Incomplete oblique projections method for solving regularized least-squares problems in image reconstruction

In this paper we improve on the incomplete oblique projections (IOP) method introduced previously by the authors for solving inconsistent linear systems, when applied to image reconstruction problems. That method uses IOP onto the set of solutions of the augmented system Ax - r = b, and converges to a weighted least-squares solution of the system Ax=b. In image reconstruction problems, systems are usually inconsistent and very often rank-deficient because of the underlying discretized model. Here we have considered a regularized least-squares objective function that can be used in many ways such as incorporating blobs or nearest-neighbor interactions among adjacent pixels, aiming at smoothing the image. Thus, the oblique incomplete projections algorithm has been modified for solving this regularized model. The theoretical properties of the new algorithm are analyzed and numerical experiments are presented showing that the new approach improves the quality of the reconstructed images.Material digitalizado en SEDICI gracias a la Biblioteca de la Facultad de Ingeniería (UNLP).Facultad de Ciencias Exacta

On the incomplete oblique projections method for solving box constrained least squares problems

The aim of this paper is to extend the applicability of the incomplete oblique projections method (IOP) previously introduced by the authors for solving inconsistent linear systems to the box constrained case. The new algorithm employs incomplete projections onto the set of solutions of the augmented system Ax − r = b, together with the box constraints, based on a scheme similar to the one of IOP, adding the conditions for accepting an approximate solution in the box. The theoretical properties of the new algorithm are analyzed, and numerical experiences are presented comparing its performance with some well-known methods.Facultad de IngenieríaFacultad de Ciencias Exacta

X-ray computed tomography of pipe sections by discrete tomography and total variation minimization

Inversio-ongelmien tutkimus on soveltavan matematiikan ala jossa tutkitaan tuntemattoman funktion palauttamista joukosta funtiosta tehtyjä mittauksia, jotka saattavat olla epätäydellisiä ja kohinaisia. Esimerkki käytännön sovelluskohteesta inversio- ongelmien alalla on röntgentomografia, missä yritetään nähdä jonkin kohteen sisäinen rakenne ottamalla useita röntgenprojektiokuvia eri puolilta tutkittavaa kohdetta. Tomografinen rekonstruktiotehtävä on erittäin huonosti asetettu ongelma Hadamardin määritelmän mukaan, mikäli käytettävissä on vain rajoitettu määrä projektiokuvia. Esimerkkejä rajoitetun datan tomografiasta ovat harvan kulman tomografia ja rajoitetun kulman tomografia. Harvan kulman tomografiassa projektioiden määrää on rajoitettu tutkittavan kohteen säteilyaltistuksen rajoittamiseksi tai kuvausprosessin nopeuttamiseksi, ja rajoitetun kulman tomografiassa projektioita on saatavilla vain tietystä kulmasta. Rajoitetun datan tomografiaan on esitetty monia algoritmeja, joista monet usein hyödyntävät ennalta tunnettua a priori- informaatiota tutkittavasta kohteesta. Tällaisia algoritmeja ovat totaalivariaatiominimointi ja diskreetin tomografian algoritmit, joita molempia tutkitaan tässä tutkielmassa. Totaalivariaatiominimoinnissa oletetaan että rekonstruoitavan kuvan totaalivariaatio on pieni. Diskreetin tomografian algoritmeissa oletetaan että kohdeobjekti koostuu rakenteeltaan homogeenisista materiaaleista, joita on vain rajoitettu määrä. Eräs modern diskreetin tomografian algoritmi on pehmeä diskreetti algebrallinen rekonstruktiomenelmä SDART. Tässä tutkielmassa tutkitaan ongelmaa, jossa tavoitteena on palauttaa metalliputken seinämän sisällä sijaitsevien alle millim etrin kokoluokkaa olevien aukkojen koko, muoto ja sijanti röntgentomografian keinoin. Tomografiset rekonstruktiot tehtiin sekä totaalivariaatiominimointimenetelmällä että SDART-algoritmilla, ja lisäksi yhdistelmällä molempia metodeja josta tässä tutkielmassa käytetään nimeä SDART-TV. Tutkimusongelma on peräisin teräsputkien hitsisaumojen laadun tutkimuksesta röntgenkuvantamisella. Hitsausprosessin aikana saumaan saattaa monista syistä syntyä poikkeamia, jotka vaikuttavat haitallisesti sauman kestävyyteen. Monissa teollisuuden sovelluksissa on tärkeää havaita sauman mahdolliset virheet. Putkihitsisaumojen röntgentutkimusta varten kehitettyjen laitteiden kanssa käytettävät röntgenilmaisimet saattavat olla pieniä verrattuna putken halkaisijaan, mikä johtaa vaikeaan rajoitetun kulman kuvausgeometriaan. Putken geometria myös toimii a priori-informaation lähteenä jota voidaan hyödyntää algoritmeissa, sillä putkia tutkiessa usein tiedetään että putken sisäosa on tyhjä. Totaalivariaatio- ja diskreetin tomografian algoritmien toimivuutta kuvatun ongelman ratkaisemisessa tutkittiin sekä laskennallisilla simulaatioilla että kokeilla fyysisellä alumiinisella testikappaleella. Kokeiden tulokset alustavasti vittaavat että modernit röntgentomografiset etukäteistietoa soveltavat algoritmit voivat soveltua putkihitsisaumojen ei-tuhoavaan testaukseen.The applied mathematical field of inverse problems studies how to recover an unknown function from a set of possibly incomplete and noisy observations. One example of practical, real-life inverse problem is X-ray tomography, where one wishes to recover the internal composition of an object by taking several X-ray projection images from different directions around the object. The tomographic inversion task is a severely ill-posed problem in Hadamard's sense if a limited amount of measurement data is available for the reconstruction. Two examples of limited data tomography are sparse tomography and limited angle tomography. In sparse tomography, tomographic projections are taken with a sparse angular sampling, either because one wishes to reduce the radiation exposure of the object or have a faster imaging process. In limited angle tomography, projections are available only from a limited angular range. The numerous algorithms that have been proposed for solving the tomographic problem with limited data are based on taking account a priori information about the object of interest. In this thesis we study two such algorithms, total variation minimization and discrete tomography. In the total variation minimization the prior assumption is that the total variation of the reconstructed image is small. In the discrete tomography, it is assumed that the target object consists of only discrete amount of different, internally homogeneous materials. Smooth Discrete Algebraic Reconstruction Technique or SDART is a modern method for discrete tomography. In this thesis we study the problem of recovering the size, shape and the location of small submillimeter-sized voids enclosed in the wall of a metal pipe section. Tomographic reconstructions were found with the total variation minimization and SDART algorithms and also with a combination of both methods, which we call SDART-TV. The study setup is inspired by the problems involved in the radiographic inspection of steel pipe girth welds. During the welding process, various defects may form in the weld seam and affect negatively the structural properties of the weld. In industrial applications it is important to discover if such defects are present in the weld. The X-ray detectors suited for the existing devices for the radiographic pipe weld inspection can be small compared to the pipe diameter, which results in a difficult limited angle imaging geometry. The geometry of pipes also provides another source of prior information for the algorithms, as the space inside the pipe is often known to contain only air or similar matter. The performance of the total variation and algorithms for discrete tomography in the setup described above was studied with both computational simulations and experiments with an aluminium pipe phantom. The results of the experiments suggest that X-ray tomography algorithms based on a priori information could be applicable for non-destructive testing of pipe welds

Model-based X-ray CT Image and Light Field Reconstruction Using Variable Splitting Methods.

Model-based image reconstruction (MBIR) is a powerful technique for solving ill-posed inverse problems. Compared with direct methods, it can provide better estimates from noisy measurements and from incomplete data, at the cost of much longer computation time. In this work, we focus on accelerating and applying MBIR for solving reconstruction problems, including X-ray computed tomography (CT) image reconstruction and light field reconstruction, using variable splitting based on the augmented Lagrangian (AL) methods. For X-ray CT image reconstruction, we combine the AL method and ordered subsets (OS), a well-known technique in the medical imaging literature for accelerating tomographic reconstruction, by considering a linearized variant of the AL method and propose a fast splitting-based ordered-subset algorithm, OS-LALM, for solving X-ray CT image reconstruction problems with penalized weighted least-squares (PWLS) criterion. Practical issues such as the non-trivial parameter selection of AL methods and remarkable memory overhead when considering the finite difference image variable splitting are carefully studied, and several variants of the proposed algorithm are investigated for solving practical model-based X-ray CT image reconstruction problems. Experimental results show that the proposed algorithm significantly accelerates the convergence of X-ray CT image reconstruction with negligible overhead and greatly reduces the noise-like OS artifacts in the reconstructed image when using many subsets for OS acceleration. For light field reconstruction, considering decomposing the camera imaging process into a linear convolution and a non-linear slicing operations for faster forward projection, we propose to reconstruct light field from a sequence of photos taken with different focus settings, i.e., a focal stack, using an alternating direction method of multipliers (ADMM). To improve the quality of the reconstructed light field, we also propose a signal-independent sparsifying transform by considering the elongated structure of light fields. Flatland simulation results show that our proposed sparse light field prior produces high resolution light field with fine details compared with other existing sparse priors for natural images.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108981/1/hungnien_1.pd

Sparse MRI and CT Reconstruction

Sparse signal reconstruction is of the utmost importance for efficient medical imaging, conducting accurate screening for security and inspection, and for non-destructive testing. The sparsity of the signal is dictated by either feasibility, or the cost and the screening time constraints of the system. In this work, two major sparse signal reconstruction systems such as compressed sensing magnetic resonance imaging (MRI) and sparse-view computed tomography (CT) are investigated. For medical CT, a limited number of views (sparse-view) is an option for whether reducing the amount of ionizing radiation or the screening time and the cost of the procedure. In applications such as non-destructive testing or inspection of large objects, like a cargo container, one angular view can take up to a few minutes for only one slice. On the other hand, some views can be unavailable due to the configuration of the system. A problem of data sufficiency and on how to estimate a tomographic image when the projection data are not ideally sufficient for precise reconstruction is one of two major objectives of this work. Three CT reconstruction methods are proposed: algebraic iterative reconstruction-reprojection (AIRR), sparse-view CT reconstruction based on curvelet and total variation regularization (CTV), and sparse-view CT reconstruction based on nonconvex L1-L2 regularization. The experimental results confirm a high performance based on subjective and objective quality metrics. Additionally, sparse-view neutron-photon tomography is studied based on Monte-Carlo modelling to demonstrate shape reconstruction, material discrimination and visualization based on the proposed 3D object reconstruction method and material discrimination signatures. One of the methods for efficient acquisition of multidimensional signals is the compressed sensing (CS). A significantly low number of measurements can be obtained in different ways, and one is undersampling, that is sampling below the Shannon-Nyquist limit. Magnetic resonance imaging (MRI) suffers inherently from its slow data acquisition. The compressed sensing MRI (CSMRI) offers significant scan time reduction with advantages for patients and health care economics. In this work, three frameworks are proposed and evaluated, i.e., CSMRI based on curvelet transform and total generalized variation (CT-TGV), CSMRI using curvelet sparsity and nonlocal total variation: CS-NLTV, CSMRI that explores shearlet sparsity and nonlocal total variation: SS-NLTV. The proposed methods are evaluated experimentally and compared to the previously reported state-of-the-art methods. Results demonstrate a significant improvement of image reconstruction quality on different medical MRI datasets

Topics in image reconstruction for high resolution positron emission tomography

Les problèmes mal posés représentent un sujet d'intérêt interdisciplinaire qui surgires dans la télédétection et des applications d'imagerie. Cependant, il subsiste des questions cruciales pour l'application réussie de la théorie à une modalité d'imagerie. La tomographie d'émission par positron (TEP) est une technique d'imagerie non-invasive qui permet d'évaluer des processus biochimiques se déroulant à l'intérieur d'organismes in vivo. La TEP est un outil avantageux pour la recherche sur la physiologie normale chez l'humain ou l'animal, pour le diagnostic et le suivi thérapeutique du cancer, et l'étude des pathologies dans le coeur et dans le cerveau. La TEP partage plusieurs similarités avec d'autres modalités d'imagerie tomographiques, mais pour exploiter pleinement sa capacité à extraire le maximum d'information à partir des projections, la TEP doit utiliser des algorithmes de reconstruction d'images à la fois sophistiquée et pratiques. Plusieurs aspects de la reconstruction d'images TEP ont été explorés dans le présent travail. Les contributions suivantes sont d'objet de ce travail: Un modèle viable de la matrice de transition du système a été élaboré, utilisant la fonction de réponse analytique des détecteurs basée sur l'atténuation linéaire des rayons y dans un banc de détecteur. Nous avons aussi démontré que l'utilisation d'un modèle simplifié pour le calcul de la matrice du système conduit à des artefacts dans l'image. (IEEE Trans. Nucl. Sei., 2000) );> La modélisation analytique de la dépendance décrite à l'égard de la statistique des images a simplifié l'utilisation de la règle d'arrêt par contre-vérification (CV) et a permis d'accélérer la reconstruction statistique itérative. Cette règle peut être utilisée au lieu du procédé CV original pour des projections aux taux de comptage élevés, lorsque la règle CV produit des images raisonnablement précises. (IEEE Trans. Nucl. Sei., 2001) Nous avons proposé une méthodologie de régularisation utilisant la décomposition en valeur propre (DVP) de la matrice du système basée sur l'analyse de la résolution spatiale. L'analyse des caractéristiques du spectre de valeurs propres nous a permis d'identifier la relation qui existe entre le niveau optimal de troncation du spectre pour la reconstruction DVP et la résolution optimale dans l'image reconstruite. (IEEE Trans. Nucl. Sei., 2001) Nous avons proposé une nouvelle technique linéaire de reconstruction d'image événement-par-événement basée sur la matrice pseudo-inverse régularisée du système. L'algorithme représente une façon rapide de mettre à jour une image, potentiellement en temps réel, et permet, en principe, la visualisation instantanée de distribution de la radioactivité durant l'acquisition des données tomographiques. L'image ainsi calculée est la solution minimisant les moindres carrés du problème inverse régularisé.Abstract: Ill-posed problems are a topic of an interdisciplinary interest arising in remote sensing and non-invasive imaging. However, there are issues crucial for successful application of the theory to a given imaging modality. Positron emission tomography (PET) is a non-invasive imaging technique that allows assessing biochemical processes taking place in an organism in vivo. PET is a valuable tool in investigation of normal human or animal physiology, diagnosing and staging cancer, heart and brain disorders. PET is similar to other tomographie imaging techniques in many ways, but to reach its full potential and to extract maximum information from projection data, PET has to use accurate, yet practical, image reconstruction algorithms. Several topics related to PET image reconstruction have been explored in the present dissertation. The following contributions have been made: (1) A system matrix model has been developed using an analytic detector response function based on linear attenuation of [gamma]-rays in a detector array. It has been demonstrated that the use of an oversimplified system model for the computation of a system matrix results in image artefacts. (IEEE Trans. Nucl. Sci., 2000); (2) The dependence on total counts modelled analytically was used to simplify utilisation of the cross-validation (CV) stopping rule and accelerate statistical iterative reconstruction. It can be utilised instead of the original CV procedure for high-count projection data, when the CV yields reasonably accurate images. (IEEE Trans. Nucl. Sci., 2001); (3) A regularisation methodology employing singular value decomposition (SVD) of the system matrix was proposed based on the spatial resolution analysis. A characteristic property of the singular value spectrum shape was found that revealed a relationship between the optimal truncation level to be used with the truncated SVD reconstruction and the optimal reconstructed image resolution. (IEEE Trans. Nucl. Sci., 2001); (4) A novel event-by-event linear image reconstruction technique based on a regularised pseudo-inverse of the system matrix was proposed. The algorithm provides a fast way to update an image potentially in real time and allows, in principle, for the instant visualisation of the radioactivity distribution while the object is still being scanned. The computed image estimate is the minimum-norm least-squares solution of the regularised inverse problem

System Characterizations and Optimized Reconstruction Methods for Novel X-ray Imaging

In the past decade there have been many new emerging X-ray based imaging technologies developed for different diagnostic purposes or imaging tasks. However, there exist one or more specific problems that prevent them from being effectively or efficiently employed. In this dissertation, four different novel X-ray based imaging technologies are discussed, including propagation-based phase-contrast (PB-XPC) tomosynthesis, differential X-ray phase-contrast tomography (D-XPCT), projection-based dual-energy computed radiography (DECR), and tetrahedron beam computed tomography (TBCT). System characteristics are analyzed or optimized reconstruction methods are proposed for these imaging modalities. In the first part, we investigated the unique properties of propagation-based phase-contrast imaging technique when combined with the X-ray tomosynthesis. Fourier slice theorem implies that the high frequency components collected in the tomosynthesis data can be more reliably reconstructed. It is observed that the fringes or boundary enhancement introduced by the phase-contrast effects can serve as an accurate indicator of the true depth position in the tomosynthesis in-plane image. In the second part, we derived a sub-space framework to reconstruct images from few-view D-XPCT data set. By introducing a proper mask, the high frequency contents of the image can be theoretically preserved in a certain region of interest. A two-step reconstruction strategy is developed to mitigate the risk of subtle structures being oversmoothed when the commonly used total-variation regularization is employed in the conventional iterative framework. In the thirt part, we proposed a practical method to improve the quantitative accuracy of the projection-based dual-energy material decomposition. It is demonstrated that applying a total-projection-length constraint along with the dual-energy measurements can achieve a stabilized numerical solution of the decomposition problem, thus overcoming the disadvantages of the conventional approach that was extremely sensitive to noise corruption. In the final part, we described the modified filtered backprojection and iterative image reconstruction algorithms specifically developed for TBCT. Special parallelization strategies are designed to facilitate the use of GPU computing, showing demonstrated capability of producing high quality reconstructed volumetric images with a super fast computational speed. For all the investigations mentioned above, both simulation and experimental studies have been conducted to demonstrate the feasibility and effectiveness of the proposed methodologies