Pan Liu

Sidy Fall

Olivier Baledent

Flow 2.0 -a flexible, scalable, cross-platform post-processing software for realtime phase contrast sequences

Introduction (850 words to conclusion 816/850)

With further development of imaging techniques and instrumentation, real-time phase contrast sequences (RT-PC) have become possible [START_REF] Chen | Dynamics of respiratory and cardiac CSF motion revealed with real-time simultaneous multi-slice EPI velocity phase contrast imaging[END_REF] . Compared to conventional phase-contrast sequences, its most significant feature is the quantification of cerebral blood flow or cerebrospinal fluid (CSF) in "real-time".

Although not yet applied in clinical practice, recent studies have shown that RT-PC has a great potential for non-invasive quantification of the effects of respiration on cerebral fluids circulation [START_REF] Aktas | Spinal CSF flow in response to forced thoracic and abdominal respiration[END_REF][START_REF] Balédent | Real-time phase contrast magnetic resonance imaging for assessment of cerebral hemodynamics during breathing[END_REF][START_REF] Lloyd | Respiratory cerebrospinal fluid flow is driven by the thoracic and lumbar spinal pressures[END_REF] . However, the fast imaging speed that impairs RT-PC image quality, and a large number of images make it more difficult to post-process RT-PC data. In general, the post-processing of RT-PC data often requires a lot of manual operations and several software to perform the different steps such as region of interest (ROI) segmentation, background field correction, anti-aliasing correction, extraction of flow signals, and finally quantify the effect of respiration on blood flow by comparing respiration and flow signals. To our knowledge, there is no post-processing software specifically dedicated to RT-PC data. Based on the original version [START_REF] Balédent | Cerebrospinal fluid dynamics and relation with blood flow: a magnetic resonance study with semiautomated cerebrospinal fluid segmentation[END_REF] , we have developed Flow 2.0 software integrating all the above-mentioned features and allowing a high level of automation for end-to-end quantification of the effect of respiration on cerebral blood flow.

Methods

Flow 2.0 was developed using IDL (Interactive Data Language) and was compiled into a 150Mb executable installation-free program which is available on Windows, Linux and Mac systems. Figure 1 shows the postprocessing workflow diagram of the software, it contains two major post-processing steps, the flow quantification (Figure 2) and the quantification of respiratory effects (Figure 3).

Flow quantification

Data extraction, can import the RT-PC images series and the parametric data from the DICOMDIR file according to the conformance statements of the different manufacturers. ROI segmentation, is consisted of two semi-automatic algorithms. One is based on region growth and active contour algorithms to segment Digital poster 2772 vessels with large deformations or displacements during acquisition such as arteries. A second is based on a fully automatic segmentation algorithm in the frequency domain that can easily identify pixels in which dynamic signals are synchronized with the cardiac frequency. This one is used for ROIs with insignificant displacements such as CSF and venous sinuses. Background field correction, is a necessary step to correct eddy current-induced velocity quantization errors [START_REF] Holland | Baseline correction of phase-contrast images in congenital cardiovascular magnetic resonance[END_REF] . Typically, a ROI must be drawn manually in the stationary tissue. Our software includes a background field correction algorithm that selects automatically the stationary tissues around the selected ROI. It used the characteristics of the phase and amplitude in the pixel to recognize static tissue. Curves flows, are calculated and displayed in the viewing interface. The main key-parameters of the flow curve (flow rate, stroke volume, areas of the ROI, maximum velocities, …) can be automatically calculated and presented in the viewing interface as well as with ECG and respiratory signals if recorded. In addition, the software also includes anti-velocity aliasing [START_REF] Balédent | Cerebrospinal fluid dynamics and relation with blood flow: a magnetic resonance study with semiautomated cerebrospinal fluid segmentation[END_REF] and image denoising functions.

A frequency domain signal processing tool interface that is useful for other processing functions was also developed. An Export function can save all the results in a text file, easily importable by other software.

Quantification of respiratory effects

The first step is to separate all the individual Cardiac Cycle Flow Curve (CCFC) of the total flow calculated in the ROI and across the entire images of series. This Signal segmentation, automatically finds the maximum flows systolic points, uses them to define the period and allows to cut the continuous flow rate signal into multiple independent CCFCs. Signal reconstruction, the CCFCs are either labelled as expiratory interval or inspiratory interval based on the recorded respiratory signal. Two average expiratory and inspiratory CCFCs are calculated and used to compare the two respiratory periods. DiffEx-In (parameter, delay) signal, allows to compare each parameter of CCFC (amplitude, mean flow, stroke volume, cardiac period) between the expiratory and the inspiratory period. We hypothesis that breathing impacts blood and CSF flows with a possible delay between the flow curves and the signal recorded by the respiratory belt sensor. This delay was calculated by applying a phase shift on the respiratory signal interval that produce the maximum DiffEx-In for the considered parameter of interest. Extraction results, uses the maximum value of DiffEx-In (parameter, delay) to indicate the intensity and delay of the effect of respiration on this parameter of blood or CSF flow.

Results

Figure 4 shows an example of an in vivo application of the software. The automatic algorithm accurately segmented the CSF Roi in the spinal canal. We can see the flow rate signal and the respiratory quantification results.

Discussion & Conclusion

We have developed a post-processing software for RT-PC that integrates the required functions, with a high degree of automation, with low requirements for a priori knowledge of anatomy, high robustness and supporting functional extensions. The software is compatible with the DICOM files generated by the MRI vendors. Our development makes the post-processing of RT-PC substantially easier, while also providing a multivariate approach to analyze the effects of respiration on cerebral fluids circulation.

Figures (4/5)

Figure 1 :

 1 Figure 1: RT-PC post-processing workflow diagram of Flow 2.0. The RT-PC sequence images can be automatically extracted by reading the DICOMDIR file, and the flow rate signal can be extracted by the flow quantification processing phase (green box & figure2). The flow rate signal and the respiratory signal can be used as input parameters to quantify the effects of respiration on the flow signal (purple box & figure 3).

Figure 2 :

 2 Figure 2: Flow quantification workflow diagram. 400 phase images and 400 amplitude images of the extracranial section were used as input parameters, and the corrected continuous flow rate signal of internal carotid artery (ICA) was used as the output result. Flow 2.0 can extract multiple signals from the ROI, and only the flow rate signal is shown in this figure. The stationary area around the ROI (red contour line) selected by the automatic background field correction algorithm.

Figure 3 :

 3 Figure 3: Quantification of respiratory effects workflow diagram. Flow rate signal of ICA and respiratory signal were used as input parameters. A DiffEx-In (parameter, delay) signal is generated for each parameter (amplitude, mean flow, stroke volume, cardiac period) and its maximum value was taken to represent the intensity and delay of the effect of respiration on this parameter.

Figure 4 :

 4 Figure 4: An in vivo application of Flow 2.0 to quantify the effect of respiration on cerebrospinal fluid (CSF) in the extracranial section. VENC = 5cm/s, imaging speed = 96ms/image, Number image = 300. The ROI of CSF can be accurately defined by a fully automated fixed segmentation algorithm, flow rate signal obtained after background field correction (green box). Combining the respiratory signal, the effect of respiration on the four parameters of CSF flow can be obtained (purple box).

Acknowledgements

This research was supported by equipex FIGURES (Facing Faces Institute Guilding Research), European Union Interreg REVERT Project, Hanuman ANR-18-CE45-0014 and Region Haut de France.

Thanks to the staff members at the Facing Faces Institute (Amiens, France) for technical assistance.

Thanks to David Chechin from Phillips industrie for his scientific support.


Pan Liu

Sidy Fall

Serge Metanbou

Olivier Balédent Real

Olivier Baledent

Real-Time Phase Contrast MRI to quantify Cerebral arterial flow change during variations breathing

The effect of respiration on cerebral arterial blood flow is not very well understood [START_REF] Bari | Nonlinear effects of respiration on the crosstalk between cardiovascular and cerebrovascular control systems[END_REF][START_REF] Ogoh | Interaction between the respiratory system and cerebral blood flow regulation[END_REF] . Conventionally, cardiac gated phase contrast sequences are used for the quantification of cerebral blood flow [START_REF] Nayler | Blood flow imaging by cine magnetic resonance[END_REF][START_REF] Balédent | Cerebrospinal fluid dynamics and relation with blood flow: a magnetic resonance study with semiautomated cerebrospinal fluid segmentation[END_REF][START_REF] Korbecki | Imaging of cerebrospinal fluid flow: fundamentals, techniques, and clinical applications of phase-contrast magnetic resonance imaging[END_REF][START_REF] Lokossou | Extracranial versus intracranial hydrohemodynamics during aging: a PC-MRI pilot cross-sectional study[END_REF] , however, it cannot be applied to studies in this area because it only provides an average Cardiac Cycle Flow Curve (CCFC) [START_REF] Spijkerman | Phase contrast MRI measurements of net cerebrospinal fluid flow through the cerebral aqueduct are confounded by respiration[END_REF] .

With real-time phase contrast (RT-PC) sequences, it has currently been shown that there is an effect of deep breathing or coughing on cerebral blood flow and cerebrospinal fluid [START_REF] Balédent | Real-time phase contrast magnetic resonance imaging for assessment of cerebral hemodynamics during breathing[END_REF][START_REF] Yildiz | Quantifying the influence of respiration and cardiac pulsations on cerebrospinal fluid dynamics using real-time phase-contrast MRI[END_REF][START_REF] Chen | Dynamics of respiratory and cardiac CSF motion revealed with real-time simultaneous multi-slice EPI velocity phase contrast imaging[END_REF][START_REF] Lagana | Arterial, venous and cerebrospinal fluid flow oscillations in real-time phase contrast MRI: type of breathing matters[END_REF] . However, it is currently not well understood whether and to what extent normal respiration affects cerebral arteries blood flow, in this study, we used RT-PC to quantify this effect using a novel time-domain analysis method.

Methods

-Image acquisition 10 participants (19 ~ 35 age; 4 female) were examined using a clinical 3T scanner, maximum gradients 80 mT/m Rate of gradient increase 120 mT m -1 ms -1 and a 32 channels head coil.

The RT-PC used in this study was a multi-shot, gradient-recalled echo-planar imaging sequence with a Cartesian trajectory. Using the parameter presented in Figure 1, the time resolution was 75ms/image, a respiratory monitoring belt was used for recording the respiratory signal. Participants were not asked to perform a specific respiratory pattern during the data collection. RT-PC was performed for each participant to quantify CABF at the intracranial and extracranial levels (Figure 1).

-Image Processing

The images of RT-PC were processed with a homemade software -Flow [START_REF] Fall | A Semi-automatic Software for Processing Real-Time Phase-Contrast MRI Data[END_REF] to extract the CABF signal. The region of interest (ROI) was defined by a semi-automatic dynamic segmentation algorithm and can be also corrected manually. The software then used an automatic background field correction algorithm to find the stationary tissue around the ROI, thus correcting for velocity errors due to eddy currents. Anti-aliasing Oral # 0321 algorithm are also supported in the software. After the segmentation and calibration, the continuous flow rate signal of RT-PC was obtained (Figure 1. E).

CABFextra was obtained by summing arterial flows in the extracranial level, and CABFintra was obtained by summing arterial blood flows in the intracranial level.

-

Quantification of the effect of respiration on arterial blood flow

The quantification process is shown in Figure 2. Signal segmentation: the software automatically segmented the continuous flow rate signal into multiple independent CCFCs. In each CCFC three parameters were calculated (mean flow, stroke volume, cardiac period). Definition of respiratory interval: the CCFCs were labelled as in expiratory interval or in inspiratory interval based on the respiratory signal. Two average expiratory and inspiratory CCFCs were calculated and used to compare the two respiratory periods. DiffEx-In (parameter, delay) signal: the percentage difference DiffEx-In was calculated for each parameter between expiratory CCFC and inspiratory CCFC. We hypothesis that respiration impacts blood with a possible delay between the flow curves and the signal recorded by the respiratory belt sensor. This delay was calculated by applying a phase shift on the respiratory interval that produce the maximum DiffEx-In for the considered parameter of interest. Extraction results, use the maximum value of DiffEx-In (parameter, delay) to indicate the intensity and delay of the effect of respiration on this parameter of CABF. Also, the ratio of the corresponding delay to the average respiratory cycle time was calculated and recorded as delay%.

-Statistical analysis Some arterial flows were excluded due to the poor quality of the signal acquisition. Finally, quantification of the effect of respiration was performed on 58 arteries. The correlation between DiffEx-In and delay% of each parameter was also tested using Spearman's correlation test.

Results

Figure 3 shows the DiffEx-In (parameter, delay) signal of the carotid arteries of two participants.

Figure 4 shows the results of CABF quantification and respiratory impact quantification. The cerebral blood flows were equal to 740±74 ml/min at the extracranial level and 679±119 ml/min at the intracranial level. The DiffEx-In were significantly smaller without considering the delay. In the case of considering delay, the delay% and DiffEx-In of mean flow rate are smaller than that of both stroke volume and cardiac period (p < 0.01). There was also a correlation between the delay% and DiffEx-In of mean flow rate and of cardiac period (Figure 5).

Discussion

In this study, the cerebral arterial blood flow obtained using RT-PC was accordance with the results previously obtained using cardiac gated phase contrast [START_REF] Balédent | Cerebrospinal fluid dynamics and relation with blood flow: a magnetic resonance study with semiautomated cerebrospinal fluid segmentation[END_REF][START_REF] Balédent | Real-time phase contrast magnetic resonance imaging for assessment of cerebral hemodynamics during breathing[END_REF] . It was also found that under normal respiration, cerebral artery blood flow increased by 4.4±1.7% during the expiratory interval compared to the inspiratory interval.

With increasing delay%, the DiffEx-In of the cardiac period was positively correlated, while the DiffEx-In of the mean flow was negatively correlated, which is a very interesting phenomenon and seems to be a positive response to stabilize stroke volume. This can be further confirmed in a subsequent study.

Conclusion

Although there are some artifactual effects, RT-PC is very useful for investigating cerebral blood flow in real time within a few seconds, without any synchronizer.

RT-PC combined with this novel method allows us to quantify the effect of respiration on multiple parameters of blood flow or cerebrospinal fluid, and the delay% parameter provides a new analytical perspective. It provides a technical support for subsequent studies on the effect of respiration on cerebral fluids circulation.

Figures (5/5)

Oral # 0321 Oral # 0321

  Phase Contrast MRI to quantify Cerebral arterial flow change during variations breathing Introduction (850 words to conclusion 849/850)

Figure 1 :

 1 Figure 1: Location of the acquisition planes (A) and arteries (C&D), protocol of RT-PC (B) and the image processing flowchart (E). Right & left internal carotid artery (ICA) and right & left vertebral artery (VA) in the extracranial acquisition plane; right & left internal carotid and basilar artery (BA) in the intracranial acquisition plane.

Figure 2 :

 2 Figure 2: Schematic diagram of the quantitative respiratory impact process. Define inspiratory interval (IN) and expiratory interval (EX) using respiratory signal (A). The flow rate signal is segmented into multiple independent CCFCs and the three parameters are extracted (B), then the percentage difference between expiratory CCFC and inspiratory CCFC (DiffEx-In) is calculated for each parameter by equation C. Finally, the maximum DiffEx-In can be found by shifting the respiratory interval (delay) (D).

Figure 3 :

 3 Figure 3: Example of DiffEx-In (parameter, delay) signal of the internal carotid arteries in two acquisition planes from two participants. ICAR indicates right internal carotid artery, the parameter indicates mean flow rate, stroke volume or cardiac period. The dotted line indicates an average respiratory cycle.

Figure 4 :

 4 Figure 4: Quantification of cerebral arterial blood flow and respiratory effects in 5 participants. The results of the quantification of respiratory effects include both cases, without and with consideration of delays. Using Spearman's test to detect the correlation between delay% and DiffEx-In. ** indicted p < 0.01.

Figure 5 :

 5 Figure 5: Distribution plots of DiffEx-In (A) and delay% (B) for each parameter. There is a negative correlation between the DiffEx-In and Delay% of the mean flow rate and a positive correlation between the DiffEx-In and Delay% of the cardiac period (C). ** indicates significant difference (p < 0.01, Wilcoxon signed-rank test).

Acknowledgements

This research was supported by EquipEX FIGURES (Facing Faces Institute Guilding Research), European Union Interreg REVERT Project, Hanuman ANR-18-CE45-0014 and Region Haut de France.

Thanks to the staff members at the Facing Faces Institute (Amiens, France) for technical assistance.

Thanks to David Chechin from Phillips industry for his scientific support.


Real-time phase contrast magnetic resonance imaging for assessment of cerebral haemodynamics during breathing.

Olivier Balédent , Pan Liu , Armelle Lokossou , Fall Sidy , Serge Metanbou , and Malek Makki EA 7516 Chimère, Jules Verne university, amiens, France, Image Processing, Jules Verne University Hospital, Amiens, France, Neuro Radiology, Jules Verne University Hospital, Amiens, France, MRI Research GIE-IFF CHU Amiens Picardie, Amiens, France

Synopsis

To investigate cerebral blood flows by EPI Phase Contrast (EPI-PC) and Normal Phase contrast sequences, in volunteers. Cerebral arterial and venous blood flows were calculated by homemade software. Arterial cerebral blood flows measured by EPI-PC were significantly higher than those measured by normal PC. Whereas venous cerebral blood flows were not different between the two technics. Post processing of continuous EPI-PC flow signal provides blood flows curves during inspiration and expiration. Blood flows during inspiration were respectively higher than during expiration. EPI-PC flow quantify arterial and venous intracranial blood flows in few seconds without any cardiac or respiratory gating.

INTRODUCTION:

Phase contrast MRI (normal-PC) can quantify cerebral blood flow dynamic by using cardiac synchronization. This sequence reconstruct only one mean flow curve from all the cardiac cycles of the acquisition. It is not possible to take in account the breathing impact that can affect the blood flow. New Phase contrast MRI sequence based on Echo Planar imaging (EPI-PC) can now produce continuously velocity map, more or less every 100 ms function of the machine and of the quality wanted . The aim of this work is to apply an EPI-PC protocol and Normal-PC to investigate total cerebral blood flow and potential respiratory effect on it.

METHODS:

The study was performed on a 3T Achieva dStream scanner with gradient strength of 40 mT/m and a 32 channels head coil (Figure 1). Fourteen adult volunteers in accordance with ethical procedure were included to investigate blood flow in their internal carotids and basilar arteries and in their straight and sagittal sinuses. (Figure 2)Two different acquisitions were used:

• 2D Cine Normal-PC using retrospective plethysmograph gating.

• 2D EPI-PC free of any synchronization.

Post processing was done by homemade software to calculate dynamic blood flow curve for each vessel during cardiac cycle. A background correction area was applied. Normal-PC provides only one flow curve to represent all the cardiac cycles of the acquisition whereas EPI-PC provides continuous flow dynamics curves. Respiratory physiological signals were recorded during the MRI acquisitions using a pneumatic belt sensor. From EPI-PC signal, each cardiac cycle was identified to reconstruct three mean cardiac flow curves from: all the cardiac cycles, the expiration periods and the inspiration periods. (Figure 3). Cerebral arterial and venous blood flows were calculated by summation of the arterial and venous flows. Measured Cerebral venous flow was artificially increased to equal mean arterial flow and generate arterio-venous flow rate and calculate cerebral blood volume change during cardiac cycle3 (figure 4). Statistical Pearson correlation and paired t-student tests were used to compare EPI-PC and Normal-PC results.

RESULTS:

A good correlation between Normal and EPI-PC measurements was found for all the arterial flows. A higher correlation was found for the venous blood flows. (Figure 5) The vessel areas of the internal carotid arteries measured in the EPI-PC images were significantly higher than those segmented in the Normal-PC images (25.7±6.1 mm2 versus 18.9±4.1) whereas area of the sagittal sinuses were not significantly different between the two sequences (35.9±9.9 mm2 versus 34.5±11.8 mm2). Arterial cerebral blood flows (864±370 ml/min) calculated by EPI-PC were significantly (p<0.01) higher than those (583±165 ml/min) measured by normal-PC. Whereas venous cerebral blood flows were not different between the two technic (416±147 ml/min for EPI-PC and 454±200 ml/min for normal-PC). Arterial and venous blood flows during inspiration were respectively higher than during expiration (10% and 8%). Duration of cardiac cycle during inspiration was significantly (p<0.01) smaller (0.80±0.10 sec) than during expiration (0.89±0.15 sec). Mean intracranial vascular blood volume change during cardiac cycle was not significantly different between expiration (0.64±0.21 ml) and inspiration (0.66±0.22 ml). DISCUSSION: Post processing is very important to extract quantitative and interpretable information from the large number of images and no commercial dedicated tool still exists. Due to poor spatial resolution of EPI-PC we found an important difference in the arteries area measurements between the two sequences which explain the important difference found in arterial flows between the two sequences. A better spatial resolution should limit partial volume effect in the arteries. This problem was limited in venous flows because the sinuses presented larger areas and less pulsatile flow. EPI-PC protocol presented a temporal resolution of 113 ms which is too small and explains why arterial flow curve shape was more round than normal-PC. These limitations should be partly corrected by increasing the accelerator EPI factor and decreasing the FOV. These preliminary results have shown how respiratory impact arterial and venous cerebral blood flows and the duration of the cardiac cycle. It was surprising to found that the cerebral blood volume change calculated was not impacted by breathing. But this volume is small and should be interpreted with caution.

1,2 1 1 1 3 1,4 1

CONCLUSION:

This new real time EPI-PC sequence is useful to investigate cerebral blood flow in few seconds without any cardiac or respiratory gating but needs a convivial and accurate post processing tool to be easily used in clinical practice. Such new flow investigations open new ways to better understand cerebral pathologies as idiopathic hydrocephalus, idiopathic hypertension, Chiari malformation and syringomyelia where cerebral fluids dynamics seems to be altered .

Figures

Figure 1 :

 1 Figure 1 : MRI Acquisition parameters

Figure 2 :

 2 Figure 2 : Intracranial acquisition planes selected for EPI-PC and Normal-PC flow sequences on 3D phase contrast angiography. Red arrows represent internal carotid arteries. Orange arrow represents basilar artery. Blue arrows represent Sagittal and Straight sinuses. The schema represents how cardiac and breathing impacts cranio-spinal fluids dynamics.

Figure 3 :

 3 Figure3: Post processing of an internal carotid artery of an healthy volunteer. On the left, blood flow measured by Normal PC, was calculated over the 32 phases of a mean cardiac cycle. On the right the same artery was measured by EPI PC which produces continuously in real time the blood flows over the entire acquisition duration. Each cardiac cycle can be identified (red star). A global mean cardiac cycle was calculated "all". Expiration and inspiration periods were identified by physiological signal to calculate a mean "Expiration" and "Inspiration" flow curves.

Figure 4 :

 4 Figure 4 :Example of EPI PC blood flows analysis. Internal carotids and basilar arteries were summed to calculate total cerebral arterial blood flows from "all" the cardiac cycle acquired, from only the cardiac cycles present in the "expiration" period and from only the cardiac cycles present in the "inspiration" period. In the same way sinuses were summed to calculate the measured cerebral venous flows and calculated theoretical arterio-venous flow and intracranial blood volume change during cardiac cycle during normal breathing. Arterial and venous flows increase during inspiration but no difference was present in arterio-venous and intracranial blood volume change during breathing.

Figure 5 :

 5 Figure 5 : Internal carotid and basilar arteries blood flows measured by normal PC were corelated with those measured by EP PC but some points were far the correlation line showing some limitations these measurements. The second graphic shows a better correlation of the sinuses blood flows measurements.

Acknowledgements

Thanks to :

Garance Arbeaumont Trocme and Caroline Fournier our clever MRI technicians.

David Chechin from Phillips for his scientific support.

All the volunteers who trust us and let us working with their spins

Agence national de la recherche : ANR-18-CE45-0014-04 and Région Haut de France for the financial support.


Ishak Ayad

Cécilia Tarpau

Mai K Nguyen

Ngoc Son Vu

Deep morphological network-based artifact suppression for limited-angle tomography

Keywords: Computed tomography, Image reconstruction, Limited angular projections, Mathematical morphology, Deep learning, U-Net

Computed tomography has been widely used in biomedical and industrial applications. The well-known filtered back-projection algorithm, probably the most used reconstruction technique, fails when the angular range used for data acquisition is not sufficient. As a consequence, reconstructions exhibit artifacts. In order to eliminate these artifacts, we propose in this article a new deep learning approach based on a U-net architecture which includes a morphological operation. This operation of mathematical morphology allows us to capture better some non-linear properties of the object to reconstruct. The proposed method provides good results for angular ranges of 170, 150, 130 and even 110 degrees. To the best of our knowledge, it is the first time a limited-angle artifact suppression method works with 110 projections.

Introduction

Computed tomography (CT) is one of the main imaging techniques used today in many areas such as the clinical diagnosis and material analysis. This imaging technique rests on the measurement of transmitting rays along straight lines. After being emitted by an x-ray tube, radiation is passing through the object to scan and is finally received by a detector. Such a system needs to rotate around the object to acquire a complete set of data, the data acquisition process is mathematically modelled by the classical Radon transform on lines introduced by Radon [START_REF] Radon | Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten[END_REF]. For image reconstruction, we need the inversion formula. Cormack [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications[END_REF] was the forerunner of CT image reconstruction with the proposition of an inversion formula for the Radon transform. This formula is the basis of the well-known filtered back-projection (FBP) algorithm. As with all analytical methods, the need for a high and uniform sampling rate is necessary. Consequently, the FBP algorithm requires complete data with an uniform sampling, that is for instance a consecutive 180°or respectively a 180°fan angle scan with Original object X-ray tube rotation domain Reconstructed object

Fig. 1: FBP reconstruction on a Lung with θ ∈ [20 • , 150 • ] and rotation step = 1 i.e. 130 projection views, The blue segments are the "distortion" artifacts due to limited-angle tomographic data, blue lines are the direction of the artifact 1°as a rotation step for parallel or fan beam reconstruction. However, when the projection data cover an angular range of less than 180°, we are faced to a limited-angle data problem. A limited-angle scan can arise in many situations, for instance where we scan a large size object, when we use large-pitch helical CT or we have to perform a restricted scanning. If one uses the FBP algorithm, this results in images with heavy directional artifacts and distortions. Some alternative reconstruction methods including optimization techniques have been studied in order to improve reconstruction quality, such as singular value decomposition (SVD) [START_REF] Liu | Singular value decomposition-based 2D image reconstruction for computed tomography[END_REF], total variation [START_REF] Gomi | Use of a Total Variation Minimization Iterative Reconstruction Algorithm to Evaluate Reduced Projections during Digital Breast Tomosynthesis[END_REF] and Tikhonov regularization [START_REF] Peng | A Tikhonov Regularization Method for Image Reconstruction[END_REF]. Even if the obtained results may be slightly better, these techniques cannot totally correct the artifact patterns of CT images. More recently, deep learning architectures using convolutional neural networks (CNN) have achieved overwhelming success in computer vision applications including image classification [START_REF] Krizhevsky | Imagenet Classification with Deep Convolutional Neural Networks[END_REF], denoising [START_REF] Zhang | Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising[END_REF] and segmentation [START_REF] Ronneberger | U-net: Convolutional Networks for Biomedical Image Segmentation[END_REF]. It is in that context that some studies proposed also to use deep learning architectures to tackle this type of image reconstruction problems. We review some of them in the next paragraph.

Gu and Ye [START_REF] Gu | Multi-Scale Wavelet Domain Residual Learning for Limited-Angle CT Reconstruction[END_REF] adapted the U-net architecture [START_REF] Ronneberger | U-net: Convolutional Networks for Biomedical Image Segmentation[END_REF] to learn artifacts in the wavelet domain instead of directly learning the artifact-free image. Wang et al. [START_REF] Wang | Deep learning based image reconstruction algorithm for limited-angle translational computed tomography[END_REF] proposed a U-net-based architecture to solve the problem by pre-processing the input image using the simultaneous algebraic reconstruction technique [START_REF] Andersen | Simultaneous Algebraic Reconstruction Technique (SART): A Superior Implementation of the ART Algorithm[END_REF] algorithm to reduce the limited-angle distortion. Tatiana and collaborators [START_REF] Bubba | Learning The Invisible: A hybrid deep learning-shearlet framework for limited angle computed tomography[END_REF] proposed a framework to solve the inverse problem of limited-angle CT by combining model-based sparse regularization using shearlets with a data-driven deep neural network approach. In [START_REF] Zhang | Image Prediction for Limited-angle Tomography via Deep Learning with Convolutional Neural Network[END_REF], Zhang et al. proposed an optimized deep convolutional neural network containing three layers. The objective of the first layer is to extract the feature maps. The second is a non-linear mapping which allows reducing the artifacts, and the final layer was used to combine the extracted maps, where each layer is modeled by a convolution operation with multiple kernels. One can refer also to the work of Würfl et al. [START_REF] Würfl | Deep learning computed tomography: Learning projection-domain weights from image domain in limited angle problems[END_REF], where he proposed a neural network architecture based on FBP to learn the compensation weights for limited-angle reconstruction. This particular method showed some limits for small angular ranges, e.g from 120°. In [START_REF] Wang | A deep network for sinogram and CT image reconstruction[END_REF], Wang et al. proposed a deep network for sinogram denoising and CT image reconstruction simultaneously introducing a FBP layer and combined it with two cascaded blocks.

In this paper, we concentrate on limited data coming from an incomplete angular data coverage. We give an example of such reconstruction in Fig. 1, that is, when the FBP algorithm is used to recover the object from data on a restricted angular domain of θ ∈ [20 • , 150 • ]. According to the considered angular range, some specific features are recovered correctly whereas other zones suffer from artifacts. This kind of artifacts is well characterized by microlocal analysis, see e.g. [START_REF] Frikel | Characterization and reduction of artifacts in limited angle tomography[END_REF]. In fact, streak artifacts appear when the zones of the sinogram corresponding to the features of the objects were suppressed because of limitedangle data.

To remove the limited-angle artifact in CT images, we propose here a deep network based on a U-net architecture and including the opening morphological operator and residual learning. After the learning process, we faced this architecture to a validation test, different from training data, in order to prove that the model is not overfitting. The reconstruction quality is quantitatively evaluated with the peak signal noise to ratio and the structural similarity index. In addition, a comparison of our results with Tikhonov regularization and the architecture proposed by Zhang [START_REF] Zhang | Image Prediction for Limited-angle Tomography via Deep Learning with Convolutional Neural Network[END_REF] is proposed.

The outline of this paper is as follows. In section 2, we present the proposed architecture suitable for artifact reduction. The obtained results are then presented in section 3. Then, Section 4 ends the paper with a discussion about the obtained results and some concluding remarks.

Materials and Methods

The proposition of an architecture convenient for limited-angular data problem

The proposed architecture (see Fig. 2) is a U-net with an encoder path made of 4 blocks. Each of the blocks has respectively 8, 16, 32, and 64 kernel depth and contains two successive convolutional layers with 3×3 filter with ReLu activation followed by another convolutional layer with 1×1 filter with ReLu activation and a batch normalization to avoid overfitting. A bottleneck is added with 128 filters and two convolutional layers with 3 × 3 filter with ReLu activation followed by another convolutional layer with 1 × 1 with ReLu activation. Each block consists of two operations, an extraction and a nonlinear mapping, see for instance Figure 6.

The decoder path is also made of 4 up blocks. Each block has respectively 64, 32, 16, and 8 kernel depth, and each is related to the same resolution block Fig. 2: Structure of the proposed deep convolutional network for artifact suppression of the encoder via a horizontal skip connection which skips some layer in the neural network and feeds the output of one layer as the input to the next layers. A skip connection provides an alternative path for the gradient (with back propagation), and gives the localization, where each block performs the same operations as the same resolution block of the encoder. In the end block of the architecture, a skip connection between the input and the output of the U-net is also added. Thus, this allows the end block to learn the difference between the input and the output. Moreover, in order to remove noise we added, to each block of the decoder, a mathematical morphology layer which uses the opening operator (composition of erosion and dilatation [START_REF] Franchi | Deep morphological networks[END_REF][START_REF] Velasco-Forero | Morpholayers[END_REF]). In fact, the opening operator removes small particles (smaller than the size of the structuring element "kernel", see Figure 3 for example, where the opening operation implements the channel-wise max/min-plus convolution).

The choice of the number of filters is important, for the first block we have chosen 8 filters in order to have very few feature maps that contains artifacts (see Fig. 6). Furthermore, we can observe that using the average pooling we are able to eliminate the artifacts on 6 feature maps and that we still have one with artifacts that will be eliminated thanks to the following blocks. Then, the decoder will not only increase the resolution of the image but also correct the shape of the object thanks to the mathematical morphology layer.

Dataset

To validate our architecture for artifact reduction and compare it to existing methods, we performed an experiment on 7300 images of the standard clinical → Fig. 3: (5×5) opening noise removal using the opening operator, where the noise is the small square, situated in the left-top corner of the image dataset called PhantomFDA [START_REF] Gavrielides | FDA phantom ct database: a resource for the assessment of lung nodule size estimation methodologies and software development[END_REF] downloaded from TCIA [START_REF] Clark | The cancer imaging archive (tcia): Maintaining and operating a public information repository[END_REF], The size of the images is 256 × 256 pixels. Figure 5 shows an example of used images.

Note that this dataset proposes only image reconstructions from complete set of projections. As far as the author's knowledge, no dataset providing limited angle reconstructions is available. We construct our own set of reconstructed images with different angular ranges (see Table 1) from perfect reconstructions using radon and iradon functions given in the skimage Python library [START_REF] Van Der Walt | scikit-image: image processing in python[END_REF].

Number of angular projections Angular range

170 [10°,180°]-[0°,170°] 150 [30°,180°]-[20°,170°] [10°,160°]-[0°,150°] 130 [50°,180°]-[40°,170°] [30°,160°]-[20°,150°] [10°,140°]-[0°,130°]
Table 1: Angular range used to build the dataset

The input images for training and testing our algorithm (as well as the ones used to train Zhang architecture) is the FBP reconstruction.

Data augmentation

To avoid the problem of overfitting we perform some data augmentation, where 4% of the original data was taken and we have performed one of the three different transformations on each CT image randomly (see Fig. 4).

In addition to this, since the used angular range is also chosen randomly, it can be considered also as a type of data augmentation, which reinforce the robustness of the proposed reconstruction algorithm. In the proposed implementation, the probability of performing twice a reconstruction of the same image for a different angular range is 0.4.

Used libraries

We used the Tensorflow library [START_REF] Abadi | TensorFlow: Large-scale machine learning on heterogeneous systems[END_REF] to generate our U-net architecture. This library offers us a simple manipulation of the model parameters. The global implementation of the architecture was implemented using the colab research google platform, which offers free GPU usage to accelerate the learning process.

Quantitative image analysis

To evaluate the image quality quantitatively, we used two metrics, the Peak Signal Noise to Ratio (PSNR) and the Structural Similarity Index Metric (SSIM). where the Mean Squared Error (MSE) is defined as:

MSE = 1 N 2 × N i=1 N j=1 [I ref (i, j) -I res (i, j)] 2 (2)
N is the length/width of the image and max(I ref ) the maximum pixel value of the reference image.

In our case max(I ref ) = 1 because all images are values-normalized between 0 and 1, hence

PSNR = -10 × log 10 (MSE) (3)
According to the literature, a value between 30 and 40 dB defines a good quality image.

Furthermore, the SSIM is calculated as follows:

SSIM = (2I res I ref + c 1 )(2Cov(I ref , I res ) + c 2 ) (I 2 res + I 2 ref + c 1 )(σ 2 res + σ 2 ref + c 2 ) ( 4
)
where I is the mean value for I, σ 2 the variance, and covariance of intensities is defined as follows: 2 are two variables to stabilize the division with weak denominator and L is the dynamic range of the pixel-values (typically this is 2 #bits per pixel -1). k 1 = 0.01 and k 2 = 0.03 by default. A SSIM value closer to one indicates a higher degree of similarity between the predicted image and the reference image.

Cov(I ref , I res ) = 1 N -1 N i=1 N j=1 [I res (i, j) -I res ] × [I ref (i, j) -I ref ] c 1 = (k 1 L) 2 , c 2 = (k 2 L)

Results

To validate and evaluate the performance of the proposed architecture, we used test images that were not given to the architecture during the training process. Furthermore, we tested our model using an image reconstructed from only 110 projections to show that the model is not overfitting on the training data. In addition to quantitative inspection of the results, the mean PSNR and SSIM of the predicted images with respect to the angular range projections are calculated and summed up in Table [START_REF] Cormack | Representation of a function by its line integrals, with some radiological applications[END_REF][START_REF] Liu | Singular value decomposition-based 2D image reconstruction for computed tomography[END_REF]. Reconstruction results show the superiority of the proposed method, compared to the existing ones. Comparison between the input and the predicted images obtained with Tikhonov regularization, the architecture proposed by [START_REF] Zhang | Image Prediction for Limited-angle Tomography via Deep Learning with Convolutional Neural Network[END_REF] and the proposed method.

Number of angular

Discussion and Conclusions

The deep learning framework including mathematical morphology efficiently enhances the limited-angle artifact removal with significantly fewer parameters than other methods. Testing our method on images reconstructed from 130, 150 or 170 projections gives very good results, even from 110 projections without any training on this kind of data (see figure 7). Future work will focus on adapting our method to other kinds of CT artifacts such as those from sparse-view projections, low-dose radiation. Moreover, the study of the effectiveness of the proposed method directly on signograms (instead of the images reconstructed by FBP) will also be of interest since the cause of these different artifacts is indeed the lack of data on sinograms. The study of the robustness of such architecture on more realistic data containing Poisson noise will also be a focus for our future research.

1 )Fig. 4 :Fig. 5 :

 145 Fig. 4: Example of obtained images by data augmentation. (b) is the result of applying a cropping with (200, 200) as a crop size, (c) a cropping followed by a translation of 25 shifted pixels and (d) results in the rotation of a random angle between 10°and 180°from the original FBP reconstructed image (a) with 130 angular projections

Fig. 6 :

 6 Fig. 6: Result of the 1 st block of the proposed model

Table 2 :

 2 Mean PSNR value for 110, 130, 150 and 170 angular range projections. Comparison between the input and the predicted images obtained with Tikhonov regularization, the architecture proposed by[START_REF] Zhang | Image Prediction for Limited-angle Tomography via Deep Learning with Convolutional Neural Network[END_REF] and the proposed method.

		Method	Input images output images
	projections		mean PSNR mean PSNR
		Tikhonov	18.7	24
	170	Zhang algorithm[13]	-	30.7
		Proposed method	-	34.5
		Tikhonov	15.5	23
	150	Zhang algorithm[13]	-	27.3
		Proposed method	-	33.5
		Tikhonov	13.5	20
	130	Zhang algorithm[13]	-	24
		Proposed method	-	32.2
		Tikhonov	10.4	18
	110	Zhang algorithm[13]	-	19.9
	Used only for tests Proposed method	-	28
	Number of angular	Method	Input images output images
	projections		mean SSIM mean SSIM
		Tikhonov	0.42	0.43
	170	Zhang algorithm[13]	-	0.98
		Proposed method	-	0.99
		Tikhonov	0.37	0.4
	150	Zhang algorithm[13]	-	0.94
		Proposed method	-	0.97
		Tikhonov	0.35	0.55
	130	Zhang algorithm[13]	-	0.87
		Proposed method	-	0.96
		Tikhonov	0.17	0.28
	110	Zhang algorithm[13]	-	0.74
	Used only for tests Proposed method	-	0.81

Table 3 :

 3 Mean SSIM value for 110, 130, 150 and 170 angular range projections.

Original image

110°130°150°170°F BP Tikhonov Zhang [START_REF] Zhang | Image Prediction for Limited-angle Tomography via Deep Learning with Convolutional Neural Network[END_REF] Proposed algorithm