State-of-the-art reconstruction algorithms for medical helical cone-beam Computed Tomography (CT) are of type non-exact Filtered Backprojection (FBP). They are attractive because of their simplicity and low computational cost, but they produce sub-optimal images with respect to artifacts, resolution, and noise. This thesis deals with possibilities to improve the image quality by means of iterative techniques. The first algorithm, Regularized Iterative Weighted Filtered Backprojection (RIWFBP), is an iterative algorithm employing the non-exact Weighted FilteredBackprojection (WFBP) algorithm [Stierstorfer et al., Phys. Med. Biol. 49, 2209-2218, 2004] in the update step. We have measured and compared artifact reduction as well as resolution and noise properties for RIWFBP and WFBP. The results show that artifacts originating in the non-exactness of the WFBP algorithm are suppressed within five iterations without notable degradation in terms of resolution versus noise. Our experiments also indicate that the number of required iterations can be reduced by employing a technique known as ordered subsets. A small modification of RIWFBP leads to a new algorithm, the Weighted Least Squares Iterative Filtered Backprojection (WLS-IFBP). This algorithm has a slightly lower rate of convergence than RIWFBP, but in return it has the attractive property of converging to a solution of a certain least squares minimization problem. Hereby, theory and algorithms from optimization theory become applicable. Besides linear regularization, we have examined edge-preserving non-linear regularization.In this case, resolution becomes contrast dependent, a fact that can be utilized for improving high contrast resolution without degrading the signal-to-noise ratio in low contrast regions. Resolution measurements at different contrast levels and anthropomorphic phantom studies confirm this property. Furthermore, an even morepronounced suppression of artifacts is observed. Iterative reconstruction opens for more realistic modeling of the input data acquisition process than what is possible with FBP. We have examined the possibility to improve the forward projection model by (i) multiple ray models, and (ii) calculating strip integrals instead of line integrals. In both cases, for linearregularization, the experiments indicate a trade off: the resolution is improved atthe price of increased noise levels. With non-linear regularization on the other hand, the degraded signal-to-noise ratio in low contrast regions can be avoided. Huge input data sizes make experiments on real medical CT data very demanding. To alleviate this problem, we have implemented the most time consuming parts of the algorithms on a Graphics Processing Unit (GPU). These implementations are described in some detail, and some specific problems regarding parallelism and memory access are discussed.

DIRA-3D is an iterative model-based reconstruction method for dual-energy helical CT whose goal is to determine the material composition of the patient from accurate linear attenuation coefficients (LACs). Possible applications are, for example, to aid in calculations of radiation transport and dose calculations in brachytherapy with low energy photons, and in proton therapy. There was a need to replace the current image reconstruction method, the PI-method, with a weighted filtered backprojection (wFBP) algorithm for image reconstruction, since wFBP is used for image reconstruction in Siemens's CT-scanners. The new DIRA-3D algorithm implemented the program take for cone-beam projection generation and the FreeCT wFBP algorithm for image reconstruction. Experiments showed that the accuracies of the resulting LACs for the DIRA-3D algorithm using wFBP for image reconstruction were comparable to the one using the PI-method for image reconstruction. The relative LAC errors reached a value below 0.2% after 10 iterations.

We propose an exact filtered backprojection algorithm for inversion of the cone beam data in the case when the trajectory is composed of a distorted circle and a line segment. The length of the scan is determined by the region of interest , and it is independent of the size of the object. With few geometric restrictions on the curve, we show that we have an exact reconstruction. Numerical experiments demonstrate good image quality.

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Une des thématiques abordée par l'équipe Image et Vidéo-Communication est la reconstruction tomographique discréte à l'aide de la transformée Mojette. Ma thèse s'inscrit dans le cadre de la reconstruction tomographique médicale. La transformée Mojette est une version discrète exacte de la transformée de Radon qui est l'outil mathématique permettant la reconstruction tomographique. Pour évaluer la qualité des reconstructions, nous avons utilisé des fantômes numériques 2D simples (objet carré, rond) en absence puis en présence de bruit. Le coeur de mon travail de thèse est la reconstruction d'un objet à l'aide d'un algorithme de rétroprojection filtrée exacte Mojette en absence de bruit s'appuyant sur la géométrie discrète. Pour un nombre fini de projections dépendant de la taille de l'objet à reconstruire la reconstruction est exacte. La majorité des tomographes industriels utilisent l'algorithme de rétroprojection de projections filtrées (Filtered Back Projection ou FBP) pour reconstruire la région d'intérêt. Cet algorithme possède deux défauts théoriques, un au niveau du filtre utilisé, l'autre au niveau de la rétroprojection elle-même. Nous avons pu mettre au point un algorithme de Mojette FBP. Cet algorithme fait partie des méthodes directes de reconstruction. Il a aussi été testé avec succès en présence de bruit. Cet algorithme permet une équivalence continu-discret lors de la reconstruction. L'étape de projection/rétroprojection Mojette présente la particularité intéressante de pouvoir être décrit par une matrice Toeplitz bloc Toeplitz. Pour utiliser cette propriété nous avons mis en oeuvre un algorithme de gradient conjugué.

In this thesis an algorithm is demonstrated for the reconstruction of hard-field Tomography images through localized block areas, obtained in parallel and from a multiresolution framework. Block areas are subsequently tiled to put together the full size image. Given its properties to preserve its compact support after being ramp filtered, the wavelet transform has received to date much attention as a promising solution in radiation dose reduction in medical imaging, through the reconstruction of essentially localised regions. In this work, this characteristic is exploited with the aim of reducing the time and complexity of the standard reconstruction algorithm. Independently reconstructing block images with geometry allowing to cover completely the reconstructed frame as a single output image, allows the individual blocks to be reconstructed in parallel, and to experience its performance in a multiprocessor hardware reconfigurable system (i.e. FPGA). Projection data from simulated Radon Transform (RT) was obtained at 180 evenly spaced angles. In order to define every relevant block area within the sinogram, forward RT was performed over template phantoms representing block frames. Reconstruction was then performed in a domain beyond the block frame limits, to allow calibration overlaps when fitting of adjacent block images. The 256 by 256 Shepp-Logan phantom was used to test the methodology of both parallel multiresolution and parallel block reconstruction generalisations. It is shown that the reconstruction time of a single block image in a 3-scale multiresolution framework, compared to the standard methodology, performs around 48 times faster. By assuming a parallel implementation, it can implied that the reconstruction time of a single tile, should be very close related to the reconstruction time of the full size and resolution image.

Filtered backprojection (FBP) reconstruction algorithms are very popular in the field of X-ray computed tomography (CT) because they give advantages in terms of the numerical accuracy and computational complexity. Ramp filter based fan-beam FBP reconstruction algorithms have the position dependent weight in the backprojection which is responsible for spatially non-uniform distribution of noise and resolution, and artifacts. Many algorithms based on shift variant filtering or spatially-invariant interpolation in the backprojection step have been developed to deal with this issue. However, these algorithms are computationally demanding. Recently, fan-beam algorithms based on Hilbert filtering with inverse distance weight and no weight in the backprojection have been derived using the Hamaker’s relation. These fan-beam reconstruction algorithms have been shown to improve noise uniformity and uniformity in resolution. In this thesis, fan-beam FBP reconstruction algorithms with inverse distance back-projection weight and no backprojection weight for 2D image reconstruction are presented and discussed for the two fan-beam scan geometries -equi-angular and equispace detector array. Based on the proposed and discussed fan-beam reconstruction algorithms with inverse distance backprojection and no backprojection weight, new 3D cone-beam FDK reconstruction algorithms with circular and helical scan trajectories for curved and planar detector geometries are proposed. To start with three rebinning formulae from literature are presented and it is shown that one can derive all fan-beam FBP reconstruction algorithms from these rebinning formulae. Specifically, two fan-beam algorithms with no backprojection weight based on Hilbert filtering for equi-space linear array detector and one new fan-beam algorithm with inverse distance backprojection weight based on hybrid filtering for both equi-angular and equi-space linear array detector are derived. Simulation results for these algorithms in terms of uniformity of noise and resolution in comparison to standard fan-beam FBP reconstruction algorithm (ramp filter based fan-beam reconstruction algorithm) are presented. It is shown through simulation that the fan-beam reconstruction algorithm with inverse distance in the backprojection gives better noise performance while retaining the resolution properities. A comparison between above mentioned reconstruction algorithms is given in terms of computational complexity. The state of the art 3D X-ray imaging systems in medicine with cone-beam (CB) circular and helical computed tomography scanners use non-exact (approximate) FBP based reconstruction algorithm. They are attractive because of their simplicity and low computational cost. However, they produce sub-optimal reconstructed images with respect to cone-beam artifacts, noise and axial intensity drop in case of circular trajectory scan imaging. Axial intensity drop in the reconstructed image is due to the insufficient data acquired by the circular-scan trajectory CB CT. This thesis deals with investigations to improve the image quality by means of the Hilbert and hybrid filtering based algorithms using redundancy data for Feldkamp, Davis and Kress (FDK) type reconstruction algorithms. In this thesis, new FDK type reconstruction algorithms for cylindrical detector and planar detector for CB circular CT are developed, which are obtained by extending to three dimensions (3D) an exact Hilbert filtering based FBP algorithm for 2D fan-beam beam algorithms with no position dependent backprojection weight and fan-beam algorithm with inverse distance backprojection weight. The proposed FDK reconstruction algorithm with inverse distance weight in the backprojection requires full-scan projection data while the FDK reconstruction algorithm with no backprojection weight can handle partial-scan data including very short-scan. The FDK reconstruction algorithms with no backprojection weight for circular CB CT are compared with Hu’s, FDK and T-FDK reconstruction algorithms in-terms of axial intensity drop and computational complexity. The simulation results of noise, CB artifacts performance and execution timing as well as the partial-scan reconstruction abilities are presented. We show that FDK reconstruction algorithms with no backprojection weight have better noise performance characteristics than the conventional FDK reconstruction algorithm where the backprojection weight is known to result in spatial non-uniformity in the noise characteristics. In this thesis, we present an efficient method to reduce the axial intensity drop in circular CB CT. The efficient method consists of two steps: the first one is reconstruction of the object using FDK reconstruction algorithm with no backprojection weight and the second is estimating the missing term. The efficient method is comparable to Zhu et al.’s method in terms of reduction in axial intensity drop, noise and computational complexity. The helical scanning trajectory satisfies the Tuy-smith condition, hence an exact and stable reconstruction is possible. However, the helical FDK reconstruction algorithm is responsible for the cone-beam artifacts since the helical FDK reconstruction algorithm is approximate in its derivation. In this thesis, helical FDK reconstruction algorithms based on Hilbert filtering with no backprojection weight and FDK reconstruction algorithm based on hybrid filtering with inverse distance backprojection weight are presented to reduce the CB artifacts. These algorithms are compared with standard helical FDK in-terms of noise, CB artifacts and computational complexity.

This work focuses on using the linear prediction based projection completion for the fan-beam and cone-beam reconstruction algorithm with no backprojection weight. The truncated data problems are addressed in the computed tomography research. However, the image reconstruction from truncated data perfectly has not been achieved yet and only approximately accurate solutions have been obtained. Thus research in this area continues to strive to obtain close result to the perfect. Linear prediction techniques are adopted for truncation completion in this work, because previous research on the truncated data problems also have shown that this technique works well compared to some other techniques like polynomial fitting and iterative based methods. The Linear prediction technique is a model based technique. The autoregressive (AR) and moving average (MA) are the two important models along with autoregressive moving average (ARMA) model. The AR model is used in this work because of the simplicity it provides in calculating the prediction coefficients. The order of the model is chosen based on the partial autocorrelation function of the projection data proved in the previous researches that have been carried out in this area of interest. The truncated projection completion using linear prediction and windowed linear prediction show that reasonably accurate reconstruction is achieved. The windowed linear prediction provide better estimate of the missing data, the reason for this is mentioned in the literature and is restated for the reader’s convenience in this work. The advantages associated with the fan-beam reconstruction algorithms with no backprojection weights compared to the fan-beam reconstruction algorithm with backprojection weights motivated us to use the fan-beam reconstruction algorithm with no backprojection weight for reconstructing the truncation completed projection data. The results obtained are compared with the previous work which used conventional fan-beam reconstruction algorithms with backprojection weight. The intensity plots and the noise performance results show improvements resulting from using the fan-beam reconstruction algorithm with no backprojection weight. The work is also extended to the Feldkamp, Davis, and Kress (FDK) reconstruction algorithm with no backprojection weight for the helical scanning geometry and the results obtained are compared with the FDK reconstruction algorithm with backprojection weight for the helical scanning geometry.

L'utilisation de protons dans le traitement du cancer est largement reconnue grâce au parcours fini des protons dans la matière. Pour la planification du traitement par protons, l'incertitude dans la détermination de la longueur du parcours des protons provient principalement de l'inexactitude dans la conversion des unités Hounsfield (obtenues à partir de tomographie rayons X) en pouvoir d'arrêt des protons. La tomographie proton (pCT) est une solution attrayante car cette modalité reconstruit directement la carte du pouvoir d'arrêt relatif à l'eau (RSP) de l'objet. La technique pCT classique est basée sur la mesure de la perte d'énergie des protons pour reconstruire la carte du RSP de l'objet. En plus de la perte d'énergie, les protons subissent également des diffusions coulombiennes multiples et des interactions nucléaires qui pourraient révéler d'autres propriétés intéressantes des matériaux non visibles avec les cartes de RSP. Ce travail de thèse a consisté à étudier les interactions de protons au travers de simulations Monte Carlo par le logiciel GATE et d'utiliser ces informations pour reconstruire une carte de l'objet par rétroprojection filtrée le long des chemins les plus vraisemblables des protons. Mise à part la méthode pCT conventionnelle par perte d'énergie, deux modalités de pCT ont été étudiées et mises en œuvre. La première est la pCT par atténuation qui est réalisée en utilisant l'atténuation des protons pour reconstruire le coefficient d'atténuation linéique des interactions nucléaires de l'objet. La deuxième modalité pCT est appelée pCT par diffusion qui est effectuée en mesurant la variation angulaire due à la diffusion coulombienne pour reconstruire la carte de pouvoir de diffusion, liée à la longueur de radiation du matériau. L'exactitude, la précision et la résolution spatiale des images reconstruites à partir des deux modalités de pCT ont été évaluées qualitativement et quantitativement et comparées à la pCT conventionnelle par perte d'énergie. Alors que la pCT par perte d'énergie fournit déjà les informations nécessaires pour calculer la longueur du parcours des protons pour la planification du traitement, la pCT par atténuation et par diffusion donnent des informations complémentaires sur l'objet. D'une part, les images pCT par diffusion et par atténuation fournissent une information supplémentaire intrinsèque aux matériaux de l'objet. D'autre part, dans certains des cas étudiés, les images pCT par atténuation démontrent une meilleure résolution spatiale dont l'information fournie compléterait celle de la pCT par perte d'énergie. / The use of protons in cancer treatment has been widely recognized thanks to the precise stopping range of protons in matter. In proton therapy treatment planning, the uncertainty in determining the range mainly stems from the inaccuracy in the conversion of the Hounsfield units obtained from x-ray computed tomography to proton stopping power. Proton CT (pCT) has been an attractive solution as this modality directly reconstructs the relative stopping power (RSP) map of the object. The conventional pCT technique is based on measurements of the energy loss of protons to reconstruct the RSP map of the object. In addition to energy loss, protons also undergo multiple Coulomb scattering and nuclear interactions which could reveal other interesting properties of the materials not visible with the RSP maps. This PhD work is to investigate proton interactions through Monte Carlo simulations in GATE and to use this information to reconstruct a map of the object through filtered back-projection along the most likely proton paths. Aside from the conventional energy-loss pCT, two pCT modalities have been investigated and implemented. The first one is called attenuation pCT which is carried out by using the attenuation of protons to reconstruct the linear inelastic nuclear cross-section map of the object. The second pCT modality is called scattering pCT which is performed by utilizing proton scattering by measuring the angular variance to reconstruct the relative scattering power map which is related to the radiation length of the material. The accuracy, precision and spatial resolution of the images reconstructed from the two pCT modalities were evaluated qualitatively and quantitatively and compared with the conventional energy-loss pCT. While energy-loss pCT already provides the information needed to calculate the proton range for treatment planning, attenuation pCT and scattering pCT give complementary information about the object. For one, scattering pCT and attenuation pCT images provide an additional information intrinsic to the materials in the object. Another is that, in some studied cases, attenuation pCT images demonstrate a better spatial resolution and showed features that would supplement energy-loss pCT reconstructions.