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

Keywords: Lattice Boltzmann method

Lattice Boltzmann Method (LBM) is a numerical tool working with a methodology based on microscopic particle models and mesoscopic kinetic equations. As a numerical tool LBM has proved its capability to simulate complex fluid flow behaviours and more recently to process medical images. In the framework of image analysis, LBM is implemented to perform denoising operation, image boundary detection and image segmentation. In addition, the lattice Boltzmann method possesses the feature of strong amenability to parallel computing, especially on low-cost powerful graphics hardware (GPU).

The main purpose of this thesis is to develop a general segmentation algorithm described by four main parts. The first part proposes an original segmentation methodology created by combining a direct volume rendering technique (2D histogram based clustering method) and a Lattice Boltzmann 3D method. The applied strategy allows to segment medical images sequences by grey-level values and gradient of grey-level values. In this part, the results are compared with the k-means clustering method, and the manually segmentation results show that the algorithm can improve both on segmentation accuracy and denoising performances. But applying this method on more complex data sets results in poor accuracy. Hence, in the second part introduces multiple thresholds based on LBM and clustering algorithm. The LBM of the first part is basically realized by solving the diffusion reaction equation. This equation takes into account a threshold that is usually calculated by the Otsu method, thereby solving the diffusion reaction equation. An innovative multi-threshold collision function is elaborated, and a new formulation for the lattice Boltzmann method is established for the clustering of 3D images. Moreover, the thresholds are fixed by the k-means clustering method. Under this new concept, 2D images from the Berkeley segmentation database and 3D volume data-set including cerebral aneurysm sequences are segmented through the new approach. The performances are evaluated with the Dice similarity coefficient, Relative volume difference, and Hausdorff distance. Then, in order to

extend the universality of the algorithm, more potential geometry and statistic properties are introduced in the LBM-based segmentation method. In addition, an innovative multivariate clustering-based collision function is introduced, which causes a new formulation of the lattice Boltzmann method for the clustering of 3D volume data-set. The centroids in the collision function are still being fixed by the k-means method. The performances are also validated by segmenting threedimensional (3D) tomography angiography images, and also evaluated with the Dice similarity coefficient and Hausdorff distance. The last part investigates the LBM-based segmentation method on parallel computing. In order to deal with large scale image, we take advantage of parallel LBM by using Graphic Processors Units (GPU) architectures to improve the performance of LBM. It has been proved efficiency can be improved by segmenting aneurysms and parent vessel walls, whole-brain data sets and stent-assisted aneurysms. The parallel segmentation method has been operated on NVIDIA graphic cards and demonstrates that the proposed method achieves at least 131 speedups under the same precision.

Keywords: segmentation algorithm, cerebral aneurysm segmentation, lattice Boltzmann method, clustering algorithm, stent-assisted aneurysm, 2D histogram GPU acceleration. 4 Fei GE Après le chapitre 1 qui donne l'état de l'art sur les travaux scientifiques portant à la fois sur le rendu volumique direct [START_REF] Drebin | Volume rendering[END_REF] et la méthode de Boltzmann sur réseau [START_REF] Mcnamara | Use of the boltzmann equation to simulate lattice-gas automata[END_REF], le chapitre 2 présente les automates cellulaires de type gaz sur réseau (Lattice Gas Cellular Automata -LGCA) précurseurs de la méthode de Boltzmann sur réseau, tels que le modèle Hardy-Pomeau-Pazzis (HPP) [Raabe, 2004] et le modèle Frisch-Hasslacher-Pomeau (FHP) [START_REF] Frisch | Lattice-Gas Automata for the Navier-Stokes Equation[END_REF], ce dernier permettant de résoudre les équa- [Sørensen, 1948], d'une différence de volume relatif (RVD) [START_REF] Altman | Measurement in medicine: the analysis of method comparison studies[END_REF]] et de la distance de

Résumé

Hausdorff [START_REF] Rockafellar | Variational analysis[END_REF]. Ensuite, la méthode de Boltzmann sur réseau est proposée dans une formulation multivariée de la fonction de collision, conduisant à une nouvelle formulation de partitionnement de volumes de données 3D. Cette formulation permet de prendre en compte des paramètres caractéristiques inusités tels que courbure, opérateur Laplacien. . . qui sont implémentés dans la fonction de collision proposée.

Le chapitre 6 étudie la mise en oeuvre de l'algorithme proposé au chapitre 5 sur carte graphique (GPU) et d'un algorithme de calcul parallèle de la méthode de

Introduction

Dans le domaine de la segmentation d'images, les équations aux dérivées partielles jouent un rôle important dans l'opération de segmentation d'images. La première utilisation d'une fonction de diffusion permettant de réduire le bruit dans une image est celle de Koenderink [Koenderink, 1984]. La segmentation d'images (2D) ou de données volumétriques (3D) en imagerie médicale nous a conduit à élaborer une méthode de segmentation d'images 3D couplant une technique de rendu volumique direct [Max, 1995] Cependant, la fonction de diffusion anisotrope P-M ou modèle P-M, met en évi-

f i ( r + ∆r, t + ∆t) = g i ( r){f i ( r) 1 τ [f eq i ( r, t) -f i ( r, t)] + (1 -g i ( r)){f i ( r + ∆r, t) + 1 τ [f eq i ( r + ∆r, t) -f i ( r + ∆r, t)]} (
f i ( r + ∆r, t + ∆t) = g i ( r){f i ( r, t) + 1 τ [f eq i ( r, t) -f i ( r, t)]} + (1 -g i ( r)){f i ( r + ∆r, t) + 1 τ [f eq i ( r + ∆r, t) -f i ( r + ∆r, t)]} + a∆t[T -ρ( r)]
(2) où f i est une fonction de distribution de particules et correspond ici au niveau de gris en chaque noeud lié à un pixel de l'image traitée. f i ( r + ∆r, t + ∆t) est la fonction de distribution des particules au noeud r + ∆r, et au temps t + ∆t, ∆r et ∆t désignent respectivement l'espacement du réseau et le pas de temps. f eq i est la fonction de distribution d'équilibre. Dans cette équation ∆t correspond à un pas de temps, ∆r à un pas spatial permettant de parcourir la distance ∆r pendant ∆t, et T à un seuil de segmentation, a est une constante. Cette équation a été proposée par Yan Wang qui permet de segmenter la lumière de l'anévrisme issue de l'image originale (Fig. 3) en utilisant la méthode de Otsu [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF].

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Fei GE Puis en 2014, Yu Chen a proposé la méthode LBGM (LBM Geodesic active contour method) dans laquelle la fonction d'évolution GAC du LBM est résolue [START_REF] Chen | Segmentation of the thrombus of giant intracranial aneurysms from CT angiography scans with lattice Boltzmann method[END_REF] Initialement, avec l'ensemble des données sur le volume, la valeur du niveau de gris de chaque pixel est recueillie, et le gradient est calculé pour chaque voxel ou pixel avec la fonction de gradient dans l'Eq.4:

f (x 0 , x 1 , ..., x n ) = ( ∂f ∂x 0 , ..., ∂f ∂x j , ..., ∂f ∂x n ) (4)
dans l'algorithme proposé, la norme du vecteur gradient est calculé via Eq.5:

| f | = ( ∂f ∂x 0 ) 2 + ... + ( ∂f ∂x j ) 2 + ( ∂f ∂x n ) 2 . ( 5
)
La construction de l'histogramme 2D IGM est réalisée à partir de deux ensem-31 Fei GE

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f i ( r + ∆r, t + ∆t) = f i ( r + ∆r, t) + 1 τ [f eq i ( r + ∆r, t) -f i ( r + ∆r, t)] + R(ρ, T j ) (6)
où f i est une fonction de distribution de particules. f i ( r + ∆r, t + ∆t) est la fonction de distribution des particules au noeud r + ∆r, et au temps t + ∆t, ∆r et ∆t désignent respectivement l'espacement du réseau et le pas de temps. f eq i est la fonction de distribution d'équilibre. ρ représente la densité et correspond ici au niveau de gris en chaque noeud lié à un pixel de l'image traitée. τ est le temps de relaxation. Dans le chapitre précédent, la méthode de segmentation implémentée avec un seuil n'est pas idéale pour le traitement de séquences d'images complexes.

En conséquence la fonction de collision du LBM est proposée dans une version spécifique, où R(ρ, T j ) représente un terme d'extra force supplémentaire (cf. Eq.6) dans le modèle LBM. Dans ce nouvel algorithme, un nouveau terme de collision à plusieurs seuils est proposé suivant l'Eq.7 , T j étant le critère principal pour réguler ρ.

R(ρ, T

j ) = α(T m -ρ), ∃m : |T m -ρ| = min|T j -ρ| (7)
où T m est le seuil le plus proche du niveau de gris pour chaque pixel. α = 2 1+ 1 1+∇ρ est une fonction non décroissante pour assurer que la procédure de propagation améliore le contraste à la frontière, représentée sur la Fig. 13.

Pour obtenir une segmentation automatique sous différentes formes, nous obtenons T j par l'algorithme de partitionnement par k-moyennes, qui est représentatif des méthodes de partitionnement de fonctions objectives basées sur des prototypes.

Le partitionnement par la méthode K-moyennes est ici utilisé. L'algorithme de partitionnement est décrit suivant l'équation 9 [MacQueen, 1967]: deuxièmes images sont les résultats de segmentation par la méthode de segmentation Chan-Vese, les troisièmes images sont les résultats de la méthode proposée.

arg min T k j=1 ρ∈T j |ρ -m j | 2 , T j = m j (
Les dernières images sont étiquetées à la main. (1) cluster 1 (2) cluster 2

(3) cluster 3 (4) cluster 4

Introduction

Image segmentation as a post processing method is becoming an important tool in many research fields such as medical imaging, computer vision, biology, etc.

This topic is to separate the region of interest (ROI) from the 2D image or 3D volumetric data-set. It is still challenging because of the increasing image varieties and the imaging devices for segmentation. The motivation is to elaborate an efficient general 3D image segmentation method. We propose a novel segmentation framework that is based on clustering algorithm [Maciejewski et al., 2013] and Lattice Boltzmann Method (LBM) within a diffusion-reaction version [START_REF] He | Theory of the lattice boltzmann method: From the boltzmann equation to the lattice boltzmann equation[END_REF] for image segmentation. The main two research fields of previous image segmentation algorithms are either difficult to fine-tune to apply to limited tasks, or difficult to have ideal computational efficiency [START_REF] Sonka | Image Processing, Analysis, and Machine Vision[END_REF]. The partial differential equation (PDE) plays an important role in the image segmentation. As a numerical method for solving PDE, the lattice Boltzmann method (LBM) is a low-order method. Compared to other solutions of PDE, it also has the advantage of being inherently adapted to the parallelization of the implemented algorithm.

In addition, the relationship between segmentation codes and the computer hardware is becoming much more closed and integrated, and the LBM can illustrate this evolution. The lattice Boltzmann method has the capability to simulate complex fluid flow behaviors [Succi, 2001] and treat medical images [START_REF] Chen | An anisotropic diffusion model for medical image smoothing by using the Lattice Boltzmann method[END_REF] [ Wang et al., 2016] [Ge andCourbebaisse, 2019] with the optimized time of computation more recently. In fact, the implementation of parallel computation code is simple for LBM, because it only locally operates the gray-level value on each node of the 3D lattice at each step for the particle propagation, without data exchange between neighbor voxels. Hence, the calculation code can be easily mapped and optimized on the thread of GPUs.

In a recent work [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF], the LBM was used to segment the thrombus part of aneurysm data-sets. The experiments and results on computed tomography angiography (CTA) images demonstrated better segmentation accuracy compared with the Chan-Vese method, Sen's model, and Luca's model. In this LBM-based segmentation method, the Otsu method [Otsu, 1979] was used to find one threshold as the diffusion penalty value in the collision function of the considered version of the lattice Boltzmann method. Although their method improved the accuracy of aneurysm segmentation, this lack of robustness of the LBM make it only used on limited applications. In this thesis, a universal LBM-based segmentation framework is proposed, which has been implemented with the 2D histogram based clustering method [Cai et al., 2013] combined with parallel lattice Boltzmann method for performing 2D images and 3D volume data-sets segmentation algorithm. In particular, the LBM within a diffusion-reaction version of the collision function [START_REF] He | Theory of the lattice boltzmann method: From the boltzmann equation to the lattice boltzmann equation[END_REF]] is dedicated to the proposed framework.

Thesis Organization

The thesis is organized as follows. After the presentation of the state of the art in chapter 1, chapter 2 presents a short history of the lattice Boltzmann method,

Introduction

Understanding the image as well as extracting image information to conduct certain tasks is a crucial application area of digital image technology. Image segmentation is expected as the first step to understand the image. In the medical field, images are acquired via various modalities such as magnetic resonance imaging (MRI), computed tomography (CT), ultrasound, etc. It can be applied as an effective tool to check the inside of the human body and allows doctors to make a more accurate diagnosis. Extracting useful information from images has become an essential task in both fields. Image segmentation can be used as an important post processing method various fields such as medical imaging, computer vision, bioinformatics, etc. The main topic of this thesis is to separate the region of interest (ROI) from the 2D image or 3D volumetric data, which is still challenging because of the increase in image types and the segmentation of imaging equipment.

The motivation of this thesis is the elaboration of a high efficiency and general 3D image segmentation method. Also, in the domain of 3D medical images segmentation, projecting 3D data to a 2D screen by rendering volume data using volume rendering [START_REF] Drebin | Volume rendering[END_REF] dividing the image into sets or clusters of pixels that are highly similar in element space. The basic operation is to examine each pixel and assign it to a cluster of feature vector values that perfectly represent its feature of interest. Regional growth is another class of region segmentation algorithms that assign pixels or adjacent regions to the identical segment if their image values are close enough via a pre-selected criterion of proximity.

The strategy of edge-based segmentation algorithms is to find the object boundaries and segment regions delineated by these boundaries. These algorithms are generally applicable to board magnitude and/or phase images produced by an on-board operator adapted to the expected image characteristics. For example, most gradient operators such as Prewitt, Kirsch or Roberts are based on the presence of an ideal gradient edge. Other contour-based segmentation techniques are composed of graph searching and contour tracking.

The following section will introduce some classic algorithms by medical imaging.

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Thresholding algorithm

Several thresholding techniques have been developed [Davis, 1975] [Fu andMui, 1981]. Typical thresholding method is called Otsu method, which [START_REF] Sezgin | Survey over image thresholding techniques and quantitative performance evaluation[END_REF]] that an optimum threshold is established by minimizing the weighted sum of within-class variances of the foreground and background pixels, it returns a single intensity threshold that separate pixels into two classes. Recall that minimization of within-class variances is tantamount to the maximization of between-class scatter. This method results in satisfactory results when the numbers of pixels in each class are close to each other, the algorithm is described as follows:

g = ω 0 1 -ω 0 × (u 0 -u) 2 (1.1) 53 Fei GE
where g is inter-class variance, the threshold is computed by maximizing inter-class intensity variance; ω 0 is foreground pixel occupancy ratio; u 0 refers to mean greylevel value of foreground; u represents mean grey-level value of image. However, it is assumed that the image is bimodal (i.e., two classes) and the method breaks down when two classes are extremely unequal (i.e., the classes possess huge different sizes) in this method. Fig. 1.2 shows that segmentation of vessel and cranial CT scan, especially on cranial, the segmentation result remains one class.

(1) Original CT scan of aneurysm, (2) segmentation result by the Otsu method.

(3) Original CTA of cranial, (4) segmentation result by the Otsu method.

Edge-based algorithm

An edge or boundary on an image can be defined by the local pixel intensity gradient. A gradient is an approximation of the first-order derivative of the image function. For a given image f (x, y) we can calculate the magnitude of the gradient can be calculated as 54 Fei GE

(1) Original CTA of blood vessels, (2) edge magnitude image with Sobel mask.

|G| = [G 2 x + G 2 y ] = [( ∂f ∂x ) 2 + ( ∂f ∂y ) 2 ].
(1.

2)

The direction is described as:

D = tan -1 ( G y G x ) (1.3)
where G x and G y stand for gradients in the x and y directions, respectively. Because the discrete nature of the digital image does not allow the direct application of continuous differentiation, the gradient can be calculated by differentiating [START_REF] Gonzalez | Digital Image Processing[END_REF]]. The amplitude and direction of the gradient can be displayed as images. The magnitude image will have a gray-level value that is proportional to the magnitude of the local intensity changes, while the direction image will have a gray-level value that indicates the direction of the maximum local gradient in the original image.

Most gradient operators in digital images involve the calculation of convolutions, such as weighted summaries of pixel intensities in local neighborhoods. The weights could be listed in the form of a digital vector, whose form corresponds to a local nearby area of the image (also known as a mask, window, or kernel). For example, in the case of a Sobel, there are two 363 masks in Eq.1.4:

G x =     -1 0 1 -2 0 2 -1 0 1     G y =     1 2 1 0 0 0 -1 -2 -1     (1.4)
The first mask is used to compute G x while the second is applied to compute 55 Fei GE G y . The gradient magnitude image is generated by combining G x and G y via Eq.3.21. Fig. 1.3 shows the edge magnitude image obtained by the sobel operator.

Region-growing algorithm

While thresholding focuses on the difference of the pixel intensity, the regional growth method searches groups of pixels with similar intensities. The growth of a region, also called region fusion, generally begins with a pixel or a group of pixels (called seeds) that belong to the interest structure. The seeds can be selected by an operator or offered by an automatic sowing procedure finishing. In the next step, the neighboring pixels are examined one by one and then added to the growth region, if they are sufficiently similar according to a uniformity test (also called the homogeneity criterion). The procedure continues until no more pixels can be added. Then, the object can be represented by all accepted pixels [START_REF] Shen | Detection and classification of mammographic calcifications[END_REF] [Torre and Poggio, 1986]. An example of the uniformity test is to compare the difference between the pixel intensity value and average intensity value over a region. If the difference is less than a predefined value, like two standard deviations of intensity in the region, the pixel will be included in the region, while it is considered as an edge pixel. The results of the regional growth depend strongly on the choice of the homogeneity criterion. If it is not correctly chosen, regions spread into or merge with other regions that do not belong to the object of interest.

Another issue related to the growth of regions is that distinct starting points may not transform into identical regions. One of advantages of regional cultivation is that It allows the spatially separated regions with the same attributes to be correctly segmented. Another advantage is that it generates connected regions.

Advanced segmentation method

Variational approaches, such as Mumford Shah model and variational level set approaches (Li et al., 2005;Bara et al., 2011;Zhang et al., 2010), provide relevant results in the medical image segmentation. The idea is to minimize an energy function via a level set evolution process, the variational approaches can evolve the curves stopping at the local or global minima of the energy function.

Some classical Level Set methods, for example the narrow band Level Set method (Lefohn et al., 2004) and Geodesic Active Contour (GAC) model (Caselles et al., 1997;Bresson et al., 2011), can detect the local gradient maximum. How-

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Algorithm 1: Region Growing Algorithm.

Input: A level-line field LLA, a seed pixel P, an angle tolerance τ , and a status variable for each pixel. Output: A set of pixels: region.

1 Add P → region 2 θ region ← cos(θ region ) 3 S x ← cos(θ region ) 4 S y ← sin(θ region ) 5 foreach pixel P ∈ region do 6 foreach pixel Q ∈Neighbourhood(P)andStatus(Q) = USED do 7 if AngleDiff(θ region , LLA(Q)) < τ then 8 Add Q → region 9 Status(Q) ← U SED 10 S x ← S x + cos(LLA(Q)) 11 S y ← S y + sin(LLA(Q))
12 θ region ← arctan(S y /S x ) 13 end 14 end 15 end 16 ever, due to their sensitivity to the strong noise in the CTA image and large time-costing in the evolution, the segmentation of medical images resulting from these methods are not acceptable. To overcome this defect, the LBM method is proposed in the framework of Chen's research [Chen, 2009].

Volume rendering algorithm

The key element of volume rendering [Max, 1995] is utilizing the analysis of image space and data space to assist rendering, which defined different tissues in medical images. Based on the mechanism design, the design of transfer function can be classified into two main areas image-centric and data-centric [Cai et al., 2013].

Data-centric methods are able to extract various features from the volume data and then assign optical properties. Image-centric methods are generally more intuitive.

There is no need for users to go through the complex and tedious trial-and-error process of manipulating the transfer function space. Instead, they directly search for the most appealing rendered image in the space of renderings.

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Image-centric transfer function

The design of TF needs to manipulate the mapping space, which makes the visualization process not intuitive. In image-centric algorithms, the user interface allows to directly interact with the rendering result to alter the optical properties [START_REF] Wu | Interactive transfer function design based on editing direct volume rendered images[END_REF]. The first image-centred TF design was proposed by [START_REF] He | Generation of transfer functions with stochastic search techniques[END_REF],where the genetic algorithm was used to produce the desired TFs based on the evaluation of images rendered by users. [START_REF] Guo | Wysiwyg (what you see is what you get) volume visualization[END_REF]

Data-centric transfer function

Data features are introduced to data-centric algorithms. For the first time, [START_REF] Bajaj | The contour spectrum[END_REF] proposed the contour spectrum to assist the user to choose relevant isovalues.This improves the feature-recognition ability of TF by extending or manipulating the feature space [START_REF] Maciejewski | Structuring feature space: A non-parametric method for volumetric transfer function generation[END_REF] including visibility [Cai et al., 2013], information divergence [Ruiz et al., 2011], feature clustering [Zhang et al., 2016] and clustering segmentation method [Ulen et al., ] based TF have intrinsic difficulties in visualizing data sets when a data are related to multiple volumetric objects. This limitation can be resolved by extending the scope of the 58 Fei GE TF domain to form 2D TFs. The intensity gradient magnitude (IGM) is the most commonly used 2D TF space [START_REF] Kindlmann | Semiautomatic generation of transfer functions for direct volume rendering[END_REF]]. As shown in Fig. 1.5, the gradient magnitude is introduced as a second dimension to ensure a better separation of the object boundaries. More sophisticated features like curvature [START_REF] Hladuvka | Curvature-based transfer functions for direct volume rendering[END_REF], classification certainty [START_REF] Lundstrom | Local histograms for design of transfer functions in direct volume rendering[END_REF], distance to a reference object [START_REF] Tappenbeck | Distance-based transfer function design: Specification methods and applications[END_REF], entity size [START_REF] Wesarg | 3d visualization of medical image data employing 2d histograms[END_REF] have been reported as a second dimension for constructing a 2D mapping space with a scalar value. The inclusion of these features provides interesting new separability of volumetric structures. The discriminatory power of TF can be further greatly enhanced by increasing its dimension beyond two. However, the difficulty of visualizing 3D and larger TF spaces makes it virtually impossible to design an effective user interface. In this regard, the community has made many attempts to develop various methods, including cluster space, machine learning, parallel coordinate plotting (PCP) and dimensional reduction.

The cluster space-based approach has spatially classified large features and transformed the volume data into a representation of the cluster space that can be manipulated by users. Clustering algorithms that have been used to classify highdimensional volumetric data contain ISODATA (Iterative Self-Organizing Data Analysis Technique) [START_REF] Tzeng | A cluster-space visual interface for arbitrary dimensional classification of volume data[END_REF]. k-means [START_REF] Caban | Texturebased transfer functions for direct volume rendering[END_REF],

density estimation [START_REF] Linsen | Surface extraction from multi-field particle volume data using multidimensional cluster visualization[END_REF] and modified dendrogram [START_REF] Wang | Modified dendrogram of attribute space for multidimensional transfer function design[END_REF].

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Conclusion

However, previous work is either difficult to fine-tune and thus cannot be applied to restricted tasks, or it is difficult to achieve the desired efficiency. As a numerical approach to solve PDE, the LBM is a low-order method with the advantage of parallelization compared with other solution of PDE. The relationship between the segmentation algorithm and hardware is becoming more integrated. Moreover, the implementation of parallel LBM is easy, because simulating the particle propagation performed by LBM only need to compute with the local operation without exchanging data between voxels. And the LBM clearly illustrates this evolution.

LBM has the capability to simulate complex fluid flow behaviors [Succi, 2001] and to treat medical images with optimized time of computation [START_REF] Chen | An anisotropic diffusion model for medical image smoothing by using the Lattice Boltzmann method[END_REF]] [Wang et al., 2016] [Ge and Courbebaisse, 2019]. The LBM within a diffusion reaction version of the collision function [START_REF] He | Theory of the lattice boltzmann method: From the boltzmann equation to the lattice boltzmann equation[END_REF]] is responsible for image processing. In a recent work [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF], LBM was used to segment the thrombus part of aneurysm data-sets, and achieve relatively better results, in which it use the Otsu method [Otsu, 1979] to find one threshold as the diffusion penalty value in collision function of LBM. The methods improved the accuracy of aneurysm segmentation. However, it should be noted that the lack of transferability is the main challenge for the LBM segmentation algorithm.

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Résumé

La méthode de Boltzmann sur réseau est développée à partir des automates à gaz en réseau (LGA) qui ont été proposés pour la première fois en 1973 [START_REF] Hardy | Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions[END_REF]

Lattice-gas cellular automata

The lattice-gas cellular automata (LGCA) model is an approach to simulate fluid flow, and firstly proposed in 1973 by Hardy, de Pazzis and pomeau (HPP), which is the original form of lattice LGCA with Boolean pseudo-fluid particles residing on a discrete two-dimensional quadratic grid [Raabe, 2004].

HPP lattice-gas cellular automata

The HPP model is the simplest LGCA, which is a 2-Dimensional cellular LGCA model over a square lattice. At each node there are four cells each associated to a link with the nearest neighbor, each cell may occupied by at most one particle. In order to solve the problems of constructing cellular automate, the update procedure is splitted into 2 parts, collision and propagation (streaming), as shown in Fig. 2.1. The HPP assume that, in the each node of the lattice there are four cells connecting the nearest neighbors which is lattice vectors or lattice velocities, each cell may be empty or occupied by at most one particle, all particles have the same mass, and after the collision the particles will change their velocity direction. The streaming and collision can be written in Eq.2.1.

n i ( r + e i , t + ∆t) = n i ( r, t) + ∆ i (2.1)
where n i ( r, t) denote the Boolean field at the discrete direction r and time t; e i is the unit lattice velocity vectors; ∆ i is collision function which can be given by:

∆ i = n i+1 n i+3 (1 -n i )(1 -n i+2 ) -n i n i+2 (1 -n i+1 )(1 -n i+3 ). (2.2) 63 Fei GE

FHP model

Frish found that it has some deviationwhile solving partical differential equations by HPP model. In 1986 Frisch, Hasslacher and Pomeau (FHP) published a hexagonal grid LGCA model which yield the incompressible Navier-Stokes equation in the macroscope limit [START_REF] Frisch | Lattice-Gas Automata for the Navier-Stokes Equation[END_REF]]. The FHP model has essential properties:

1 At each node the lattice vector has six directions which correspond to six neighbor nodes.

2 All particles have the same mass, and can move along the directions i to neighbor cell.

3 Cell can be empty or occupied by at most one particles.

4 The particle will only locally collide in the each cell.

The collision and steaming of FHP model is shown in Fig. 2.2 and its mathematical model is the same with HPP model, but has six directions in Eq.2.3. of grid, each discrete velocity may be taken at most one particle.

n i ( r + e i , t + ∆t) = n i ( r, t) + ∆ i (2.3)
(2) Higher order correlations between particles are neglected.

The Boltzmann equation

Recently, LBM has become a common tool for the numerical simulation of fluid flows [Succi, 2001]. LBM is used for the simulation of complex systems and evolves to specific domains such as image processing [START_REF] Chen | An anisotropic diffusion model for medical image smoothing by using the Lattice Boltzmann method[END_REF]. A key feature is the ability to link the interaction term in the model with a physical model. , 1988] so as to reduce the numerical noise. The Boltzmann equation uses the distribution f describe high level system, FHP model by substituting

n i → f, c i → v, ∆ → Q
gives the Boltzmann equation shown in Eq.2.4:

∂ρ ∂t + v f = Q(f, f ).
(2.4)

Q(f,

The Lattice Boltzmann equation

The Boltzmann equation has the the problem of complicated nature of collision integral, therefore Bhatnagar, Goss and Krook (BGK) has proposed an alternative expression in 1954 [START_REF] Bhatnagar | A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems[END_REF]. It simplified the collision function Q(f, f ), and distribution function changes with the collision and steaming. The LBM BGK approximation is shown in Eq.2.5.

∂f ∂t + v f = ω[f ( r, t) -f eq ( r, t)]. (2.5)
ω is a relaxation frequency which controls the rate of approaching the equilibrium. The distribution function f depends on space, velocity and time. v is discretized by set of velocities v i , so the discrete Boltzmann equation is:

65 Fei GE ∂f i ∂t + v i f i = ω[f i ( r, t) -f eq i ( r, t)] (2.6)
where f i presents the discrete distribution function, and f eq i ( r, t) is the discrete equilibrium distribution function at node r at time t:

f eq i = ω i ρ[1 + e i u c 2 s + ( e i u) 2 2c 4 s - u 2 2c 2 s ] (2.7)
where ω i is constant, ρ is mass density, c s is constant, e i is lattice velocity, u is velocity of fluid. Then keep the parameter length scale reference density, reference speed and time between particle collisions, the non-dimensionalized discrete Boltzmann equation is :

∂f i ∂t + c i f i = ω[f i ( r, t) -f eq i ( r, t)].
(2.8)

The discrete form of Eq.2.8 is:

f i (x, t + ∆t) -f i (x, t) ∆t + c ix f i (x + ∆x, t + ∆t) -f i (x, t) ∆x +c iy f i (x + ∆y, t + ∆t) -f i x, t ∆y + ... = ω(f i -f eq i )
(2.9)

While selecting the lattice spacing as ∆x, the lattice velocities at each direction is c i = ∆x/∆t. ω = 1 τ , where τ is a relaxation time. Then the lattice Boltzmann equation is:

f i (x + c i ∆t, t + ∆t) -f i (r, t) = - 1 τ [f i -f eq i ].
(2.10)

At each simulation step, iteration is mainly separated into two steps: the collision step and the streaming step:

f i ( r, t + ∆t) = f i (x, t) - 1 τ (f i (( r, t) -f eq i (( r, t)) (2.11) f i (( r + e i ∆t, t + ∆t) = f i (( r, t).
(2.12)

Lattice Boltzmann Method in 1D

After overcome all of the problems of lattice-gas cellular automata, Qian [START_REF] Qian | Lattice bgk models for navier-stokes equation[END_REF] proposed DdQm model in 1992 is LBM classic model, the simplest discrete form is in 1-dimensional space, Eq.2.13 is LBM D1Q3 model.

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e = [0, 1, -1], c s = c √ 3 , w i =        2 3 , e 2 i = 0 1 6 , e 2 i = c 2 (2.13)
where w i and c s is determined by the geometry of the lattice and isotropy the directions. D1Q3 (one dimension and three directions at each node) geometry model is shown in Fig. 2.3. We take D1Q3 as an example, particles can only reside on the nodes and move to their nearest neighbors along links. There are three types of moving particles. Particles of type 1 with lattice velocity e 2 i = 0 stay on the nodes, particles of type 2 move along the axes with lattice velocity e 2 i = c. Here is some models and its parameters.

w i =                  1 2 , e 2 i = 0 1 6 , e 2 i = c 2 1 12 , e 2 i = 4c 2
(2.14) 68 Fei GE

Lattice Boltzmann Method in 3D

LBM 3D models are introduced in this section. The LBM 3D models are cubic lattice in 3 dimensions in which velocities connecting the nodes. The D3Q9, D3Q15, D3Q19 and D3Q27 model are shown in Eq.2.17, Eq.2.18, Eq.2.19 and Eq.2.20, the lattice structures are shown in Fig. 2.7, Fig. 2.8, Fig. 2.9 and Fig. 2.10.

c 0 = (0, 0, 0) c 1 = (1, 1, 1) c 2 = (1, -1, 1) c 3 = (-1, -1, 1) c 4 = (-1, 1, 1) c 5 = (1, 1, -1) c 6 = (1, -1, -1) c 7 = (-1, -1, -1) c 8 = (-1, 1, 1)
(2.17)

Conclusion

Two of the simplest models of lattice gas automata (HPP and FHP) are presented in this chapter. The connection between the lattice automata equation and the lattice Boltzmann equation gives us a general idea of the lattice Boltzmann method.

In addition, different discrete speed models dedicated to more complex situations in 2D and 3D are also exhibited in this chapter.

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Chapter 3

Lattice Boltzmann method dedicated to image processing

Diffusion equation on image smoothing

Diffusion equation as a kind of PDE has become an important research topic for image processing. Koenderink [Koenderink, 1984] applied the diffusion function to improve the image in 1984, which is shown in Eq.3.1,

∂ρ = α 2 ρ. (3.1)
In diffusion function, ρ is the density of the diffusion material. However, Koenderink set K to the 'reference luminance' of the image. The derived image can be obtained by convolution with the Gaussian kernel:

ρ(r, r ) = e -(r-r )/4t 4πt . (3.2)
However, while denoising the boundary edges has become the blurring edges, Perona and Malik [START_REF] Perona | Scale-space and edge detection using anisotropic diffusion[END_REF] proposed the anisotropic diffusion function (P-M model) in 1990 which is expressed as:

∂ ∂t [c(x, y, t)I x ] = div[c(x, y, t)I x I] (3.3)
where c is the diffusion coefficient. the coefficient will descent while the gradient ascents, and vice versa.

Solving the diffusion equation on image smoothing by LBM

To solve the diffusion equation by LBM, the LBM equation is described as follows:

f i ( r + ∆r, t + ∆t) = f i ( r, t) + 1 τ [f eq i ( r, t) -f i ( r, t)] (3.4)
where f i presents the discrete distribution function, and f eq i ( r, t) is the discrete equilibrium distribution function at node r at time t. τ is a relaxation time.

To initialize the image, the original image is suitable as the initial value of the proposed method, and the initial values of the particle distribution function f i ( r, 0) and equilibrium function f eq i ( r, 0) are expressed as follows:

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Solving the diffusion equation on image smoothing by LBM

f i ( r, 0) = f eq i ( r, 0) = β i ρ( r, 0) (3.5)
where β is: where M ∈ [0, 1], and ρ is:

β i =                 
ρ( r, t) = i f i ( r, t) (3.7)
where f i in Eq.(3.7) gives the density map, which represents the grey value at each lattice node as well as the next iteration. f i is the density map at each lattice node and at the former iteration. f eq i represents the local equilibrium distribution which is the volume grey-level value distribution function. β i and e i are LBM D3Q19 models and represent the direction and lattice velocities are shown in Fig. 3.1. They can be expressed as follows : e 0 = (0, 0) e 1,2 , e 3,4 , e 5,6 = (±1, 0, 0), (0, ±1, 0), (0, 0, ±1) e 7,...,10 , e 11,...,14 , e 15,...,18 = (±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1).

(3.8)

The evolution function can be written as follows:

f i ( r + ∆t c i , t + ∆t) = f i ( r, t) + 1 τ [f eq i ( r, t) -f i ( r, t)].
(3.9) τ denotes as relaxation time. It shows that the time efficiency of the density distribution tends to be balanced. Moreover, it has been proved that even though the fluid is uniform and τ is larger than 0.5, the LBM still remains steady [START_REF] Sterling | Stability analysis of lattice boltzmann methods[END_REF]]. The distribution function can be expanded by multi-scale technique (Chapman-Enskog expansion) as follows: From the Eq.3.9, we calculate an approximation of f eq i by expanding the left part of Eq.3.9 by Taylor expansion in time and space:

f i = f (0) i + f (1) i + 2 f (2) i + ... (3.10) 76 Fei GE
f i ( r, t) + c i ∆t + ∂ ∂t ∆t = f i ( r, t) + 1 τ [f i ( r, t) -f eq i ( r, t)] + o 2 . (3.11)
Then based on Eq.3.10 and derivation, we find that the equation is similar to diffusion function as follow:

c i + ∂f i ∂t + f (1) i τ ∆t = 0 (3.12) then, f (1)
i = τ ∆t∂ r f i + τ ∆t∂ t f i . (3.13)
According to the conservation relation, the formula can be written as :

i [f i ( r + ∆t c i , t + ∆t) -f i ( r, t)] = 0. (3.14)
The Taylor expansion is then inserted to the left part of Eq.3.14,

i (∂ r c i f i + ∂ t f i + o 2 ) = 0 (3.15)
and therefore, 77 Fei GE

∂ t ρ = - i ∂ r c i f i . (3.16)
Next, Chapman-Enskog expansion is inserted to the equation:

∂ t ρ = - i ∂ r ( c i f (0) i + c i f (1) i ) (3.17)
the component velocities c i obey the following equations:

i c i = 0 i c i f (0) i = 0 (3.18)
and therefore,

∂ t ρ = - i ∂ r c i f (1) i (3.19)
based on Eq.3.13, we derive as follow:

∂ t ρ = - i ∂ r c i (τ ∆t∂ r f i + τ ∆t∂ t f i ) = -τ ∆t i ∂ rα ∂ r β c iα c i β f i .
(3.20)

In the Eq.3.20, we can see that

i ∂ rα ∂ r β c iα c i β f i → 2 ρ (3.21)
and finally we have

∂ρ ∂t = -τ ∆t 2 ρ. (3.22)
In LBM-BGK, the diffusion coefficient τ is an adjustable parameter range, which is τ > 0.5. According to Eq.3.11 -3.22, the LBM method offers a solution for diffusion function, which is the main application of PDE for image segmentation.

The key features of LBM are the stability and efficiency to resolve the PDE. While the LBM is difficult on the generalized application mainly due to the implicit parameter. This indicates that it is difficult for the LBM method to find the penalty threshold for volume data-sets. To obtain it, we proposed a novel collision function in the next chapter.

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LBM on image smoothing

Anisotropic diffusion function has both forward and backward diffusion processes.

The backward diffusion process of the P-M model results in a stair-casing effect which demonstrates that the grey-level value rapidly rises at the large grey-level value pixel. In 2008, Yu applied LBM for image smoothing [START_REF] Chen | An anisotropic diffusion model for medical image smoothing by using the Lattice Boltzmann method[END_REF]].

It revealed that the LBM evolution equation only has a forward diffusion process by Chapman-Enskog expansion, which can effectively overcome the stair-casing effect. The anisotropic diffusion model based on LBM is described in Eq.3.23:

f i ( r + ∆r, t + ∆t) = g i ( r){f i ( r, t) + 1 τ [f eq i ( r, t) -f i ( r, t)]} + (1 -g i ( r)){f i ( r + ∆r, t) + 1 τ [f eq i ( r + ∆r, t) -f i ( r + ∆r, t)]} (3.23)
where f i represents the discrete distribution function and corresponds to a greylevel value at each node linked to image pixel, and f i ( r + ∆r, t + ∆t) is particle distribution function in each node r+∆r, and at time t+∆t; ∆r and ∆t correspond to space and time steps. f eq i ( r, t) is the discrete equilibrium distribution function at node r at time t. τ is the relaxation time.

A semi-permeable membrane g i acting as a selective filter, is introduced to allow certain molecules to diffuse through the membrane, thereby causing it to diffuse anisotropically. g i is the possibility of crossing the membrane, and the D2Q9 model is used to show the evolution equation in Fig. 3.2.

A CT scan of the aneurysm is used to compare the detail effect between the 79Fei GE

(1) P-M method result (2) LBM based smoothing result

(3) Zoom of the P-M method result (4) Zoom of the proposed method P-M method and LBM method. The result displayed in Fig. 3.3 shows that the proposed method can diffuse images while sharpening the boundary information.

However, while utilizing the P-M anisotropic diffusion function, the P-M model clearly shows a stair-casing effect compared with the LBM method in Fig. 3.

3[3][4].

LBM anisotropic diffusion function on image segmentation

In 2009, Yu [Chen, 2009] proposed image segmentation based on LBM anisotropic diffusion function. As shown in Fig. 3.2, he used the same D2Q9 model. In order to enhance the image, a diffusion resource a∆t[T -ρ( r)] was added in the model.

The LBM evolution equation is expressed as follows :

80 Fei GE f i ( r + ∆r, t + ∆t) = g i ( r){f i ( r, t) + 1 τ [f eq i ( r, t) -f i ( r, t)]} + (1 -g i ( r)){f i ( r + ∆r, t) + 1 τ [f eq i ( r + ∆r, t) -f i ( r + ∆r, t)]} + a∆t[T -ρ( r)] (3.24)
where f i represents the discrete distribution function and corresponds to a greylevel value at each node linked to image pixel, and f i ( r + ∆r, t + ∆t) is particle distribution function in each node r+∆r, and at time t+∆t; ∆r and ∆t correspond to space and time steps. f eq i ( r, t) is the discrete equilibrium distribution function at node r at time t. τ is the relaxation time. Yan [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF] proposed a segmentation threshold T , calculated by the Otsu method, segmented the lumen portion in Fig. 3.4 from the original image. Thereafter, the gray-level values of the lumen region were replaced by the mean value outside it, LBM was applied again on it. Next, level-set was used to optimize the expanding disk in Eq.3.25 in order to define the boundary of thrombus: (1) Initialize the values of the LBM grid as the gray-level value of the image.

s = disk ∩ LBM image disk (3.25)
(2) Particle collision and streaming according.

(3) Calculate gradient flow and discrete gradient flow. (4) Update the image according to (3). ( 5) Proceed to step

(1).

The results are acceptable, but the accuracy of this model still needs to be enhanced.

Conclusion

In

Introduction

In this chapter, the most common lattice Boltzmann model is applied. The previous research on LBM dedicated to the image or volume data-set segmentation mainly possess two problems. One problem is the restriction of the Otsu method [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF] that is to locate the threshold value where the sum of foreground and background spreads reaches its minimum, which is difficult to handle complex data-sets. Another problem is the time efficiency of calculating the large amount of volume data-set. Meanwhile, LBM costs a large amount of memory footprint. Therefore, the threshold calculating from the 2D histogram based kmeans clustering method is performed as the key parameter to segment volume data-set to address the first problem.

LBM methodology and parameter

In this section, the most common lattice Boltzmann model is used. It is based on a linearized BGK collision term, as shown in Eq.4.1 [START_REF] Ge | Volume rendering and lattice-boltzmann method[END_REF]:

f i ( r + ∆r, t + ∆t) -f i ( r, t) = Ω i (4.1)
and with BGK expression of Ω i

f i ( r + ∆r, t + ∆t) -f i ( r, t) = - 1 τ [f i ( r, t) -f eq i ( r, t)] (4.2)
where f i represents the discrete distribution function and corresponds to a greylevel value at each node linked to image pixel, and f i ( r + ∆r, t + ∆t) is particle distribution function in each node r+∆r, and at time t+∆t; ∆r and ∆t correspond to space and time steps. f eq i ( r, t) is the discrete equilibrium distribution function at node r at time t. τ is the relaxation time. LBM is an iterative method performed via successive steps. The first one is the streaming of distribution of particles f i and the second is the collision phase of f i at each node of the lattice.

Anisotropic diffusion equation plays an important role in the application of the partial differential equation for image processing. However, there is rare reports about LBM for resolving the anisotropic diffusion equation.The mathematical deduction in chapter 3 shows that the global equation of this model and the diffusion reaction equation are very similar. In this chapter, a lattice Boltzmann anisotropic diffusion model with a specific diffusion resource is proposed. The specific diffusion 85 Fei GE resource is shown in Eq.4.3 where the collision function Ω i is implemented as a reaction-diffusion equation [Chen, 2009]:

f i ( r + ∆ e i r, t + ∆t) = f i ( e i r, t) + 1 τ [f i ( e i r, t) -f eq i ( e i r, t)] + R(T, ρ).

2D histogram based LBM segmentation method

In the context of TF, the successive iterations of the algorithm lead to finding the appropriate configuration for volume rendering. These one-dimensional TFs solely consider the intensity values of the image, which indicates that all voxels sharing the same intensity values are rendered in the same way. In order to find a way to distinguish those voxels, additional image properties can be taken into account for generating multi-dimensional TFs. One such property is gradient magnitude which can be easily derived from the original image data. How to select voxels with a specific combination of intensity and gradient magnitude values remains challenging. To this end, the 2D Intensity Gradient Magnitude (IGM) histogram [Cai et al., 2013] should reveal several singularities to make it easier to cluster the considered image. Typically, the x-coordinate is intensity; the y axis gradient presents magnitude, and the 2D bin brightness of the 2D histogram is the frequency showing the number of voxels with a specific combination of intensity and gradient size. They are commonly used when manual labelling of a data-set becomes expensive.

Finally, unsupervised classification currently called clustering consists of defining classes from the data without knowing the class labels. The objective is to identify classes of objects or clusters, which are more similar to each other than other clusters. Such an approach to data analysis is closely related to the creation of a data model. That is to say, defining a simplified set of attributes can provide an intuitive explanation of the relevant aspects of the data set.. Clustering methods [START_REF] Guha | Cure: an efficient clustering algorithm for large databases[END_REF] are generally more demanding than supervised approaches [Jing et al., 2007] [Camastra andVerri, 2005], but they provide more information on complex data. Current work generally focuses on the clustering method [Lloyd, 2006].

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k-means clustering

The K-means clustering is a hard-clustering unsupervised algorithm [MacQueen, 1967], which represents a typical strategy of the prototype-based objective function clustering methods. K-means clustering is utilized for importance curves clustering [START_REF] Wang | Importance-driven time-varying data visualization[END_REF]. In our algorithm, the K-means clustering method is used to

Segmentation algorithm methodology

Medical imaging makes great contributions to the detection and analysis of brain aneurysm of patients. In light of this, computed tomography angiography (CTA) can provide a diagnosis for the care and follow-up of patients with brain aneurysms.

In this chapter, the studied aneurysm is recorded on 464 slices with a 3D CTA scan (Fig. 4.4) and issued from the data base of the Thrombus project (FP7-269966).

A brain aneurysm is a vascular disorder as a small hernia due to weakened blood vessel walls, which occur mainly at the bifurcation of blood vessels. An aneurysm Using the gray value of each pixel in the image or volume data set, the gradient of each pixel or voxel can be calculated using the gradient function in Eq.4.6 :

f (x 0 , x 1 , ..., x n ) = ( ∂f ∂x 0 , ..., ∂f ∂x j , ..., ∂f ∂x n ). (4.6)
In the proposed algorithm, the norm of the gradient vector at each voxel is computed by Eq.4.7:

| f | = ( ∂f ∂x 0 ) 2 + ... + ( ∂f ∂x j ) 2 + ( ∂f ∂x n ) 2 . (4.7)
The 2D IGM histogram is constructed with two synthetic data-sets, which are gradient and grey-level values. Then, using the density map generated in each LBM iteration, the K-means clustering algorithm clusters the classes in the volume and then calculates the optimal threshold to distinguish the living tissues.

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Conclusion

This chapter presents a new segmentation algorithm in which reconstruct the reaction diffusion equation at a mesoscopic scale via the LBM method leads to the optimization of a consistent volume rendering method in regard to the complexity of the cranial medical images with cerebral aneurysms. The promising potential of the proposed method lies in the using of several thresholds, updated along with the successive iterations of the LBM code, which leading to the concept of dynamic clustering of medical images sequences and improving the calculation efficiency.

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Chapter 5

Lattice Boltzmann multi-thresholds method design

Abstract

In the image processing domain, the Lattice Boltzmann method is ba-

Introduction

In this chapter, a LBM-based segmentation algorithm is proposed using multiple thresholds calculated by the K-means clustering method. Firstly, the methodology pipeline of Lattice Boltzmann multi-thresholds method algorithm (LBMTM) is presented in Section 5.2.1. Secondly, the design of the multi-thresholds clustering collision function is given in Section 5.

Segmentation framework design

While applying the proposed method in Chapter 4 to segment a more complex volumetric data-set, it is found that the method in the last chapter does not show ideal accuracy and computational performance. There are two mainly purposes of our method:

• The definition of nearby diffuse objects (lumen, stent, boundary and the whole aneurysm).

• The segmentation result should have high contrast in the whole volume.

Segmentation methodology pipeline

However, the two purposes conflict with each other when dealing with diffuse objects. After applying LBM segmentation on the whole volume, it is impossible 97 Fei GE to segment the diffuse objects especially after the placement of the stent. Herein, a Lattice Boltzmann multi-threshold method is introduced to increase the contrast among the diffuse objects and segment the different tissues from volume data-set.

Fig. 5.1 exhibits the proposed method pipeline for image segmentation. By introducing the clustering algorithm, thresholds are continually optimized to differentiate living tissues. These thresholds are addressed to the LBM block (Fig. 5.1) which is implemented for dealing with a reaction-diffusion equation at a microscopic scale for the purpose of image processing. This innovative method continuously iterates and modifies the data-set. With the refined data-set, the IGM is clustered into several regions, and each region corresponds to a specific tissue.

Subsequently, the refined IGM and volumetric data-set are applied to render the volume and generate the segmentation result.

Multi-thresholds clustering collision function design

To remove the staggered invariants in LBM models, Qian [Qian, 1997] proposed to use f ractional propagation as an effective strategy for suppressing undesired invariants. During the propagation step, the updating procedure is described as follows:

f i ( r + ∆r, t + ∆t) = (1 -g i ( r))f i ( r + ∆r, t) + g i ( r)f i ( r, t) (5.1)
where g i ,the possibility of one voxel to go through the encountered medium, describes the f ractional propagation, which is based on encountered medium values from 0 to 1. In this algorithm, g i = 1. [Chen, 2009] suggested a diffusion resource based on Eq.5.1, and the LBM evolution equation is expressed as Eq.5.2,

f i ( r + ∆r, t + ∆t) = f i ( r, t) + 1 τ [f eq i ( r, t) -f i ( r, t)] + R(ρ, T ) (5.2)
where ρ represents the grey-value on each node. In chapter 5, only one threshold [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF] calculated and fit into the LBM-based segmentation framework.

Especially when segmenting the complex data-set (data-sets with multiple tissues), the result is not ideal. Herein, the collision function of LBM proposed in a specific version, R(ρ, T ) represents the new collision term (the extra force) in the LBM model [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF]. In the novel algorithm, multiple threshold value T j is replaced the original T , as shown in Eq.5.2.

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f i ( r + ∆r, t + ∆t) = f i ( r + ∆r, t) + 1 τ [f eq i ( r + ∆r, t) -f i ( r + ∆r, t)] + R(ρ, T j ) (5.3)
where f i ( r + ∆ r, t + ∆t) corresponds to the grey value distribution function at r + ∆ t at time t + ∆t. ∆ r and ∆t are sampling interval and time step, respectively. τ is relaxation time (1/τ is collision frequency). f eq i is local equilibrium distribution which represents the volume grey-level value distribution function. In this novel algorithm, according to Eq.5.3, a new multi-thresholds collision term is proposed, as shown in Eq.5.4. In terms of reaction function in Eq.5.3, T j is used as multi-thresholds as the main condition criterion to regulate ρ. However, multiple thresholds T j are introduced into the algorithm. A new scheme which finds the nearest threshold value is designed in Eq.5.4.

R(ρ, T j ) = α(T m -ρ), ∃m : |T m -ρ| = min|T j -ρ| (5.4)
where T m is nearest grey-level value threshold for each pixel. To achieve automatically segmentation at different forms, T j by histogram-based K-means hardclustering algorithm is obtained, which is a representative of the prototype-based objective function clustering methods. And α = 2 1+ 1 1+∇ρ is a non decreasing function to ensure the propagation procedure enhance the contrast at boundary as shown in Fig. 5.2. 2D histogram-based K-means clustering is utilized for the importance curve clustering [START_REF] Wang | Importance-driven time-varying data visualization[END_REF]. In the proposed method, the clustering method aims to compute the thresholds for clustering tissues in volume data-set.

The clustering algorithm is described as follows,

arg min T k j=1 ρ∈T j |ρ -m j | 2 , T j = m j (5.5)
where ρ is one voxel of volume data-set; m j is the center value of class T j . (ρ and m j are both multidimensional variables). T j is the thresholds that are used as parameters of LBM segmentation. 100 Fei GE

Experiments and results

Application data

The proposed segmentation method was performed on 2D images and 3D volume data-sets, the description are as follows, and the 2D images and slices of 3D volume are shown in Fig. 5.3 and Fig. 5.4:

2D images: The 2D images we use for validating our proposed segmentation algorithm are derived from the Berkeley segmentation data-set [START_REF] Martin | A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics[END_REF] that is an open source data-set with hand-labeled segmentation.

Data_ 1: The whole cardiac volume data-set aims to validate the proposed method on different types of medical images. In this chapter, the studied cardiac data-set has been recorded on 320 slices with a 3D CTA scan.

Data_ 2: The whole CT brain volume aims to test our method on the segmentation of the entire brain. In the light of this, computed tomography angiography (CTA) plays a crucial role in providing diagnosis for the care and follow-up of patients with cerebral aneurysms. In this chapter, the studied aneurysm has been recorded on 464 slices with a 3D CTA scan, and it is issued from the data base of the Thrombus project (FP7-269966).

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Data_ 3: A brain aneurysm is a vascular disorder as a small hernia due to weakened blood vessel walls, which occur mainly at the bifurcation of blood vessels. An aneurysm may rupture and thus lead to a subarachnoid hemorrhage with the consequence of high mortality rates sometimes. The aneurysm has been recorded on 131 slices with a 3D CTA scan.

Data_ 4: One of the intra-cranial aneurysm treatment methods is to use tubular support(stent) placed temporarily inside a blood vessel, or canal, which can facilitate to aid healing or relieve an obstruction. In this chapter, for precisely locating the vascular stent of the stent-assisted intra-cranial aneurysm, we use patient data which is recorded on 512 slices of 3D CTA scan. 102 Fei GE

Evaluation Function

In this chapter, Dice similarity coefficient (DSC) [Sørensen, 1948], Relative volume difference (RVD) [START_REF] Altman | Measurement in medicine: the analysis of method comparison studies[END_REF] and Hausdorff distance [START_REF] Rockafellar | Variational analysis[END_REF] are applied to assess the accuracy of our method [START_REF] Bernard | Deep Learning Techniques for Automatic MRI Cardiac Multi-structures Segmentation and Diagnosis: Is the Problem Solved[END_REF]. For data set segmentation results V sr and references V ref , the DSC and RV D are defined as: 5.8) where sup represents the upper bound(supremum) and inf represents the lower bound (infimum). In this paper we use 3D d H for evaluate the segmentation result in order to avoid management of missing segmentation problem on end slices.

DSC = 2(|V sr ∩ V ref |) (|V ref , | + |V ref |) (5.6) RV D = V sr -V ref V ref ( 5
d H (V sr , V ref ) = max{ sup Vsr∈V inf V ref ∈V d(V sr , V ref ), sup V ref ∈V inf Vsr∈V d(V sr , V ref )} (

Segmentation Result

2D images

In this section, the proposed method is compared with the Chan-Vese [START_REF] Chen | Segmentation of the thrombus of giant intracranial aneurysms from CT angiography scans with lattice Boltzmann method[END_REF] segmentation method from 106

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3D volume data-sets

Then, the proposed method on 3D cardiac volume data-set is applied. The implemented parameters of reaction-diffusion function R(T j , ρ) are thresholds T j calculated from the 2D histogram with the clustering algorithm in Eq.5.5. Based on all accomplished parameters, the histogram is computed for assessing the amount of the data. Then, for counting the intensity of the data-set, bins of the 2D histogram is computed by the grey-level value and the gradient magnitude calculated on each voxel. X-coordinate is gradient magnitude value, and Y-coordinate is grey-level value in Fig. 5.11(1). Then, using the 2D histogram as the input of the the proposed method, the 2D histogram is clustered into ths classes by clustering method in Eq.5.5, as showed in Fig. 5.11(2). Next, thresholds are used as the key parameters of the multi-thresholds collision function Eq.5.2 and Eq.5.4. Based on iter iterations, it can be clearly found that the whole volume is clustered into ths clusters. And compared with other parts, the outline of the aneurysm has a high contrast, and it is not difficult to identify nearby diffuse objects.

(1) IGM histogram (2) Clustered IGM histogram Here, a short implementation of each step on pipeline is constructed to show how we manipulate 2D histogram and apply it for segment 3D cardiac image. The 5 clusters are shown in Fig. 5.12(1)-( 5), and the corresponding 4 grey-level value thresholds of volume data are 1.3, 5.7, 28.7 and 125.0, respectively. 107 Fei GE

(1) Cluster_1 (2) Cluster_2

(3) Cluster_3 (4) Cluster_4

(5) Cluster_5 The parameters that depend on the histogram and histogram are used as inputs for a multi-threshold LBM. At each iteration of our method, we continuously optimize the grey value of each voxel through multi-thresholds collision function in Eq. (5.4). Hence we can obtain the region of interest(ROI). In Fig. 5.14, it can be seen that the aneurysm frame is segmented from the CT scan of cranial. After segmenting the aneurysm contour based on the method in the cranial part, we focus our work on the aneurysm local part. In Fig. 5.16, we applied our method on the aneurysm part. At the end of the algorithm, we obtain 109 Fei GE the contour by using a non-incremental convex hull algorithm. Then, we applied 110 Fei GE the above method to a stent-assisted aneurysm in Fig. 5.18. It is noticed that parts of the stent are not that clear especially for the aneurysm part, because the stent and vascular are close to the neighbor tissues in this region.

113

Fei GE for image segmentation [Li andTian, 2007] [Zhang andWu, 2009]. Given that, it also should consider to integrate the advantages of two or more feature spaces [START_REF] Liu | Image segmentation framework based on multiple feature spaces[END_REF]. But how to combine different features for image segmentation is still a challenging issue.

Two feature spaces are integrated into the LBMTM algorithm and propose a multivariate collision function in the LBM framework. The gradient and curvature of the volume data-set are considered to help the algorithm segment image, which aims to elaborate a universal image segmentation approach.

The subject of this section is to further introduce the new approach LBMTM and apply it to the volumes with more complex compositions. In the LBMTM algorithm, the gradient value calculated in 2D histogram effectively improved the segmentation result. Therefore, other potential properties are introduced to the algorithm. For testing the performance of the LB multi-variable segmentation algorithm, two volume data-sets are segmented. The results demonstrate that the multivariate clustering-based collision function of the LBM can greatly improve the segmentation result.

LBM multivariate clustering-based collision function

A new method for image segmentation is proposed in this section by fitting a multivariate feature space and a clustering method integrated with LBM. Fig. 5.21 114 Fei GE exhibits the pipeline of the proposed method, which includes five steps as follows:

Step.1 Based on the volume data-set, we can calculate the corresponding multiple variables at each voxel.

Step.2 Construct 2D histogram with the corresponding variable magnitude.

Step.3 The K-means clustering method is used to classify the 2D histogram and compute the optimal centroids.

Step.4 Then, the thresholds are addressed to the LBM block for anisotropic diffusion.

Step.5 Back to Step.1 until the segmentation results meet the convergence condition.

The proposed method continuously iterates and refines the data-set. Meanwhile, the 2D density histogram is also computed at each iteration and clustered into regions, and each region corresponds to a kind of tissue by K-means algorithm. as a reaction-diffusion equation [START_REF] Qian | Lattice bgk models for navier-stokes equation[END_REF]:

f i ( r + ∆ r, t + ∆t) = f i ( r, t) + 1 τ [f eq i ( r, t) -f i ( r, t)] + R(C j , ρ) (5.9)
where R(C j , ρ) is a diffusion term. ρ represents the grey-value on each node.

Herein, a new multivariate clustering-based collision function is proposed, as shown in Eq.5.10, in which C j is performed as the geometric information and statistics criterion to regulate ρ. Currently, the gradient, curvature [Koenderink and van Doorn, 1992] and the frequencies of them in terms of the multiple threshold values are considered in Chapter 5.

R(C

j , ρ) = α(C m -ρ), ∃m : |C m -ρ| = min|C j -ρ| (
arg min C l m w 1 (v 1,1 -c 1 ) 2 + ... + w k (v l,m -c j ) 2 (5.11) 116 Fei GE
where v is the multivariate derived from the voxels of volume data-set; w refers to the weight coefficient of each variate; c j represents the average value of class C (cluster center). The C j denotes as the centroids that are used as the main parameter in the proposed collision function.

Experimental result

Segmentation result

In this section the Data_1 and Data_2 are segmented to perform the proposed method. Firstly, the proposed method calculates the magnitude of the considered Finally, with the gray value as the main criterion, the gray value in the LBM method is adjusted until the convergence condition is satisfied, which includes the tolerance and max iteration time. We applied the proposed method on the whole cranial volume data-set, and the clustered 2D histograms are displayed in Fig. 5.23.The whole cranial is segmented into four parts, and each color on the 2D histogram reflects a cluster. Fig. 5.24 reveals all of the corresponding clusters . (1) cluster 1 (2) cluster 2

(3) cluster 3 (4) cluster 4

Figure 5.24: The cranial segmentation result of each cluster in the 3D view.

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Fei GE

Validation

For this purpose, we use Dice similarity coefficient (DSC) and Hausdorff distance are conducted to assess the accuracy of our method [START_REF] Bernard | Deep Learning Techniques for Automatic MRI Cardiac Multi-structures Segmentation and Diagnosis: Is the Problem Solved[END_REF].With respect to data-set segmentation results of V sr and references of V ref , the DSC is defined as:

DSC = 2(|V sr ∩ V ref |) (|V ref , | + |V ref |) (5.12)
where V sr represents the volume data-set of segmentation results, and V ref is the volume data-set of references which are the volumes manually segmented by clinical doctors. The DSC is a magnitude value from 0 to 1 (no match to complete match). Meanwhile, Hausdorff distance d H is also calculated to measure the distance between two subsets, and it is defined as follows:

d H (V sr , V ref ) = max{ sup Vsr∈V inf V ref ∈V d(V sr , V ref ), sup V ref ∈V inf Vsr∈V d(V sr , V ref ) (5.13)
where sup represents the upper bound (supremum), and inf represents the lower bound (infimum). In this section, 3D d H is used to evaluate the segmentation result in order to avoid the management of missing segmentation problems on the end slices. While dealing with volume data-set (whole cranial) with several compositions, the result in Tab.5.2 show that the proposed method is more accurate than that of the k-means and Chan-Vese method in DSC.

119 Fei GE

Conclusion

This chapter proposed a novel multi-threshold LBM framework to solve a reactiondiffusion equation at a macroscopic scale via the multi-thresholds collision function. This framework can effectively optimize a consistent segmentation method in regard to the complexity of the cerebral medical images with cranial, cerebral architectures. We operated four volume data-set on our GPU cluster 'Saki', where the computation was completely GPU-based (data write and read once). The results show that the parallel computational scheme is suitable for GPU computing, which can greatly improve the method efficiency. This scheme is also applicable for the large data-set segmentation. The method of parallel segmentation has been executed

Introduction

The relationship between the segmentation algorithm and hardware is becoming much more integrated. The LBM has the capability to simulate the fluid flow behavior [Wolf-Gladrow, 2000] and can be used for image processing recently [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF]. It is easy to implement the parallel computation because of the local operation of LBM that uses the gray-level value on each node of the volume lattice for propagation without data transmission. Therefore, it is feasible to map and calculate the lattices on the thread of GPU. For solving the Computational Fluid Dynamics (CFD) field [Zhao, 2007] and image processing field [START_REF] Hagan | Parallel 3d image segmentation of large data sets on a gpu cluster[END_REF], its parallel computing ability on GPU has been proved. However, because of the complexity of the medical image, the high-performance medical image segmentation still is a challenge. While dealing with large scale volumetric dataset, it has been revealed that the proposed method is not ideal for computational performance. LBM as a numerical approximation method exhibit the following advantages:

• Easy to implement with the core program only with a few lines of code;

• Implicit computation of curvatures;

• Flexible control of generating smooth segmentation results;

• Strong amenability to parallel computing, especially on low-cost, powerful graphics hardware (GPU) there are already some LBM implementations of CFD on GPUs, but the implementation of LBM on image segmentation is still difficult. In 2009, Aaron used LBM to solve the level-set method [START_REF] Hagan | Parallel 3d image segmentation of large data sets on a gpu cluster[END_REF], in which the proposed scheme was not ideal for complex data-set. Hence, in this section, the proposed LBMTM is implemented on the GPU cluster by CUDA-10.0 in order to accelerate its computation efficiency.

Differences between CPUs and GPUs

The design of GPU and CPU is different. The reason behind the discrepancy in floating-point capability between the CPU and GPU is that GPU is dedicated to 123 Fei GE highly parallel computing and accurately achieves the purpose of graphics rendering. Therefore, it is designed so that more transistors are dedicated to data processing than data caching and flow control [NVIDIA, 2019]. Thus, the complexity is greatly reduced. For example, the GPU does not have CPU intelligence, no prefetching, no branch prediction, etc. The GPU dedicates most of its transistors to the processing in Fig. 6.1. This makes GPUs ideal for compute-intensive, highly parallel computation. More specifically, GPUs excel at data-parallel computations. Just like the AVX (SIMD) units in CPUs. GPUs have a reasonable amount of High Bandwidth Memory (HBM) onboard. This memory has greater bandwidth than server RAM, while the drawback is less of it. GPUs attach to a server via the PCIe bus except for NVLink power systems, which ultimately limits communication speed between a CPU and GPU.

GPU memory architecture

The LBMTM algorithm is mainly implemented on NVIDIA GPU Tesla P100, and the memory architecture is shown in Fig. 6.2.GPU is a hardware device that contains multiple small hardware units called Streaming Multiprocessors (SM).

Each SM can execute many threads concurrently. Each one of these SM exists a cache called L1 that is much smaller than that on CPU, but the bandwidth is much larger. As seen in Fig. 6.2, all GPUs have a cache called L2 cache, and its on GPU is much smaller than the size of L2 or L3 cache on CPU. But again, the bandwidth of L2 cache in GPU is much larger than that of L2 cache on CPU. On Tesla P100, this GPU Memory has 16 gigabytes memory, which is used to store the volume data-set and particle distribution and equilibrium distribution.

LBM-based segmentation algorithm

LBM is derived from the cellular automaton scheme, which models particles on a discrete grid. Each point of the grid contains a specific network structure and is linked to its neighbors. Lattice structures we proposed in this section are defined by D3Q7 and D3Q19 which refers to the dimensions and number of connected links between the lattice and its neighbors. Fictitious particles moving along the links and their averaging behaviors are firstly used to simulate traditional fluid dynamics. Based on the numerical computation process derived from microscopic statistical physics, it recovers the Naiver-Stokes equations that govern flow behavior. The independent variables of the LBM equation are particle distribution functions of each link from a grid point to one of its neighbors. Particle distribution functions model the probability that a particle packet will flow through a lattice link to its corresponding neighbor. Between two consecutive steps of the continuous calculation, the function is modified by performing a local relaxation that models inter-particle collisions. This section will introduce a complete physical description and its application in visual simulations.

The first step of the algorithm is to initialize the model which discretes the 125 Fei GE simulation space to a grid, and each node of the grid generates a lattice structure.

Each node of the grid is regarded as a voxel of the volume data-set. The particle distribution function and equilibrium function are show in Eq.6.1 and Eq.6.2.

f i ( r, 0) = f 0 i ( r, 0) = β i ρ( r, 0) (6.1)

where ρ is the volume data-sets and β i is given by: Then, at each simulation step, iteration is mainly separated into two phases that are the steaming phase in Eq.6.3 and collision phase in Eq.6.4.

β i =                 
f i (( r + e i ∆t, t + ∆t) = f i (( r, t) (6.3) f i ( r, t + ∆t) = f i (x, t) -1 τ (f i (( r, t) -f eq i (( r, t)) + R(ρ, T j ) (6.4)

where R(ρ, T j ) represents the collision term(the extra force) in the LBM model [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF]: R(ρ, T j ) = α(T m -ρ), ∃m : |T m -ρ| = min|T j -ρ|. (6.5)

At a given time step t, each particle distribution function f i , along one link vector e i at a lattice point. r is updated by a relaxation process with respect to f eq i . The collision process is controlled by a relaxation parameter τ . τ controls the rate at which the equation approaches an equilibrium state. α∆t(T m -ρ) is an additional external force, in which T m is the nearest threshold from the k-means clustering method. The particle distribution function f i is updated according to Eq.6.4. Then, ρ and equilibrium function f eq i will be updated in Eq.6.6 and Eq.6.7.

ρ( r, t) = f eq i ( r, t + ∆t) = β i ρ( r, t + ∆t) (6.7)

where β i is the same formula in Eq.6.2. With the updated ρ, T j is re-calculate by the 2D histogram based k-means clustering algorithm in section 4.3.1.

The segmentation algorithm from Eq.6.1 to Eq.6.7, it reveals that if we have enough memory to store the distribution data-set, the calculation at each node of the grid and at each iteration is independent. This makes the LBM-base segmentation algorithm suitable for carrying into GPUs.

6.5 Parallel LBMTM algorithm

Computational procedure

The LBMTM image segmentation algorithm is implemented in the following steps: 127 Fei GE f i ( r, 0) = f eq i ( r, 0) (6.10) f i ( r, 0) = f eq i ( r, 0) = β i ρ( r, 0) (6.15)

7 β i =                 

Performance result

Parallel computing can be regarded as one of the main advantages of LBM. At first, the proposed method is implemented on a NVIDIA graphic card of the laptop, the Quadro M1000M with 4 gigabytes of GPU memory and programming by python and CUDA-10.0. However, it is found that when the large scale data-set is segmented by LBMTM, the memory of GPU is not enough. Hence, the LBM method is then implemented on NVIDIA graphic card Tesla-P100 with 16GB memory in the cluster "saki" of CREATIS. The results show that when the aneurysm of 128 × 128 × 128 is segmented by proposed method under 15 iterations, it costs 0.24 s on GPU in terms of 31.50 s running on CPU. In order to assess the performance of the proposed method, a comparison is made by running the proposed code on one thread of CPU i7-6820HQ CPU at 2.70GHz. Tab.6.1 compares the performance between CPU and GPU. The result shows that the proposed method on GPU exhibits efficiency (at least 131 times) and accuracy (Tab.5.1).

130

Fei GE Then we found that while segmenting large scale volume data-set, the performances of the proposed method were not ideal, so we propose and implement a parallel LBM based segmentation algorithm implemented on GPU. Concretely the parallel segmentation method has been run on nVIDIA graphic card Tesla P100, the experiment demonstrate that the proposed method accelerates the calculation on GPU by at least a factor of 131 speedup with the same precision reached with CPU, and also accompanied by the characteristic which is the larger the volume data-set, the greater the relative computational efficiency.

Perspective

Scientific researches based on the LBM contributing to enlarge the landscape of image processing are very rare to find; In this context, this thesis proposes an framework that combines the LBM and clustering method. The results demonstrate its great adaptability to the segmentation of both 2D images and 3D volume datasets. In future works, the LBM-based segmentation algorithm could be universally used in the image processing field as a specific way to characterize properties of image. And it may also contribute to provide learning matrix or dictionaries of machine learning system. And the other advantage of the proposed framework is its intrinsic adaptability to parallel computation, the LBM-based framework also has the superiority to segment or process 4D volumes of data-set or 3D real-time segmentation [START_REF] Wang | Multilevel segmentation of intracranial aneurysms in ct angiography images[END_REF].

Author's publications

  La méthode de Boltzmann sur rseau (LBM) est un outil numérique avec une méthodologie basée sur des modèles microscopiques de particules et des équations cinétiques mésoscopiques. Elle repose sur des mécanismes de propagation et de collision au cours du temps de distributions de particules se propageant sur un réseau régulier. Si les lois de conservation sont imposées en chaque noeud du réseau, alors la solution générée correspondra à la modélisation de phénomènes physiques à l'échelle macroscopique. Dans ce contexte la méthode de Boltzmann est tout à fait adaptée pour résoudre un problème de mécanique des fluides qui équivaut à résoudre indirectement l'équation de Navier-Stokes. Récemment des travaux en traitement d'images ont été réalisés en adaptant la méthode de Boltzmann sur réseau pour des opérations de segmentation, de débruitage,... LBM a également l'avantage d'être très adapté au calcul parallèle, notamment sur du matériel graphique puissant et peu coûteux (GPU).Le but principal de cette thèse est de développer un algorithme général de segmentation. Cette thèse se compose de quatre parties principales. La première partie propose une méthodologie de segmentation originale créée en couplant une technique de rendu volumique direct (méthode de clustering à base d'histogrammes 2D) et une méthode de Boltzmann sur réseau 3D. La stratégie réside dans la possibilité de segmenter des séquences d'images médicales par niveaux de gris et gradient de niveaux de gris. Dans cette partie, les résultats sont comparés avec la méthode de classification par k-moyennes et les résultats de segmentation manuelle montrent que l'algorithme a amélioré à la fois la rapidité du traitement et la limitation du bruit de calcul numérique. Mais nous avons constaté que la méthode peut s'appliquer à des ensembles de données plus complexes, mais que la précision des résultats n'est pas suffisante. La deuxième partie propose donc un processus de multi-euillages basé sur la LBM et un algorithme de partionnement. LBM de la première partie est essentiellement mis en oeuvre pour la segmentation en résolvant une équation de réaction-diffusion prenant en compte un seuil communément cal-culé par la méthode de Otsu. Une fonction innovante de collision multi-seuils est élaborée, conduisant à une nouvelle formulation de la méthode de Boltzmann sur réseau pour le partitionnement d'images 3D, les seuils étant fixés par la méthode de partitionnement par k-moyenness. Dans le cadre de ce nouveau concept, des images 2D de la base de données de segmentation de Berkeley et des données en volume 3D comprenant des séquences d'anévrismes cérébraux traités par endoprothèse sont segmentées par la nouvelle approche. Les performances sont évaluées via le coefficient de similarité de Dice, la différence de volume relative et la distance de Hausdorff. Ensuite, afin d'étendre l'universalité de l'algorithme basée sur les propriétés potentielles de la géométrie et de données statistiques sont introduits dans la méthode de segmentation basée sur la LBM, et une fonction de collision innovante basée sur un partitionnement multivarié est élaborée conduisant à une nouvelle formulation de la méthode de Boltzmann pour le traitement de données volumiques 3D. Les performances sont également validées par la segmentation d'images tomographiques tridimensionnelles (3D) d'angiographie. La dernière partie examine la méthode de segmentation basée sur une version parallélisée de LBM. Pourour traiter les images à grande échelle, nous tirons parti de la LBM parallèle en utilisant des architectures de processeurs graphiques (GPU) afin d'améliorer la performance de la LBM. Nous avons prouvé l'efficacité du LBM parallèle en segmentant la paroi d'un anévrisme traité pas stent et des vaisseaux sanguins parents. La méthode de segmentation parallèle a été testée sur une carte graphique nVIDIA et démontre que la méthode proposée reduit les temps de calcul d'un facteur supérieur à 100. Mots-clés: algorithme de segmentation, segmentation de l'anévrisme cérébral, méthode de Boltzmann sur réseau, algorithme de clustering, anévrisme traité par stent , histogramme 2D, accélération GPU. Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 State-of-the-art 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 General segmentation algorithm . . . . . . . . . . . . . . . . . . . . 1.2.1 Thresholding algorithm . . . . . . . . . . . . . . . . . . . . . 1.2.2 Edge-based algorithm . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Region-growing algorithm . . . . . . . . . . . . . . . . . . . 1.3 Advanced segmentation method . . . . . . . . . . . . . . . . . . . . 1.4 Volume rendering algorithm . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Image-centric transfer function . . . . . . . . . . . . . . . . 1.4.2 Data-centric transfer function . . . . . . . . . . . . . . . . . 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Lattice Boltzmann method 2.1 Lattice-gas cellular automata . . . . . . . . . . . . . . . . . . . . . 2.1.1 HPP lattice-gas cellular automata . . . . . . . . . . . . . . . 2.1.2 FHP model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Lattice Boltzmann equation . . . . . . . . . . . . . . . . . . . . Contents 6 Parallel LBM-based algorithm design 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Differences between CPUs and GPUs . . . . . . . . . . . . . . . . . 6.3 GPU memory architecture . . . . . . . . . . . . . . . . . . . . . . . 6.4 LBM-based segmentation algorithm . . . . . . . . . . . . . . . . . . 6.5 Parallel LBMTM algorithm . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Overview of implementation . . . . . . . . . . . . . . . . . . 6.5.2 Computational procedure . . . . . . . . . . . . . . . . . . . 6.6 Performance result . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion and perspective 7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Author's publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Effet d'escalier estompé via la méthode de Boltzmann sur réseau. . 3 Lumière et Thrombus de l'anévrisme cérébral. . . . . . . . . . . . . 4 CTA de l'anévrisme cérébral: (1) CTA du cerveau, (2) CTA d'anévrisme. 30 5 Pipeline d'algorithmes k-moyennes. . . . . . . . . . . . . . . . . . . 6 Méthode de couplage de la méthode de Boltzmann sur réseau et de la méthode de partitionnement par k-moyennes. . . . . . . . . . . . 7 Résultat de segmentation CTA de l'anévrisme cérébral du premier patient, qui prend six coupes d'anévrisme cérébral comme exemples de segmentation 2D. La partie rouge correspond aux frontières vasculaires, la partie blanche est l'intérieur du vaisseau. Les parties noires et grises sont les autres tissus. . . . . . . . . . . . . . . . . . 8 Résultat de la segmentation CTA de l'anévrisme cérébral du deuxième patient, qui prend six coupes d'anévrisme cérébral comme exemples de segmentation 2D. La partie rouge correspond aux frontières vasculaires, la partie blanche est l'intérieur du vaisseau. Les parties noires et grises sont les autres tissus. . . . . . . . . . . . . . 9 2D IGM histogramme initial (sans LBM). . . . . . . . . . . . . . . 10 Segmentation résultats: la gauche: vaisseau parent (rouge), la droite: phase sanguine (jaune). . . . . . . . . . . . . . . . . . . . . . . . . . 11 2D IGM histogramme par la méthode proposée. . . . . . . . . . . . 12 Segmentation par la méthode proposée: la gauche: vaisseau parent (rouge), la droite: phase sanguine (jaune). . . . . . . . . . . . . . . 13 La fonction non-décroissante du paramètre α. . . . . . . . . . . . . 14 Résultat de segmentation: (1) Image 2D (2) Méthode Chan-Vese (3) Méthode proposée (4) Référence. . . . . . . . . . . . . . . . . . 15 Résultat de segmentation: (1) Image 2D (2) Méthode Chan-Vese (3) Méthode proposée (4) Référence. . . . . . . . . . . . . . . . . . 16 Résultat de segmentation : (1) Image 2D (2) Méthode Chan-Vese (3) Méthode proposée (4) Référence. . . . . . . . . . . . . . . . . . 17 Résultat de la segmentation -crâne segmenté en utilisant iter = 2, ths = 7. (1) Données du volume d'origine en vue 3D, (2) données "demi-volume" en vue 3D, (3) résultat de la segmentation en vue 3D, (4) résultat de la segmentation en vue 2D. . . . . . . . . . . . . 18 Résultat de la segmentation -anévrisme segmenté en utilisant iter = 3, ths = 4.(1) Données du volume d'origine en vue 3D, (2) données "demi-volume" en vue 3D, (3) résultat de la segmentation en vue 3D, (4) résultat de la segmentation en vue 2D. . . . . . . . . . . . . 19 Résultat de la segmentation -anévrisme traité par stent, segmenté en utilisant iter = 3, ths = 5. (1) Données du volume d'origine en vue 3D, (2) données "demi-volume" en vue 3D, (3) résultat de la segmentation en vue 3D, (4) résultat de la segmentation en vue 2D. 20 Histogramme 2D multi-varié du crâne. . . . . . . . . . . . . . . . . 21 Résultat de la segmentation pour chaque cluster en vue 3D. . . . . 22 Pseudo code du noyau sur le GPU. . . . . . . . . . . . . . . . . . . 1.1 An example of histogram based thresholding method is used to segment CT angiography data-set. . . . . . . . . . . . . . . . . . . . . 1.2 An example of segmentation of blood vessel and cranial by the Otsu method, (1) and (3) are the original CT scan images, (2) and (4) are the segmentation results by the Otsu method. . . . . . . . . . . 1.3 Edge detection by the Sobel operator. . . . . . . . . . . . . . . . . . 1.4 WYSIWYG system interface [Guo et al., 2011]. . . . . . . . . . . . 1.5 Intensity gradient magnitude histogram: original IGM histogram(left), cluster IGM histogram (right). . . . . . . . . . . . . . . . . . . . . . 2.1 HPP Collision configuration and its propagation, after the collision the two cells will occupy the formerly empty cells. . . . . . . . . . . 2.2 FHP model: two possible results of head on particles collision, showing the regular of hexagonal symmetry. . . . . . . . . . . . . . . . . 2.3 LBM D1Q3 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 LBM D1Q5 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 LBM D2Q7 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 LBM D2Q9 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 LBM D3Q9 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . D3Q15 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 LBM D3Q19 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 LBM D3Q27 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Particles directions of D3Q19. . . . . . . . . . . . . . . . . . . . . . 3.2 LBM D2Q9 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Overcoming the stair-casing effect of classic anisotropic diffusion method in 2D view. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The lumen part and thrombus part of cerebral aneurysm. . . . . . . 4.1 Cerebral aneurysm 2D IGM histogram: (1) first aneurysm 2D slice, (2) IGM histogram of the first aneurysm , (3) second aneurysm 2D slice, (4) IGM histogram of the second aneurysm. . . . . . . . . . . 4.2 K-means algorithm pipeline. . . . . . . . . . . . . . . . . . . . . . . 4.3 Two cerebral aneurysm data-set clustered 2D IGM histogram example.4.4 CT scan of cranial (left) and cerebral aneurysm (right). . . . . . . . 4.5 Matching LBM and k-means clustering method. . . . . . . . . . . . 4.6 The cerebral aneurysm CTA segmentation result of the first patient, which takes six slices of cerebral aneurysm as examples of 2D segmentation. The red part is the vascular borders, the white part refers to the interior of the vessel. The black and gray parts correspond the other tissues. . . . . . . . . . . . . . . . . . . . . . . 4.7 The cerebral aneurysm CTA segmentation result of the second patient, which takes six slices of cerebral aneurysm as examples of 2D segmentation. The red part is the vascular borders, the white part refers to the interior of the vessel. The black and gray parts correspond the other tissues. . . . . . . . . . . . . . . . . . . . . . . 4.8 IGM histogram clustering result without LBM, which is classified to 2 cluster (red part and yellow part). . . . . . . . . . . . . . . . . 4.9 Clustering result without LBM, the parent vessel (red) and lumen part of a blood vessel (yellow). . . . . . . . . . . . . . . . . . . . . . 4.10 IGM histogram clustering result with LBM, which is classified to 2 cluster (red part and yellow part). . . . . . . . . . . . . . . . . . . 4.11 Clustering result with LBM, the parent vessel (red) and lumen part of a blood vessel (yellow). . . . . . . . . . . . . . . . . . . . . . . . 5.1 Main pipeline of the proposed algorithm. . . . . . . . . . . . . . . . Stent-assisted aneurysm segmentation results: the segmented stentassisted aneurysm by using iter = 3, ths = 5. (1) Original volume data-set, (2) half-volume data-set in 3D, (3) segmentation result in 3D, (4) segmentation result in 2D. . . . . . . . . . . . . . . . . . . . 5.20 Hausdorff distance values of 512 stent-assisted aneurysm slices. . . . 5.19 Dice similarity coefficient by choosing the different iterations. . . . .

I

  am also pleased to thank Patrick CLARYSSE and Mathilde GI-AGALONE, who made it easier for me to visit research institutes and laboratory and gave an opportunity to become a member of their teams. It is impossible to conduct this research without their precious support. They all really mean a lot to me. I also would like to remember my fellow labmates for the fun-time we spent together, precious experience when we cooperate with each other to complete tasks before deadlines and discussion problems. And I feel fairly happy to work along with these dear friends from CREATIS: Mathilde Giacalone, Pierre-Antoine Ganaye, Sarah Leclerc, Anchen Chai, Noelie Debs, Yunyun Sun, Bingqing Xie, Pei Niu, Yuemeng Feng, and Hoai-Thu Nguyen. Thanks for your inspirations and help for my research, the Monday and Friday cake and the "soire", especially for all the moments that we shared together. Finally, I would like to thank my wife Bowen ZHNAG and my family for their continuous support and priceless encourage throughout my thesis. Synthèse La segmentation d'images comme méthode de post-traitement est une étape importante dans de nombreux domaines de recherche tels que l'imagerie médicale, la vision par ordinateur, la biologie. . . L'objectif de ce post-traitement est de prélever une (ou des) région(s) d'intérêt (RDI) dans une image (2D -composée de pixels) ou dans des données volumétriques (3D -composée de voxels). Cette opération est toujours un défi parce que la variété des images à segmenter et des dispositifs d'imagerie, ne cesse d'évoluer. En définitive la motivation principale de cette thèse est l'élaboration d'une méthode de segmentation d'image 3D généralisée efficace. Un modèle original de segmentation multi-seuils basé sur l'algorithme des k-moyennes couplé au calcul d'un histogramme 2D et sur la méthode de Boltzmann sur réseau dans une version 'réaction-diffusion' adaptée à la segmentation d'image, est ici proposé comme candidat à la problématique posée. Pour présenter les développements théoriques et les résultats des calculs numériques obtenus pendant la durée de la thèse, le manuscrit est organisée comme suit.

  Partant de ce concept, la méthode de Boltzmann sur réseau (Lattice Boltzmann method -LBM) est ici utilisée en tant que méthode numérique pour résoudre des équations aux dérivées partielles (EDP) à l'échelle mésoscopique, et qui présente l'avantage d'être adaptée naturellement à la parallélisation en comparaison d'autres méthodes de résolutions des EDP. Par ailleurs, les codes de segmentation sont de plus en plus ancrés au système informatique sur lequel ils sont implémentés. La méthode de Boltzmann sur réseau (Lattice Boltzmann method -LBM) illustre bien cette évolution et a prouvé sa capacité à simuler des comportements d'écoulement de fluides complexes et plus récemment à traiter des images médicales avec des temps de calcul optimisés. Également, les implémentions de codes parallèles sont directes en raison des opérations locales effectuées sur chaque noeud du réseau régulier sur lequel des distributions de particules se propagent et entrent en collision et ceci en fonction du 23 Fei GE temps. En effet sur un domaine partitionné en plusieurs sous-domaines, chaque partition peut-être implémentée sur cartes graphiques ou processeurs graphiques (Graphical Processing Unit en anglais -GPU), pour du calcul parallèle et seules les distributions de particules en bordures de ces partitions sont à propager, ce qui se traduit par une optimisation des temps de calcul. En conséquence, un code de segmentation optimisé a été mis en oeuvre sur la base de l'algorithme des k-moyennes dont les conditions initiales sont données via un histogramme 2D et la méthode de Boltzmann sur réseau dans une version où la fonction de collision est construite pour résoudre une EDP de réaction-diffusion adaptée pour prendre en compte les seuils déterminés via la méthodes des k-moyennes . Cette méthode originale est ici appliquée pour réaliser la segmentation de séquences d'images médicales traitant d'anévrismes cérébraux. Dans des travaux récents de Yan[Wang et al., 2016], la méthode de Boltzmann sur réseau a été utilisée pour segmenter la lumière et le thrombus d'anévrismes partiellement thrombosés, débouchant sur des résultats sensiblement meilleurs que ceux obtenus par les méthodes classiques (Chen-Vase, Otsu. . . ). Il faut noter qu'il s'agit d'une méthode dans laquelle la méthode de Otsu calcule un seuil influant sur un terme de diffusion de la fonction de collision. Dans le continuité des travaux de Yan Wang et Yu Chen, l'originalité de cette thèse réside dans la création d'une méthode de segmentation à seuils multiples dont l'implémentation sur carte graphique conduit à des temps de calcul réduits d'un facteur supérieur à cent.

Figure 1 :

 1 Figure 1: LBM D2Q9 lattice.

Figure 3 :

 3 Figure 3: Lumière et Thrombus de l'anévrisme cérébral.

Figure 4 :

 4 Figure 4: CTA de l'anévrisme cérébral: (1) CTA du cerveau, (2) CTA d'anévrisme.

Figure 5 :

 5 Figure 5: Pipeline d'algorithmes k-moyennes.

Figure 6 :

 6 Figure 6: Méthode de couplage de la méthode de Boltzmann sur réseau et de la méthode de partitionnement par k-moyennes.

Figure 7 :

 7 Figure 7: Résultat de segmentation CTA de l'anévrisme cérébral du premier patient, qui prend six coupes d'anévrisme cérébral comme exemples de segmentation 2D. La partie rouge correspond aux frontières vasculaires, la partie blanche est l'intérieur du vaisseau. Les parties noires et grises sont les autres tissus.

Figure 9 :

 9 Figure 9: 2D IGM histogramme initial (sans LBM).

Figure 10 :

 10 Figure 10: Segmentation résultats: la gauche: vaisseau parent (rouge), la droite: phase sanguine (jaune).

Figure 11 :

 11 Figure 11: 2D IGM histogramme par la méthode proposée.

Figure 12 :

 12 Figure 12: Segmentation par la méthode proposée: la gauche: vaisseau parent (rouge), la droite: phase sanguine (jaune).

  8) où p est un voxel de l'ensemble de données de volume ; m i est la valeur moyenne de la classe T j . (p et m i sont deux variables multidimensionnelles). T j sont les seuils qui sont utilisés comme paramètres de segmentation LBM. Puis la méthode proposée est comparée avec la méthode de segmentation Chan-Vese dans les Fig.14 à Fig.16. Les premières images sont des images originales, les 36Fei GE

Figure 13 :

 13 Figure 13: La fonction non-décroissante du paramètre α.

Figure 14 :

 14 Figure 14: Résultat de segmentation: (1) Image 2D (2) Méthode Chan-Vese (3) Méthode proposée (4) Référence.

Figure 16 :

 16 Figure 16: Résultat de segmentation : (1) Image 2D (2) Méthode Chan-Vese (3) Méthode proposée (4) Référence.

Figure 17 :

 17 Figure 17: Résultat de la segmentation -crâne segmenté en utilisant iter = 2, ths = 7.(1) Données du volume d'origine en vue 3D, (2) données "demi-volume" en vue 3D, (3) résultat de la segmentation en vue 3D, (4) résultat de la segmentation en vue 2D.

Figure 18 :

 18 Figure 18: Résultat de la segmentation -anévrisme segmenté en utilisant iter = 3, ths = 4.(1) Données du volume d'origine en vue 3D, (2) données "demi-volume" en vue 3D, (3) résultat de la segmentation en vue 3D, (4) résultat de la segmentation en vue 2D.

Figure 21 :

 21 Figure 21: Résultat de la segmentation pour chaque cluster en vue 3D.

Figure 22 :

 22 Figure 22: Pseudo code du noyau sur le GPU.

  including lattice-gas cellular automata, HPP model, FHP model, Boltzmann equation, and Lattice Boltzmann models. Then, the models with different distributions of velocities are introduced in 1D, 2D or 3D versions.Chapter 3 introduces the image processing form based on lattice Boltzmann method, in which diffusion equations form collision terms is to achieve image smoothing. The diffusion equation is solved by the mean of the lattice Boltzmann method. In the same spirit, an anisotropic diffusion function is implemented to segment images.Chapter 4 combines the lattice Boltzmann method and direct volume rendering technique (2D histogram based clustering method). The strategy classifies images sequences by considering both the grey-level value of voxels and the gradient magnitude calculated on each voxel, which could be used as second property for segmentation. Chapter 5 extends the method introduced in chapter 4, and gives an overview of the proposed framework for the 3D multi-thresholds segmentation method. Multiple threshold methods are considered to perform a more sophisticated classification of volumetric data and thus deliver better separated volumetric objects or structures. This gives the algorithm the flexibility to subdivide related clusters, such as the structure corresponding to a particular tissue or volume dataset. Then, the lattice Boltzmann multi-thresholds clustering method for the segmentation of 46 Fei GE 2D images and 3D volume data-sets with complex compositions is implemented and operated with adapted parameters. To identify the accuracy of this new approach and versatility of the methodology, three evaluation methods involving the Dice similarity coefficient (DSC), Relative volume difference (RVD) and Hausdorff distance are selected to compare the results obtained with conventional methods. Then, under the proposed framework, other potential properties (geometry information) are integrated into the segmentation method in order to improve the adaptability of the algorithm. According to this new concept and to evaluate performance, the cranial volume data-set and brain aneurysm sequence are segmented. The experimental validation is performed by the Dice similarity coefficient and Hausdorff distance. Chapter 6 studies the implementation of the framework proposed in Chapter 5 on Graphic processing units (GPU) by taking advantage of local propagation of LBM. The results show that the parallel computational scheme is suitable for both GPU computing as well as the large data-sets segmentation. Chapter 7 concludes the thesis and proposes some promising future perspec--the-art Abstract In this chapter, we introduce the general segmentation methods, meanwhile we describe a state of art review of volume rendering transfer function design. The segmentation techniques can be divided into classes in many ways, depending on the classification scheme. The most common used segmentation techniques is classified into three board categories in this chapter including thresholding algorithm, edge-based algorithm and region-growing algorithm. The transfer function design mainly has two categories, image-centric transfer function design and data-centric transfer function design. Résumé Dans ce chapitre, nous reprenons les méthodes générales de segmentation. Egalement, nous présentons un état d'art du rendu de volumes et du concept de fonction de transfert. Les techniques de segmentation peuvent être divisées en classes de plusieurs façons, selon le schéma de classification. Les techniques de segmentation les plus utilisées sont classées en trois catégories: l'algorithme de seuillage basic, l'algorithme de détection de contours et l'algorithme de croissance de région. La fonction de transfert est également utilisée dans deux versions, fonction de transfert centrée sur l'image et fonction de transfert centrée sur les données. 50 Fei GE

  They are neither based on the of the image nor local properties, such as the local mean value and standard deviation, or the local gradient. The global thresholding is regarded as the most intuitive approach. When a single threshold is selected for the entire image, according to the image , the thresholding is called global. If the threshold depends on the local properties of certain regions of the image like the local mean grey-level value, the thresholding is called local. If the local thresholds are selected independently for each pixel (or group of pixels), the thresholding is called dynamic or adaptive. Fig.1.1 displays original volume of a 3D aneurysm data-set of CT angiography and a 1D histogram. In order to segment the aneurysm, T is the threshold needed to separate the data-set into two classes.(1) Original CT angiography aneurysm data-set. (2) 1D histogram of the data-set.

Figure 1

 1 Figure 1.1: An example of histogram based thresholding method is used to segment CT angiography data-set.

Figure 1

 1 Figure 1.2: An example of segmentation of blood vessel and cranial by the Otsu method, (1) and (3) are the original CT scan images, (2) and (4) are the segmentation results by the Otsu method.

Figure 1

 1 Figure 1.3: Edge detection by the Sobel operator.

  proposed the What You See Is What You Get (WYSIWYG) system that enabled users to directly interact with visualization parameters to find the ROI, as shown in Fig.1.4. For this type of algorithms, users have to manipulate the visual properties to identify the important voxel in volume data-set, which is time-consuming and difficult to fine tune.

Figure 1

 1 Figure1.4: WYSIWYG system interface[START_REF] Guo | Wysiwyg (what you see is what you get) volume visualization[END_REF].

Figure 1 . 5 :

 15 Figure 1.5: Intensity gradient magnitude histogram: original IGM histogram(left), cluster IGM histogram (right).

Figure 2

 2 Figure 2.1: HPP Collision configuration and its propagation, after the collision the two cells will occupy the formerly empty cells.

Figure 2

 2 Figure 2.2: FHP model: two possible results of head on particles collision, showing the regular of hexagonal symmetry.

  f ) is the collision integral for 2 particles, and this equation considers the following approximations: (1) Only two-particle collisions; (2) The velocities of the two colliding particles are uncorrelated before collision; (3) External forces do not influence the local collision dynamics.

Figure 2

 2 Figure 2.3: LBM D1Q3 lattice.

Figure

  Figure 2.4: LBM D1Q5 lattice.

Figure

  Figure 2.7: LBM D3Q9 lattice.

Figure

  Figure 2.8: LBM D3Q15 lattice.

Figure

  Figure 2.9: LBM D3Q19 lattice.

Figure 3 . 1 :

 31 Figure 3.1: Particles directions of D3Q19.

Figure 3

 3 Figure 3.2: LBM D2Q9 lattice.

Figure 3 . 3 :

 33 Figure 3.3: Overcoming the stair-casing effect of classic anisotropic diffusion method in 2D view.

Figure 3 . 4 :

 34 Figure 3.4: The lumen part and thrombus part of cerebral aneurysm.

ρ

  represents the density of the fluid with ρ = i f i on each node. e i is the LBM DdQm model which uses the form of the equilibrium distribution function.There are some differences in the LBM model in 2D as mentioned in section 3 of Chapter 2. T is the threshold. Briefly, the reaction-diffusion LBM solves a reaction-diffusion equation at a microscopic scale. R is diffusion resource in Eq.4.4: R(T, ρ) = a∆t[T -ρ( r)] (4.4) where R(T, ρ) represents the reaction-diffusion resource term(the extra force) in the model [Wang et al., 2016]; T is the threshold calculated from 2D histogrambased k-means clustering method; the algorithm of computing the threshold value T is presented in Section 4.3.

  Fig.4.1 shows two examples of 2D IGM histograms of two aneurysms.86Fei GE

Figure 4 . 1 :

 41 Figure 4.1: Cerebral aneurysm 2D IGM histogram: (1) first aneurysm 2D slice, (2) IGM histogram of the first aneurysm , (3) second aneurysm 2D slice, (4) IGM histogram of the second aneurysm.

  compute the threshold for clustering tissues in volume data-set. The basic k-means clustering pipeline is shown in Fig.4.2. Eq.4.5 is the k-means formula of minimizing within-cluster distance.

Figure 4

 4 Figure 4.2: K-means algorithm pipeline.

Figure 4 . 3 :

 43 Figure 4.3: Two cerebral aneurysm data-set clustered 2D IGM histogram example.

  may rupture and thus lead to a subarachnoid hemorrhage with the consequence of high mortality rates sometimes. Many projects work on the analysis and treatment of brain aneurysms, and specific needs are required to estimate the evolving of an aneurysm after endovascular treatment. In this framework, image processing techniques such as volume rendering bring essential means of patient follow-up.

Figure 4

 4 Figure 4.4: CT scan of cranial (left) and cerebral aneurysm (right).

Fig. 4

 4 Fig.4.5 exhibits a synoptic of our LBM-based k-means clustering algorithm design method for medical image segmentation. Two major algorithms in our segmentation algorithm are LBM and K-means clustering methods.

Figure 4 . 5 :

 45 Figure 4.5: Matching LBM and k-means clustering method.

  These thresholds are addressed to the LBM block in Fig.4.5 for solving a reactiondiffusion equation at a microscopic scale. This innovative segmentation scheme continuously iterates and modifies the data-set. With the refined data-set, the 2D IGM histogram is computed at each iteration until meeting the iteration convergence. Next, the IGM is clustered into several regions, and each region corresponds to a specific tissue by K-means the algorithm. Subsequently, the refined IGM and volumetric data-set are applied to render the volume and generate the segmenta-segmentation result is shown in Fig.4.6 and Fig.4.7. The red contours are superimposed on the original image, which is the boundary of the vessel. The proposed algorithm in this section is used to segment the two patient CTA with 131 slices and 128 slices, respectively, and the results contain ROI. The proposed algorithm and k-means clustering method is applied to segment 3D cerebral aneurysm sequence in Fig.4.8 and Fig.4.9. Fig.4.8 (top) displays the IGM histogram calculation results of the K-means clustering algorithm (without LBM). It can be seen that the K-means algorithm clusters IGM into two parts, in which the red part is the vessel boundary and the yellow part is the blood within intra-cranial aneurysms and parent blood vessel.

Figure 4 . 6 :

 46 Figure 4.6: The cerebral aneurysm CTA segmentation result of the first patient, which takes six slices of cerebral aneurysm as examples of 2D segmentation. The red part is the vascular borders, the white part refers to the interior of the vessel. The black and gray parts correspond the other tissues.

Figure 4

 4 Figure 4.7: The cerebral aneurysm CTA segmentation result of the second patient, which takes six slices of cerebral aneurysm as examples of 2D segmentation. The red part is the vascular borders, the white part refers to the interior of the vessel. The black and gray parts correspond the other tissues.

Figure 4

 4 Figure 4.8: IGM histogram clustering result without LBM, which is classified to 2 cluster (red part and yellow part).

Figure 4 . 9 :

 49 Figure 4.9: Clustering result without LBM, the parent vessel (red) and lumen part of a blood vessel (yellow).

Fig. 4 .

 4 Fig.4.10 displays the IGM histogram of the proposed method with LBM. It can be seen that the data-set is less sensitive to noise than without LBM compared to the results in Fig.4.11. The volume rendering results are more distinctive and clearer than Fig.4.9. The underlying objective is to reveal arches within 2D IGM histogram in order to detect different living tissues in medical images via LBM [Wu and Qu, 2007]. Based on one threshold T issued from K-means, the solving of the reaction-diffusion equation via LBM can reveal several arches. In this example, the yellow arches correspond to the blood, and the red arches represent the parent vessel.

Figure 4 .

 4 Figure 4.10: IGM histogram clustering result with LBM, which is classified to 2 cluster (red part and yellow part).

Figure 4 .

 4 Figure 4.11: Clustering result with LBM, the parent vessel (red) and lumen part of a blood vessel (yellow).

  sically implemented for segmentation by solving a reaction-diffusion equation which only considers one threshold. In this chapter, an innovative multi-thresholds collision function is elaborated, a new formulation of the lattice Boltzmann method for 2D image and 3D volume data-set clustering is proposed, where the threshold is fixed by the clustering method. In order to test its effectiveness, 2D natural images and 3D medical volume data-sets are segmented and the performances are validated by the Dice similarity coefficient, Relative volume difference and Hausdorff distance. Compared to the 3D visual perception and quantitatively, the results of the proposed method are superior to those of the existing methods for dealing with the stent-assisted intra-cranial aneurysm. As an alternative, a LBM-based segmentation method is constructed through an innovative multivariate collision function for clustering 3D volume data-sets. The centroids of the collision function are still determined by 2D histogram-based k-means clustering algorithm. According to this new concept, the cranial volume data-set and brain aneurysm sequence are segmented to evaluate the performances. The experimental validation of this concept is performed by the Dice similarity coefficient and Hausdorff distance. 95 Résumé Dans le domaine du traitement d'images, la méthode de Boltzmann sur réseau est essentiellement mise en oeuvre pour l'opération de segmentation en résolvant une équation de réaction-diffusion prenant en compte un seuil couramment calculé via la méthode d'Otsu. Dans cette thèse, une fonction innovante de collision multi-seuils est élaborée, qui aboutit à une nouvelle formulation de la méthode de Boltzmann sur réseau pour le partitionnement d'images 3D, les seuils étant fixés par la méthode de partitionnement. Les performances sont validées en segmentant des images angiographiques par tomographie tridimensionnelle (3D), et sont évaluées par calcul du coefficient de similarité de Dice, la différence de volume relatif et la distance de Hausdorff. En comparant à partir de la perception visuelle 3D et quantitativement, les résultats de la méthode proposée sont supérieurs aux méthodes existantes pour traiter les anévrismes intracrâniens traités par stent. Comme alternative, une fonction de collision multivariée innovante est développée, contribuant à une nouvelle formulation de la LBM pour partitionner des données volumiques 3D. Les centres de gravité de la fonction de collision sont déterminés par l'algorithme de partitionnement par k-moyennes via un histogramme 2D. Selon ce nouveau concept, l'ensemble des données sur le volume du crâne et la séquence de l'anévrisme cérébral sont segmentés pour évaluer la performance. La validation expérimentale de ce nouveau concept est effectuée en calculant le coefficient de similarité de Dice et la distance de Hausdorff.96Fei GE

  2.2. Thirdly, data sources consist of 2D images data-set and 3D volume data-sets segmentation experiments, and the results are presented in Section 5.3.1. Thereafter, evaluation methods are introduced to validate the segmentation results by the proposed algorithm, which are used to calculate the differences between the segmentation result and reference. To test the accuracy of the new approach, three types of evaluation function, including Dice similarity coefficient (DSC), Relative volume difference (RVD) and Hausdorff distance, are discussed in Section 5.3.2. Afterwards, the LBMTM is applied to segment data source including 2D images and 3D volume data-set, and the segmentation result is presented in Section 5.3.3. In addition, the proposed method is compared with the Chan-Vese algorithm and K-means clustering methods in Section 5.3.2 by the segmentation result of 3D volume data-set, respectively. This contains cranial, cerebral aneurysm and stent-assisted aneurysm, and is presented at the end of Section 5.3.3.

Figure 5 . 1 :

 51 Figure 5.1: Main pipeline of the proposed algorithm.

Figure 5 . 2 :

 52 Figure 5.2: The α non-decreasing function.

Figure 5 .

 5 Figure 5.3: The 2D images data-set.

Figure 5 .

 5 Figure 5.4: Three phases of the treatment of an aneurysm. Left: the CT scan of the head; Middle: locate the position of an aneurysm; Right: stent-assisted aneurysm.

  .7) where V sr represents the volume of segmentation results, and V ref denotes the volume of references which are manually segmented by clinical doctors. DSC is a magnitude value from 0 to 1 (no match to complete match), and smaller RV D indicates high accuracy. Meanwhile, Hausdorff distance d H is also calculated to measure the distance between two subsets. The equation for Hausdorff distance is defined as:

Fig. 5 .

 5 5 to Fig.5.10. The first images are the original one; the second images are segmentation results of the Chan-Vese segmentation method; the third images are the results of the proposed method. The last images are hand-labeled segmentation.103Fei GE

Figure 5 . 5 :

 55 Figure 5.5: 2D image segmentation results: (1) 2D origin image (2) the Chan-Vese method (3) the proposed method (4) reference.

Figure 5 .

 5 Figure 5.6: 2D image segmentation results: (1) 2D origin image (2) the Chan-Vese method (3) the proposed method (4) reference.

Figure 5 .

 5 Figure 5.8: 2D image segmentation results: (1) 2D origin image (2) the Chan-Vese method (3) the proposed Method (4) reference.

Figure 5 .

 5 Figure 5.10: 2D image segmentation results: (1) 2D origin image (2) the Chan-Vese method (3) the proposed method (4) reference.

Figure 5 .

 5 Figure 5.11: Cardiac IGM histogram (left) and clustered IGM histogram of cardiac volume data-set (right), each color corresponds to a cluster.

Figure 5 .

 5 Figure 5.12: Cardiac volume data-sets segment to five clusters, which correspond to the clusters in Fig.5.11.

Figure 5 .

 5 Figure 5.13: Cerebral data-set presented in the IGM Histogram (left) and 2D multithreshold clustered histogram (right).

Fig. 5 .Fig. 5 .

 55 Fig.5.18 correspond to three patient data, respectively. The first images are the origin 3D volumes data-set rendering by hundreds of CT scans slices. The second images are the half-result volumes data-set to make the position comparison with original volume. The third images are the 3D segmentation results, and the fourth images are the slices from 3D segmentation results.

Figure 5 . 14 :

 514 Figure 5.14: Cranial segmentation results: CTA of segmented cranial by using iter = 2, ths = 7. (1) Original volume data-set, (2) half-volume data-set in 3D, (3) segmentation result in 3D, (4) segmentation result in 2D.

Figure 5 .

 5 Figure 5.15: Aneurysm presented in the IGM histogram (left) and 2D multi-threshold histogram (right).

Figure 5 .

 5 Figure 5.16: Aneurysm segmentation results: segmented aneurysm by using iter = 3, ths = 4. (1) Original volume data-set, (2) half-volume data-set in 3D, (3) segmentation result in 3D, (4) segmentation result in 2D.

Figure 5 .Figure 5 .

 55 Figure 5.17: Stent-assisted aneurysm presented in the IGM histogram (left) and 2D multi-threshold histogram (right).

Figure 5 .

 5 Figure 5.20: Hausdorff distance values of 512 stent-assisted aneurysm slices.

Figure 5 .

 5 Figure 5.19: Dice similarity coefficient by choosing the different iterations.

Figure 5 .

 5 Figure 5.21: Main pipeline of the LBM multivariate method.

  decreasing function to ensure the propagation procedure enhance the contrast at boundary. When handling volume data-set with multiple compositions, we still utilize the k-means clustering algorithm to obtain the C j by segmenting the images shown in Eq.5.11:

  variable and constructs the 2D histogram in Fig.5.22. Then, based on the 2D histogram from the k-means clustering method, the centroids of all variables are obtained. Because of the limitation of the 2D histogram, only two features can be considered. Next, based on the clustered 2D histogram and centroids from the previous step, the corresponding grey-level value can be derived.

Figure 5 .

 5 Figure 5.22: Multivariate 2D histogram of cranial; Left: Grey-level Value with gradient magnitude. Right: Clustered Gradient-Grey-level 2D histogram.

  aneurysms and stent-assisted intra-cranial aneurysm. By introducing the clustering algorithm and 2D IGM histogram, the potential of the proposed framework depends on updating with several thresholds and successive iterations of the LBM code, leading to the concept of dynamic clustering of medical image sequences.Then in order to demonstrate the potential possibility of the LBM based segmentation frame work, the lattice Boltzmann multi-variate segmentation algorithm is proposed, in which a reaction-diffusion equation at a macroscopic scale is solved via the clustering-based multi-variate collision function. The promising potential of the proposed method lies in the fitting of derived geometric properties and statistic information of optimized images along with the successive iterations of the LBM framework, which results in the concept of dynamic clustering of medical images sequences. Two data-sets are utilized in experiments to verify the validity and applicability of the proposed method. The segmentation result is compared with the Chan-Vese algorithm as well as k-means clustering method. , a parallel scheme for large scale volume data-set segmentation is proposed. This scheme originates from the LBM-based framework in chapter 5, which takes advantage of local propagation of LBM. The method is implemented on graphics processing unit (GPU)

Figure 6

 6 Figure 6.1: GPU devotes more transistors to Data processing[NVIDIA, 2019].

Figure 6

 6 Figure 6.2: GPU memory architecture.

  ρ and β i the initial values of the particle distribution and equilibrium function can be calculated.

6. 5 . 1

 51 Overview of implementationThe main implementation procedure of parallel computing is to design the kernel function of GPU. The kernel function of LBM is the concrete implementation of the parallel computing, which assigns LBM cell (voxels of volume data) on each thread of GPU, as shown in Fig.6.3. In this method, because the LBM D3Q19 model is utilized, a 4-D matrix is applied on GPU to store the LBM Data in the kernel, including 3-D volumetric data and its corresponding 19 velocities data.

Figure 6 . 3 :

 63 Figure 6.3: Assignment of volume data on GPU.

9

  Figure 6.4: Pseudo kernel code on GPU.

  The central theme of this thesis is the LBM for image segmentation. The LBM operates at a mesoscopic scale dealing with distributions of particles in order to simulate macroscopic phenomenon, it is also a numerical tool for solving partial differential equation. The challenge of this thesis is the transformation and adaptation of the LBM to image processing treatment such as image segmentation operation. The main contributions of this thesis are synthesized as follow.First an original segmentation methodology created by coupling a 2D histogram based clustering method and 3D LBM is constructed, the strategy lies in the opportunity to cluster medical images sequences by taking into consideration not only the grey-level value of voxels but also some geometric characteristics such as curvature index. Under this new concept, a cerebral aneurysm sequence has been processed with the objective of segmenting the wall of a patient specific aneurysm and the associated parent blood vessels. Then this thesis follows for overcoming the disadvantage of large volume dataset containing multiple components, the lattice Boltzmann multi-thresholds segmentation algorithm is proposed. A novel multi-thresholds collision function (LBM-TM) is elaborated leading to a new formulation of the LBM for the clustering of 3D images, the thresholds being fixed by a 2D histogram-based clustering method. In order to test the performance, 2D images from Berkeley segmentation open source database and volume data-set including a cerebral aneurysm medical imaging sequence, and more challenging a deployed stent in the parent vessels of the cerebral aneurysm are segmented and compared. The results are quantified in term of Dice similarity coefficient, Relative volume difference and Hausdorff distance. 133 Thirdly based on the segmentation framework, the 2D histogram based clustering method opens the opportunity to integrate other geometric or statistic properties into the method. The first results obtained with this concept are very fruitful and open new perspectives of research in the domain.

Figure

  Figure A.1: 2D view and parameter input widget of the software.

Figure

  Figure A.2: 3D view of the software.

  5.21 Main pipeline of the LBM multivariate method. . . . . . . . . . . . 5.22 Multivariate 2D histogram of cranial; Left: Grey-level Value with gradient magnitude. Right: Clustered Gradient-Grey-level 2D histogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	In the beginning, I would like to appreciate my supervisor Guy COURBEBAISSE
	for guiding me well throughout the research work, including the title selection
	5.23 Multivariate 2D histogram of cranial; Left: curvature-grey-level his-and the result finding. His immense knowledge, motivation and patience always
	15	16 18	Fei GE Fei GE Fei GE

togram. Right: clustered curvature-grey-level histogram. . . . . . .

5.24

The cranial segmentation result of each cluster in the 3D view. . . . 6.1 GPU devotes more transistors to Data processing[NVIDIA, 2019]. . 6.2 GPU memory architecture. . . . . . . . . . . . . . . . . . . . . . . . 6.3 Assignment of volume data on GPU. . . . . . . . . . . . . . . . . . 6.4 Pseudo kernel code on GPU. . . . . . . . . . . . . . . . . . . . . . . A.1 2D view and parameter input widget of the software. . . . . . . . . A.2 3D view of the software. . . . . . . . . . . . . . . . . . . . . . . . . motivate me to complete research work. Without his help, I could not finish this research work. He is my mentor and a better advisor for my doctorate study beyond the imagination. Apart from my Supervisor, I would like to appreciate Yu CHEN, Laurent NAVARRO, and Romain NOEL for giving encouragement and sharing insightful suggestions. They all play crucial roles in polishing my research writing skills.

  pour segmenter le thrombus d'anévrismes cérébraux en 3D.

	Contributions
	En raison de la complexité des images CTA (Computed Tomography Angiogra-
	phy) et du temps nécessaire à la segmentation des images médicales résultant des
	méthodes actuelles, cette thèse propose une solution originale basée sur la méthode
	de Boltzmann en réseau dédiée au traitement d' images. Pour pallier ce défaut,
	une version originale de la méthode de Boltzmann sur réseau est proposée dans la
	continuité des recherches de Chen [Chen, 2009].
	Une stratégie originale de segmentation multi-seuils via la méthode de Boltz-
	mann est proposée et appliquée à des données 2D et 3D. L'élaboration d'une fonc-
	tion de collision originale couplée à un algorithme des k-moyennes réalisant une
	division en "K" partitions ("clusters") des niveaux de gris de l'image considérée,
	permet une segmentation efficace à seuils multiples. La précision et l'efficacité
	de la solution proposée ont été validées sur des images de références et sur des
	séquences d'imagerie médicale traitant d'anévrismes cérébraux.
	Dans ce dernier cas d'applications, l'imagerie médicale apporte une contribu-
	tion essentielle à la détection et à l'analyse de l'anévrisme cérébral de patients.
	Dans ce contexte, l'angiographie par tomodensitométrie (CTA) joue un rôle cru-
	cial dans le diagnostic et le suivi des patients chez qui un anévrisme cérébral a été
	détecté. Les anévrismes étudiés ont été enregistrés sur 464 coupes avec un CTA
	scan 3D (Fig. 4) et sont issus de la base de données du projet Thrombus (FP7-
	269966). Un anévrisme cérébral est un trouble vasculaire assimilable à une petite
	hernie due à un affaiblissement de la paroi d'un vaisseau sanguin et se produit
	notamment à la bifurcation de vaisseaux sanguins. Un anévrisme peut se rompre,
	entraînant parfois une hémorragie sous-arachnoidienne ayant pour conséquence
	des taux de mortalité élevés. De nombreux projets portent sur le traitement des
	anévrismes cérébraux et leur évolution après un traitement endovasculaire (traite-
	ment par stent). Aussi un besoin spécifique d'analyse de formation d'un thrombus
	conduisant à la guérison s'est fait sentir. Dans ce cadre, les techniques de traite-
	ment d'images apportent des moyens essentiels pour le diagnostic, le traitement et
	le suivi des patients.
	29	Fei GE

Résultat de la segmentation CTA de l'anévrisme cérébral du deuxième patient, qui prend six coupes d'anévrisme cérébral comme exemples de segmentation 2D. La partie rouge correspond aux frontières vasculaires, la partie blanche est l'intérieur du vaisseau

  . Les parties noires et grises sont les autres tissus.

	bles de données synthétiques, que sont le gradient et la valeur de niveau de gris.
	Ensuite, avec la carte de densité produite à chaque itération LBM, l'algorithme
	de partitionnement à k-moyennes regroupe les classes du volume, puis calcule les
	seuils optimaux dans le but de différencier différents tissus vivants rencontrés dans Figure 8:
	le domaine médical. Ces seuils sont adressés au bloc LBM de la Fig. 6 qui est im-
	plémenté pour résoudre une équation de réaction-diffusion. L'histogramme IGM
	2D est calculé à chaque itération pour obtenir le seuil optimal. Le résultat de la
	segmentation CTA 2D est représenté Fig.7 et Fig.8 [GE et al., 2017]. Les con-
	tours rouges se superposent à l'image originale, et correspondent à la limite du
	vaisseau. L'algorithme de segmentation proposé dans cette section est appliqué à
	deux séquences d'images médicales CTA respectivement de 131 et 128 coupes et
	dont la taille des images correspond aux régions d'intérêt pertinentes pour l'analyse
	des données.
	Nous avons appliqué l'algorithme à toute la séquence de l'anévrisme cérébral.
	Le résultat montre que la méthode proposée est comparable à la méthode de
	partitionnement par K-moyennes. Les Fig.9 et Fig.10 affichent les résultats du
	32	Fei GE

  . LBM est un modèle qui peut décrire le débit du fluide. En outre, LBM peut être considéré comme le format discret de l'équation de Boltzmann continue. D'une autre manière, la LBM est une méthode mathématique mésoscopique qui simule un phénomène macroscopique. Dans ce chapitre, les modèles généraux HPP, FHP et LBM sont présentés.

	62	Fei GE

  The numerical LBM scheme is developed by discretizing the continuity Boltzmann equation [Wolf-Gladrow, 2000]. In 1988 McNamara and Zanetti proposed to use mean occupation numbers instead of Boolean fields [McNamara and Zanetti

  this chapter, several LBM-based algorithms for image processing are presented, including image smoothing, image denoising, as well as segmentation. However, it is demonstrated that the advantages of the LBM have not been fully investigated in two aspects. One is flexible control to produce smooth segmentation results, and the other is strong amenability to parallel computing, especially on low-cost, powerful graphics processor units (GPU).

	Résumé
	Une méthodologie de segmentation originale créée en couplant une
	technique de rendu volumique direct (méthode de partitionnement
	basée sur des histogrammes 2D) et une méthode de Boltzmann sur Chapter 4 réseau 3D est proposée dans ce chapitre. La stratégie sous-jacente ré-
	side dans la possibilité de partitionner les séquences d'images médicales
	en tenant compte non seulement de la valeur des niveaux de gris des Segmentation Algorithm coupling voxels mais également le gradient de niveau de gris calculé sur chaque
	voxel utilisé comme deuxième paramètre pour la segmentation. Selon LBM and k-means clustering ce nouveau concept, l'objectif du traitement de séquence d'anévrisme
	cérébral est de segmenter la paroi de l'anévrisme et les vaisseaux san-method guins parents.
	Abstract
	An original segmentation methodology created by coupling a direct
	volume rendering technique (2D histogram based clustering method)
	and a Lattice Boltzmann 3D method is proposed in this chapter. The
	underlying strategy lies in the opportunity to cluster medical image
	sequences by considering not only the grey-level value of voxels. More-
	over, the gradient magnitude calculated on each voxel could be used
	as the second property for segmentation. Under this new concept, the
	purpose of processing a cerebral aneurysm sequence is to segment the
	aneurysm wall and parent blood vessels.
	82 84	Fei GE Fei GE

Table 5 .

 5 1: Values of RV D, DSC and d H from the Chan-Vese method, K-means clustering method and the proposed method.The other main problem of image segmentation is difficult because different regions have distinct feature patterns. Features like color, texture, intensity and edge are some widely used in image segmentation. A variety of methods have been proposed

	Data-sets Evaluation Values Chan-Vese K-means Proposed method
		DSC	0.8905	0.9479	0.6732
	Data_1	RV D	0.0010	0.0221	0.0183
		d h	32.2491	61.0164	64.5058
		DSC	0.0551	0.7922	0.8254
	Data_2	RV D	0.0583	0.0933	0.0684
		d h	34.2491	46.4004	12.0830
	5.4 Clustering with curvature information
	5.4.1 Objective

  Subsequently, the refined 2D density histogram and volumetric data-set are applied to render the volume and locate the clusters in image or volume data-set. This clustered IGM histogram is applied as a "reference page" to calculate the corresponding thresholds of the LBMTM algorithm. Nevertheless, the gradient also restricts the transferability of the proposed algorithm. It is found that other geometry or statistic properties of the volume data-set can be served as features of the k-means clustering method to increase segmentation accuracy. Therefore, the collision function in LBMTM is reconstructed which is the multivariate clusteringbased collision function. In this section, the curvature is selected as an attempt in this direction. After the computation of the gradient and curvature, the proposed method is recursively applied onto the 2D histogram, and a different color is assigned to different features in the 2D histogram. Segmentation can be performed on the 3D volume to the exact different regions of the volume data-set. Herein, the LBM is used in a specific version where the collision function Ω i is implemented

		approximates a spatially connected structure by the k-means clustering algorithm.
		The grey-level value and gradient value are collected to construct a 2D his-
	togram. Moreover, the IGM histogram is segmented into several regions, and each
	115	Fei GE

Table 5 .

 5 2: The Dice similarity coefficient and Hausdorff Distance in different segmentation methods.

		Validation method Cerebral Aneurysm
	Chan-Vese	DSC	0.8905
		d H	32.2491
	K-means	DSC	0.9479
		d H	61.0164
	Proposed method	DSC	0.9557
		d H	56.5509

  Parallel LBMTM algorithm part 2. Calculate f i ( r, t) and construct 2D histogram and equilibrium function with:

	Algorithm 3: 1 Collision with function
	2
	f i ( r, t + δt) = f i (x, t) -	1 τ	(f i (( r, t) -f eq i (( r, t)) + α∆t(T m -ρ)	(6.13)
	3 Update the voxel with:
	4
				19
	ρ( r, t + ∆t) =	f i ( r, t + ∆t)	(6.14)
				i
	6
				k
		arg min C	j=1 ρ∈C j
					(6.12)
	128				Fei GE

6 Set the maximum iteration time N; 7 while iteration <N do 8 Calculate T j with function |ρ -m j | 2 , T j = m j (6.11) 9 Steaming with function:

f i (( r + e i ∆t, t + ∆t) = f i (( r, t)

5

Table 6 .

 6 1: Performance comparison between GPU and CPU.

	Speed-up-ratio	131.24	240.44	615.35	670.17
	time per slice	on GPU	0.13 ms	0.10 ms	0.48 ms	0.68 ms
	time per iteration	on GPU	16.0 ms	20.1 ms	246.0 ms	349 ms
	Calculation time	on GPU	240 ms	310 ms	3690 ms	5329 ms
	3D Volume Volume size Calculation time on CPU	Aneurysm_1 128*128*128 31498 ms	Aneurysm_2 201*201*131 75157 ms	Brain/head 512*512*464 2270650 ms	Stent-assisted 512*512*512 3571335 aneurysm ms

  F. Ge, R. Noël, L. Navarro, and G. Courbebaisse, Volume Rendering and Lattice-Ge, and G. Courbebaisse. A novel parallel lattice Boltzmann method on large scale medical image segmentation. International Conference on Biomedical & Health Informatics, IEEE EMBS BHI 2019, USA, Chicago, May 2019. F. GE, and G. Courbebaisse. Multi-Thresholds Image Segmentation coupling Lattice Boltzmann Method and clustering Method, J. of Computer Physics Communication, Elsevier Ed., Submitted in December 2019.

	F. 135	Boltzmann Method, GRETSI 2017, ID168, Juan Les Pins (France), September 2017. 134 Fei GE Fei GE 136 Fei GE 144 Fei GE

R. Noël, F. Ge, Y. Zhang, L. Navarro, and G. Courbebaisse, Lattice Boltzmann method for modelling of biological phenomena, the 25th European Signal Processing Conference EUSIPCO, Greek island of

Kos, pp. 2654Kos, pp. -2658Kos, pp. , 2017. .

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2020LYSEI012/these.pdf © [F. Ge], [2020], INSA Lyon, tous droits réservés

Fei GECette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2020LYSEI012/these.pdf © [F. Ge], [2020], INSA Lyon, tous droits réservés

La méthode de Boltzmann sur réseau (LBM) est un outil numérique utilisé pour résoudre les équations aux dérivées partielles à l'échelle mésoscopique, permettant ainsi de simuler des phénomènes à l'échelle macroscopique. La thèse porte sur l'étude de la méthode de Boltzmann sur réseau pour la segmentation d'images.Les principales contributions sont les suivantes.Une méthodologie de segmentation originale créée en couplant une méthode de partitionnement à base d'histogrammes 2D et la méthode de Boltzmann sur réseau 3D est proposée; la stratégie réside dans la réalisation d'une classification de séquences d'images médicales en prenant en compte non seulement la valeur des niveaux de gris des voxels, mais aussi des gradients de niveaux de gris existants entre voxels voisins, et qui dans notre application médicale sont liés à la loi de comportement des tissus vivants à caractériser. Dans le cadre de ce nouveau concept, une séquence d'anévrisme cérébral est traitée dans le but de segmenter la paroi de l'anévrisme et les vaisseaux sanguins parents.Il résulte de ce travail de thèse la conception originale d'une méthode de Boltzmann sur réseau multi-seuils, dans laquelle la résolution d'une équation de réactiondiffusion à l'échelle mésoscopique est réalisée via l'élaboration d'une fonction de collision multi-seuils. Ceci conduit à l'optimisation d'une méthode de rendu de volume cohérente en rapport avec la complexité des images médicales cérébrales pour l'étude d'anévrismes cérébraux traités entre autres par stent (prothèse endovasculaire). En introduisant l'algorithme de partitionnement et l'histogramme 2D IGM, un nouveau concept est proposé et réside dans l'utilisation de plusieurs

Acknowledgments

Acknowledgments 20 20

Fei GE 132 Fei GE

Algorithm 2: Parallel LBMTM algorithm part 1.

Input: The volume data-set ρ , LBM model D2Q9 or D3Q19, relaxation time τ , maximum iteration time N , center number m, Output: Volume data-set

1 Initialize the f i ( r, t) on each node of the grid from the grey-level value;

2 Initialize the value of the ρ( r, 0); 3 Initialize the f eq i according to the two functions as follows:

)