-ray attenuation measurement in CT. The x-ray photons emitted by the source pass through the object which is usually a patient for clinical applications. The detector measures an intensity of x-ray quantum, spatially resolved over the x-ray fanbeam. Note that a bowtie filter in front of the X-ray tube before irradiating any object is usually applied to modulate the x-ray flux.

General introduction to spectral x-ray CT

Figure 1.1 illustrates the setup of a x-ray attenuation measurement in general x-ray computed tomography (CT). The source emits a flux of x-ray quantum at an object. The object consists of x-ray-attenuating materials that are described by their linear attenuation coefficient µ( x, E) at position x. The detector registers the quantum passing through the object.

The main difference between spectral CT and conventional energy integrating CT or dual-energy CT is the detector. For conventional CT or dual-energy CT, detectors only count photons of different energy in one (Figure 1.2a) or two channels (Figure 1.2b). While for spectral CT, the new photon-counting detector (PCD) can register photons in multiple channels (Figure 1.2c). In the following, we briefly review the spectral characteristics of the source, attenuations in the object and detectors for different CT imaging.

Source

The x-ray sources are resulted from an electron beam striking a metal target. The electrons are accelerated in a vacuum tube with a high voltage U acc , and then bombard a metal target (usually tungsten, alloy of rhenium and tungsten, molybdenum or copper) with a high velocity. The deceleration of electrons will lead to the emission of Bremsstrahlung, where the lost kinetic energy is converted into photons with continuous spectrum. On the other hand, the incident electrons can also interact with the orbital electrons. When an orbital electron is knocked out of the inner shell of the target metal atom, the electron from higher energy levels will fill the inner shell vacancy, resulting in emission of x-ray.

Usually the transitions are from upper shells into K shell (called K lines), into L shell (called L lines). The wavelength of X-rays emitted through this process depends on the type of metal material and the energy gap between the respective electron orbital shells. Thus the X-rays have several specific discrete energies, called characteristic X-ray. Meanwhile, bowtie filters are widely applied in current x-ray CT to modulate the output of the radiation source. Filtering the x-ray beam can reduce the radiation dose to the patient and help to minimize image artifact. The filters reduce the range of x-ray energies that reach the patient by removing the low-energy x-rays. Low-energy (or 'soft') x-rays are readily absorbed by the patient, therefore they do not contribute to the CT image but do contribute to the radiation dose to the patient. In addition, creating a more uniform beam intensity improves the CT image by reducing artifacts that result from CHAPTER 1. PRINCIPLES OF SPECTRAL X-RAY CT beam hardening [Mail et al., 2008, Heismann, Björn J., Bernhard T. Schmidt andFlohr., 2012].

The physical unit of this spectrum is quanta per energy, i.e. the number of quantum per kiloclectron volts (keV). The maximum of the emitted quantum energy E is limited by the voltage used for the acceleration of the x-ray-generating electrons. Figure 1.3 shows a x-ray source spectrum (filtered by a bowtie filter) for the tube voltage 120 kV. The Bremsstrahlung emission and characteristic X-rays can be observed in this spectrum. Figure 1.3: X-ray tube spectrum for the tube acceleration voltage U =120 kV. The spectrum is mainly composed of a broad quantum distribution caused by Bremsstrahlung in the xray tube anode material. The characteristic lines are due to inner electron shell transitions during a photoelectric effect event in the anode material atoms. Measured on Philips spectral CT prototype [START_REF] Cormode | Multicolor spectral photon-counting computed tomography: in vivo dual contrast imaging with a high count rate scanner[END_REF], Si-Mohamed et al., 2018] The intensity of emitted X-rays are determined by photon flux and the energy of photons. The photon flux is defined as the number of photons per second per unit area and it is proportional to the tube current. In clinical applications, it is important to evaluate the dose of x-ray. Multiple metrics has been proposed for the quantitative measurement of xray dose, e.g. water equivalent diameter (D W ), CT dose index (CT DI vol ) and size-specific dose estimation SSDE.

Attenuation

X-ray attenuation is characterized by Lambert-Beer's Law and is primarily a function of beam energy and the density and composition of the material of the object being scanned.

GENERAL INTRODUCTION TO SPECTRAL X-RAY CT

For a monochromatic X-ray beam of energy E, Lambert-Beer's law gives the beam intensity value I after crossing a thickness ∆x of matter as:

I = I 0 e -L(u) µ( x)dx , (1.1)
where I 0 is the initial X-ray intensity, µ( x, E) the linear attenuation coefficient at position or pixel x for energy E, in units of cm -1 , and L(u) the u-th ray. In the case of polychromatic x-ray beam, equation (1.1) becomes:

I = E max E min I 0 (E)e -L(u) µ( x,E)dx dE, (1.2)
where [E min , E max ] is the energy range of the x-ray spectrum.

The linear attenuation coefficient µ of all materials depends on the photon energy of the beam and the atomic number of elements in the material. Since it is the mass of the material itself that is providing the attenuation, linear attenuation coefficients are often characterized by µ/ρ, the mass attenuation coefficient µ m in units of cm 2 /g, as:

µ m (E) = µ/ρ, (1.3)
where ρ is the mass density.

Physical mechanisms of x-ray attenuation

The mass attenuation coefficient µ m (E) is a characteristic function for all chemical elements indexed by the atomic number Z in the periodic table of elements. In the clinical range below 200 keV, µ m is determined by three mechanisms: coherent (Rayleigh) scatter, photoelectric absorption, and Compton scatter. Figure 1.4 shows the relative strengths of these three interactions (and their total) over the diagnostic range of energies for water and lead.

Coherent or Rayleigh scattering is the apparent deflection of x-ray beams caused by atoms being excited by the incident radiation and then re-emitting waves at the same wavelengths. This phenomenon is useful in x-ray diffraction studies, where the x-ray energies are of the order of a few kiloelectron volts and thus the wavelengths arc the same order of magnitude as atomic dimensions. It is relatively unimportant in the energies used in diagnostic radiology, as is seen by the plots of µ r /ρ in Figure 1.4.

Photoelectric effect is the absorption of a photon by a tightly bound electron, as seen by the plots of µ p /ρ in Figure 1.4. The electron is ejected from the atom with enough kinetic energy, then a vacancy left behind in the shell. Often an electron in a outer shell falls into the inner shell to fill the vacancy, releasing energy in the form of a photon. This is accompanied by the emission of characteristic x-ray photons called fluorescence radiation. Lower-energy excitation is absorbed in the M and L shells, while higher-energy radiation is absorbed in the inner K shell. For X-ray energies, electrons in the K shell are predominantly involved. Especially, the attenuation coefficient undergoes a sharp increase in the energy region corresponding to the K shell. This is known as the K absorption edge (i.e. K-edge). This absorption becomes particularly important with the high-atomic-number (high-Z) materials because their binding energies are closer to x-ray energy range. For the lower-atomic-number elements, such as are found in water and organic material, this

GENERAL INTRODUCTION TO SPECTRAL X-RAY CT

K edge occurs below the diagnostic energy spectrum being used. However, for high-Z materials, such as lead shown in Figure 1.4b, this K edge occurs within the spectrum of interest.

Compton effect consists of a collision between an x-ray photon and either a free or a loosely bound electron in an outer shell. It occurs when the incident photon has energy that is much more than the binding energy. As seen in Figure 1.4a, Compton scattering (µ c /ρ) is the largest component for water in most of the energy range. In clinical applications, Compton scattering is the most significant and most troublesome source of tissue attenuation.

Higher atomic-number elements will experience primarily photoelectric absorption, while those of lower atomic numbers will be dominated by Compton scattering. Since each interaction is independent, the total attenuation coefficient is the sum of that due to photoelectric, Rayleigh, and Compton coefficients, as is seen by the plots of µ T /ρ in Figure 1.4.

Mass attenuation coefficients of mixtures

For mixtures, the mass attenuation coefficient is given by:

µ m = Σ i w i µ i ρ i , (1.4)
where ρ is the density of the mixture, µ i the linear attenuation coefficient of element i, ρ i the density of element i, and w i the fraction by weight of the element in the material. Precise measurement data of the mass attenuation coefficient is available in numerous literature sources. The U.S. National Institute of Standards and Technology (NIST) hosts the constantly updated XCOM source at www.nist.gov [START_REF] Saloman | X-ray attenuation coefficients (total cross sections): Comparison of the experimental data base with the recommended values of Henke and the theoretical values of Scofield for energies between 0[END_REF]. Figure shows four categories of materials existed in human body: soft tissue, bone material, contrast agents and metallic implants, and associated materials.

Soft tissue has elemental compositions close to water, i.e., it mainly contains hydrogen and oxygen atoms. Soft tissue has typical densities between 0.9 and 1.1 g/cm 3 . Healthy and pathogenic tissue types usually differ by 10 to 20 mg/cm 3 in density. Note that µ m of water is very similar with soft tissue, which are usually treated in one category.

Bone material as the second group contains the bone mineral hydroxyapatite in varying concentrations. It generally has significantly higher densities than soft tissue, around 1.5 to 2 g/cm 3 . Both the elevated density and the calcium and phosphorus mass contributions lead to a comparably strong x-ray attenuation of bone material, in particular for low energies. CT imaging benefits greatly from the generally high contrast between bone material and soft tissue.

Contrast agents containing iodine, gadolinium etc, which has K-edge within the clinical x-ray tube spectrum. In conventional CT, these agents are administered to the patient to increase contrast in soft tissue exams or CT angiograms. Statistically, contrast agents are used in more than 50% of CT exams. In most cases, iodine (I) based contrast agents are used. Gadolinium (Gd) based substances, which are the standard contrast agent in MRI, are rarely administered in conventional CT exams. The injection of the contrast agent into a blood vessel generates a so-called contrast bolus that travels through the CHAPTER 1. PRINCIPLES OF SPECTRAL X-RAY CT blood vessel system. The contrast between blood and surrounding tissue in simultaneously acquired CT images is strongly increased. CT angiography, the representation of the human blood vessel system, is performed in this fashion. Unlike conventional CT, sCT uses energy-resolving detectors that can simultaneously sample the energy spectrum at multiple regions. This allows a higher spectral resolution, enabling the identification of material-specific spectral characteristics, such as the K-edge signature of the contrast agent. This is particularly interesting as the K-edge energies of contrast agents that contain heavy elements such as gadolinium, ytterbium, bismuth or gold nanoparticles (AuNPs) are within the clinical x-ray tube spectrum [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography[END_REF], Pan et al., 2010, Schirra et al., 2014, Pan et al., 2014[START_REF]Evaluation of spectral photon counting computed tomography K-edge imaging for determination of gold nanoparticle biodistribution in vivo[END_REF], Si-Mohamed et al., 2018].

Metallic implants such as iron (Fe), titanium (Ti) hip joint replacements or gold (Au) tooth fillings might be present in some scanning regions and patients. They might be a source of strong streak-like artifacts in CT images since the localized strong decrease in projection signal emphasizes the limits of linear projection mathematics. Metallic implants are usually not part of diagnostic tasks and are neglected in the following discussion.

GENERAL INTRODUCTION TO SPECTRAL X-RAY CT

Detectors

Detectors of conventional CT and dual-energy CT

Conventional CT devices are equipped with energy integrating detectors (EIDs) in which the detected signal is proportional to the total energy deposited by all photons without specific information about an individual photon or its energy. EIDs use a two-step process to convert the incoming X-ray intensity into an electronic signal, as shown in Figure 1.6. First, the X-rays are converted into visible light in a scintillator layer. Below, the figure shows a photodiode array that converts the emitted light into an electric current that is digitized in dedicated application-specific integrated circuit (ASIC). The scintillator layer is made of a ceramic material that is mechanically structured into pixels. Septa of finite width separate the individual pixels in order to suppress optical cross-talk. The effective detector pixel sizes range from 0.5 to 0.625 mm at the isocenter for the commercial EIDs. In dual-energy CT, an additional attenuation measurement is obtained at a second energy. A number of technical approaches have been developed to realize dual-energy CT: e.g. two temporally sequential scans, rapid switching of x-ray tube voltage, dual x-ray sources and dual-layer detector, etc, as shown in Figure 1.7.

Sequential acquisition: The approach that requires the least hardware effort is the sequential acquisition of two datasets at different tube voltages. Sequential acquisition can be achieved either as two subsequent helical scans or as a sequence with subsequent rotations at alternating tube voltages and stepwise table feed. This approach may make sense in systems with broad detectors, but a disadvantage is the rather long delay between both acquisitions. The delay is too long to preclude artifacts from cardiac or respiratory motion or changes in contrast material opacification.

Rapid voltage switching (Figure 1. is rapid voltage switching. With this method, the tube voltage alternates between a high value and a low value, and transmission data are collected twice for every projection or, in practice, for immediately adjacent projections. However, the tube current could not be increased quickly enough for the low tube potential measurements to achieve comparable noise levels in both the low and high tube potential data sets. This difference in noise limited extension of the technique beyond bone densitometry applications. Still, the spectral resolution of rapid voltage-switching systems remains limited and usually requires additional dose.

Dual-source CT (Figure 1.7b): Dual-source CT is a CT system where two x-ray sources and two data acquisition systems are mounted on the same gantry, positioned orthogonally to one another. This setup requires nearly twofold investments in hardware but offers important advantages: the voltage, current, and filter can be chosen independently for both tubes to achieve an optimal spectral contrast with sufficient transmission and the least overlap. The data are acquired simultaneously by both orthogonal systems. However, because both tubes are simultaneously energized, scattered radiation whose original primary photon came from one tube may be detected by the detector for the other tube, and vice versa. This degrades spectral separation and requires the implementation of an appropriate scatter-correction algorithm.

Dual-layer detector (Figure 1.7c): Another approach for acquiring du-al-energy CT projection data uses a single high tube potential beam and layered or "sandwich" scintillation detectors, as shown in Figure 1.8. The low-energy data are collected from the front or innermost detector laye+r and the high-energy data are collected from the back or outermost detector layer. An advantage to this approach is that the low-and high-energy data sets are acquired simultaneously, and the data from the inner and outer detector layers are recorded at all times. whereas, with this setup, the scintillator materials determine 1.1. GENERAL INTRODUCTION TO SPECTRAL X-RAY CT the spectral resolution, and the sensitivity profiles of the available materials have a rather broad overlap. Therefore, the contrast of the spectral information is limited or requires a relatively high additional dose. Each photons can be detected individually and its energy can be measured.

Spectral CT imaging is realized by photon-counting detectors (PCD). This new generation of detectors measures and analyzes incident photons individually and produces a digital measure of photon energy through pulse height analysis. Each photon impacting the detector is counted and its energy determined. As shown in Figure 1.9, semiconductor materials such as cadmium telluride (CdTe) are able to convert X-rays directly into electric signal pulses. Each incoming X-ray quantum generates a number of electronhole pairs; then the electrons and the holes move to the opposite directions, the anode and the cathode respectively, under a strong electrical field inside the detector material; the moving charge carriers induce the signals on the electrodes. Highly integrated circuits transform these charge pulses into voltage pulses of a few nanoseconds duration that can be counted digitally.

The direct conversion of the photon signal into a digital measurement eliminates electronic noise, which offers the opportunity to acquire data at lower noise levels and in turn to use lower radiation doses. Meanwhile, instead of being less weighted, low-energy photons can be counted in order to bring valuable information, e.g. more contrast to the image. Furthermore, PCDs can eliminate the septa between adjacent detector pixels required by EIDs to avoid cross-talk and maintain spatial resolution. This feature leads to a PCD detector size of 0.25 mm at the isocenter (compared with 0.5-0.6 mm for EIDs) [START_REF] Zhou | Comparison of a Photon-Counting-Detector CT with an Energy-Integrating-Detector CT for Temporal Bone Imaging: A Cadaveric Study[END_REF].

However, PCDs are not flawless, the measurement of PCDs may deviate from the truth for several reasons [START_REF] Taguchi | Vision 20/20: Single photon counting x-ray detectors in medical imaging[END_REF]]:

• Pulse pileup. When several photon interactions occur at the same time, multiple pulses may be piled up and be counted as one event, resulting in wrong recorded energy and counts loss.

GENERAL INTRODUCTION TO SPECTRAL X-RAY CT

• Charge sharing. When the generated electron charge cloud reaches the anode near the pixel boundary, they maybe divided and counted by the neighboring pixels to a smaller energy level.

• K-escape X-rays. X-ray fluorescence or Auger electrons may be generated during the photoelectric effect. The fluorescent X-ray photons can either be absorbed by the PCD pixel with the primary interaction again, be detected by an adjacent pixel, or leave the PCD completely.

• Compton scattering. The photon energy is not fully transmitted to electrons, resulting in energy loss.

• Charge trapping. The electron or hole is trapped by a trapping center and is thermally re-emitted with a delay. This will result in lower-energy counting.

• Polarization and long-term reliability.

As a consequence, two adjacent energy bins are not sharply separated by the energy threshold at their common boundary. Their sensitivities are described by so-called detector response function d(E) and the effects contributing to the degradation of the energy resolution lead to an overlap of the detector response function. d(E) yields the relative signal amount generated by a specific x-ray quantum of energy E. For an ideal integrating detector, often d(E) = E is assumed. However, considering the aforementioned physical effects, d(E) has a more sophisticated structure. Figure 1.10 shows the experimented d(E) function, which measured by illuminating PCD under constant flux conditions with monochromatic photons of energy E. Considering the detector response function, a more general form of Lambert-Beer's law, equation (1.2), is given by:

Īi (u) = R I 0 (E)d(E)e -L(u) µ( x,E)dl dE, (1.5)
where Īi (u) is the mean measured signal (number of photons penetrating materials) recorded by a PCD for the u-th ray within the i-th energy bin, R the energy range of the i-th bin, and d(E) the detector response function. Meanwhile, sCT suffers from severe quantum noise problem because of the limited photon flux, as an immediate consequence of preventing the pileup effect. Specifically, the limited photons can lead to worse signal-to-noise ratio (SNR) in sCT images than in traditional CT images, considering image noise is inversely related to the square root of the total number of photons used for reconstruction [START_REF] Leng | Noise reduction in spectral CT: Reducing dose and breaking the trade-off between image noise and energy bin selection[END_REF]. Noises at different energy bins are independent when the charge sharing effect of PCD can be ignored. Due to the severe quantum noise, the measured signal is assumed to be corrupted by independent Poisson:

I i (u) = P (λ = Īi (u)), (1.6)
where P(λ) denotes the Poisson distribution of mean λ, and I i (u) the measured number of photons for the u-th ray in the i-th energy bin.

SPECTRAL CT IMAGE RECONSTRUCTION

Spectral CT image reconstruction

The aim of reconstruction is to obtain the linear attenuation coefficients at each energy bin:

µ( x, i) = argmin µ D(I i (u), µ( x, i)), (1.7)
where µ( x, i) denotes the reconstructed linear attenuation coefficients at the i-th energy bin, which also represents the reconstructed spatial image at the i-th energy bin, and D the discrepancy function.

For sCT, there are two strategies to reconstruct the spatial images: separately reconstruct image at each bin (separate reconstruction), or jointly reconstruct by exploiting the information of all bins (joint reconstruction).

Separate reconstruction

Separate reconstruction is to independently form images at each energy bin using common reconstruction methods. In traditional CT, image reconstruction is an important step to retrieve the cross sectional attenuation coefficients from the measured X-ray data, which is the transmitted beam intensity along a line between X-ray source and detector. Specifically, reconstruction aims to reconstruct the spatial distributed µ( x, E) from its line integrals L(u) µ( x, E)dx.

For monochromatic radiation of energy E = E 0 , the line integral given by the ratio of the two intensities I and I 0 as:

ln(

I 0 I ) = L(u) µ( x, E 0 )dx, (1.8)
which is equivalent to a Radon transform. However, x-ray spectrum in practice is usually polychromatic, which means the line integral is not directly measured and the resultant behavior is nonlinear. Generally, the reconstruction model of polychromatic radiation assumes that we can calculate the an effective value of µ( x, Ē), where Ē is the average energy on the range of the x-ray beam. As a consequence, the line integral is:

ln(

I 0 I ) = L(u)
µ( x, Ē)dx.

(1.9)

This assumption yields some artifacts in the CT reconstruction due to beam hardening. In this case, the x-ray photons passing through the object are considered to have an effective detected energy. When quantum pass through thick or high-atomic-number-object regions such as bone, the average energy of the detected quantum increases due to the lower energy photons are more likely to be absorbed. In other words, the beam spectrum is 'hardened.' As a result, the model separate reconstruction is formulated as:

µ( x, i) = argmin µ D(ln( I i (u) R n 0 (E)d i (E)dE ), µ( x, i)).
(1.10)

To solve the minimization problem of (1.10), we briefly introduce two common reconstruction methods in traditional CT: filtered back-projection (FBP) method and iterative reconstruction method.

CHAPTER 1. PRINCIPLES OF SPECTRAL X-RAY CT

Filtered back-projection reconstruction method

The filtered back-projection algorithm is one of the most commonly used algorithms for tomographic image reconstruction. As the name implies, there are three steps in the filtered back-projection algorithm: the transform part which transforms projection data into frequency domain with Fourier transform; the filtering part, which can be visualized as a simple weighting of each projection in the frequency domain; the back-projection part which consists of projecting back each view along a line corresponding to the direction in which the projection data were collected, as:

1 Calculate Fourier transform of line integral -ln( I 0 I ), get F(u, v);

2 Multiply F(u, v) with transfer function of ramp filter |ω|, get Q β ;

3 Back-project Q β , get image µ(x, y);

In details, we generalize µ(x, y) with f (x, y), the path that the X-ray crosses in the object can be given by:

x cos β + y sin β = t, (1.11) in which β and t are polar coordinates of the path that the X-ray beam crosses through the object. Given F(u, v) the Fourier transform of the projection data (line integral) with u and v representing two spatial frequencies, the object function f (x, y) may then be obtained fromF(u, v) by the inverse Fourier transform:

f (x, y) = +∞ -∞ +∞ -∞
F(u, v)e 2πi(ux+vy) du dv.

(1.12)

Knowing that the Fourier slice theorem relates the parallel projections to object data which are collected in the polar coordinates, we change in equation (1.12) from Cartesian coordinates (u, v) to polar coordinates (β, ω) as follows:

   u = ω cos β v = ω sin β dudv = ωdωdβ, (1.13) to get f (x, y) = 2π 0 +∞ 0 F(ω cos β, ω sin β)e 2πωi(
x cos θ+y sin θ) ω dω dβ.

(1.14)

We substitute the two-dimensional Fourier transform F(ω cos β, ω sin β) of object with the Fourier transform of the projection data at angle β, and the above equation can be expressed as:

f (x, y) = π 0 +∞ 0 F(ω cos β, ω sin β)we 2πi(x cos β+y sin β) dw dβ = π 0 +∞ -∞ F(ω cos β, ω sin β)|w|e 2πi(x cos β+y sin β) dw dβ = π 0 Q β (x cos β + y sin β) dβ,
(1.15)

SPECTRAL CT IMAGE RECONSTRUCTION

where

Q β (x cos β + y sin β) = Q β (t) = +∞ -∞
F(ω cos β, ω sin β)|w|e j2πt dw.

(1.16)

As we can see, this algorithm consists of three main steps. The first step is to do a Fourier transform to projection data (line integral). Then, to filter the projections by a kernel whose frequency domain response is |ω| and known as the ramp filter. At last, these filtered projections Q β are back-projected to recover the object function f (equation (1.15)).

In practical applications, the ramp filter significantly amplifies the statistical noise existing in the projection data. To minimize this problem, the ramp kernel is usually associated with an apodization window. The most common window functions used are: i) sinc (Shepp-Logan filter), ii) cosine, iii) Hamming and iv) Hanning window functions.

Iterative reconstruction method

Iterative reconstruction method is another category of image reconstruction method for X-ray CT that begins with an image assumption, and compares it to real time measured values while making constant adjustments until the two are in agreement, by the following different steps:

1 Set the reconstruction image an initial value f (0) ; 2 Calculate the forward projection of the f (k-1) (for the first iteration k = 1, the second iteration k = 2, and so on);

3 Compare the difference between real measured data and calculated projection by step 2 ; 4 Correct reconstruction image by back-projection and obtain f (k) ;

5 Increase k by 1 (k = k + 1) and repeat step 2 , 3 and 4 until the stopping criterion is satisfied ;

The initial image f (0) is arbitrary, but usually we make it 0, or an image reconstructed by another method such as FBP. The stopping criterion can either be the maximum number of iterations, or the small enough update of current image estimate , or the predefined quality criterion [START_REF] Beister | Iterative reconstruction methods in X-ray CT[END_REF]. A set of iterative reconstruction methods has been proposed, such as the algebraic reconstruction technique (ART) [START_REF] Gordon | Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography[END_REF], simultaneous iterative reconstruction technique(SIRT) [Gilbert, 1972], simultaneous algebraic reconstruction technique (SART) [START_REF] Andersen | Simultaneous Algebraic Reconstruction Technique (SART): A Superior Implementation of the Art Algorithm[END_REF], maximum likelihood expectation-maximization (ML-EM) algorithm [START_REF] Shepp | Maximum Likelihood Reconstruction for Emission Tomography[END_REF], and so on. Here we introduce in detail the most basic and popular ART method.

In ART reconstruction, the projection process of CT imaging system can be represented by:

b i = ∑ j a i,j f j , (1.17)
where b i is the integration data measured with the i th ray (total number of rays M =No.rotation angles × No.detectors); a i,j is the weighting factor that represents the contribution of the j th voxel to ray i; f j is the attenuation coefficient of voxel j in our case. Therefore, we have:

CHAPTER 1. PRINCIPLES OF SPECTRAL X-RAY CT A f = b, (1.18)
where A = a i,j is the system matrix, f = [ f 1 , f 2 ..., f N ] is the image vector where N is the total number of voxels in the reconstruction image, and b = [b 1 , b 2 ..., b M ] is the measure- ment vector. To solve f , the Kaczmarz method [KACZMARZ, 1993] is applied and the following iterative formula is obtained:

f (k) = f (k-1) + λ k-1 b i -f (k-1) a T i a i a T i a i , (1.19)
where k is the number of iterations and λ k-1 is the relaxation parameter, i = k mod M + 1, a i is a vector made of the i th row of A, .

Compared to analytical FBP method, iterative reconstruction has advantage to reduce noise and improve image quality by integrating various physical models and incorporating prior information. It outperforms FBP method especially with sparse and incomplete data. Specifically, additional regularization term can be processed using iterative reconstruction method, as

µ( x, i) = argmin µ D(ln( I i (u) R n 0 (E)d i (E)dE ), µ( x, i)) + R, (1.20)
where R denotes the regularization term. Typically, total variation (TV) methods [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] have demonstrated their power in CT reconstruction with only a few x-ray projections [START_REF] Laroque | Accurate image reconstruction from few-view and limited-angle data in diffraction tomography[END_REF], Song et al., 2007, Sidky and Pan, 2008, Chen et al., 2008] and low-dose CT [START_REF] Tian | Low-dose CT reconstruction via edge-preserving total variation regularization[END_REF]. In such approaches, the energy function of a TV form is minimized subject to a data fidelity condition posed by the x-ray projections. Since this energy term corresponds to image gradient, minimizing it will effectively remove those high spatial gradient parts such as noise and streaking artifacts in the reconstructed CT images. However, the most serious disadvantage of iterative reconstruction is its huge computational costs, since each iteration involves forward projections.

Joint reconstruction

The main objective of joint reconstruction is to mitigate artifacts by exploiting the information of multi bins. Specifically, the combination of multi-bin information can be realized during or after reconstruction (post-process). For methods during reconstruction, additional prior was exploited, such as sparsity and/or low-rank ( [START_REF] Gao | Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM)[END_REF]), full-spectrum image reconstructed using conventional filtered back-projection [START_REF] Yu | Spectral prior image constrained compressed sensing (spectral PICCS) for photon-counting computed tomography[END_REF], Liu et al., 2016], or features learned by tensor-based dictionary learning method [START_REF] Zhang | Tensor-Based Dictionary Learning for Spectral CT Reconstruction[END_REF].

For post-process, redundant information embedded in reconstructed sCT images was utilized. For example, S. Leng proposed a method to denoise at each single bin with the help of a low-noise composite image which is produced by averaging images of all energy bins [START_REF] Leng | Noise reduction in spectral CT: Reducing dose and breaking the trade-off between image noise and energy bin selection[END_REF]. Z. Li proposed a multi-energy non-local means method to exploit spatio-spectral features in order to achieve noise reduction [START_REF] Li | An effective noise reduction method for multi-energy CT images that exploit spatio-spectral features[END_REF] 1.3. APPLICATIONS OF SPECTRAL CT IMAGING

Applications of spectral CT imaging

SCT can count the number of photons in multi separated energy bins with one single exposure, which is conceptually advantageous when compared to dual-energy CT. Dualenergy CT, at present, has a greater range of clinical applications, e.g. virtual monoenergetic imaging, automated bone/plaque removal in CT angiography, virtual noncontrast-enhanced/non-calcium imaging, kidney stone characterization, and discrimination of gout from pseudo gout [START_REF] Kalender | An algorithm for noise suppression in dual energy CT material density images[END_REF], Wu et al., 2009, Matsumoto et al., 2011,Yu et al., 2012,Mendonca et al., 2014,McCollough et al., 2015,Zhao et al., 2016,Iyama et al., 2018]. Although sCT is not currently used in clinical practice, it provides promising perspectives for many vitally important applications and opens up many new application possibilities. The main advantage of sCT technology is better sampling of the spectral information from the transmitted spectrum. Among the applications of using spectral information, quantitative material decomposition is the hottest topic, including K-edge imaging of contrast materials and selective imaging of other previously indistinguishable materials. In addition, virtual monochromatic imaging is another method of using spectral information to improve image quality. Other improvements of sCT compared to conventional CT, such as higher spatial resolution, lower radiation exposure and lower noise are also expected to provide benefits for diagnostic imaging.

Quantitative material decomposition

The attenuation of x-rays by matter is dependent on the energy of the photons. SCT can count the number of photons in different energies (i.e. separated energy bins) with one single exposure. Thanks to this advantage, it becomes possible to efficiently decompose different effects or materials (so-called basis materials) present in a pixel of spectral CT images.

Material decomposition can be realized by the means of separating different effects. For DECT, Alvarez and Macovski proposed to extract energy-dependent information by represent the total attenuation with functions correspond to two effects, photoelectric and Compton effect, and the energy-independent parts of the two effects were then reconstructed [START_REF] Alvarez | Energy-selective reconstructions in X-ray computerised tomography[END_REF]. The reconstructed photoelectric image provides composition information of object while the reconstructed Compton image provides mass density information of object. Although the Alvarez-Macovski dual-energy approach successfully offers a quantitative method for the characterization of tissue composition and eliminates beam-hardening, the benefits from the additional spectral information are limited. With sCT, whereas, more than two effects can be separated. Roessl and Proksa developed this idea by adding the K-edge terms on the photoelectric and Compton effect decomposition formula and gadolinium was discriminated from normal tissues in the thorax phantom in their simulation study [START_REF] Roessl | K-edge imaging in x-ray computed tomography using multi-bin photon counting detectors[END_REF]. Their results were further validated by CT system experiments where the contrast agents (gadolinium and iodine) were well quantified [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography[END_REF]. This technique was named K-edge imaging and it enables multi-agent imaging [START_REF] Cormode | Atherosclerotic Plaque Composition: Analysis with Multicolor CT and Targeted Gold Nanoparticles[END_REF],Pan and Yang, 2010, Panta et al., 2015, Si-Mohamed et al., 2017a, Lee et al., 2016a, Lee et al., 2016b].

Meanwhile, material decomposition can also be realized by directly recognizing different materials. A pixel can contain several materials. To decompose these materials, the CHAPTER 1. PRINCIPLES OF SPECTRAL X-RAY CT materials in the pixel can be considered as a mixture and the latter can be decomposed into different basis materials. Basis materials here and 'materials' in general are not the same notion. For example, water and PMMA (water-like materials) can be considered a whole and is identified as the basis material water; calcium (Ca) in the spine can be identified as the basis material iodine under certain energy-bin setting, e.g. 30-50 kev, 51-61 kev, 62-71 kev, 72-82 kev and 83-130 kev. e et al. proposed a calibrated least squares fitting technique to decompose hydroxyapatite and iodine by a simulation study [START_REF] Le | Least squares parameter estimation methods for material decomposition with energy discriminating detectors[END_REF]. Based on this work, Alessio et al. made further improvement and their decomposition results with carotid endarterectomy specimens suggested the presence of water, lipid, and calcium deposits in the plaque walls [START_REF] Alessio | Quantitative material characterization from multi-energy photon counting CT[END_REF]. Zainon et al. used small animal spectral micro-CT with Medipix3 detector (thresholds: 10, 16, 22 and 28keV) to scan the carotid atherosclerotic plaque; water-like, calcium-like and lipid-like tissues were well separated according to histology comparison, but iron was not distinguished from calcium [Zainon, 2012].

Virtual monochromatic imaging

Virtual monochromatic spectral images (VMS) or virtual mono-energetic images (Mo-noE) are reconstructed from a accurately decomposed material images and mass attenuation coefficients [START_REF] Matsumoto | Virtual Monochromatic Spectral Imaging with Fast Kilovoltage Switching: Improved Image Quality as Compared with That Obtained with Conventional 120-kVp CT[END_REF], Wu et al., 2009], which can be calculated in the projection domain or in the image domain. For a given kilo-electron voltage, the object is depicted as if it is imaged with a monochromatic beam at the same kilo-electron voltage. VMS imaging has the potential to reduce beam-hardening artifacts and improve contrast to noise ratio (CNR). Hard X-rays attenuate less than soft X-rays, thus VMS images of higher energy have the potential to reduce beam-hardening [START_REF] Neuhaus | Improvement of Image Quality in Unenhanced Dual-Layer CT of the Head Using Virtual Monoenergetic Images Compared With Polyenergetic Single-Energy CT[END_REF], Yu et al., 2012] and metal artifacts [START_REF] Wang | Metal artifacts reduction using monochromatic images from spectral CT: Evaluation of pedicle screws in patients with scoliosis[END_REF]. Such VMS images also have the advantage to improve CNR, since the low-energy attenuation carries more contrast [START_REF] Zhao | Improving image quality in portal venography with spectral CT imaging[END_REF] and contrast agents at energy around absorption edge (i.e. k-edge in clinical applications) change greatly [START_REF] Yu | Virtual monochromatic imaging in dual-source dual-energy CT: radiation dose and image quality[END_REF].

High spatial resolution imaging

PCDs can provide improved spatial resolution and reduced radiation dose compared to conventional CT, because of small charge cluster size and the absence of electronic noise allowing reduced pixel size, the size compared to scintillator and photodiode conventional CT detectors, the absence of dead space between detectors [START_REF] Iwanczyk | Photon Counting Energy Dispersive Detector Arrays for X-ray Imaging[END_REF]. Consequently, this is leading to sharper edges and better delineation of structures in reconstructed images, with the additional value of decreasing the partial volume effects from small objects [Si-Mohamed et al., 2017a].

High contrast imaging

sCT is expected to require less contrast material to be administered to patients than the currently used amounts due to a better CNR (particular at low current dose) [Taguchi andIwanczyk, 2013, Schirra et al., 2014]. Importantly, sCT provides additional energy information and will allow enhanced contrast of different materials in the body.

SUMMARY

There are mainly three enhancement cases in CT imaging: integrating, counting and energy weighting. The first case uses weighting factors that are proportional to the energy of photon; the second case sums up the number of interacting photons with constant weighting factor; and in the third case, the energy-bin data are weighted by energy dependent factor to obtain better CNR and signal-to-noise Ratio (SNR) [START_REF] Carramate | Energy weighting technique in Quantum Computed Tomography using a MPGD[END_REF]. By giving lower-energy data more weight, the dose efficiency is effectively improved and enables a higher CNR at a constant patient dose, or a lower dose at a constant CNR [START_REF] Berglund | Energy weighting improves dose efficiency in clinical practice: implementation on a spectral photon-counting mammography system[END_REF]. Energy weighting can be performed both in projection domain or image domain data. In the research of Giersch el al., by using weighting factor E -3 , the SNR can be enhanced by up to from 1.3 to 1.9 or reduce the dose by factor of 2.5 without losing image quality [START_REF] Giersch | The influence of energy weighting on X-ray imaging quality[END_REF]]. An "image-based" weighting method by Schmidt shows that the CNR is improved by factors of 1.0 to 1.3 compared to photon counting weighting with negligible beam-hardening artifacts [Schmidt, 2009].

Summary

This chapter presents the general principles of spectral CT, including the following sections:

1. Spectral CT imaging system. Spectral CT system contains three parts: source emitting a flux of x-ray quantum at an object, object interacting with X-ray and detector counting the quantum. The main difference between spectral CT and conventional energy integrating CT or dual-energy CT is the detector. For conventional CT or dual-energy CT, detectors only count photons of different energy in one or two channels. While for spectral CT, the new photon-counting detector (PCD) can register photons in multiple channels.

2. Spectral CT images reconstruction. Reconstruction aims to obtain the spatial-dependent linear attenuation coefficients, which can be performed independently at each bin (separate reconstruction) or jointly by exploiting all spectrum information (joint reconstruction).

3. Applications of spectral CT. Spectral CT employs PCDs and is able to obtain spectral information of several energy bins with single acquisition. PCDs are semiconductor based detectors that have the ability to discriminate transmitted photons corresponding to selected energy bins. Spectral CT imaging enables to overcome many limitations of the previous techniques and opens up many new applications, such as quantitative material decomposition, virtual monochromatic imaging, high spatial resolution imaging and high contrast imaging.

CHAPTER 2. MODEL OF MATERIAL DECOMPOSITION AND MATHEMATICAL FOUNDATIONS

State of the art

Compared to conventional energy-integrating CT or dual energy CT, sCT can count the number of photons in multiple energy bins with one single exposure, i.e. utilizing spectral information. This advantage enables efficient material decomposition that aims to quantitatively separate different materials present in a pixel. Different approaches were developed to realize material decomposition: decompose projection data acquired at different energy bins into different material sinograms, each of which corresponds to a material (i.e. the so-called basis material), or different physical effects (e.g. photoelectric effect, Compton effect and extended K-edge components) based on their energy-dependent characteristics, and then reconstruct individually each spatial material image containing one single material (projection-domain approach) [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography[END_REF], Zimmerman and Schmidt, 2015, Ducros et al., 2017]; or firstly reconstruct the spatial image from each energy bin sinogram and then decompose the reconstructed spatial images corresponding to different energy bins into spatial material images (imagedomain approach) [Le Huy and Molloi, 2011, Taguchi et al., 2007, Li et al., 2015, Li et al., 2017]; or directly reconstruct spatial material images from projection data (one-step approach) [Foygel [START_REF] Barber | An algorithm for constrained one-step inversion of spectral CT data[END_REF], Mechlem et al., 2018, Hou et al., 2018].

There is actually another commonly used standard to categorize the material decomposition methods for spectral CT. According to the type of decomposition basis, material decomposition can be divided into two categories: effect based and material based methods. Effect based approaches aims to decompose data into different physical effects (Figure 1.4a), where the attenuation is considered as a linear combination of the photoelectric effect, Compton effect and extended K-edge components if there are materials with their K-edge within the CT energy range. Material based method is to decompose data into different basis materials (Figure 1.4b), which describes the linear attenuation coefficient by the contributions of a basis of materials.

Projection-domain material decomposition

For CT images, the object model of material decomposition is described by:

µ( x, E) = M ∑ α=1 f α (E)p α ( x), (2.1)
where M denotes the number of decomposition basis in use, p α ( x) the physico-chemical or quantity property (depending on the concrete formulation of Equation (2.1)) of the unknown object, and f α (E) a known function that varies with energy.

Projection-domain effect based method (k-edge imaging)

For decomposition using basis effects, p α ( x) is the physico-chemical property and f α (E) the energy-dependent characteristic of different physical effect. In the work of [START_REF] Roessl | K-edge imaging in x-ray computed tomography using multi-bin photon counting detectors[END_REF], µ( x, E) is taken as the combination of photoelectric absorption, Compton scattering and extra K-edge components if there is any K-edge material in the object:

µ( x, E) = f ph (E)p ph ( x) + f KN (E)p Co ( x) + M-2 ∑ α=1 µ mα (E)ρ α ( x), (2.2)

STATE OF THE ART

where p ph ( x) and p Co ( x) represent location dependency of photoelectric absorption and Compton scattering, respectively. f ph (E) denotes the cross section of photoelectric absorption that can be approximated by E -3 . f KN (E) is the cross section of Compton scattering, namely Klein-Nishina function. ρ α ( x) denotes the density of K-edge material α at point x and µ mα (E) the mass attenuation coefficient of K-edge material α at energy E.

(M -2) represents the number of K-edge materials inside the object.

Considering both the measurement process described by Beer-Lambert law and Equation (2.2), we have

n i (p α ) = E f (i) ∑ E=E s (i) d(E)n 0 (E) exp[- M ∑ α=1 f α (E)p α ( x) ds],
(2.3)

where n i denotes the expected number of photons at the i-th energy bin, d(E) the detector response function, n 0 the number of photons in the initial spectrum at energy E, and [E s (i), E f (i)] the start and final energies of the i-th bin, respectively. Equation (2.3) can be rewritten as:

n i (P α ) = E f (i) ∑ E=E s (i) d(E)n 0 (E) exp[- M ∑ α=1 f α (E) p α ( x) ds] = E f (i) ∑ E=E s (i) d(E)n 0 (E) exp[- M ∑ α=1 f α (E)P α (sx, sy)] (2.4)
= F (P), (2.5) with P α (sx, sy) = p α ( x) ds, (2.6)

where P α (sx, sy) is the line integral of location-dependent parameter p α ( x) along the measured projection path, (sx, sy) the location of the measured projection in the sinogram, with sx denoting detector pixels and sy denoting measurement angles, and F (P) the forward model of projection-domain decomposition model. To estimate P α (sx, sy), maximum-likelihood (ML) method is used in [Roessl andProksa, 2007, Schlomka et al., 2008]. Firstly suppose the detected number of photons M i form a set of independent Poisson random variables,

M i ∼ Poisson(mean = n i (P α )), (2.7)
then the likelihood function as the possibility of measurement result (M 1 = m 1 , M 2 = m 2 , ... M N = m N , ) can be calculated:

P(m 1 , ..., m N |n 1 (P α ), ..., n N (P α )) = N ∏ i=1 n i (P α ) m i m i ! e -n i (P α ) .
(2.8) For computational convenience in seeking the extremum, Equation (2.7) is simplified by applying a negative logarithm operator on both sides and dropping the term that is 57 CHAPTER 2. MODEL OF MATERIAL DECOMPOSITION AND MATHEMATICAL FOUNDATIONS unrelated to P α . Then calculating the negative logarithm likelihood function, we will obtain:

L(P α ) = -ln [P(m 1 , ..., m N |n 1 (P α ), ..., n N (P α )] = N ∑ i=1 [n i (P α ) + ln m i ! -m i ln n i (P α )] ∼ = N ∑ i=1 [n i (P α ) -m i ln n i (P α )] .
(2.9) By minimizing the above negative log-likelihood function, the line integral of object density P α can be solved. As we can see in Equation (2.9), P α is the line integral of p α , thus p α can be obtained by applying a conventional reconstruction on P α .

The K-edge imaging technology enables specific imaging of high-Z K-edge materials, such as iodine bismuth, gold and gadolinium. These materials are good candidates of contrast agents. In the work of Feuerlein, a partially occluded stent was simulated by using a calcified plaque isoattenuated to a surrounding gadolinium chelate solution. By K-edge imaging, the separated gadolinium image clearly shows perfused lumen of vessel [START_REF] Feuerlein | Multienergy Photon-counting K-edge Imaging: Potential for Improved Luminal Depiction in Vascular Imaging[END_REF]. In the work of Pan et al., fibrin-specific bismuth-enriched K-edge nanocolloid (nanoK (Bi)) particles are well enhanced in both spectral CT scans of a blood vessel phantom and carotid artery endarterectomy specimens [START_REF] Pan | A Survey on Transfer Learning[END_REF]. Cormode et al. made in vivo experiments using rabbits after injections of gold nanoparticles and iodinated contrast agent, their results show that gold and iodine distributions are clearly differentiated [START_REF] Cormode | Multicolor spectral photon-counting computed tomography: in vivo dual contrast imaging with a high count rate scanner[END_REF].

Projection-domain material based method

We have introduced that the attenuation coefficient can be considered as the linear combination of the basis material mass attenuation coefficients weighted by their densities. So we have

µ( x, E) = µ m1 (E)ρ 1 ( x) + µ m2 (E)ρ 2 ( x) + ... + µ mM (E)ρ M ( x) = M ∑ α=1 µ mα (E)ρ α ( x),
(2.10) where ρ α ( x) denotes the density of material α at point x. Material α is not exclusive to K-edge materials, but also applicable for lighter materials. µ mα (E) designates the mass attenuation coefficient of material α at energy E, which can be found in NIST database [START_REF] Saloman | X-ray attenuation coefficients (total cross sections): Comparison of the experimental data base with the recommended values of Henke and the theoretical values of Scofield for energies between 0[END_REF]]. M should be smaller than or equal to N, otherwise the problem will be under-determined and an infinite number of solutions exist if there are no other equality constraints. The types of materials chosen for decomposition depend on the application and prior knowledge of the scanned objects. Material based decomposition can be solved with the same method of effect based decomposition.

Image-domain material decomposition

Image-domain material decomposition works on conventionally reconstructed images µ(i, x) for each energy bin B i , which represents energy-resolved linear attenuation co-

STATE OF THE ART

efficient. The reconstructed multi-energy images are decomposed into the linear combination of mass attenuation coefficients weighted by the corresponding mass density, described by:

µ( x, i) = μm1 (i)ρ 1 ( x) + μm2 (i)ρ 2 ( x) + ... + μmM (i)ρ M ( x) = M ∑ α=1 μmα (i)ρ α ( x), i = 1, ..., B, (2.11)
where μmα (i) designates the calculated effective mass attenuation coefficient of the α-th basis material at the i-th energy bin, M the total number of basis materials, B the total number of energy bins, and ρ α ( x) the mass density of the α-th basis material at pixel x.

The aim of material decomposition is to solve ρ α from the Equation (2.11).

  

µ( x 1 , 1) . . . µ( x N P , 1) . . . . . . . . .

µ( x 1 , B) • • • µ( x N P , B)    Y =    μm1 (1) . . . μmM (1) . . . . . . . . . μm1 (B) • • • μmM (B)    M •    ρ 1 ( x 1 ) . . . ρ 1 ( x N P ) . . . . . . . . . ρ M ( x 1 ) • • • ρ M ( x N P )    X , (2.12)
where M denotes the decomposition matrix, each column of which represents the effective mass attenuation coefficients of one basis material for the B energies, Y ∈ R B×N P and X ∈ R M×N P the reconstructed multi-energy spatial images containing linear attenuation coefficients µ and the mass densities ρ of basis materials with N p indicating the total number of pixels or voxels, respectively. Furthermore, considering the noise, Equation (2.12) can be rewritten as:

Y = MX + N, = F (X), (2.13)
where N denotes the noise and F (X) the forward model of image-domain decomposition. Theoretically, the decomposition matrix can be initialized by the effective mass attenuation coefficients, calculated using [Le Huy and Molloi, 2011]:

μmα (i) = E∈E i n 0 (E)d(E)µ mα (E)dE E∈E i n 0 (E)d(E)dE , i = 1, ..., B, (2.14)
where µ mα (E) is the theoretical mass attenuation coefficient at energy E retrieved from NIST [START_REF] Saloman | X-ray attenuation coefficients (total cross sections): Comparison of the experimental data base with the recommended values of Henke and the theoretical values of Scofield for energies between 0[END_REF], and E∈E i n 0 (E)dE the total number of incident photons belonging to the i-th energy bin of width E i . In other words, μmα (i) represents the averaged value of all the theoretical mass attenuation coefficients inside each single energy bin, which is an estimate of the true mass attenuation coefficient corresponding to that energy bin.

Material-based decomposition has wider applications and also presents the potential to be optimized. Le and Molloi used a calibrated least squares fitting technique in image domain to quantify calcium and iodine [Le Huy and Molloi, 2011] for breast imaging. To assess tissue composition of atherosclerotic plaque, Alessio and and MacDonald used CHAPTER 2. MODEL OF MATERIAL DECOMPOSITION AND MATHEMATICAL FOUNDATIONS weighted least-squares method, which assumes that the materials are limited at a certain location to only those contained in one of the pre-selected classes. Therefore, this method firstly segments the image into pre-selected classes, then decomposes the pixel into corresponding materials contained in the class [START_REF] Alessio | Quantitative material characterization from multi-energy photon counting CT[END_REF]. Zeng et al. proposed a reconstruction method that uses penalized weighted least-squares scheme incorporating the structure tensor total variation regularization to gain better material decomposition quality [START_REF] Zeng | Iterative image reconstruction for multienergy computed tomography via structure tensor total variation regularization[END_REF]. Ducros et al. uses a regularized weighted least squares Gauss-Newton algorithm to decompose the spectral X-ray projection images of a thorax phantom into soft tissue, bone and gadolinium bases [START_REF] Ducros | Regularization of nonlinear decomposition of spectral x-ray projection images[END_REF].

One-step material decomposition

Recently, several methods have been proposed which reconstruct material-specific volumes directly from the photon counts. They are commonly referred to as "one-step inversion", or simply "one-step" methods [START_REF] Cai | A full-spectral Bayesian reconstruction approach based on the material decomposition model applied in dual-energy computed tomography[END_REF], Long and Fessler, 2014, Weidinger et al., 2016, Foygel Barber et al., 2016, Mechlem et al., 2018]. All of these methods are iterative: there is currently no analytical inversion formula for the material decomposition problem, let alone for one-step inversion. They consist in combining the forward models of the tomographic reconstruction and the material separation inverse problems, yielding a single (but more complex) forward model, which takes as input a set of material volumes, and yields photon counts as output. In discrete form, the forward model reads

n = E f (i) ∑ E=E s (i) d(E)n 0 (E) exp[-∑ j a ij M ∑ α=1 µ mα (E)ρ α ( x)], = F (ρ), (2.15)
where a ij is the coefficient of the forward projection matrix A at row i and column j (discussed in Equation (1.12)), and F (ρ) the forward model of one-step decomposition. One-step inversion then means finding ρ α ( x) for which the forward model yields photon counts n as similar as possible to those measured by the scanner. By construction, the drawbacks of two-step methods are circumvented, and there is no risk of losing information. One-step methods therefore have the potential to yield higher image quality than their two-step counterparts, but they are also slower to converge. This is likely to prevent their use in many applications, as it happened for regularized iterative reconstruction techniques on standard CT. Since in the multi-energy CT case the amount of data contained in a single acquisition is multiplied by the number of energy bins, and the one-step problem is more complex than the reconstruction problem alone, the need for fast methods is even more acute than for standard CT.

Inverse problem of image-domain material decomposition

In this work, we mainly focus on the image-domain material based material decomposition. The problem of solving the image-domain material based material decomposition described in Equation (2.13) is an ill-posed linear inverse problem. A number of methods to solve this kind of problem were reported in the literature. They can be categorized into

INVERSE PROBLEM OF IMAGE-DOMAIN MATERIAL DECOMPOSITION

two main types: optimization without and with regularization, as:

X = argmin X D(Y, X), X = argmin X D(Y, X) + λR, (2.16)
where Y ∈ R B×N P denotes the reconstructed multi-energy spatial images with N P indicating the total number of pixels or voxels and B the total number of energy bins, M ∈ R B×N M the decomposition matrix with N M the total number of basis materials, X ∈ R N M ×N P the decomposed basis material images of mass densities ρ, D the discrepancy function, R the regularization term, and λ a global regularization parameter. Generally, D is calculated using Euclidean distance:

D = Y -F (X) 2 F , (2.17)
where • F denotes Frobenius norm for Euclidean distance. The Frobenius norm is matrix norm of a m × n matrix A defined as the square root of the sum of the absolute squares of its elements:

A F = m ∑ i=1 n ∑ j=1 |a ij | 2 .
(2.18)

Note that for projection-domain material decomposition, some literatures reported that D can use weighted Euclidean distance adapted to data corrupted by Poisson noise:

D = n i -F (P) 2 W .
(2.19)

Considering the more complex noise condition of reconstructed sCT images, we choose Euclidean distance in our proposed image-domain material decomposition.

The pseudo-inverse method with singular value decomposition (SVD) is a usual optimization method without regularization [START_REF] Faby | CT calibration and dose minimization in image-based material decomposition with energy-selective detectors[END_REF], as:

X = M + Y, (2.20)
where M + is the pseudo-inverse of M solved by SVD. The SVD of a matrix A is the factorization of A into the product of three matrices A = UDV T where the columns of U and V are orthonormal and the matrix D is diagonal with positive real entries. The following formula can be used to find the pseudo-inverse of A: A + = VD + U T .

In the optimization method with regularization, sparse representation was often employed as that multi-energy data often has high correlation. The major motivation of sparse representation is to make use of the sparsity nature of multi-dimensional data. An appropriate sparse representation can give an accurate material decomposition with few energy bins. Typically, sparse representation via L1 norm minimization is considered efficient in many areas [Donoho, 2006, Candes et al., 2006] to prevent oversmoothing,

Y -F (X) 2 F + X 1 , (2.21) where • 1 is L1 norm, i.e. X 1 = m ∑ i=1 n ∑ j=1 |x ij |.
Other methods based on sparse representation were also studied for material decomposition, including gradient sparsity via bilateral total variation (BTV) [START_REF] Touch | Machine Learning-based Techniques to Address Spectral Distortions in Photon Counting X-ray Computed Tomography[END_REF] and sparse transform via tight frame [START_REF] Gao | Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM)[END_REF].

CHAPTER 2. MODEL OF MATERIAL DECOMPOSITION AND MATHEMATICAL FOUNDATIONS

Unreliability in image-domain material decomposition

The performance of the image-domain material decomposition (2.15) is influenced by both the quality of the reconstructed spatial images Y and the separation ability of M.

The quality of spatial images Y is deteriorated by data inconsistency stemming from the aforementioned PCD limitations and artifacts associated with reconstruction algorithm. Reconstruction being a preliminary step to image-domain material decomposition, errors in the reconstructed spatial images can degrade the subsequent decomposition, e.g. severe noise and beam-hardening artifacts, etc [START_REF] Elbakri | Statistical image reconstruction for polyenergetic X-ray computed tomography[END_REF]. The separation ability of M is limited by the interference between its basis materials, which is caused by the presence of similar basis materials having close µ m and their low estimation precision. The data inconsistency causes false pixel similarity, called µ-similarity in spatial images, while the basis material interference causes false basis materials similarity, called µ m -similarity. More precisely, µ-similarity implies that distinction between different mixtures in spatial images can be rendered difficult by data inconsistency. To explain the similarity between basis materials, we illustrate in Fig. 2.1 two linear attenuation coefficient (µ) curves corresponding respectively to mixtures 1 and 2 (which have different components or different mass densities; for instance, in this figure, mixtures 1 and 2 correspond respectively to the dilutions of gadolinium (mass density 2 mg/cc) and iodine (1 mg/cc)). The five circle points indicate the theoretical values at five different energy bins (i.e. 30-50 keV, 51-61 keV, 62-71 keV, 72-82 keV and 83-130 keV). Even in the ideal case, mixtures 1 and 2 cannot be totally distinguished due to their low mass densities. Moreover, data inconsistency, related to the strong quantum noise, can make that the measured µ value is not exactly equal to the theoretical one at a given energy bin. On the contrary, the measured value for the same mixture at the same energy bin can vary with pixel position. In other words, in the same image region containing the same mixture, different pixels that contain the same mixture can have different µ values. This is reflected in Fig. 2.1 by the observation that the µ value (blue or red points) at a given energy bin slips vertically around the theoretical value, thus creating an uncertainty interval. In the interval where blue and red points overlap, we can no longer distinguish which µ value corresponds to which mixture. Such µ-similarity between different mixtures (materials) can degrade the subsequent decomposition.

µ m -similarity results from material interference between the basis materials inside M, which makes it difficult to reliably separate basis materials having close µ m . Moreover, the interference is more severe for image-domain material decomposition considering that the effective mass attenuation coefficient μm involved in (2.7) is an averaged value and the low precision can magnify the similarity between basis materials. It is then necessary to obtain μm from calibration phantom containing materials of known concentrations [START_REF] Clark | Spectral diffusion: an algorithm for robust material decomposition of spectral CT data[END_REF], which is adopted in the present study. However, while this approach helps to improve the precision of μm , the material interference stemming from close µ m is still obvious. Actually, µ m -similarity leads material decomposition to an ill-conditioned problem. To cope with this ill-conditioned problem, existing basis material optimization methods, e.g. µ m -preconditioning methods, utilize linear combination of original basis materials to create synthetic materials, such that M = MP, where P represents the transform matrix and M the synthetic materials. µ m -preconditioning strategy aims to improve the overall condition of material decomposition to increase the algorithm efficiency [Foygel [START_REF] Barber | An algorithm for constrained one-step inversion of spectral CT data[END_REF] rather than the reliability of material decomposition.

INVERSE PROBLEM OF IMAGE-DOMAIN MATERIAL DECOMPOSITION

In short, the unreliability of material decomposition resulting from both µ-and µ msimilarity still remains a persistent problem.

In this thesis, we proposed three different methods for reliable and accurate material decomposition, respectively discussed in Chapter III, IV and V.

Optimization method of material decomposition

All of the proposed methods are solved by alternating direction method of multipliers (ADMM) [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF], a simple but powerful algorithm that is well suited to distributed convex optimization, and in particular to problems arising in applied statistics and machine learning.

ADMM is an algorithm that is intended to blend the decomposability of dual ascent with the superior convergence properties of the method of multipliers. The algorithm solves problems in the form

minimized f (x) + g(x), s.t. Ax + Bz = c, (2.22)
with variable x ∈ R n and z ∈ R m , where A ∈ R p×n , B ∈ R p×m , and c ∈ R p . We will assume that f and g are convex. The optimal value of the problem (2.22) will be denoted

CHAPTER 2. MODEL OF MATERIAL DECOMPOSITION AND MATHEMATICAL FOUNDATIONS by

p * = inf{ f (x) + g(x)|Ax + Bz = c}. (2.23)
As in the method of multipliers, we form the augmented Lagrangian

L p (x, z, y) = f (x) + g(x) + y T (Ax + Bz = c) + (ρ/2) Ax + Bz = c 2 2 .
(2.24)

ADMM consists of the iterations

x k+1 = argmin x L p (x, z k , y k ),
(2.25)

z k+1 = argmin z L p (x k+1 , z, y k ),
(2.26)

y k+1 = y k + ρ(Ax k+1 + Bz k+1 -c), (2.27)
where ρ. The algorithm is very similar to dual ascent and the method of multipliers: it consists of an x-minimization step (2.25), a z-minimization step (2.26), and and a dual variable update (2.27). As in the method of multipliers, the dual variable update uses a step size equal to the augmented Lagrangian parameter ρ. In ADMM, x and z are updated in an alternating or sequential fashion, which accounts for the term alternating direction.

Summary

This chapter presents the principles of material decomposition in spectral CT, including the following sections:

1. State of the art of material decomposition. Different approaches were developed to realize material decomposition: projection-domain approach -decompose projection data acquired at different energy bins into different material sinograms, each of which corresponds to a material (i.e. the so-called basis material), or different physical effects (e.g. photoelectric effect, Compton effect and extended K-edge components) based on their energy-dependent characteristics, and then reconstruct individually each spatial material image containing one single material; image-domain approach -firstly reconstruct the spatial image from each energy bin sinogram and then decompose the reconstructed spatial images corresponding to different energy bins into spatial material images; one-step approach -directly reconstruct spatial material images from projection data.

2. Inverse problem of image-domain material decomposition. Solving material decomposition is an inverse problem. Common methods were introduced, e.g pseudoinverse method with singular value decomposition, method with sparsity regularization, and method with total variation (TV) regularization, etc. Meanwhile, the bottlenecks of image-domain material decomposition were introduced, including µ-similarity and µ m -similarity. At last, the optimization method utilized in our works was discussed.

Introduction

sCT has recently received increasingly intensive attention, due to the new advances in photon-counting detector technology. Compared to conventional energy integrating CT or dual-energy CT, spectral CT can count the number of photons in separated energy bins with one single exposure [START_REF] Faby | Performance of today's dual energy CT and future multi energy CT in virtual non-contrast imaging and in iodine quantification: A simulation study[END_REF]. Thanks to this advantage, it becomes possible to decompose efficiently different materials (basis materials) present in the same pixel of spectral CT images, which provides promising perspectives for both medical applications and industrial nondestructive evaluation.

According to the type and number of basis materials, material decomposition in spectral CT can be an ill-posed, or even ill-conditioned inverse problem. Such problem can arise before [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography[END_REF], Zimmerman and Schmidt, 2015, Ducros et al., 2017], during [Foygel Barber et al., 2016,Mechlem et al., 2018,Hou et al., 2018], or after [START_REF] Taguchi | Image-domain material decomposition using photon-counting CT[END_REF], Le Huy and Molloi, 2011, Faby et al., 2014, Li et al., 2015, Li et al., 2017] image reconstruction process. A number of material decomposition methods were reported in the literature. They can be categorized into two main types: optimization without and with regularization. The pseudo-inverse method with singular value decomposition (SVD) is a usual optimization method without regularization [START_REF] Faby | CT calibration and dose minimization in image-based material decomposition with energy-selective detectors[END_REF]. In the optimization method with regularization, sparse representation was often employed as that multi-energy data often has high correlation. The major motivation of sparse representation is to make use of the sparsity nature of multi-dimensional data. An appropriate sparse representation can give an accurate material decomposition with few energy bins. Typically, sparse representation via L1-norm minimization is considered efficient in many areas [START_REF] Candes | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information[END_REF],Donoho, 2006] to prevent oversmoothing. Other methods based on sparse representation were also studied for material decomposition, including gradient sparsity via bilateral total variation (BTV) [START_REF] Clark | Spectral diffusion: an algorithm for robust material decomposition of spectral CT data[END_REF] and sparse transform via tight frame [START_REF] Gao | Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM)[END_REF]. All the above-cited methods were developed in overdetermined cases; no work is available for material decomposition in underdetermined cases where the number of basis materials is greater than that of energy bins.

Actually, in practice an object always contains multiple materials and we do not know a priori what exactly the number of materials in it is. Therefore, material decomposition in underdetermined cases remains a practical problem. The fact that basis materials with close atomic numbers have similar mass attenuation coefficients comes to worsen the problem. To cope with these problems, we propose to use the joint sparsity [START_REF] Nie | Efficient and robust feature selection via joint L2, 1-norms minimization[END_REF] characteristics of both spectrum and space information. Moreover, since neighboring pixels have similar materials, we exploit the joint sparsity in local small patches. Meanwhile, we also make use of the structural low-rank property that the decomposed materials in the corresponding local small patch should share the same edge, due to the local property that, in a local patch, a decomposed material is either present or absent at each pixel of the patch. Finally, since the mass density of materials has always limited value, we introduce a bound constraint on mass density. All that leads us to propose a material decomposition method based on bounded mass Density, local joint Sparsity and structural low-Rank (DSR).

MATERIAL DECOMPOSITION USING MULTIPLE CONSTRAINTS

Material decomposition using multiple constraints

For spectral CT images, basis materials with close atomic numbers have similar mass attenuation coefficients, and this is the source of ill-conditioned problems and constitutes the main bottleneck that limits the performance of material decomposition methods. To cope with this problem, we propose to find the main features by exploiting the similarities in multi-dimensional data. To this end, we use a sparse representation method that allowing processing the high correlation of both spectral and spatial dimensions. Such joint sparsity of spectral and spatial dimensions would be interesting for decomposing similar materials in a robust way.

Such joint sparsity of spectral and spatial dimensions would be interesting for decomposing similar materials in a robust way. We then adopt the mixed L2,1-norm that is particularly suitable for small patches to combine the information. The norm L2,1 is defined by:

X 2,1 = I ∑ i=1 J ∑ j=1 x 2 ij , (3.1)
where x ij is the mass density of the i-th basis material at the j-th pixel in X.

L2,1-norm can enforce a small number of nonzero rows in X, compared to usual L1norm that tends to give sparse solution pixel by pixel. In other words, L1-norm limits the number of basis materials exiting in one pixel, while L2,1-norm enforces pixels to share the same atoms of basis materials in each local small patch. For the present application of spectral CT images, the L2,1-norm allows us to combine space (rows in X) and spectrum (columns in X) information.

At the same time, in addition to the spectral and spatial similarities, multi-energy images also present the structural coherence, which reflects the fact that the decomposed materials should share the same edges in the same small patch due to their local property in the image. Such edge information in a material image is also sparse and can be captured by gradient operator. To process the edge similarities, we utilize low-rank representation on the gradient matrices of all the decomposed materials. The low-rank constraint is realized via nuclear norm that is the convex envelope of matrix rank [START_REF] Candès | Robust principal component analysis[END_REF]. As a result, the structural low-rank (SL) is calculated using:

X * = I ∑ i=1 σ i (∆X), (3.2)
where X is the gradient matrix of X and X * is the nuclear norm equal to the sum of its singular values σ i (∆X). Actually, this kind of rank-sparsity encouraging norm has also been discussed in color image denoising, with the name total nuclear variation (TNV) [Holt, 2014]. However, in the present study, it is utilized to process the structural similarity that existing material images Xi (nonzero rows in X of equation ( 2)) should share the same edges. A cost function containing the joint sparsity and low-rank regularizations and a data-fidelity term may give an optimal solution to the problem of material decomposition. We then formulate our material decomposition method based on local

F + λ 1 X P 2,1 + λ 2 X P * , (3.3)
where Y P and X P denote the patches extracted respectively from the reconstructed images and the decomposed material images λ 1 and λ 2 > 0 are the scalar regularization parameters. At the same time, since the mass density of materials always has limited value, we introduce the bound constraints into equation (3.3). In other words, the mass densities of materials should be lower bounded by a nonnegative number and upper bounded by a reasonable limit. Thus, after taking into account the multiple constraints of bounded mass Density, local joint Sparsity and structural low-Rank (DSR), our material decomposition model becomes:

argmin X 1 2 Y P -MX P 2 F + λ 1 X P 2,1 + λ 2 X P * + bound l (X P ) + bound u (X P ), (3.4)
where bound l (X P ) and bound u (X P ) denote the lower and upper mass density bounds, respectively.

DSR: solution algorithm by ADMM

The minimization problem of (3.4) is solved by the alternating direction method of multipliers (ADMM) [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF]. Firstly, we form the augmented Lagrangian for problem (3.4):

L λ (V 1 , V 2 , V 3 , V 4 , X) = arg min V 1 ,V 2 ,V 3 ,V 4 ,X 1 2 Y P -V 1 F 2 + λ 1 V 2 2,1 + λ 2 V 3 * + bound l (V 4 ) + bound l (V 5 ) + τ 2 MX P -V 1 -D 1 2 F + τ 2 X P -V 2 -D 2 2 F + τ 2 ∇X P -V 3 -D 3 2 F + τ 2 X P -V 4 -D 4 2 F + τ 2 X P -V 5 -D 5 2 F , (3.5)
where τ > 0 is the augmented Lagrangian parameter.

EXPERIMENTS AND RESULTS

ADMM consists of the following iterations:

X P t+1 ← arg min X P τ 2 MX P -V 1 t -D 1 t 2 F + τ 2 X P -V 2 t -D 2 t 2 F + τ 2 ∇X P -V 3 -D 3 2 F + τ 2 X P -V 4 t -D 4 t 2 F + τ 2 X P -V 5 t -D 5 t 2 F , (3.6) V 1 t+1 ← arg min V 1 1 2 Y P -V 1 2 F + τ 2 MX P t -V 1 -D 1 t 2 F , (3.7) V 2 t+1 ← arg min V 2 λ 1 V 2 2,1 + τ 2 X P t -V 2 -D 2 t 2 F , (3.8) V 3 t+1 ← arg min V 3 λ 2 V 3 * + τ 2 ∇X P -V 3 -D 3 2 F , (3.9) V 4 t+1 ← arg min V 4 bound l (V 4 ) + τ 2 X P t -V 4 -D 4 t 2 F , (3.10) V 5 t+1 ← arg min V 5 bound u (V 5 ) + τ 2 X P t -V 5 -D 5 t 2 F . (3.11)
Equations (3.6) to (3.11) aim to solve primal variables X P , V 1 , V 2 , V 3 , V 4 and V 5 . Equation (3.8) is processed by the algorithm FOCUSS [Kowalski et al., 2009, Rao andKreutz-Delgado, 1999], and equation (3.9) by the singular value thresholding (SVT) method [START_REF] Cai | A Singular Value Thresholding Algorithm for Matrix Completion[END_REF]. The solutions of X P , V 1 , V 2 , V 3 , V 4 and V 5 are given by:

X P t+1 = (M T M + ∇ T ∇ + 3I) -1 [A T (V 1 t + D 1 t ) + (V 2 t + D 2 t ) + ∇ T (V 3 t + D 3 t ) + (V 4 t + D 4 t ) + (ub -V 5 t -D 5 t )],
(3.12)

V 1 t+1 = 1 1 + τ Y P + τ(MX P t -D 1 t ) , (3.13) V 2 t+1 = (X P t -D 2 t ) • (I - λ 1 /τ X P t -D 2 t 2
) + , (3.14) (3.17) where ub is the upper bound vector and (x) + = max(x, 0). For clarity, the detail of ADMM for the DSR decomposition method is given in Algorithm 1. The iteration will stop when the criterion of ADMM is satisfied, e.g. the residuals are small enough.

V 3 t+1 = SVT(∇X P -D 3 , λ 2 τ ), (3.15) V 4 t+1 = (X P t -D 4 t ) + , (3.16) V 5 t+1 = (ub -X P t -D 5 t ) + ,

Experiments and results

Two methods were compared with the proposed DSR method: (a) pseudo-inverse with SVD, (b) regularization method with L1-norm constraint (L1). For clarity, we rewrite the models of the three methods as below:

λ 1 , λ 2 , τ, V 1 , V 2 , V 3 , V 4 , V 5 , D 1 , D 2 , D 3 , D 4 , D 5 Repeat: Calculate: X P t+1 = (M T M + ∇ T ∇ + 3I) -1 [A T (V 1 t + D 1 t ) + (V 2 t + D 2 t ) + ∇ T (V 3 t + D 3 t ) + (V 4 t + D 4 t ) + (ub -V 5 t -D 5 t )] Calculate: V 1 t+1 = 1 1+τ Y P + τ(MX P t -D 1 t ) Calculate: V 2 t+1 = (X P t -D 2 t ) • (I - λ 1 /τ X P t -D 2 t 2 ) + Calculate: V 3 t+1 = SVT(∇X P -D 3 , λ 2 τ ) Calculate: V 4 t+1 = (X P t -D 4 t ) + Calculate: V 5 t+1 = (ub -X P t -D 5 t ) + Update Lagrange multipliers: D 1 , D 2 , D 3 , D 4 , D 5 Output: X P • Pseudo-inverse with SVD: X = M + Y, • L1: argmin X 1 2 Y -MX 2 F + λ X 1 + bound l (X P ) + bound u (X P ),
• DSR: argmin X 1 2 Y P -MX P 2 F + λ 1 X P 2,1 + λ 2 X P * + bound l (X P ) + bound u (X P ).

Image quality metrics

The performance of the proposed material decomposition was quantitatively evaluated using different metrics. The normalized Euclidean distance was utilized for the accuracy [START_REF] Ducros | Regularization of nonlinear decomposition of spectral x-ray projection images[END_REF], considering its better evaluation for various concentrations compared to the common metric mean squared error. Given both the m-th decomposed basis materials x m and ground-truth x gt m , the normalized Euclidean distance is:

error m = x m -x gt m 2 x gt m 2 . (3.18)
The smaller the normalized Euclidean distance, the more accurate the decomposition precision.

The performance of the proposed DSR method was evaluated on both digital and physical data.

Digital data

The projection data of spectral CT was simulated using INSA software Virtual X-ray Imaging (VXI) [START_REF] Duvauchelle | Computer code to simulate X-ray imaging techniques[END_REF] and then reconstructed by the method FBP. The reconstructed phantom has 800 * 800 pixels and contains five materials: water, Polymethyl Methacrylate (PMMA), gadolinium (Gd), iodine (I) and iron (Fe), as shown in figure 3.1. The number on each disk designates the concentration of materials (mg/cc).

Note that the disk with # stands for mixture inserts that contain three basis materials (Gd, I and Fe) with the same concentration in each column. The X-ray energy bins were set as: 30-40kev, 40-50kev, 50-60kev, 60-70kev and 70-80kev. The number of energy bins 5 are selected in terms of the setting of Philips prototype.

To fully evaluate the impacts of noise, we list the results of material decomposition based on reconstructed image without and with Poisson noise. Meanwhile, decomposition on different basis materials were tested, in terms of the enormous effects of category and number of basis materials. Furthermore, the impacts of low-energy bins were evaluated, on which the decomposition results are shown.

Without Poisson noise

The decomposition results of five existing basis materials (water, PMMA, iron, iodine and gadolinium) are shown in figure 3.2. Note that in the present study, the lower mass density bound is set as 0 and the upper bound as 10 times of the maximum mass density value of existing materials. The method pseudo-inverse with SVD can not quantify the 5 existing basis materials. It can be partly explained by the condition number κ(A) of matrix, which is equal to the ratio of the largest to smallest singular value, and is used to measure how sensitive a function is to errors in the input [Vacca, 1981]. A problem with a high condition number is ill-conditioned. In our experiments, there are 20 similar basis materials, leading to a big condition number κ(A) = 1.7 × 10 5 , which implies that the result of usual pseudo-inverse with SVD is not reliable for our ill-condition problem. Therefore, we will focus on the other two methods. As can be seen in figure 3.2, the proposed DSR method has higher decomposition accuracy compared to the L1 method. Even for the material without k-edge (Fe), DSR shows good ability in detection and quantification. For more quantitative analysis, we compare the mass density of Fe, I and Gd in the corresponding rows. The results indicate that DSR has good decomposition ability for all of iron, iodine and gadolinium, as shown in figure 3.3. In contrast, L1 yields much lower density of iron and higher density of iodine compared to the theoretical values.

CHAPTER 3. MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT USING MULTIPLE CONSTRAINTS IN IMAGE DOMAIN

Effects of basis materials

To evaluate the impacts of basis materials for the proposed decomposition methods, we considered 20 materials as basis materials divided into 4 categories, including the five materials (bold ones) present in the phantom, as listed in Table 3.1. Because most elements in the human body have relatively small atomic numbers, we put low-Z elements: sodium (Na), magnesium (Mg), aluminum (Al), silicon (Si), phosphorus (P), sulfur (S), chlorine (Cl), argon (Ar) and potassium (K) into the basis materials as category 1. Manganese (Mn), cobalt (Co), copper (Cu) and zinc (Zn) were selected for comparison with Fe, as category 2. To test the accuracy of I and Gd, another common contrast agent xenon (Xe) was also added, as category 3. Bone was selected to test the accuracy of water and PMMA because of their similar attenuation coefficients. The mass attenuation coefficients were retrieved from National Institute of Standards and Technology (NIST) [START_REF] Saloman | X-ray attenuation coefficients (total cross sections): Comparison of the experimental data base with the recommended values of Henke and the theoretical values of Scofield for energies between 0[END_REF].

To make a comprehensive evaluation of the proposed DSR, the results of material decomposition for the remaining categories of basis materials in Table 3.1 are given in Figure 3.5: The MAEs of materials in Table 3.1.

EXPERIMENTS AND RESULTS

Na

Effects of low-energy bins

To evaluate the impacts of bins with lower energy, decomposition performance in terms of different number of energy bins, we add the 6-th energy bin: 20-30 keV. The decomposition results of five existing basis materials are shown in figure 3.6. Visually, there is no obvious difference with 5 bins for the methods SVD and L1. For the proposed DSR method, decomposition result of iron and gadolinium has more cross-contamination with water and PMMA. To further quantitatively evaluate the impacts of the 6-th bin, we compare the mass density of Fe, I and Gd in the corresponding rows. For iodine, the results of DSR method is overestimated for all the concentrations, while for iron and gadolinium underestimated. The results indicate that with the 6-bin, DSR method has worse stability in terms of the decomposition results of 5-bins were always underestimated for all the three materials (as shown in figure 3.3).

With Poisson noise

The decomposition results of five existing basis materials with Poisson noise are shown in figure 3.8. Even for the material without k-edge (Fe), DSR shows good ability in detection and quantification, which is similar to the results without Poisson noise. Visually, there are obviously more errors for water-like materials than for iron, iodine and gadolinium.

Fortunately in practical applications, the quantitative information of the last three materials are more useful, as a result of which we will pay more attention to the materials of interest: iron, iodine and gadolinium. As an illustration, we list error m of iron, iodine and gadolinium for the three methods in Table 3.2. Clearly, DSR method has smaller error m for all the three materials compared to L1 method.

Physical data

The physical phantom was acquired on a Phillips sCT prototype [START_REF] Si-Mohamed | Multicolour imaging with spectral photoncounting CT: a phantom study[END_REF], Cormode et al., 2017] Meanwhile, we have another in vivo data to further evaluate the proposed method. Reconstructed images of a rabbit 15 min after injection of gadolinium are shown in Fig. 3.10. Note that calcium (Ca) in the spine is decomposed on the basis material iodine, due to their similarity of mass attenuation coefficients in our energy thresholds setting. For gadolinium, although the ground-truth is unknown in the in vivo data, tubes of gadolinium dilutions nearby the rabbit were evaluated alternatively.

Figure 3.10: The reconstructed image of in vivo data at the first energy bin 15 min after the injection of gadolinium. The results of material decomposition on the physical phantom are shown in Fig. 3.11. In terms of the worse ability of separating water and PMMA and less interests of them in clinical applications, we only considered three basis materials, water, iodine and gadolinium, in the physical phantom. The proposed DSR material decomposition method has clearly better performance in detection and quantification compared to SVD and L1 methods. Firstly, our method shows much better morphological accuracy, especially for gadolinium which are obviously overwhelmed for both SVD and L1 methods but are more accurate for DSR method. Secondly, more quantitatively as shown in Table 3.3, DSR method has clearly smaller normalized Euclidean distance. Specially, gadolinium exhibits higher accuracy compared to iodine with DSR method. For in vivo data, the decomposition results of different methods are shown in Fig. 3.12. Obviously, both SVD and L1 method have worse detection ability for both iodine and gadolinium. On the contrary, the proposed DSR method has higher accuracy for both iodine and gadolinium. It should be noted that all the three methods cannot detect gadolinium of 2 mg/cc in the in-vivo data, which may ne resulted from the precision of effective decomposition matrix M.

EXPERIMENTS AND RESULTS

Gadolinium Iodine Water+PMMA

Discussion and conclusions

We have proposed a material decomposition method for spectral CT by simultaneously exploiting the limited value of mass density, the multi-dimensional nature and high correlation of multi-energy data and the local property of neighboring pixels. To get rid of the difficulties due to similarities among basis materials, spectral and spatial redundancies are eliminated through using joint sparsity and structural low-rank. At the same time, we also add bound constraint on mass density, which is naturally true in practi-3.5. DISCUSSION AND CONCLUSIONS cal applications. Preliminary results revealed that the proposed DSR method has a good material decomposition ability even for very poorly conditioned problems.

The performance of material decomposition is influenced simultaneously by multiple effects, such as quantum noise, the number of energy bins and the set of basis materials. Actually, sCT material decomposition suffers from strong quantum noise at each single bin stemming from narrow bin-width. As shown in figure 3.2 (without Poisson noise) and figure 3.8 (with Poisson noise), the Poisson noise obviously degrades the decomposition performance. Theoretically, more bins can bring more spectral information. However, sCT material decomposition suffers from strong quantum noise at each single bin stemming from narrow bin-width. In other words, the energy resolution of sCT is limited by a trade-off between the total number of energy bins and the image quality at each bin. As a result, in practice, the number of energy bins of sCT is always constrained, e.g. 5 bins for Phillips sCT prototype (the setting in our experiments) [START_REF] Si-Mohamed | Multicolour imaging with spectral photoncounting CT: a phantom study[END_REF], Cormode et al., 2017]. Meanwhile, the bins of too lower energy can further worsen the decomposition. As shown in figure 3.7, material decomposition using 6 bins, with the additional bin of 20-30 kev, has more errors for all the three methods. For the set of basis materials, the decomposition is dramatically degraded by large number of basis materials, especially by the materials with similar mass attenuation coefficients more basis materials, as shown in figure 3.4. Nevertheless, even in this condition, the proposed DSR method still has better performance, as illustrated in figure 3.5.

The regularization in the proposed DSR method is a convex problem. We have used an iterative algorithm ADMM to solve the proposed model. To accelerate convergence, the initialization of variables in the ADMM was set as the results of pseudo-inverse with SVD. Although there are four regularizations in the cost function, only the joint sparsity term and the structural low-rank term have the regularization parameters λ 1 and λ 2 . The results of material decomposition are clearly influenced by the regularization parameters. This is in fact a persistent problem, since solving cost function with more than one regularization parameter is still a challenging problem [START_REF] Clark | Spectral diffusion: an algorithm for robust material decomposition of spectral CT data[END_REF],Gao et al., 2011, Zhang et al., 2016, Ding et al., 2018]. The scalar regularization parameters λ 1 and λ 2 influence respectively the level of sparsity and low-rank. The augmented Lagrangian parameter τ that is equal to the step size of iteration has also an obvious impact on convergence speed. In practice, to make performance less dependent on the initial choice, we have updated τ in each iteration by the way discussed in [Boyd et al., 2010, WANG and[START_REF] Wang | [END_REF](try to keep the primal and dual residual norms in the ADMM within a factor as they both converge to zero). Finally, due to the use of local property, the choice of regularization parameters λ 1 , λ 2 and τ is also influenced by the size of patch. Generally, a smaller size of patch should have the correspondingly smaller regularization parameters, otherwise obvious errors will occur near the edges of patches in the decomposed material images. Note that neither too large patches nor too small patches accords with the local property or result in an accurate decomposition, and it should be set according to the property of each spectral CT data.

In conclusion, the proposed DSR method exhibits good performance in simultaneously separating and quantifying two contrast agents, e.g. iodine and gadolinium. In particular, the good ability of DSR to distinguish the contrast agents (e.g. iodine, gadolinium and xenon) suggests its interesting potential use for industrial and medical applications. In the future work, the collaborative joint sparsity and structural low-rank should CHAPTER 3. MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT USING MULTIPLE CONSTRAINTS IN IMAGE DOMAIN be further investigated to make full use of the sparsity and low-rank property hidden in the spectral CT data. Finally, a more robust strategy for the choice of regularization parameters could also be investigated.

Introduction

SCT is a new kind of multi-energy X-ray CT that offers new possibilities for getting insights into material components in an object, thanks to the advances in photon-counting detector (PCD). Compared to conventional energy-integrating CT or dual energy CT, sCT can count the number of photons in multiple energy bins with one single exposure, i.e. utilizing spectral information. This advantage enables efficient material decomposition that aims to quantitatively separate different materials (at least three) present in a pixel.

However, the reliability and accuracy of the decomposition approaches (which were discussed in Chapter 2 and 3) are always impacted by unavoidable similarity between linear attenuation coefficients of basis materials [START_REF] Si-Mohamed | Multicolour imaging with spectral photoncounting CT: a phantom study[END_REF], Si-Mohamed et al., 2017a] (more details were discussed in Chapter 2). Such similarity makes it difficult to separate the basis materials. The difficult basis material separation is further worsened by strong quantum noise. The latter also limits the ability of PCD to recognize photons between adjacent energy bins [START_REF] Schlomka | Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography[END_REF], which renders the basis material separation still more difficult. Meanwhile, the performance of image-domain material decomposition is also susceptible to the quality of reconstructed sCT images including in particular the impact of beam hardening.

To improve the reliability and accuracy of material decomposition, it is important to make full use of more information beside the aforementioned spectral information [START_REF] Zhang | Spectral CT Reconstruction With Image Sparsity and Spectral Mean[END_REF], Li et al., 2017, Zhang et al., 2017, Fadili et al., 2010, Xia et al., 2019]. A straightforward way of realizing this is to exploit morphological information embedded in the reconstructed sCT images. Image-domain material decomposition is then an approach of choice. In this chapter, we investigate a novel image-domain material decomposition method by directly decreasing the impacts of similarity between basis materials with the help of multiple features extracted from the reconstructed multi-energy spatial sCT images. The idea is to exploit the abundant information and high correlations in sCT images suffering from serious reconstruction errors and artifacts. To do that, we perform basis material optimization by selecting basis materials according to their spatio-energy similarity in segmented region of interests (ROIs) of multi-energy sCT images, thus leading to so-called ROI-wise material decomposition. To our knowledge, this is the first work to improve the mathematical condition of material decomposition through optimizing basis materials by means of spatio-energy segmentation.

The rest of this chapter is organized as follows. In Section 4.2, we describe the proposed method of ROI-wise material decomposition. Section 4.3 presents experiments and results on both simulations and real data. Finally, Sections 4.4 is given discussion and conclusion.

ROI-wise material decomposition

To make material decomposition more reliable and accurate, we propose to fully exploit their spatio-energy similarity in ROIs of multi-energy sCT images and inter-bin uncorrelation and material-dependent characteristics of noise in sCT images. To do that, we perform a basis material optimization through reducing the impact of material similarity by means of spatio-energy segmentation. Meanwhile, we construct a noise-reduced composite image by introducing a so-called energy averaging [START_REF] Leng | Noise reduction in spectral CT: Reducing dose and breaking the trade-off between image noise and energy bin selection[END_REF], in which images at different energy bins are mixed but noise is reduced. Another characteristic of noise in sCT is that it is material-dependent, meaning that noise shows different impacts on materials according to their categories and mass densities, e.g. water-like materials (water, Poly(methyl methacrylate) (PMMA) and soft issue, etc.) with smaller mass attenuation coefficients and materials with lower densities are much easier to be influenced by noise compared to the dense materials of interest. It was reported in [START_REF] Ducros | Regularization of nonlinear decomposition of spectral x-ray projection images[END_REF], Si-Mohamed et al., 2018] that dilutions of contrast agents (e.g. iodine, gadolinium and gold, etc.) with lower concentrations have smaller contrast to noise ratio. The resulting noise-reduced composite image is then utilized in a material-wise way in the fine material decomposition. The global diagram of the proposed ROI-wise material decomposition is illustrated in figure 4.1.

ROI-WISE MATERIAL DECOMPOSITION

Basis material optimization is first processed based on the spatio-energy segmentation that separates multi-energy spatial images into different ROIs. Next, the basis materials in M are selected by performing a coarse material decomposition followed by a relative population thresholding. The part of basis material optimization is boxed in blue in figure 4.1 and more details will be discussed in Section 4.2.1.

The noise-reduced composite image is constructed by averaging, in the direction of energy, multiple spatial images corresponding to different energy bins (the green part in figure 4.1 and details presented in Section 4.2.2).

The basis material optimization result and the noise-reduced composite image are finally fed in the fine material decomposition block to obtain the final material images.

Basis material optimization

As the core of our ROI-wise material decomposition, basis material optimization comprises two steps: spatio-energy segmentation and basis material selection. The first step aims to separate the reconstructed spatial images into different ROIs. The second one deletes irrelevant basis materials using a coarse material decomposition and a relative population thresholding.

We first obtain the ROIs of multi-energy images by spatio-energy segmentation, where each ROI represents a homogeneous area containing similar materials. To achieve such spatio-energy segmentation, morphological and spectral features of multi-energy images are used. figure 4.2 illustrates the segmentation of two ROIs in the multi-energy spatial images. Spectral features are extracted by regrouping the sCT images at all the energies as a single three-dimensional (3-D) image in which the energy is taken as a third dimension. Each pixel in the 3-D image has multi-energy values (energy-dependent µ). Pixels having different µ curves, i.e. different spectral features, belong to different ROIs, as shown in the upper-middle part of figure 4.2. Morphological features of sCT images are in fact energydependent because the characteristics of materials in sCT images are energy-dependent. As a result, structures imperceptible at certain energy bins may be distinguished more easily at another bin, depending on the physical characteristics of materials. Therefore, we take the morphological features from the energy bin having the most reliable segmentation by evaluating a pre-processing segmentation based on Gaussian mixture model (GMM) at each single energy bin. Then, the extracted structures are treated as the common constraint for images at all energy bins, as shown in the lower-middle part of figure 4.2, where Y s denotes the multi-dimensional image containing common structures. The spectral and morphological features are then exploited jointly for segmentation based on clustering.

In details, we use the kernel k-means method for the fusion of spectral and morphological features. The main advantage of kernel k-means is that it can make full use of kernel properties, which provides the ability to combine different features [START_REF] Dhillon | Kernel k-means[END_REF], Fauvel et al., 2012]. More precisely, we utilize the kernel k-means method to automatically segment the pixels in the 3-D image:

D({π k } K k=1 ) = K ∑ k=1 ∑ y∈π k Φ(y) -m k 2 2 , s.t. m k = Φ(y) π k 1 , (4.1)
where Φ represents a non-linear transform function, {π k } K k=1 the partitioning of multienergy pixel values y ∈ R B into different clusters π k , K the total number of clusters (i.e. the total number of ROIs and is initialized by the total number of basis materials (M)), m k the ideal cluster values, and • 2 and • 1 denote L-2 and L-1 norm, respectively. As an enhanced algorithm of normal k-means, kernel k-means can separate vectors in a highdimensional feature space based on the non-linear transform. The non-linear transform can be calculated with a convenient kernel, and new kernel can be constructed by linearly adding two basic kernels, denoted by K 1 and K 2 respectively, i.e. θ 1 K 1 + θ 2 K 2 , which is still a kernel. We utilize this property to combine spectral and spatial features which are separately selected by two kernels, named as K spectrum , K space . The new kernel is then

K σ,λ = θK spectrum σ + (1 -θ)K space σ , ( 4.2)
where 0 ≤ θ ≤ 1 and 1θ designate the weights for spectral and spatial kernels, respectively. θ controls the relative impacts of spectral and spatial features on the final kernel. The kernel for spectral features is a Gaussian radial basis kernel:

(K spectrum σ ) ij = κ δ (y i , y j ) = exp(- y i -y j 2 2 2σ 2 ), (4.3)
where κ denotes the Gaussian radial basis kernel and σ the variance of the corresponding Gaussian distribution. The kernel for spatial features is also a Gaussian radial basis kernel:

(K space σ ) ij = κ δ (y si , y sj ) = exp(- y si -y sj 2 2 2σ 2 ). (4.4)
However, the pixel values

y si ∈ R B calculated in K space σ
are those of the pre-processed 3-D image Y s containing the 'labels' of morphological information. The pre-processing aims to detect spatial features from the image having the most reliable morphological information. We assume that all the data points are generated from a mixture of a limited number CHAPTER 4. ROI-WISE MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT of Gaussian distributions. Although materials have very different attenuation properties at different bins, the pixel values at each bin still obey the Gaussian distribution, of which the parameters and weights are different at different bins. More precisely, we firstly divide pixels in each spatial image by a classical classification method involving GMM, which is a probabilistic model that assumes that all the data points are generated from a mixture of a limited number of Gaussian distributions:

p M (y b ) = K ∑ i=1 α i p(y b |m G i , σ i ), (4.5)
where p(y b |m G k , σ i ) represents the i-th Gaussian distribution with means m G i and covariance σ i , α i the corresponding weights, K the total number of components, and y b the b-th bin value of pixel y in the 3-D image. The models are known a priori; they are Gaussian. But, their parameters and mixture weights are unknown. The unknown parameters (mean m G , covariance σ) in Gaussian models as well as the weights α were automatically estimated via the expectation maximization (EM) algorithm [START_REF] Mclachlan | The EM algorithm and extensions[END_REF]. Meanwhile, the parameters are calculated independently at different bins.

Then, the spatial image at each energy bin is segmented into K areas by classification. In our case the probability density function p M (y b ) of GMM is used to estimate the reliability of segmentation results. More precisely, the reliability of segmentation is evaluated by the optimal value of the loglikelihood cost function:

LL(Y b ) = ln N P ∏ i=1 p M (y b i ) , (4.6)
where N P denotes the total number of pixels in the two-dimensional (2-D) image at each bin. The segmentation at certain energy bin with larger loglikelihood value is believed to be more reliable, and then the detected edges in the image at that energy bin are taken as the common edges for all the spatial images at different bins. Note that the images at different bins should be normalized (pixel value/maximum value at that bin) before comparing loglikelihood values. After that, the mean value of pixels inside each segmented area at each bin is calculated:

ȳb {π k } = 1 N k N k ∑ i=1 y b i , s.t. y b i ∈ {π k }, (4.7)
where y b i ∈ {π k } represents the i-th pixel value in the k-th segmented area at the b-th bin, and N k the total number of pixels insider the k-th area. Then, a new 3-D image Y s (the subscript 's' refers to 'spatio') is produced by assigning its pixel value the corresponding mean value ȳb {π k } :

y b si = ȳb {π k } , s.t. y b si ∈ {π k }, (4.8)
where y b si ∈ {π k } represents the i-th pixel value of Y s in the k-th segmented area at the b-th bin. In other words, a 'label' is attributed to each pixel associated with its spatial features. Therefore, each multi-dimensional pixel is now assigned to two different values: the original value (y) containing spectral feature and the mean value (y s ) containing 4.2. ROI-WISE MATERIAL DECOMPOSITION morphological feature, of which the features can be extracted by different kernels and fused together by our final kernel in equation (4.2).

We have in total three k-means clustering procedures in the spatio-energy segmentation. Both spectral and morphological features are learned using clustering methods. Specifically, the spectral features are learned based on the vector classification (i.e. to classify vectors containing energy information), while the morphological features are learned based on the classification of mixture of Gaussian distributions (from the bin with the highest likelihood). Obviously, the two clustering procedures utilize different features. Finally, the morphological and spectral features are combined together using kernel kmeans. As illustrated in figure 4.3, there are three clusterings in the second step of basis material optimization: clustering of spectral features (vector clustering), clustering of spatial features at each bin for the calculation of the highest likelihood, and clustering based on both spectral and spatial kernels. The spectral features are learned from vector information, while the spatial features utilize bin-by-bin information. The second step of basis material optimization is to update the initialized decomposition matrix under the principle of keeping the minimum needed number of basis materials (with respect to ground-truth) while excluding as many as possible. To do this, we introduce a coarse material decomposition in each ROI, which exploits the sparse nature by L-1 norm (lasso [Tibshirani, 1996]):

argmin X 1 2 Y -MX 2 F + λ X 1 , (4.9)
where 1 2 Y -MX 2 F is the data fidelity term calculated in terms of Frobenius Norm • F and λ the Lagrange multiplier. The coarse material decomposition method is subject to obvious unreliability of detecting materials and poor accuracy [START_REF] Xie | Material Decomposition in X-ray Spectral CT Using Multiple Constraints in Image Domain[END_REF]. It can nevertheless help indicate us which basis materials should be considered nonexistent, while the others are more plausible under certain selection criterion. Therefore, we propose a relative population thresholding (RPT) method to determine which basis material deserves to be selected. The RPT method is based on the population percentage defined as the ratio of the number of pixels containing certain decomposed material N x to the total number of pixels N ROI in each ROI (i.e. one pixel 'contains' a material means the decomposed mass density of this material is a positive number at the pixel); only the materials with percentage above a threshold will be preserved in each ROI. The physical meaning of RPT is partly involved with the aforementioned local property in both spectral and morphological domains. Local property limits the distribution of basis materials, which means that a corresponding minimum population percentage for all the CHAPTER 4. ROI-WISE MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT ROIs should exist. In other words, the threshold of population percentage is associated with the aggregation degree of materials. After basis material optimization, we obtain a new M from M in the k-th ROI, as:

Mk ← (M) α , i f N xαk N ROIk ≥ T s.t. argmin X k 1 2 Y k -MX k 2 F + λ X k 1 , (4.10)
where Mk ∈ R B×M denotes the optimized decomposition matrix containing M entries in the k-th ROI, (M) α the α-th basis material in decomposition matrix M, ← the operator of assigning certain material from M to M, N xαk the number of pixels containing the α-th basis material in the k-th ROI, T the population threshold, and Y k and X k the multi-energy spatial images and coarsely decomposed materials in the corresponding ROI, respectively.

Noise-reduced composite image construction

Noises at different energy bins are independent when the charge sharing effect of PCD can be ignored. Thus, by averaging multi-energy spatial images in the dimension of energy, we obtain a composite image where noise is reduced and information is mixed:

Y c = 1 B B ∑ b=1 Y b , (4.11)
where Y b denotes the spatial image at the b-th energy bin.

Fine material decomposition

Fine material decomposition is the last step in the proposed ROI-wise material decomposition method, which involves three terms: data fidelity term based on the optimized decomposition matrix M, denoising regularization term involving the noise-reduced composite image Y c , and sparsity regularization term. Especially, the denoising regularization term is devoted to exploiting the inter-bin uncorrelation and material-dependent characteristics of noise in sCT images owning to the presence of Y c . To prevent water-like materials (e.g. water, PMMA and soft tissue, etc.) from degrading materials of interest (e.g. contrast agents), we propose to utilize the noise-reduced composite image Y c in a material-wise way. More precisely, we introduce a weight vector w ∈ R 1×M associated with relative mass attenuation coefficients w α , w = [w α μmα ], (4.12) where μmα is the mean value of effective mass attenuation coefficients of the α-th basis material for all the energy bins, μmα = 1 B ∑ B b=1 μmα (b), and w α = μmα / μmB with μmB designating the mean value of effective mass attenuation coefficients of water-like material. Actually, w highlights the importance of materials of interest by decreasing the disturbance from water-like materials, i.e. in a material-wise way. Hence, the denoising regularization term involving composite spatial image can be formulated as Y c -wX 2 TV , where • TV represents TV-norm.

EXPERIMENTS AND RESULTS

Mathematically, we formulate the fine material decomposition as:

argmin X ∑ k 1 2 Y k -Mk X k 2 F + λ 1 X k 1 + λ 2 Y ck -wX k 2 TV = argmin X 1 2 Y -MX 2 F + λ 1 X 1 + λ 2 Y c -wX 2 TV . (4.13)
where Y -MX 2 F denotes the data fidelity term, X 1 the sparsity regularization term, and Y c -wX 2 TV the denoising regularization term. The above fine material decomposition model is solved via the alternating direction method of multipliers (ADMM) iteration method [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF], Jiao et al., 2016].

Experiments and results

The performance of the proposed ROI-wise material decomposition method was evaluated on both digital and physical phantom data.

Image quality metrics

The performance of the proposed material decomposition was quantitatively evaluated using different metrics. The normalized Euclidean distance that utilized in Chapter 3 was also utilized for accuracy. Meanwhile, to emphasize the reliability of proposed method, we introduce two new criteria: the false positive (FP) and false negative (FN). The FP or FN rate is calculated as the ratio of the number of wrongly recognized pixels N FP (i.e. for materials inexistent in ground-truth but in decomposition results) or unrecognized N FN (i.e. for materials existent in ground-truth but not decomposed in results) to the total number of pixels in all the ROIs N ROIs :

FP = N FP N ROIs ; FN = N FN N ROIs . (4.14)
For sCT material decomposition, a smaller FP rate means smaller errors of confusing different materials, while a larger FN occurs when existing materials cannot be recognized.

In other words, the smaller the FP and FN, the more reliable the material decomposition.

Digital data

The digital phantom is the same with the one in Chapter 3. Two other methods were compared to the proposed ROI-wise material decomposition method: (a) the common TV method as:

argmin X 1 2 Y -MX 2 F + λ X TV ; (4.15)
(b) the Coarse method in (4.9). Note that the 'Coarse' method is also an intermediate step of the ROI-wise method, which can be utilized to evaluate the impacts of the involved basis material optimization and energy averaging-based denoising. For clarity, we denote the proposed ROI-wise material decomposition by 'ROI' method in the Figures. The decomposition results of five basis materials (K=5: water, PMMA, iron, iodine and gadolinium) are shown in figure 4.4. In contrast to the other two methods, the proposed ROI-wise material decomposition method shows better detection ability in terms of edge-preserving performance. The images of water and PMMA are important, e.g., for non-contrast imaging. Visually, there are obviously more errors in water-like materials (e.g., water and PMMA, etc) than in iron, iodine and gadolinium for all the three methods. However, compared to TV and Coarse method, the proposed ROI-wised method has better edge-preserving performance for water-like materials. In practical applications, the quantitative information of the last three materials are more useful, as a result of which we will pay more attention to the materials of interest: iron, iodine and gadolinium.

As an illustration, we list error m of iron, iodine and gadolinium for the three methods in Table 4.1. Clearly, ROI-wise method has the smallest error m for all the three materials compared to TV or Coarse method. We plot in figure 4.5 both measured and true concentrations of each vial and the corresponding linear regressions in the simulation experiment. As observed, the ROI-wise method has a constant behavior, namely the concentration of different inserts is always (except at the lowest concentration of 5 mg/cc) underestimated. In contrast, the other two methods present unpredictable behavior by either underestimating the concentration (for iron and gadolinium) or overestimating the concentration (for iodine). The results in terms of FP and FN are given in Table 4.2. The proposed ROI-wise material decomposition method has a much smaller FP compared to the TV and Coarse methods. While for FN, the three methods exhibit similar performance for iodine and gadolinium, while for iron, ROI-wise method leads to obvious larger errors.

Physical data

We show the results on both physical phantom and in vivo data which are the same with in Chapter 3. The decomposition results of three basis materials (K=3: water-like, iodine and gadolinium) on the physical phantom are shown in figure 4.6. The proposed ROI-wise material decomposition method has clearly better performance in detection and quantification compared to TV and Coarse methods. Firstly, our method shows much better morphological accuracy, even for water-like materials. The selected areas for water-like materials using both TV and ROI-wise methods, shown in figure 4.7, illustrate that the edges of the selected areas are obviously blurred for both TV and Coarse methods but are more accurate for ROI-wise method. As observed, none of the three methods has accurately decomposed iodine with concentration 1 mg/cc. For example, ROI-wise method was not able to recognize iodine inside the disk, while the other two methods were not able to separate the iodine with water or gadolinium of 2 mg/cc. 4.3, ROI-wise method has clearly smaller normalized Euclidean distance. Specially, ROI-wise method has higher accuracy for gadolinium than for iodine. We also list the mean error in each disk (mean of each disk -ground-truth of each disk) in Table 4.4 to further evaluate the decomposition performance of different concentrations. The mean errors of the ROI-wise method are always negative (i.e. underestimated) whatever the inserts, while the other two methods lead to either positive (i.e. overestimated) or negative mean errors, depending on inserts. 4.5 lists both FN and FP for different materials using the three methods among which the proposed ROI-wise method exhibits the best accuracy. Especially for FP rate, the proposed ROI-wise method can have even 88% improvement for iodine and 95% for gadolinium compared to the Coarse method. Concerning the influence of important kernel parameters including the weight of spatial kernel θ and the variance of the Gaussian distribution σ, the decomposition performance of iodine with different [θ σ 2 ] is listed in Table 4.6 (noting that gadolinium has

EXPERIMENTS AND RESULTS

Gadolinium

EXPERIMENTS AND RESULTS

the same trend for [θ σ 2 ], no longer listed here). A too small or big θ can lead to larger normalized Euclidean distance error m . An intermediate value of θ = 0.2 was selected in the experiment. σ shows less impact on the decomposition, as a result of which we chose σ 2 = 0.5 according to its overall smaller error m for different θ. λ is 1e-5 for both equation 4.10 and 4.13, with which the sparsity constraint is relatively small. To evaluate the impact of lambda on the decomposition, we give more details about the material decomposition with different lambda in Coarse method, and the results are shown in figure 4.8. The results show that the RMSE of decomposition varies relatively smoothly with the sparsity constraint when lambda is smaller than 1e-5. Specifically, the sparsity using L1-constraint (Coarse) has stable and relatively smaller influence on the decomposition results. As a result, the improvement of our results should mainly come from the coarse-fine procedure. Moreover, the influence of the relative population threshold T is illustrated in figure 4.9. Both a smaller or bigger T can lead to larger error m . However, the error varies relatively smoothly with T, especially for gadolinium. In our setup, we chose T = 0.4 that corresponds to the smallest normalized Euclidean distance. The decomposition results of the in vivo data are shown in figure 4.10. Both TV and L1 methods produce noisier results for both iodine and gadolinium. On the contrary, the proposed ROI-wise method has better detection ability for both iodine and gadolinium. Specifically, although all the three methods can detect iodine in the spine, the ROI-wise method gives the highest accuracy considering it reduce obviously noise in the background. For gadolinium, Table 4.7 lists the decomposed concentration of each tube. Clearly, the ROI-wise method presents the highest accuracy (the smallest error) for all the measured concentrations.

Discussion and conclusion

We have proposed a ROI-wise material decomposition method. The proposed method enables materials to be more reliably and accurately decomposed. This is mainly due to the introduction of spatio-spectral segmentation that allows pertinent features encoded in multi-energy sCT images to be extracted for basis materials optimization.

Generality of the method

The proposed ROI-wise method is generic in the sense that it provides a framework for obtaining material decomposition from spatial reconstructed images. It provides promising perspectives for both medical applications and industrial nondestructive evaluation, considering its high accuracy and reliability. In addition to contrast agent imaging applications, our method can also be used for non-contrast imaging in medical applications. Indeed, in this work, we focused on five basis materials: water, PMMA, iron, iodine and gadolinium. Although none of the three methods can separate water and PMMA, the proposed ROI-wised method has nevertheless better edge-preserving performance for both water and PMMA, which implies that it can improve the performance of noncontrast imaging.

Reliability and sensitivity

The results show that the ROI-wise material decomposition method favors more reliability while the sensitivity of detecting materials is somewhat sacrificed as the trade-off. This may partly explain why only materials with a concentration over certain limit could be accurately decomposed. This is the case for iodine with concentration 1 mg/cc com-CHAPTER 4. ROI-WISE MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT pared to higher concentrations (figure 4.6). These results are consistent with the high improvement of FP (88% improvement for iodine and 95% for gadolinium compared to the Coarse method on physical phantom) while similar performance of FN (even worse compared to Coarse). Although Coarse or TV method shows better FN for certain materials (e.g. Coarse and TV for iron on digital phantom, and Coarse for gadolinium on physical phantom), the two methods cannot separate materials of small concentrations or water-like materials. In other words, Coarse or TV method produced smaller FN at the cost of confusing materials of interest with water-like materials. For example, smaller FN of Coarse method compared to ROI method on physical phantom is ascribed to the fact that Coarse method was not able to detect gadolinium of 2 mg/cc from water or iodine of 1 mg/cc. The above detection limit of the proposed method is due to multiple factors, such as the performance of reconstruction and ROI segmentation. In clinical applications, although both FN and FP are preferred to be as small as possible, the focus depends on concrete situations. For example, false positive diagnoses of potential cancers create clinical uncertainty and often lead to multiple unnecessary biopsies or in certain cases surgical management of low-grade and low-volume disease [START_REF] Quon | False positive and false negative diagnoses of prostate cancer at multi-parametric prostate MRI in active surveillance[END_REF]. Conversely, false negative diagnoses in terms of missing or delaying a diagnosis can also lead to real risks [START_REF] Verbeek | What is an acceptable false negative rate in the detection of prostate cancer[END_REF].

The trade-off between sensitivity and reliability of the proposed ROI-wise method is regularized by the threshold T in the relative population thresholding. As shown in figure 4.9, too small or too large T induced the increase of decomposition errors, because smaller T leads to poor reliability (but high sensitivity). In other words, when T is too small, noise and reconstruction errors will have a strong impact on decomposition accuracy. On the opposite, excessive T will degrade the decomposition ability for materials of small concentration. Fortunately, the results show a relatively large range for the choice of T around the optimal value, which implies that the proposed method is relatively little sensitive to T.

Influence of image reconstruction quality

Finally, it is worth noting that image reconstruction quality has dramatical influence on the performance of material decomposition in image domain. Because of excessively low dose allocated to detectors, sCT reconstruction at each energy bin is a problem of low-dose CT reconstruction, which is also a challenging problem. A worse image reconstruction quality may deteriorate the performance of ROI segmentation, which in return may influence the accuracy of ROI-wise method, especially for images containing small structures (e.g., small blood vessels). We have chosen a common but efficient algorithm in the field of low-dose CT reconstruction (SART-TV) to reconstruct sCT images. Note that there are many other methods that jointly reconstruct sCT image to mitigate artifacts [START_REF] Gao | Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM)[END_REF], Yu et al., 2016, Liu et al., 2016, Zhang et al., 2017, Leng et al., 2011, Li et al., 2017], which will in turn improve the performance of image-domain material decomposition. The results of reconstruction using SART-TV still show obvious artifacts and noise. Nevertheless, even in this situation, the materials were still correctly decomposed on both digital and physical phantoms, which demonstrates the reliability of the proposed ROI-wise method.

In conclusion, we have proposed a ROI-wise material decomposition method for sCT 4.4. DISCUSSION AND CONCLUSION by jointly optimizing basis materials and reducing noise. This is achieved through spatioenergy segmentation and exploiting both morphological and spectral information in the sCT images. The results on digital and physical phantoms showed that the ROI-wise material decomposition method presents clearly higher accuracy and reliability compared to common decomposition methods based on TV or lasso regularization. In the future work, the ability of detecting low-concentration materials will further be investigated to improve the sensitivity of the method while maintaining reliability. Meanwhile, the ability of identifying small structures having small material concentration will also be investigated.

Introduction

It is a huge challenge to obtain high accuracy of material decomposition in clinical applications, especially for low-concentration materials [START_REF] Chandarana | Iodine quantification with dual-energy CT: Phantom study and preliminary experience with renal masses[END_REF], Jacobsen et al., 2019]. This is linked to the principle itself of sCT. With respect to conventional CT or dual-energy CT to some extent, sCT divides the whole energy range into so-called energy bins. More there are energy bins, the higher energy resolution (narrower energy-bin width) and more accurate material decomposition. However, more energy bins means at the same time less photons in each energy bin and consequently higher quantum noise. In other words, energy resolution of sCT is limited by a trade-off between the total number of energy bins and the image quality at each bin, which restricts the accuracy of material decomposition and ability of detecting low-concentration materials.

Most decomposition methods try to improve performance of material decomposition by exploiting the high correlation between spectral and spatial features embedded in the multi-energy images. However, almost no work can be found in the literature to enhance energy resolution of sCT by software.

In this chapter, we propose to introduce the notion of super-energy-resolution (SER) for sCT in order to improve material decomposition accuracy. The idea is to learn the relationship between spectral features of simulated data and those of physical data, then applying it to other physical data. To do it, theoretical sCT images containing abundant spectrum information, designated as SER images, are simulated according to physical phantoms. The mapping between actual sCT images and SER images are learned by utilizing a coupled dictionary learning (CDL) method in a pixel-wise way. SER images synthesized based on the learned mapping are then used for material decomposition in image-domain.

Trade-off between number of energy bins and undesired factors such as random noise

The measured signals are assumed to be corrupted by independent Poisson noise, discussed in Equation (2.7), listed again here:

M i ∼ Poisson(mean = n i (P α )), (5.1)
Noise of X-ray CT image is inversely related to the square root of the total number of photons used for reconstruction [START_REF] Leng | Noise reduction in spectral CT: Reducing dose and breaking the trade-off between image noise and energy bin selection[END_REF]. In other words, sCT suffers from stronger quantum noise for narrower bin. Meanwhile, as discussed in Section 2.2, material decomposition in image-domain can be formulated in matrix form as:

Y = MX + N, (5.2)
where Y ∈ R B×N P denotes the reconstructed multi-energy spatial images of µ with N P indicating the total number of pixels or voxels, M ∈ R B×M the decomposition matrix of µ m , X ∈ R M×N P the decomposed basis material images of mass densities ρ and N the 5.3. SUPER-ENERGY-RESOLUTION IMAGING USING PIXEL-WISE LEARNING noise. Generally, atoms in the decomposition matrix can be initialized by the effective mass attenuation coefficients [Le Huy and Molloi, 2011] according to:

μmα (i) = E∈E i n 0 (E)d i (E)µ mα (E)dE E∈E i n 0 (E)d i (E)dE , i = 1, ..., B, (5.3)
where µ mα (E) is the theoretical mass attenuation coefficient at energy E retrieved from NIST [START_REF] Saloman | X-ray attenuation coefficients (total cross sections): Comparison of the experimental data base with the recommended values of Henke and the theoretical values of Scofield for energies between 0[END_REF], and E∈E i n 0 (E)dE the total number of incident photons belonging to the i-th energy bin of width E i . μmα (i) represents the averaged value of all the theoretical mass attenuation coefficients inside the entire energy bin, which is an estimate of the true mass attenuation coefficient corresponding to that energy bin. As a result, thinner bins can produce more precise μmα (i), i.e. more precise M.

As a result, the aforementioned trade-off between the number of energy bins and noise level leads to limited energy resolution in sCT. In practice, the number of energy bins of sCT is always constrained, e.g. 5 bins for Phillips sCT prototype [START_REF] Si-Mohamed | Multicolour imaging with spectral photoncounting CT: a phantom study[END_REF], Cormode et al., 2017].

Super-energy-resolution imaging using pixel-wise learning

This section firstly presents super-energy-resolution (SER) imaging in sCT. Then, the SER image synthesis method is introduced. At last, SER-based material decomposition is described in detail.

SER imaging in sCT

We propose to improve the energy resolution by means of fully exploiting both simulation data and physical data. To do it, SER is introduced in sCT.

SER images are virtual multi-energy images with much more bins but noise-reduced compared to the actual sCT images. figure 5.1 illustrates the relationship between actual sCT, denoted by low-energy-resolution (LER) imaging, and SER imaging by respectively plotting the multi-energy values of corresponding pixels in the LER and SER images. For one pixel in the reconstructed LER images (e.g. yellow asterisks in the left of figure 5.1, designated as y L ∈ R B L ), it has 5 values for 5 energy bins, i.e. B L = 5. While for the corresponding SER pixel (yellow points in the right of figure 5.1, designated as y S ∈ R B S ), it can have much more values, e.g. 50 values for B S = 50 (30 -80 keV). Note that the choice of 50 bins corresponds to the most precise µ m we can get from NIST. That is, µ is sampled every 1 keV [START_REF] Saloman | X-ray attenuation coefficients (total cross sections): Comparison of the experimental data base with the recommended values of Henke and the theoretical values of Scofield for energies between 0[END_REF].

SUPER-ENERGY-RESOLUTION IMAGING USING PIXEL-WISE LEARNING

SER image synthesis based on coupled dictionary learning

We learn the mapping between LER and SER spectral features by utilizing CDL method in a pixel-wise way.

Pixel-wise training strategy

It is inefficient to directly learn the mapping from the whole LER and SER images or patches. In general, learning morphological features of certain organ or more is another challenge that demands abundant images with various spatial information for training. It will be expensive to produce these kinds of physical data currently. Meanwhile, the complex material components can make the learning more difficult.

We solve it by training the mapping form LER and SER pixel pairs, rather than the whole images or patches. With this strategy, the training samples only contain spectral information. Correspondingly, the physical phantoms used in our work can have simple shapes, but abundant material components.

Meanwhile, to cope with the difficulty of extracting features from noisy LER pixels, we enhance the learning by adding conventional CT images. The conventional CT images of physical phantoms are measured on the same machine but reconstructed with full spectrum, thus having lower noise level compared to with single energy bin. The conventional CT and sCT pixels are then concatenated together as one LER sample for training. As a result, now y L ∈ R (B L +1) is a combined LER pixel.

Learn the mapping by CDL

CDL was used in the field of spatial super resolution [START_REF] Yang | Image superresolution via sparse representation[END_REF], Wang et al., 2012, Hu et al., 2018]. In the theory of dictionary learning, features of a pixel y can be represented as a sparse combination of atoms inside an over-complete dictionary D. That is, y = Dα, where α is the sparse codes with very few nonzero entries:

argmin α 0 , s.t. y -Dα 2 2 ≤ (5.4)
where α 0 designates L 0 norm of α. Generally, solving optimization problem in (5.4) is NP-hard, and it can be instead solved using L 1 norm, as:

argmin α y -Dα 2 2 + λ α 1 , (5.5)
where λ is the Lagrange parameter. LER and SER pixel sets can then be separately represented using a LER dictionary D L and a SER dictionary D S : argmin (5.6) where Y L ∈ R

Λ L Y L -D L Λ L 2 F + λ L Λ L 1 , argmin Λ S Y S -D S Λ S 2 F + λ S Λ S 1 ,
Λ, D Y -D Λ 2 F + λ Λ 1 , s.t. Y = [ Y S Y L ], D = [ D S D L ], Λ = [ Λ S Λ L ].
(5.7)

Meanwhile, a transform matrix is utilized to decrease the impacts of dramatic gap between simulation and physical data: (5.8) where W denotes the transform matrix that was firstly introduced in spatial super resolution for stable cross-style image synthesis [START_REF] Wang | Semi-coupled dictionary learning with applications to image super-resolution and photo-sketch synthesis[END_REF]. The enhanced training model then becomes:

Λ S = W Λ L ,
argmin D,Λ,W Y S -D S Λ S 2 F + Y L -D L Λ L 2 F + γ Λ S -W Λ L 2 F + λ W W 2 F + λ S Λ S 1 + λ L Λ L 1 .
(5.9)

We solve the above model by the alternating direction method of multipliers (ADMM) iteration method [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF], which alternately updates coupled dictionaries D, sparse codes Λ and transform matrix W, as in Algorithm 2. It should be noted that atoms in the dictionaries learned using a pixel-wise strategy are energy-dependent, in contrary to spatial features in traditional spatial-superresolution applications. figure 5.3 illustrates the difference between spatial super resolution and super energy resolution by showing the learned features. Atoms learned from a spatial image are morphological features, while for SER, atoms in the dictionary are energy-dependent and look more like µ curves.

After training, we can easily synthesize SER pixel ŷS from its LER version y L with the learned coupled dictionaries and transform matrix, by solving the following optimiza-5.3. SUPER-ENERGY-RESOLUTION IMAGING USING PIXEL-WISE LEARNING Figure 5.3: Features Learned with dictionary learning method for spatial super resolution and super energy resolution. Note that the dictionary for sCT images (lower) is the SER dictionary. tion:

SER(y L → ŷS ) : argmin ŷS ŷS -D S α S 2 F + y L -D L α L 2 F + γ α S -W α L 2 F + λ S α S 1 + λ L α L 1 .
(5.10)

Material decomposition using SER images

The synthesized SER images are then utilized for image-domain material decomposition.

A more precise decomposition matrix M suitable for the SER images is then needed. For LER, µ m of each basis material (one column of M in equation (5.2) designated as M L ) is an effective value on B L energy bins. In contrast, M for SER (designated as M S ) has much more energy bins, e.g. B S = 50. Each column of M S is a more precise value directly retrieved from NIST, as illustrated in figure 5.4. To evaluate the decomposition performance of the synthesized SER images, we choose a common image-domain decomposition method using L 1 norm (lasso) regularization.

CHAPTER 5. SUPER-ENERGY-RESOLUTION MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT USING PIXEL-WISE LEARNING

Mathematically, we formulate the SER-based material decomposition using lasso as:

x = argmin x ŷS -M S x 2 F + λ x 1 .
(5.11)

The total algorithm of SER based material decomposition is shown in Algorithm 3. Output: x

Experiments and results

The performance of the proposed ROI-wise material decomposition method was evaluated on both digital and physical phantom data.

Image quality metrics

Both SER images synthesis and material decomposition were quantitatively evaluated. Normalized Root Mean Square Error (NRMSE) (i.e. NRMSE = RMSE/ x for the image x) and peak signal-to-noise ratio (PSNR) were utilized to assess the quality of synthesized SER images. Note that NRMSE and PSNR were respectively calculated on the whole images of iodine and gadolinium, in order to show the total synthesis performance of all concentrations. The smaller the NRMSE and the higher the PSNR, the more accurate the synthesis.

To evaluate the accuracy of material decomposition, mean and standard deviation (δ) were calculated for each dilution. Meanwhile, linear regression between the nominal (y-axial) and measured concentrations (using the calculated mean values, x-axial) was performed for all dilutions, as: y = ax + b, where a and b represent the slope and yintercept, respectively. To further assess the detection ability of contrast agents, we also calculated the limit of detection (LOD) for both iodine and gadolinium. The LOD was described by the Clinical and Laboratory Standards Institute [START_REF] Moretti | CLSI EP17-A protocol: A useful tool for better understanding the low end performance of total prostate-specific antigen assays[END_REF] as: .12) where xblank denotes mean water signal detected in the images of decomposed contrast agents, δ blank the standard deviation of the measurement and LOB the limit of blank.

LOB = xblank + 1.645δ blank , LOD = LOB -b a , ( 5

EXPERIMENTS AND RESULTS

LOB denotes the highest apparent contrast agent concentration expected to be found when replicates of a sample contain no contrast agent, which was calculated using tubes of water in each phantom. LOD is taken as the lowest concentration of a contrast agent in a sample that can be detected.

Digital data

Digital phantom data generation

The data of sCT was simulated using INSA software Virtual X-ray Imaging (VXI) [START_REF] Duvauchelle | Computer code to simulate X-ray imaging techniques[END_REF]. For LER imaging, the X-ray energy bins were set as: 30-40kev, 40-50kev, 50-60kev, 60-70kev and 70-80kev, while for SER, bin interval is equally set as 1keV in the range of 30-80 kev. The detector response function was considered ideal in the simulation. In our experiments, 11 phantoms were generated, which have the same shape but different materials. The impact of phantom shapes on decomposition was considered insignificant in this work, because SER imaging focus on spectral features rather than spatial ones. Each reconstructed phantom (one slice) has 256*256 pixels and contains five materials: water, PMMA, gadolinium (Gd) and iodine (I), one of which is shown in figure 5.5. Disks in different phantoms contain mixtures of different concentration in the range of 0-20 mg/cc for both I and Gd. As a result, 655 thousands of training pairs were extracted from 10 of the 11 phantoms for training, while the left one for testing. Note that the digital phantom for testing has the same material components and concentration with the physical phantom discussed in Chapter 3 (figure 3.9).

Results

Two kinds of sCT images were utilized to evaluate the proposed SER-based material decomposition: LER images Y L and synthesized images ŶS . The corresponding results of material decomposition are respectively denoted as X L and XS .

Firstly, we show the performance of synthesized SER images based on CDL. Four atoms in the trained dictionary for SER images are shown in figure 5 The decomposition results of four basis materials are shown in figure 5.8. In contrast to LER, SER-based material decomposition shows much higher detection ability for all the basis materials and is much similar with the ground-truth. Especially, for iodine, LER-based method cannot even separate it from the other three materials, while SERbased method shows much better precision. For more quantitative analysis, we compare the mass density of Fe, I and Gd in the corresponding rows. The results indicate that DSR has good decomposition ability for all of iron, iodine and gadolinium, as shown in figure 5.9. In contrast, L1 yields much lower density of iron and higher density of iodine compared to the theoretical values. As an illustration, we list error m of water, PMMA, iodine and gadolinium in Table 5.1. Clearly, SER-based method has smaller error m for all the four basis materials.

Physical data

Physical phantom data generation

The experiments were performed on physical data from a Philips sCT prototype [START_REF] Si-Mohamed | Multicolour imaging with spectral photoncounting CT: a phantom study[END_REF],Cormode et al., 2017]. Projection data was obtained at tube voltage 120 kVp and current 220 mA. The volumetric CT dose index (CTDTI vol ) was 22.2 mGy. The scan consists of 2400 projections; each projection has 924 parallel rays; each ray contains 5 energy bins. The energy thresholds of sCT are the same for all the physical data in this work: 30, 51, 62, 72, 81 keV. The performance of the proposed SER based material decomposition method was evaluated on both physical phantoms and in vivo data. The physical phantoms are shown in figure 5.10a, the images of which are the reconstructed gadolinium at one single bin. The upper phantom without a large ring is for training, while the lower one with the large ring for testing. sCT images of phantom with large ring have stronger quantum noise and more artifacts, due to less photons that can pass the phantom and be detected. Removable tubes containing different dilutions of iodine or gadolinium are inserted in the center of phantoms, as shown in figure 5.10b. The number on each disk refers to the concentration (mg/cc) which covers from 0.1 mg/cc to 15 mg/cc. In our experiments, both iodine (I) and gadolinium (Gd) were measured for training and testing.

Meanwhile, we use another in vivo dataset to further evaluate the proposed method. Reconstructed images of a rabbit 5 and 15 min after injection of gadolinium are shown in figure 5.11, respectively. The image of 5 min after injection is a little blurred due to movement. The spatial images of all physical data were separately reconstructed at each energy bin via a common reconstruction method: conjugate gradient method (with Reconstruction Toolkit (RTK) [START_REF] Rit | The Reconstruction Toolkit (RTK), an open-source cone-beam CT reconstruction toolkit based on the Insight Toolkit (ITK)[END_REF]). All phantoms were reconstructed into 9 slices and the size of each slice is 900*900. We learned the coupled dictionaries composed of 1024 (N D ) atoms from 400K (N t ) random LER-SER pixel pairs. Training NRMSE and PSNR (dB) of synthesized SER images at each energy bin are shown in figure 5.12. SER images of iodine have relatively higher accuracy than gadolinium, e.g. smaller NRMSE and higher PSNR, although both of them have good synthesis performance. Note that low-energy bins have relatively larger NRMSE and smaller PSNR, especially for bins of k-edge (i.e. energy bins containing k-edge). Also, we need to recover more details at the first two bins: 30 -50 keV and 51 -61 keV. The material decomposition results of synthesized SER images are shown in figure 5.13. Visually, the proposed SER-based material decomposition method presents good detection ability in terms of edge-preserving performance in training. It can substantially preserve edge information even for low-concentration iodine and gadolinium e.g. 0.5 mg/cc, as illustrated in the regions-of-interests (ROIs). Note that all the ROIs are shown with grayscale images in order to highlight morphological information.

EXPERIMENTS AND RESULTS

To quantitatively assess the decomposition, linear regressions of iodine and gadolinium for all the concentrations are illustrated in figure 5.14. Both iodine and gadolinium have clearly higher accuracy with respect to theoretical concentrations. The correlations between the measured and prepared concentrations are strongly linear for all dilutions (all R 2 ≥ 0.95). The slopes are close to 1 for both iodine (a = 0.96) and gadolinium (a = 0.97). The y-intercept values are very low, e.g. b = -0.26 for iodine and -0.38 for gadolinium. Additionally, the decomposition is very reliable because both iodine and gadolinium are always underestimated and their standard deviations are small. 5.2 lists LOD for both iodine and gadolinium, to further assess the detection limit. Both iodine and gadolinium can be detected for concentrations lower than 0.5 mg/cc in training. More specifically, the proposed SER method has better detection ability for iodine than for gadolinium.

Testing

The LER-based material decomposition method using the same lasso regularization was compared to the proposed SER-based method. Both physical phantoms with large ring and in vivo data were tested.

The material decomposition results of physical phantoms with large ring are shown in figure 5.15. In contrast to LER method, the proposed SER method shows better performance in quantification and detection compared to LER method. Firstly, the proposed SER method gives much better morphological accuracy in terms of edge-preserving performance and much less noise, even for low-concentration materials, e.g. 0.5 mg/cc for both iodine and gadolinium. The edges are substantially preserved by SER method. While for LER method, the selected areas of low-concentration materials are overwhelmed by noise and cannot be detected. Secondly, more quantitatively as shown in figure 5.16, linear regressions of both I and Gd illustrate that SER-based method has much smaller errors. SER method has more precise slope (for iodine: a LER = 0.71 and a SER = 0.87; for gadolinium: a LER = 1.24 and a SER = 0.94) and y-intercept value (for iodine: b LER = 1.71 and b SER = -0.08; for gadolinium: b LER = 0.92 and b SER = -0.51). Meanwhile, SER method is more reliable with smaller standard deviations and almost all decomposed values are always underestimated. As can be observed in Table 5.2 that lists LOD values of different methods for both iodine and gadolinium, our SER method has much stronger detection ability for both iodine and gadolinium compared to LER method by producing the smaller LOD values in testing. It has better detection ability for iodine than for gadolinium, which represents the same results as in training. Meanwhile, the decomposition results of different methods on in vivo data are shown in figure 5.17. Calcium (Ca) in the spine is decomposed on the basis material iodine, due to their similarity of mass attenuation coefficients in our energy thresholds setting. For gadolinium, although the ground-truth is unknown in the in vivo data, tubes of gadolinium dilutions nearby the rabbit were evaluated alternatively. Obviously, LER method cannot separate gadolinium from iodine and has more noise for both iodine and gadolinium. On the contrary, the proposed SER method has better detection ability for both iodine (in the spine) and gadolinium (in tubes and organs). Mean and standard deviation of gadolinium in the tubes are listed in Table 5.3. Obviously, the SER method leads to more accurate decomposition for gadolinium of concentrations covering from 10 mg/cc to 2 mg/cc. 6.8 + 0.7 10.0+ 1.8 5 2.5 + 0.5 4.4 + 0.5 2 0.8 + 0.4 1.9 + 0.3

EXPERIMENTS AND RESULTS

LER

EXPERIMENTS AND RESULTS

LER

Discussion and conclusion

Our main goal is to improve the energy resolution of sCT in order to improve the decomposition accuracy and detection ability. To realize it, we have proposed a SER imaging by fully exploiting abundant spectral features embedded in both simulation and physical data. Preliminary results revealed that the proposed SER method enables accurate decomposition and high detection ability even for low-concentration materials.

The SER-based material decomposition method is reliable. Firstly, even for noisier conditions with respect to training, the proposed SER method enables high decomposition accuracy and detection ability. In our experiments, we trained the dictionaries on the phantoms without large ring, but tested on the ones with large ring which have more noise and artifacts. However, the results still show better decomposition accuracy and detection ability of the proposed method compared to the LER method, especially for iodine. Secondly, accurate decomposition of in vivo data polluted by movement (5 min after the injection) also shows the reliability of our method. Moreover, the decomposition results of both iodine and gadolinium are always underestimated for almost all concentrations.

Each single synthesized SER image is a virtual monochromatic image. Actually, for dual-energy CT, there is a similar but different imaging method called virtual monochromatic image (mono-E or VMS) [START_REF] Iyama | The Usefulness of Dual-Layer Spectral Computed Tomography for Myelography: A Case Report and Review of the Literature[END_REF], Matsumoto et al., 2011], which aims to enhance the visibility of contrast agent. A mono-E image is generally reconstructed from a pair of decomposed basis material images (e.g. a soft-tissue-like basis material and a contrast agent) in dual-energy CT [START_REF] Wu | A weighted polynomial based material decomposition method for spectral x-ray CT imaging[END_REF], McCollough et al., 2015]. The performance of mono-E is determined by the material decomposition that is processed only on actual images (LER images in the present study). In contrast, SER images are directly synthesized using spectral features learned from both simulation and physical data. As a result, our method can be regarded as a new but more accurate mono-E imaging.

In conclusion, we have proposed a SER imaging to improve the energy resolution of sCT. This is achieved through learning a mapping between simulation and physical data in a pixel-wise way and applying it to other physical data. The results on both physical phantoms and in vivo data showed that the SER-based material decomposition method presents clearly higher accurate material decomposition and detection ability compared to the LER-based method. In the future work, the performance of other image-domain decomposition methods using SER images will be investigated.

Conclusions

In this thesis, we have studied the quantitative material decomposition methods for spectral photon-counting CT. As mentioned in General Introduction, our main contribution is that we have proposed three image-domain material decomposition methods: (1) Material Decomposition in X-ray Spectral CT Using Multiple Constraints in Image Domain;

(2) ROI-Wise Material Decomposition in Spectral Photon-Counting CT; (3) Super-Energy-Resolution Material Decomposition in Spectral Photon-Counting CT Using Pixel-wise Learning.

Before presenting the achievements of the present thesis work, the basic principles of spectral CT have been reviewed in Chapter 1, including spectral CT imaging system, reconstruction methods and applications. Then, the model of material decomposition and mathematical foundations of solving the decomposition are discussed in detail.

Acquainted with the general principles of spectral CT and knowledge of material decomposition, we set up the present work following the above-mentioned three aspects:

(1) Material Decomposition in X-ray Spectral CT Using Multiple Constraints in Image Domain (Chapter 3).

We have proposed a material decomposition method for spectral CT by simultaneously exploiting the limited value of mass density, the multi-dimensional nature and high correlation of multi-energy data and the local property of neighboring pixels. To get rid of the difficulties due to similarities among basis materials, spectral and spatial redundancies are eliminated through using joint sparsity and structural low-rank. At the same time, we also add bound constraint on mass density, which is naturally true in practical applications. The results of both simulation and physical experiments revealed that the proposed DSR method has a good material decomposition ability even for very poorly conditioned problems.

(2) ROI-Wise Material Decomposition in Spectral Photon-Counting CT (Chapter 4).

We have proposed a ROI-wise material decomposition method for sCT by jointly optimizing basis materials and reducing noise. This is achieved through spatioenergy segmentation and exploiting both morphological and spectral information in the sCT images. The results on digital and physical data showed that the ROIwise material decomposition method presents clearly higher accuracy and reliability compared to common decomposition methods based on TV or lasso regularization.

(3) Super-Energy-Resolution Material Decomposition in Spectral Photon-Counting CT Using Pixel-wise Learning (Chapter 5).

We have proposed a SER imaging to improve the energy resolution of sCT. This is achieved through learning a mapping between simulation and physical data in a pixel-wise way and applying it to other physical data. The results on both digital and physical data showed that the SER-based material decomposition method presents clearly higher accurate material decomposition and detection ability compared to the LER based method.

Main Symbols

  3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i 13 Les résultats de la décomposition de matériaux sur le fantôme physique en utilisant trois méthodes. De gauche à droite: décompositions TV, Coarse et ROI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Les résultats de la décomposition de matériaux sur la base du fantôme physique. De gauche à droite: Pseudo-inverse avec TV, Coarse et ROI. . . 15 Coefficients d'atténuation linéaires: LER (gauche) et SER (droite). Les courbes de couleurs différentes indiquent des pixels différents; les pixels LER et SER correspondants ont la même couleur. . . . . . . . . . . . . . . . 16 Illustration de l'imagerie du sCT basée sur la SER. . . . . . . . . . . . . . . 17 Caractéristiques apprises avec la méthode d'apprentissage par dictionnaire pour la super résolution spatiale et la super résolution énergétique. Notons que le dictionnaire pour les images sCT (en bas) est le dictionnaire SER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Coefficients d'atténuation massique en M L et M S pour l'iode et le gadolinium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 (a) Fantômes physiques sans (gauche) et avec (droite) gros anneau: gadolinium au 4ème bande; (b) inserts: les étiquettes font référence à la concentration nominale présente dans chaque tube (mg/cc). . . . . . . . . . . . . . . 20 (a) Les résultats des différentes méthodes de décomposition de matériaux sur les fantômes physiques dans les essais; (b) les ROIs en (a) pour les matériaux à faible concentration. Les étiquettes des ROI se rapportent à la concentration nominale présente dans chaque tube (mg/cc). . . . . . . . . 21 Les régressions linéaires de(a) l'iode et (b) du gadolinium pour différentes méthodes de décomposition des matériaux dans les essais : modèle de régression linéaire, R au carré (R 2 ) et écart-type (std). . . . . . . . . . . . . 22 Les résultats des différentes méthodes de décomposition de matériaux sur les données in vivo dans les essais: a) 5 min après l'injection d'agents de contraste; b) 15 min après. Il est à noter que le calcium de la colonne vertébrale est décomposé sur la base de l'iode et que la base de gadolinium se trouve dans les tubes et les organes. . . . . . . . . . . . . . . . . . . . . . 1.1 X-ray attenuation measurement in CT. The x-ray photons emitted by the source pass through the object which is usually a patient for clinical applications. The detector measures an intensity of x-ray quantum, spatially resolved over the x-ray fan-beam. Note that a bowtie filter in front of the X-ray tube before irradiating any object is usually applied to modulate the x-ray flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Representation of the information provided by different detectors in X-ray CT. (a) Energy integrating detectors; (b) Dual-layer detectors; (c) photoncounting detectors. (Adapted form

  . . . . . . . . . . . . . . . . . . . . . . iv 1.4 Total and components of mass attenuation coefficients of (a) water and (b) lead (ρ = 11.5 g/cm 2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Mass attenuation coefficients of the chemical elements and compounds marked in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Energy integrating detectors convert x-ray into an electrical signal in a two-step process: first, a scintillator layer (GOS) converts x-ray into visible light. Photodiodes then convert light into an electrical current. . . . . 1.7 (a) Rapid kilovoltage-switching system containing only one tube and one detector. Voltage is switched rapidly between two levels; (b) dual-source CT system with two tubes and detectors mounted orthogonally in one gantry. Tubes are operated at different tube voltages; (c) dual-layer detector system with one x-ray tube running at constant voltage. Dual-energy information is derived from two layers of detector with different sensitivity profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 In a dual-layer detector, low-energy quanta are predominantly collected in the front layer. X-rays capable of passing through the front layer are predominantly of higher energy and are collected in the back detector layer. 1.9 Photon-counting detector can directly convert x-ray into an electrical current. Each photons can be detected individually and its energy can be measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 (a) Measured detector response function one threshold in one detector pixel resulting from monochromatic illumination with photon energies ranging from 25 keV to 80 keV. The count rate in the detector was about 105 photons pixel -1 s -1 . The data are normalized to an integrated count rate of 1 cts s -1 above 20 keV and (b) measurements of input monochromatic photons of 30, 40, 50, 60, 70 and 80 keV. . . . . . . . . . . . . . . . . . . . . 2.1 Similarity attenuation coefficient curves of two basis materials. The two curves represent two different mixtures, where mixture 1 and mixture 2 are respectively dilutions of gadolinium (2 mg/cc) and iodine (1 mg/cc). The circle points represent theoretical values of the mixtures at five different energy bins. At each energy bin, the blue or red points slip vertically around their theoretical value, thus creating an overlapped interval. . . . . 3.1 Digital phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The results of material decomposition based on reconstructed sTC images without Poisson noise for the three methods: 5 energy bins, 5 basis materials. Top to bottom: Pseudo-inverse with SVD, L1 and DSR. . . . . . . . . 3.3 The mass densities of iron, iodine and gadolinium: 5 energy bins. . . . . . 3.4 The results of material decomposition based on reconstructed sTC images without Poisson noise for the three methods: 5 energy bins, 20 basis materials. Top to bottom: Pseudo-inverse with SVD, L1 and DSR. . . . . . . . . 3.5 The MAEs of materials in Table 3.1. . . . . . . . . . . . . . . . . . . . . . . . 3.6 The results of material decomposition based on reconstructed sTC images without Poisson noise for the three methods: 6 energy bins, 5 basis materials. Top to bottom: Pseudo-inverse with SVD, L1 and DSR. . . . . . . . . v 3.7 The mass densities of iron, iodine and gadolinium: 6 energy bins. . . . . . 3.8 The results of material decomposition based on reconstructed sTC images with Poisson noise using the three methods: 5 energy bins, 5 basis materials. Top to bottom: Pseudo-inverse with SVD, L1 and DSR. . . . . . . . . . 3.9 (a) Physical phantom; (b) the reconstructed image of physical phantom at the first energy bin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 The reconstructed image of in vivo data at the first energy bin 15 min after the injection of gadolinium. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 The results of material decomposition based on physical phantom. Top to bottom: Pseudo-inverse with SVD, L1 and DSR. . . . . . . . . . . . . . . . 3.12 The results of material decomposition based on in vivo data. Top to bottom: Pseudo-inverse with SVD, L1 and DSR. . . . . . . . . . . . . . . . . . 4.1 Diagram of the ROI-wise material decomposition algorithm. Top to bottom: the spatio-energy segmentation based basis material optimization (blue) and noise-reduced composite image construction (green), and the fine material decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diagram of the spatio-energy segmentation. Left to right: multi-energy spatial images containing mixture 1 and mixture 2, the spatio-energy segmentation using spectral (upper-middle) and spatial (lower-middle) features, and segmented ROIs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Sketch of spatio-energy segmentation. . . . . . . . . . . . . . . . . . . . . . 4.4 The results of material decomposition using three methods on digital phantom. Left to right: TV, Coarse and ROI decompositions. . . . . . . . . . . . 4.5 The linear regressions of material decomposition using three methods for (a) iron, (b) iodine and (c) gadolinium. . . . . . . . . . . . . . . . . . . . . . 4.6 The results of material decomposition on physical phantom using three methods. Left to right: TV, Coarse and ROI decompositions. . . . . . . . . 4.7 The results of decomposed water-like materials in selected areas in figure 4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Influence of λ on the RMSE using Coarse method. . . . . . . . . . . . . . . 4.9 Influence of relative population threshold on the normalized Euclidean distance for iodine and gadolinium. . . . . . . . . . . . . . . . . . . . . . . 4.10 The results of material decomposition based on physical phantom. Left to right: Pseudo-inverse with TV, Coarse and ROI. . . . . . . . . . . . . . . . 5.1 Linear attenuation coefficients: LER (left) and SER (right). Curves with different colors denote different pixels; corresponding LER and SER pixels have the same color. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Illustration of SER based sCT imaging. . . . . . . . . . . . . . . . . . . . . . 5.3 Features Learned with dictionary learning method for spatial super resolution and super energy resolution. Note that the dictionary for sCT images (lower) is the SER dictionary. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Mass attenuation coefficients in M L and M S for iodine and gadolinium. . 5.5 Digital phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 5.6 Four atoms (the 12th, 38th, 78th and 113th) in the trained dictionary for SER images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 The performance of reconstruction: (a) RMSE; (b) PSNR . . . . . . . . . . . 5.8 The results of material decomposition based on three sCT images. Left to right: ground-truth, theoretical SER, LER and SER images . . . . . . . . . 5.9 The mass densities of iodine, gadolinium and water in corresponding dilution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 (a) Physical phantoms without (upper) and with (lower) large ring: gadolinium at the 4-th bin; (b) inserts: labels refer to the nominal concentration present in each tube (mg/cc). . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 The reconstructed image of in vivo data at the first energy bin: (a) 5 min after injection of gadolinium; (b) 15 min after. . . . . . . . . . . . . . . . . . 5.12 The (a) NRMSE and (b) PSNR of synthesized SER images for iodine and gadolinium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 The results of SER based material decomposition method on physical phantoms in training. Note that ROIs are shown in grayscale to highlight morphological information. Labels in ROIs refer to the nominal concentration present in each tube (mg/cc). . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 The linear regressions of iodine and gadolinium for SER based material decomposition in training: linear regression model, R-squared (R 2 ) and standard deviation (std). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 (a) The results of different material decomposition methods on physical phantoms in testing; (b) the ROIs in (a) for low-concentration materials. Labels in ROIs refer to the nominal concentration present in each tube (mg/cc). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16 The linear regressions of (a) iodine and (b) gadolinium for different material decomposition methods in testing: linear regression model, R-squared (R 2 ) and standard deviation (std). . . . . . . . . . . . . . . . . . . . . . . . . 5.17 The results of different material decomposition methods on in vivo data in testing: (a) 5 min after the injection of contrast agents; (b) 15 min after. Note that calcium in spine is decomposed on the basis material 'iodine', and the basis material gadolinium exists in tubes and organs. . . . . . . . vii List of Tables 1.1 Technical approaches of dual-energy CT. . . . . . . . . . . . . . . . . . . . . 3.1 Basis materials and the corresponding atomic numbers. . . . . . . . . . . . 3.2 The normalized Euclidean distance error m of different decomposition methods on digital phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The normalized Euclidean distances error m of different decomposition methods on physical phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The normalized Euclidean distance error m of different decomposition methods on digital phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The FP and FN rates of different decomposition methods on digital phantom. 4.3 The normalized Euclidean distances error m of different decomposition methods on physical phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The mean error of decomposition in each disk on physical phantom. . . . 4.5 The FP and FN rates of different decomposition methods on physical phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The normalized Euclidean distances error m of different decomposition methods on physical phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 The mean error of decomposed gadolinium in each tube on in vivo data. . 5.1 The normalized Euclidean distance error m of different decomposition methods on digital phantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The LOD of different material decomposition methods on physical phantoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The mean and standard deviation (std) of each tube on in vivo data (15 min after injection of gadolinium). . . . . . . . . . . . . . . . . . . . . . . . viii Résumé étendu en français de la thèse En 1971, Godfrey Hounsfield a construit le premier scanner de tomographie (CT-computed tomography) par rayons X. Son co-inventeur Allan McLeod Cormack, quelques années auparavant, avait utilisé la transformée de Radon et son inverse pour décrire théoriquement un appareil de tomographie radiologique et une méthode de reconstruction d'image. Pour leurs recherches, Cormack et Hounsfield ont reçu le prix Nobel de physiologie ou de médecine en 1979. Actuellement, le scanner CT classique est une technique de balayage radiologique largement utilisée dans les domaines cliniques et industriels. Il permet de répondre à des tâches diagnostiques urgentes telles que la stadification des tumeurs en oncologie, l'analyse des accidents vasculaires cérébraux aigus ou la planification de la radiothérapie, ainsi qu'à des missions industrielles telles que l'analyse de la qualité et le balayage des bagages.

  multanément l'iode et le gadolinium, surtout à faible dose de rayonnement [Si-Mohamed et al., 2017a]. De plus, de nombreux scanners DECT exposent le patient à deux faisceaux d'énergie qui peuvent entraîner une exposition au rayonnement potentiellement élevée, 1 DÉCOMPOSITION DE MATÉRIAUX DANS LE DOMAIN DE L'IMAGE DANS LA TOMOGRAPHIE SPECTRALE AUX RAYON X POUR DES

Figure 2 :

 2 Photon Counting Detectors(a)

  applications d'importance vitale et ouvre de nombreuses possibilités d'application. Le principal avantage de la technologie sCT est un meilleur échantillonnage de l'information spectrale du spectre transmis. Parmi les applications de l'utilisation de l'information spectrale, la décomposition quantitative des matériaux est le sujet le plus délicat, y compris l'imagerie du k-edge des produits de contraste et l'imagerie sélective d'autres matériaux auparavant impossibles à distinguer. De plus, l'imagerie monochromatique virtuelle est une autre méthode d'utilisation de l'information spectrale pour améliorer la qualité de l'image. D'autres améliorations du sCT par rapport au CT classique, comme une résolution spatiale plus élevée, une exposition au rayonnement plus faible et un bruit plus faible, devraient également apporter des avantages pour l'imagerie diagnostique. La première partie de ce manuscrit, qui décrit le contexte nécessaire pour les chapitres suivants: Chapitre1, intitulé " Principles of Spectral CT ", les principes du CT par rayons X commune sont présentés, y compris la source des rayons X et l'interaction des rayons X avec les matériaux; le principe de différents détecteurs pour différents types de CT par rayons X est introduit, y compris le CT classique à énergie intégrée, DECT et sCT; les applications du nouvelle sCT sont introduites. Chapitre2, intitulé " Models of Material Decomposition and Mathematical Foundations", présente un examen des méthodes de décomposition de matériaux existantes pour la tomographie spectraler par rayons-X. Les modèles de décomposition de matériaux basés sur différentes méthodes sont décrits, y compris les méthodes dans le domaine de projection, dans le domaine de l'image et à une étape; différentes méthodes d'optimisation sont examinées. La deuxième partie est consacrée aux méthodes de décomposition du matériau dans le domaine d'image: Chapitre3, intitulé "Material Decomposition in X-ray Spectral CT Using Multiple Constraints in Image Domain", en utilisant la densité de masse limitée, la parcimonie conjointe locale, et le faible rang structurel (DSR) dans le domaine de l'image, nous obtenons une INTRODUCTION GÉNÉRALE décomposition très précise de matériaux tels que le gadolinium, l'iode et le fer. Les résultats sur les données numériques et physiques du fantôme ont démontré que la méthode DSR proposée conduit à une décomposition plus précise que la méthode pseudo-inverse habituelle avec décomposition en valeur singulière (SVD-singular value decomposition) et la méthode de régularisation parcimonieuse courante avec contrainte de norme L1 (lasso).

FBP.Figure 3 :Figure 4 :Figure 6 :

 346 Figure 3: Fantôme numérique.

Figure 7 :

 7 Figure 7: L'image reconstruite des données in vivo au premier bande d'énergie 15 min après l'injection de gadolinium.

  posé. Il faut noter que ni les tailles trop grandes ni les tailles trop petites ne correspondent à la propriété locale ou n'entraînent une décomposition précise, et qu'elles doivent être réglées en fonction de la propriété de chaque donnée sCT. En conclusion, la méthode de DSR proposée présente de bonnes performances dans la séparation et la quantification simultanées de deux agents de contraste, par exemple l'iode et le gadolinium. En particulier, la bonne capacité de la DSR à distinguer des agents de contraste similaires (p. ex., l'iode, le gadolinium et le xénon) suggère son utilisation potentielle intéressante pour des applications industrielles et médicales. Dans le cadre des travaux futurs, il faudrait étudier plus à fond la propriété conjointe de sparsité et de faible rang structurel afin d'utiliser pleinement la propriété de sparsité et de faible rang cachée dans les données sCT. Enfin, une stratégie plus robuste pour le choix des paramètres de régularisation pourrait également être étudiée. Chapitre4, intitulé " ROI-Wise Material Decomposition in Spectral Photon-counting CT ", cette méthode opère par région. Elle consiste à optimiser les matériaux de base en se basant sur la segmentation spatio-énergétique des régions d'intérêt (ROI-regionsof-interests) dans les images sCT, à réduire le bruit en faisant le moyennage des images spatiales multi-énergie, et à effectuer une décomposition fine des matériaux impli-DÉCOMPOSITION DE MATÉRIAUX DANS LE DOMAIN DE L'IMAGE DANS LA TOMOGRAPHIE SPECTRALE AUX RAYON X POUR DES APPLICATIONS MÉDICALES quant une matrice de décomposition optimisée, une régularisation du débruitage et une régularisation parcimonieuse. Les résultats sur des données numériques et physiques ont montré que la méthode proposée de décomposition des matériaux ROI par ROI (ROI-wise-region-of-interests-wise) présente une fiabilité et une précision nettement supérieures à celles des méthodes de décomposition courantes fondées sur la régularisation de la variation totale (TV) ou de la norme L1. Pour rendre la décomposition de matériaux plus fiable et plus précise, nous proposons d'exploiter pleinement leur similitude spatio-énergétique dans ROIs des images sCT multi-énergie et de l'absence de corrélation inter-bandes et des caractéristiques du bruit dépendant du matériau dans les images sCT. Pour ce faire, nous effectuons une optimisation du matériau de base en réduisant l'impact de la similarité des matériaux par le biais d'une segmentation spatio-énergétique. En même temps, nous construisons une image composite à bruit réduit en introduisant un soi-disant moyennage d'énergie [Leng

Figure 9 :

 9 Figure 9: Diagramme de l'algorithme de décomposition de matériaux ROI par ROI. De haut en bas: la segmentation spatio-énergétique basée sur l'optimisation des matériaux de base (bleu) et la construction d'images composites à bruit réduit (vert), et la décomposition fine des matériaux.

Figure 11 :Figure 12 :Figure 13 :Figure 14 :

 11121314 Figure 11: Les résultats de la décomposition de matériaux à l'aide de trois méthodes sur le fantôme numérique. De gauche à droite: Décompositions TV, Coarse et ROI.

Figure 15 :

 15 Figure 15: Coefficients d'atténuation linéaires: LER (gauche) et SER (droite). Les courbes de couleurs différentes indiquent des pixels différents; les pixels LER et SER correspondants ont la même couleur.

Figure 18 :Figure 21 :Figure 22 :

 182122 Figure 17: Caractéristiques apprises avec la méthode d'apprentissage par dictionnaire pour la super résolution spatiale et la super résolution énergétique. Notons que le dictionnaire pour les images sCT (en bas) est le dictionnaire SER.

  DECT deviceswere enabled to gain information on the energy-dependent (i.e., spectral but only two) attenuation properties of the object. Different technical realizations were employed. The straightforward solution is the so-called dual-kVp technique. Two CT scans with different x-ray tube acceleration voltages are performed. The two resulting data sets contain information on the spectral x-ray attenuation characteristics of the object. These spectral data can be used to obtain additional information on the object. Nevertheless, DECT systems only perform a two-point analysis of the X-ray attenuation, which improves tissue characterization and allows quite precise iodine quantification, but is insufficient to simultaneously discriminate between iodine and gadolinium, especially at low radiation dose[Si-Mohamed et al., 2017a]. In addition, many DECT systems expose the patient to two energy beams that can result in potentially high radiation exposure, and motion can create issues for aligning the two datasets. Spectral CT (sCT) or multi-energy has recently received increasingly intensive attention, due to the new advances in photon-counting detector (PCD) technology. Compared to conventional energy integrating CT or DECT, spectral CT can count the number of photons in separated energy bins with one single exposure [Taguchi and Iwanczyk, IMAGE-DOMAIN MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT FOR MEDICAL APPLICATIONS

  Figure1.1: X-ray attenuation measurement in CT. The x-ray photons emitted by the source pass through the object which is usually a patient for clinical applications. The detector measures an intensity of x-ray quantum, spatially resolved over the x-ray fanbeam. Note that a bowtie filter in front of the X-ray tube before irradiating any object is usually applied to modulate the x-ray flux.

Figure 1

 1 Photon Counting Detectors(a)

  Figure 1.4: Total and components of mass attenuation coefficients of (a) water and (b) lead (ρ = 11.5 g/cm 2 ).

Figure 1

 1 Figure 1.5: Mass attenuation coefficients of the chemical elements and compounds marked in the legend.

Figure 1

 1 Figure 1.6: Energy integrating detectors convert x-ray into an electrical signal in a twostep process: first, a scintillator layer (GOS) converts x-ray into visible light. Photodiodes then convert light into an electrical current.

Figure 1

 1 Figure 1.7: (a) Rapid kilovoltage-switching system containing only one tube and one detector. Voltage is switched rapidly between two levels; (b) dual-source CT system with two tubes and detectors mounted orthogonally in one gantry. Tubes are operated at different tube voltages; (c) dual-layer detector system with one x-ray tube running at constant voltage. Dual-energy information is derived from two layers of detector with different sensitivity profiles.

  Figure1.8: In a dual-layer detector, low-energy quanta are predominantly collected in the front layer. X-rays capable of passing through the front layer are predominantly of higher energy and are collected in the back detector layer.

  Photon-counting detector of spectral CT

Figure 1

 1 Figure 1.10: (a) Measured detector response function one threshold in one detector pixel resulting from monochromatic illumination with photon energies ranging from 25 keV to 80 keV. The count rate in the detector was about 105 photons pixel -1 s -1 . The data are normalized to an integrated count rate of 1 cts s -1 above 20 keV and (b) measurements of input monochromatic photons of 30, 40, 50, 60, 70 and 80 keV.

Figure 2

 2 Figure 2.1: Similarity attenuation coefficient curves of two basis materials. The two curves represent two different mixtures, where mixture 1 and mixture 2 are respectively dilutions of gadolinium (2 mg/cc) and iodine (1 mg/cc). The circle points represent theoretical values of the mixtures at five different energy bins. At each energy bin, the blue or red points slip vertically around their theoretical value, thus creating an overlapped interval.

  data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.2 Physical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 80

  CHAPTER 3. MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT USING MULTIPLE CONSTRAINTS IN IMAGE DOMAIN joint sparsity and structural low-rank as:

  CHAPTER 3. MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT USING MULTIPLE CONSTRAINTS IN IMAGE DOMAINAlgorithm 1 ADMM for the DSR decomposition method Input: Y P , M Initialization:

  Figure 3.1: Digital phantom.

  Figure 3.2: The results of material decomposition based on reconstructed sTC images without Poisson noise for the three methods: 5 energy bins, 5 basis materials. Top to bottom: Pseudo-inverse with SVD, L1 and DSR.

Figure 3

 3 Figure 3.3: The mass densities of iron, iodine and gadolinium: 5 energy bins.

  The decomposition results of five existing materials are shown in figure3.4. All of the three methods has worse decomposition performance compared to the condition that only 5 basis materials exists in the decomposition matrix. Even though, our proposed DSR method shows better accuracy for gadolinium and some better detection ability of iron and iodine. For the other materials, we use the mean absolute error (MAE =∑ N i=1 |theoritical valuepratical value| i N) to quantify the accuracy. The MAEs of the other basis materials are plotted in N logarithmic coordinates, as shown in figure3.5. Compared to L1, DSR method has smaller MAE for most materials in Table3.1.

Figure 3 . 4 :

 34 Figure 3.4: The results of material decomposition based on reconstructed sTC images without Poisson noise for the three methods: 5 energy bins, 20 basis materials. Top to bottom: Pseudo-inverse with SVD, L1 and DSR.

Figure 3

 3 Figure 3.6: The results of material decomposition based on reconstructed sTC images without Poisson noise for the three methods: 6 energy bins, 5 basis materials. Top to bottom: Pseudo-inverse with SVD, L1 and DSR.

Figure 3

 3 Figure 3.8: The results of material decomposition based on reconstructed sTC images with Poisson noise using the three methods: 5 energy bins, 5 basis materials. Top to bottom: Pseudo-inverse with SVD, L1 and DSR.

  Figure 3.9: (a) Physical phantom; (b) the reconstructed image of physical phantom at the first energy bin.

Figure 3 .

 3 Figure 3.11: The results of material decomposition based on physical phantom. Top to bottom: Pseudo-inverse with SVD, L1 and DSR.

Figure 4 . 1 :

 41 Figure 4.1: Diagram of the ROI-wise material decomposition algorithm. Top to bottom: the spatio-energy segmentation based basis material optimization (blue) and noisereduced composite image construction (green), and the fine material decomposition.

CHAPTER 4 .

 4 ROI-WISE MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT

  Figure 4.2: Diagram of the spatio-energy segmentation. Left to right: multi-energy spatial images containing mixture 1 and mixture 2, the spatio-energy segmentation using spectral (upper-middle) and spatial (lower-middle) features, and segmented ROIs.

Figure 4 . 3 :

 43 Figure 4.3: Sketch of spatio-energy segmentation.

Figure 4 . 4 :

 44 Figure 4.4: The results of material decomposition using three methods on digital phantom. Left to right: TV, Coarse and ROI decompositions.

Figure 4 . 5 :

 45 Figure 4.5: The linear regressions of material decomposition using three methods for (a) iron, (b) iodine and (c) gadolinium. 93

Figure 4

 4 Figure 4.6: The results of material decomposition on physical phantom using three methods. Left to right: TV, Coarse and ROI decompositions.

Figure 4 .

 4 Figure 4.7: The results of decomposed water-like materials in selected areas in figure 4.6.

Figure 4 . 8 :

 48 Figure 4.8: Influence of λ on the RMSE using Coarse method.

CHAPTER 4 .

 4 Figure 4.9: Influence of relative population threshold on the normalized Euclidean distance for iodine and gadolinium.

Figure 4 .

 4 Figure 4.10: The results of material decomposition based on physical phantom. Left to right: Pseudo-inverse with TV, Coarse and ROI.

Algorithm 2

 2 ADMM of training the coupled dictionaries Input: Y L , Y S (training pairs) Initialization: D S , D L , Λ S , Λ L and W Training: while condition update sparse codes: Λ S , Λ L update coupled dictionaries: D S , D L update transform matrixes: W end Output: D S , D L and W

Figure 5 . 4 :

 54 Figure 5.4: Mass attenuation coefficients in M L and M S for iodine and gadolinium.

Figure 5 . 5 :

 55 Figure 5.5: Digital phantom.

  .6. The images can be represented by atoms multiplied by sparse coding coefficients. The results in terms of of RMSE and PSNR (dB) of synthesized SER images ŶH and interpolated up-sampled images ŶL are plotted in figure 5.7. ŶH have better precision in terms of smaller RMSE and CHAPTER 5. SUPER-ENERGY-RESOLUTION MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT USING PIXEL-WISE LEARNING higher PSNR. The synthesized SER images based on dictionary learning have obviously better precision of reconstruction, compared to the high-dimensional images interpolated based on pre-material-decomposition. It should be noticed that RMSE and PSNR are worse at the energy of k-edge.

Figure 5 . 6 :

 56 Figure 5.6: Four atoms (the 12th, 38th, 78th and 113th) in the trained dictionary for SER images

5. 4 .

 4 Figure 5.8: The results of material decomposition based on three sCT images. Left to right: ground-truth, theoretical SER, LER and SER images

Figure 5 . 9 :

 59 Figure5.9: The mass densities of iodine, gadolinium and water in corresponding dilution.

  Figure 5.10: (a) Physical phantoms without (upper) and with (lower) large ring: gadolinium at the 4-th bin; (b) inserts: labels refer to the nominal concentration present in each tube (mg/cc).

Figure 5

 5 Figure 5.11: The reconstructed image of in vivo data at the first energy bin: (a) 5 min after injection of gadolinium; (b) 15 min after.

Figure 5

 5 Figure 5.12: The (a) NRMSE and (b) PSNR of synthesized SER images for iodine and gadolinium.

5. 4 .

 4 Figure 5.13: The results of SER based material decomposition method on physical phantoms in training. Note that ROIs are shown in grayscale to highlight morphological information. Labels in ROIs refer to the nominal concentration present in each tube (mg/cc).

Figure 5

 5 Figure 5.14: The linear regressions of iodine and gadolinium for SER based material decomposition in training: linear regression model, R-squared (R 2 ) and standard deviation (std).

  Figure 5.15: (a) The results of different material decomposition methods on physical phantoms in testing; (b) the ROIs in (a) for low-concentration materials. Labels in ROIs refer to the nominal concentration present in each tube (mg/cc).

Figure 5

 5 Figure 5.16: The linear regressions of (a) iodine and (b) gadolinium for different material decomposition methods in testing: linear regression model, R-squared (R 2 ) and standard deviation (std).

Figure 5

 5 Figure 5.17: The results of different material decomposition methods on in vivo data in testing: (a) 5 min after the injection of contrast agents; (b) 15 min after. Note that calcium in spine is decomposed on the basis material 'iodine', and the basis material gadolinium exists in tubes and organs. 121

BX

  Number of energy bins d(E) Detector response function E Energy M Decomposition matrix n i Expected number of photons at the i-th bin P α Density integral of material α Decomposed basis material images Y Reconstructed multi-energy spatial images Z Atomic number µ Linear attenuation coefficient µ m Mass attenuation coefficient ρ Density λ i Expected number of photons in the i-th energy bin • F Frobenius norm 'UNIVERSITE DE LYON OPEREE AU SEIN DE L'INSA LYON NOM : XIE DATE de SOUTENANCE : 19/03/2020 (avec précision du nom de jeune fille, le cas échéant) Prénoms : Bingqing TITRE : Image-domain material decomposition in spectral photon-counting CT for medical applications NATURE : Doctorat Numéro d'ordre : 2020LYSEI021 Ecole doctorale : Electronique, électrotechnique, automatique (EEA) Spécialité : Traitement du Signal et de l'Image RESUME :La décomposition de matériaux est un problème fondamental et primordial dans la tomographie spectrale (sCT-spectral computed tomography) par rayons X basée sur des détecteurs à comptage de photons (PCD-photon counting detector). La présente thèse porte sur le développement de méthodes de décomposition de matériaux en utilisant des informations spectrale et morphologique encodées dans des images sCT multi-énergie. Dans ce cadre, trois méthodes ont été développées. Pour la première méthode, en utilisant la densité de masse limitée, la parcimonie conjointe locale, et le faible rang structurel (DSR) dans le domaine de l'image, nous obtenons une décomposition très précise de matériaux tels que le gadolinium, l'iode et le fer.Les résultats sur les données numériques et physiques du fantôme ont démontré que la méthode DSR proposée conduit à une décomposition plus précise que la méthode pseudo-inverse habituelle avec décomposition en valeur singulière (SVD-singular value decomposition) et la méthode de régularisation parcimonieuse courante avec contrainte de norme L1 (lasso). La deuxième méthode opère par région. Elle consiste à optimiser les matériaux de base en se basant sur la segmentation spatioénergétique des régions d'intérêt (ROI-regions-of-interests) dans les images sCT, à réduire le bruit en faisant le moyennage des images spatiales multi-énergie, et à effectuer une décomposition fine des matériaux impliquant une matrice de décomposition optimisée, une régularisation du débruitage et une régularisation parcimonieuse. Les résultats sur des données numériques et physiques ont montré que la méthode proposée de décomposition des matériaux ROI par ROI (ROI-wiseregion-of-interests-wise) présente une fiabilité et une précision nettement supérieures à celles des méthodes de décomposition courantes fondées sur la régularisation de la variation totale (TV) ou de la norme L1. Dans la troisième méthode, nous proposons la notion d'imagerie sCT à super-résolution énergétique (SER-super-energy-resolution), qui est réalisée en établissant la relation entre la simulation et les fantômes physiques au moyen d'un apprentissage par dictionnaire couplé, de manière pixel par pixel. L'efficacité de ces méthodes proposées a été validée sur des données de fantômes numériques, de fantômes physiques et in vivo. Les résultats montrent que, pour la même méthode de décomposition de matériaux utilisant la régularisation par lasso, l'imagerie à super-résolution énergétique proposée présente une précision de décomposition et un pouvoir de détection beaucoup plus élevé que ce que peut fournir la machine sCT actuelle.MOTS-CLÉS: Tomographie, Rayons X, CT spectrale, Détecteur à comptage des photons, Décomposition de matériaux, Apprentissage machine, Représentation éparse Laboratoire (s) de recherche : CREATIS Directeur de thèse: ZHU Yuemin Président de jury : Composition du jury : LAQUERRIERE Patrice, VINCENT Nicole, PIERE-SIMON Jouk, ZHU Yuemin, KAFTANDJIAN Valérie, DUVAUCHELLE Philippe

  DÉCOMPOSITION DE MATÉRIAUX DANS LE DOMAIN DE L'IMAGE DANS LA TOMOGRAPHIE SPECTRALE AUX RAYON X POUR DES APPLICATIONS MÉDICALES Le sCT peut compter le nombre de photons dans des bandes à énergie multiple séparées avec une seule exposition, ce qui est conceptuellement avantageux par rapport à DECT. A l'heure actuelle, DECT a connu une grande gamme d'applications cliniques, p. ex. imagerie virtuelle mono-énergétique, élimination automatisée des os et des plaques en angiographie par CT, imagerie virtuelle sans contraste et sans calcium, caractérisation des calculs rénaux et distinction entre la goutte et la pseudo goutte

  DANS LE DOMAIN DE L'IMAGE DANS LA TOMOGRAPHIE SPECTRALE AUX RAYON X POUR DES APPLICATIONS MÉDICALESpixels voisins ont des matériaux similaires, nous exploitons la sparsité conjointe dans de petits voisinages locaux. En même temps, nous utilisons aussi la propriété structurelle de rang inférieur selon laquelle les matériaux décomposés dans une petite région locale correspondante devraient partager le même bord ('edge'), en raison de la propriété locale selon laquelle, dans une région locale, un matériau décomposé est soit présent soit absent à chaque pixel de la région. Enfin, comme la densité de masse des matériaux a toujours une valeur limitée, nous introduisons une contrainte liée à la densité de masse. Tout cela nous amène à proposer une méthode de décomposition de matériaux basée sur la densité de masse limitée, la parcimonie conjointe locale, et le faible rang structurel.

Toutes les méthodes citées ci-dessus ont été développées dans des cas surdéterminés; aucun travail n'est disponible pour la décomposition de matériaux dans des cas sous-déterminés où le nombre de matériaux de base est supérieur à celui des bandes d'énergie.

En fait, dans la pratique, un objet contient toujours de multiples matériaux et nous ne savons pas a priori quel est le nombre exact de matériaux qu'il contient. Par conséquent, la décomposition de matériaux dans les cas sous-déterminés reste un problème pratique. Le fait que les matériaux de base à numéros atomiques proches ont des coefficients d'atténuation massique similaires vient aggraver le problème. Pour faire face à ces problèmes, nous proposons d'utiliser les caractéristiques communes de parcimonie ou sparsité

[START_REF] Nie | Efficient and robust feature selection via joint L2, 1-norms minimization[END_REF]

des informations spectrales et spatiales. De plus, comme les DÉCOMPOSITION DE MATÉRIAUX Une telle sparsité mixte de dimensions spectrale et spatiale serait intéressante pour décomposer des matériaux similaires de manière robuste. Nous adoptons alors la norme mixte L2,1 qui est particulièrement adaptée aux petites régions pour combiner les informations. La notion détaillée de la norme L2,1 est examinée au chapitre 3. La norme L2,1 peut imposer un petit nombre de lignes non nulles dans X, comparé à la norme L1

Table 1 .

 1 1 lists the compares of different technical approaches of dual-energy CT.

Table 1 .

 1 1: Technical approaches of dual-energy CT.

	Technology	Strengths	Weaknesses
	Sequential Acquisition	No special detectors required	Long acquisition time:
			sensitive to movements
			and more artifacts.
	Rapid Voltage Switching Near-simultaneous acquisition Requires specialized hardware;
		of low-and high-energy data	Noise level may differ between
			low-and high-energy images;
			Relatively high overlap of spectrum.
	Dual-Source CT	Independent tube current	Requires specialized hardware;
		and filtration optimization;	Scatter-correction.
		Relatively low degree
		of spectral overlap.
	Dual-layer Detector	Simultaneous acquisition	Requires specialized hardware;
		of low-and high-energy data	Relatively high spectrum overlap;
			Noise level may differ between
			low-and high-energy images.

Table 3 .

 3 1: Basis materials and the corresponding atomic numbers.

	Category 1	M Z	Na 11	Mg 12	Al 13	Si 14 15 16 17 18 19 P S Cl Ar K
	Category 2	M Z	Mn 25	Fe 26	Co 27	Cu Zn 29 30
	Category 3	M Z	I 53	Xe 54	Gd 64

Table 3 .

 3 2: The normalized Euclidean distance error m of different decomposition methods on digital phantom.

	Materials	Methods SVD	L1 DSR
	Iron	49.03 0.23 0.16
	Iodine	1.91 0.37 0.18
	Gadolinium	0.36 0.31 0.30

Table 3 .

 3 3: The normalized Euclidean distances error m of different decomposition methods on physical phantom.

	Materials	Methods SVD L1 (positive) DSR
	Iodine	1.96	0.46	0.36
	Gadolinium	1.03	0.98	0.21

Table 4 .

 4 1: The normalized Euclidean distance error m of different decomposition methods on digital phantom.

	Materials	Methods	TV Coarse ROI
	Iron	0.21	0.23	0.14
	Iodine	0.28	0.37	0.15
	Gadolinium	0.26	0.31	0.12

Table 4 .

 4 2:The FP and FN rates of different decomposition methods on digital phantom.

	Materials	Methods	TV Coarse ROI
	Iron		13.6	18.4	0.02
	Iodine	FP (%)	5.2	9.6	0.01
	Gadolinium		4.9	6.1	0.01
	Iron		9.1	14.8	19.5
	Iodine	FN (%)	29.3	28.5	29.9
	Gadolinium		28.0	26.3	30.3

Table 4 .

 4 3: The normalized Euclidean distances error m of different decomposition methods on physical phantom.

	Materials	Methods	TV Coarse ROI
	Iodine	0.41	0.46	0.38
	Gadolinium	0.26	0.98	0.23

Table 4 .

 4 4: The mean error of decomposition in each disk on physical phantom.

	Materials	Methods	TV Coarse ROI
		1 (mg/cc)	0.4	0.1	-1
	Iodine	3.5 7	-0.8 -3.1	-1 -3.4	-0.6 -2.7
		14	-4.1	-4.0	-4.5
		2	-0.7	1.7	-0.3
	Gadolinium	4 8	-0.9 -2.2	1.4 1.3	-0.3 -2.2
		16	-2.4	0.8	-3.0
	Table

Table 4 .

 4 5:The FP and FN rates of different decomposition methods on physical phantom.

	Materials	Methods	TV Coarse ROI
	Iodine Gadolinium	FP (%)	7.3 28.9	11.7 84.3	1.4 4.4
	Iodine Gadolinium	FN (%)	5.1 2.0	4.2 0.7	4.5 1.8

Table 4 .

 4 6: The normalized Euclidean distances error m of different decomposition methods on physical phantom.

	θ	σ s /σ	0.2	0.4	0.6	0.8	1
	0		0.26 0.26 0.26 0.26 0.26
	0.2	0.22 0.22 0.22 0.22 0.22
	0.4	0.22 0.21 0.22 0.22 0.22
	0.6	0.20 0.26 0.21 0.21 0.22
	0.8	0.21 0.22 0.22 0.38 0.22
	1		0.21 0.22 0.22 0.21 0.39

Table 4 .

 4 7: The mean error of decomposed gadolinium in each tube on in vivo data.

	True Conc.	Methods TV Coarse ROI
	10 (mg/cc)	-3.3	-3.2	-1.0
	5	-3.8	-2.7	-0.6
	2	-2.0	-1.2	-1

  (B L +1)×N t and Y S ∈ R B S ×N t respectively designate the LER and SER pixel sets with N t pixels in the training set, D L ∈ R (B L +1)×N D and D S ∈ R B S ×N D the LER and CHAPTER 5. SUPER-ENERGY-RESOLUTION MATERIAL DECOMPOSITION IN SPECTRAL PHOTON-COUNTING CT USING PIXEL-WISE LEARNING SER dictionaries containing N D atoms, Λ L and Λ S the corresponding sparse codes, and Y -DΛ 2 F the data fidelity term calculated in terms of Frobenius Norm • F . Then, the connection between LER and SER features is established by jointly training D L and D S , formulated as:

	argmin

Table 5 .

 5 1: The normalized Euclidean distance error m of different decomposition methods on digital phantom.

	Materials	Methods LER SER
	water+PMMA	0.6	0.4
	Iodine	0.6	0.1
	Gadolinium	0.5	0.1

Table 5 .

 5 2: The LOD of different material decomposition methods on physical phantoms.

		LER	SER
			Training Testing
	I	1.28	0.34	0.56
	Gd 5.70	0.45	0.84

Table 5 .

 5 3: The mean and standard deviation (std) of each tube on in vivo data (15 min after injection of gadolinium).

		LER	SER
	True Conc. (mg/cc)	mean+std
	10

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2020LYSEI021/these.pdf © [B. Xie], [2020], INSA Lyon, tous droits réservés

Enfin, il convient de noter que la qualité de la reconstruction de l'image a une influence considérable sur la performance de la décomposition de matériaux dans le domaine de l'image. En raison de la dose excessivement faible allouée aux détecteurs, la reconstruction par CT à chaque bande d'énergie est un problème de reconstruction par CT à faible dose, qui est également un problème difficile. Une qualité de reconstruction d'image plus mauvaise peut détériorer la performance de la segmentation du ROI, ce qui en retour peut influencer la précision de la méthode par ROI, particulièrement pour les images avec de petites structures (p. ex. petits vaisseaux sanguins). Nous avons choisi un algorithme commun mais efficace dans le domaine de la reconstruction par CT à faible dose (SART-TV) pour reconstruire les images du sCT. Cependant, les résultats de la reconstruction montrent encore des artefacts et du bruit évidents. Néanmoins, même dans cette situation, les matériaux étaient encore correctement décomposés sur les fantômes numériques et physiques, ce qui démontre la robustesse et la fiabilité de la méthode proposée en termes de ROI. En conclusion, nous avons proposé une méthode de décomposition de matériaux ROI par ROI pour le sCT en optimisant conjointement les matériaux de base et en réduisant le bruit. Ceci est réalisé par la segmentation spatio-énergétique et l'exploitation des informations morphologiques et spectrales dans les images sCT. Les résultats sur les fantômes numériques et physiques ont montré que la méthode de décomposition de matériaux ROI par ROI présente une précision et une fiabilité nettement supérieures à celles des méthodes de décomposition courantes basées sur la TV ou la régularisation lasso. Dans les travaux futurs, la capacité de détecter des matériaux à faible concentration sera étudiée plus en détail afin d'améliorer la sensibilité de la méthode tout en maintenant la fiabilité. Entre-temps, la capacité d'identifier de petites structures ayant une faible concentration de matériaux sera également étudiée. Chapitre5, intitulé " Super-Energy-Resolution Material Decomposition in Spectral Photoncounting CT Using Pixel-wise Learning ", nous proposons la notion d'imagerie sCT à superrésolution énergétique (SER-super-energy-resolution), qui est réalisée en établissant la relation entre la simulation et les fantômes physiques au moyen d'un apprentissage par dictionnaire couplé, de manière pixel par pixel. L'efficacité de ces méthodes proposées a été validée sur des données de fantômes numériques, de fantômes physiques et in

traste) en DECT[START_REF] Wu | A weighted polynomial based material decomposition method for spectral x-ray CT imaging[END_REF], McCollough et al., 2015]. La performance du mono-E est déterminée par la décomposition de matériaux qui est traitée uniquement sur les images réelles (images LER dans le travail présent). En revanche, les images LER sont directement synthétisées en utilisant les caractéristiques spectrales apprises à la fois des fantômes de simulation et des fantômes physiques. Par conséquent, notre méthode peut être considérée comme une nouvelle imagerie mono-E précise. En conclusion, nous avons proposé une imagerie SER pour améliorer la résolution en énergie du sCT. Pour ce faire, nous apprenons à établir une correspondance entre la simulation et les fantômes physiques au niveau du pixel et nous l'appliquons à d'autres données physiques. Les résultats sur les fantômes physiques et les données in vivo ont montré que la méthode de décomposition de matériaux basée sur le SER présente une décomposition de matériaux et une capacité de détection nettement plus précises que la méthode basée sur le LER. Dans les travaux futurs, la performance d'autres méthodes de décomposition de domaine d'image utilisant des images SER sera étudiée.

The proposed SER imaging is achieved by introducing the assumption that if containing the same basis materials and concentrations, the mapping between LER and SER images is determined and can be learned. We realize it by learning the relationship between physical data and simulation data and then applying it to other physical data, as illustrated in figure 5.2. The physical data are measured and reconstructed as LER images, while the corresponding SER versions are produced by simulation with same shapes and components but more bins. Then, LER and SER spectral features are extracted and their relationship will be learned by a mapping which can be utilized to synthesize SER images for other in vivo data.

Perspectives

With respect to the proposed material decomposition methods for spectral photon-counting CT, several potential improvements can be carried out in the future:

(1) Material Decomposition in X-ray Spectral CT Using Multiple Constraints in Image Domain. The collaborative joint sparsity and structural low-rank should be further investigated to make full use of sparsity and low-rank property hidden in the spectral CT data. Meanwhile, a more robust strategy for the choice of regularization parameters could also be investigated.

(2) ROI-Wise Material Decomposition in Spectral Photon-Counting CT. The ability of detecting low-concentration materials may further be investigated to improve the sensitivity of the method while maintaining reliability. What's more, the ability of identifying small structures having small material concentration could also be investigated.

(3) Super-Energy-Resolution Material Decomposition in Spectral Photon-Counting CT Using Pixel-wise Learning. The performance of other image-domain decomposition methods using SER images may be investigated.

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