Novel algorithms in X-ray computed tomography imaging from under-sampled data

This thesis presents novel algorithms in X-ray computed tomography imaging using limited or sparse data: I. A non-uniform rational basis splines (NURBS) curve is used to represent the boundary of a target. Markov chain Monte Carlo (MCMC) strategy is applied for estimating the unknown curve from the projection data and an attenuation value of the target. In this case, the target is assumed to be homogeneous (it contains only one material). Instead of a single output, the solution of MCMC as a Bayesian framework is a posterior distribution. In addition, the results of the method are conveniently in CAD-compatible format. II. Adaptive methods for choosing regularization parameter are proposed. The first approach is called the controlled wavelet domain sparsity (CWDS). This is based on enforcing sparsity in the two-dimensional wavelet transform domain, and the second so-called the controlled shearlet domain sparsity (CSDS) in the three-dimensional shearlet transform domain. The proposed methods offer a strategy to automatically choosing regularization parameter where the end-users could avoid manually tuning the parameters. A known {\it a priori} sparsity level calculated from some available objects/samples is required. Both algorithms above have been successfully implemented for real measured X-ray data and the results using under-sampled data outperform the baseline method. The proposed methods incur heavy computation costs, however implementing parallelization strategy could save the computation time.Tiivistelmä Tässä väitöskirjassa esitetään uusia algoritmeja röntgenkuvaukseen perustuvaan tietokonetomografiaan käyttäen harvan ja rajoitetun kulman mittausdataa. Erityisesti työssä esitetään seuraavat lähestymistavat: I. Ensimmäinen lähestymistapa perustuu NURBS (engl., non-uniform rational basis splines) –mallin käyttöön. NURBS on matemaattinen malli, jota käytetään kuvattavan kohteen reunojen esittämiseen. Soveltamalla tätä yhdessä Markovin ketju Monte Carlo –strategian (MCMC) kanssa voidaan estimoida reunan käyrä, sekä kohteen vaimenemista kuvaava arvo. Tässä lähestymistavassa kohde oletetaan homogeeniseksi eli sen oletetaan sisältävän vain yhtä ainetta. Käyttäen MCMC-mentelmää saadaan estimoitaville parametreille tilastollinen a posteriori -jakauma. II. Toinen lähestymistapa perustuu adaptiiviseen regularisointiparametrin valitsemiseen. Tätä varten kehitettiin kaksi strategiaa. Ensimmäinen näistä perustuu harvuuden vahvistamiseen ja kontrolloimiseen kaksiulotteisessa aallokemuunoksessa. Toinen taas perustuu harvuuden kontrolloimiseen nk. komiulotteisessa shearlet-sivuttaissiirtymämuunnoksessa. Molemmat menetelmät mahdollistavat regularisointiparametrin automaattisen valitsemisen ilman että loppukäyttäjän tarvitsee itse siihen puuttua. Ennakkotieto kuvattavan objektin harvuuden tasosta kuitenkin vaaditaan. Tässä väitöskirjassa molempia lähestymistapoja testattiin käytännössä käyttäen oikeaa mitattua röntgendataa. Molemmissa lähestymistavoissa uudet algoritmit toimivat paremmin kuin perinteiset vertailumenetelmät. Uudet algoritmit ovat kuitenkin laskennallisesti erittäin raskaita. Tulevaisuudessa suurteholaskennan keinoilla niihin käytettyä laskenta-aikaa voitaneen kuitenkin pienentää

Controlled wavelet domain sparsity for x-ray tomography

Tomographic reconstruction is an ill-posed inverse problem that calls for regularization. One possibility is to require sparsity of the unknown in an orthonormal wavelet basis. This, in turn, can be achieved by variational regularization, where the penalty term is the sum of the absolute values of the wavelet coefficients. The primal-dual fixed point algorithm showed that the minimizer of the variational regularization functional can be computed iteratively using a soft-thresholding operation. Choosing the soft-thresholding parameter mu > 0 is analogous to the notoriously difficult problem of picking the optimal regularization parameter in Tikhonov regularization. Here, a novel automatic method is introduced for choosing mu, based on a control algorithm driving the sparsity of the reconstruction to an a priori known ratio of nonzero versus zero wavelet coefficients in the unknown.Peer reviewe

Shearlet-based regularization in sparse dynamic tomography

Peer reviewe

An Automatic Regularization Method : An Application for 3-D X-Ray Micro-CT Reconstruction Using Sparse Data

X-ray tomography is a reliable tool for determining the inner structure of 3-D object with penetrating X-rays. However, traditional reconstruction methods, such as Feldkamp-Davis-Kress (FDK), require dense angular sampling in the data acquisition phase leading to long measurement times, especially in X-ray micro-tomography to obtain high-resolution scans. Acquiring less data using greater angular steps is an obvious way for speeding up the process and avoiding the need to save huge data sets. However, computing 3-D reconstruction from such a sparsely sampled data set is difficult because the measurement data are usually contaminated by errors, and linear measurement models do not contain sufficient information to solve the problem in practice. An automatic regularization method is proposed for robust reconstruction, based on enforcing sparsity in the 3-D shearlet transform domain. The inputs of the algorithm are the projection data and a priori known expected degree of sparsity, denoted as 0 <C-prPeer reviewe

Undersampled Dynamic X-Ray Tomography With Dimension Reduction Kalman Filter

In this paper, we propose a prior-based dimension reduction Kalman filter for undersampled dynamic X-ray tomography. With this method, the X-ray reconstructions are parameterized by a low-dimensional basis. Thus, the proposed method is computationally very light, and extremely robust as all the computations can be done explicitly. With real and simulated measurement data, we show that the method provides accurate reconstructions even with very limited number of angular directions.Peer reviewe