Iterative CT reconstruction using shearlet-based regularization

In computerized tomography, it is important to reduce the image noise without increasing the acquisition dose. Extensive research has been done into total variation minimization for image denoising and sparse-view reconstruction. However, TV minimization methods show superior denoising performance for simple images (with little texture), but result in texture information loss when applied to more complex images. Since in medical imaging, we are often confronted with textured images, it might not be beneficial to use TV. Our objective is to find a regularization term outperforming TV for sparse-view reconstruction and image denoising in general. A recent efficient solver was developed for convex problems, based on a split-Bregman approach, able to incorporate regularization terms different from TV. In this work, a proof-of-concept study demonstrates the usage of the discrete shearlet transform as a sparsifying transform within this solver for CT reconstructions. In particular, the regularization term is the 1-norm of the shearlet coefficients. We compared our newly developed shearlet approach to traditional TV on both sparse-view and on low-count simulated and measured preclinical data. Shearlet-based regularization does not outperform TV-based regularization for all datasets. Reconstructed images exhibit small aliasing artifacts in sparse-view reconstruction problems, but show no staircasing effect. This results in a slightly higher resolution than with TV-based regularization

Superiorization and Perturbation Resilience of Algorithms: A Continuously Updated Bibliography

This document presents a, (mostly) chronologically ordered, bibliography of scientific publications on the superiorization methodology and perturbation resilience of algorithms which is compiled and continuously updated by us at: http://math.haifa.ac.il/yair/bib-superiorization-censor.html. Since the beginings of this topic we try to trace the work that has been published about it since its inception. To the best of our knowledge this bibliography represents all available publications on this topic to date, and while the URL is continuously updated we will revise this document and bring it up to date on arXiv approximately once a year. Abstracts of the cited works, and some links and downloadable files of preprints or reprints are available on the above mentioned Internet page. If you know of a related scientific work in any form that should be included here kindly write to me on: [email protected] with full bibliographic details, a DOI if available, and a PDF copy of the work if possible. The Internet page was initiated on March 7, 2015, and has been last updated on March 12, 2020.Comment: Original report: June 13, 2015 contained 41 items. First revision: March 9, 2017 contained 64 items. Second revision: March 8, 2018 contained 76 items. Third revision: March 11, 2019 contains 90 items. Fourth revision: March 16, 2020 contains 112 item

Shearlet-based regularized reconstruction in region-of-interest computed tomography

Region of interest (ROI) tomography has gained increasing attention in recent years due to its potential to reducing radiation exposure and shortening the scanning time. However, tomographic reconstruction from ROI-focused illumination involves truncated projection data and typically results in higher numerical instability even when the reconstruction problem has unique solution. To address this problem, both ad hoc analytic formulas and iterative numerical schemes have been proposed in the literature. In this paper, we introduce a novel approach for ROI tomographic reconstruction, formulated as a convex optimization problem with a regularized term based on shearlets. Our numerical implementation consists of an iterative scheme based on the scaled gradient projection method and it is tested in the context of fan-beam CT. Our results show that our approach is essentially insensitive to the location of the ROI and remains very stable also when the ROI size is rather small.Peer reviewe

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