Mathematics and Algorithms in Tomography

This is the eighth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, sonar, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, vector tomography, and texture analysis

Fast Markov chain Monte Carlo sampling for sparse Bayesian inference in high-dimensional inverse problems using L1-type priors

Sparsity has become a key concept for solving of high-dimensional inverse problems using variational regularization techniques. Recently, using similar sparsity-constraints in the Bayesian framework for inverse problems by encoding them in the prior distribution has attracted attention. Important questions about the relation between regularization theory and Bayesian inference still need to be addressed when using sparsity promoting inversion. A practical obstacle for these examinations is the lack of fast posterior sampling algorithms for sparse, high-dimensional Bayesian inversion: Accessing the full range of Bayesian inference methods requires being able to draw samples from the posterior probability distribution in a fast and efficient way. This is usually done using Markov chain Monte Carlo (MCMC) sampling algorithms. In this article, we develop and examine a new implementation of a single component Gibbs MCMC sampler for sparse priors relying on L1-norms. We demonstrate that the efficiency of our Gibbs sampler increases when the level of sparsity or the dimension of the unknowns is increased. This property is contrary to the properties of the most commonly applied Metropolis-Hastings (MH) sampling schemes: We demonstrate that the efficiency of MH schemes for L1-type priors dramatically decreases when the level of sparsity or the dimension of the unknowns is increased. Practically, Bayesian inversion for L1-type priors using MH samplers is not feasible at all. As this is commonly believed to be an intrinsic feature of MCMC sampling, the performance of our Gibbs sampler also challenges common beliefs about the applicability of sample based Bayesian inference.Comment: 33 pages, 14 figure

Bayesian Analysis of ODE's: solver optimal accuracy and Bayes factors

In most relevant cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior deriving from the error in the numerical solver. To compare a numerical and the theoretical posterior distributions we propose to use Bayes Factors (BF), considering both of them as models for the data at hand. We prove that the theoretical vs a numerical posterior BF tends to 1, in the same order (of the step size used) as the numerical forward map solver does. For higher order solvers (eg. Runge-Kutta) the Bayes Factor is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error free posteriors. Two examples are presented where nearly 90% CPU time is saved while all inference results are identical to using a solver with a much finer time step.Comment: 28 pages, 6 figure

Image processing methods for limited angle tomography and sparse angle tomography

The purpose of this thesis was two-fold: Firstly, to evaluate if limited angle tomography is suitable for clinical implant planning. Secondly, to improve clinical image quality and workflow of the limited angle tomography by developing new imaging processing algorithms. Conventional computed tomography (CT) design is not optimal in the sense of cost, workflow or dose for two reasons. Firstly, CT devices are typically expensive and bulky devices because they require a stable X-ray production, rigid gantry with accurate and repeatable movements, high scanning speed, solid patient support and a low-noise X-ray detector. Secondly, current non-regularized reconstruction techniques require high dose per projection image as well as a huge number of projection image. This also limits the usage of the CT imaging to serious trauma cases and other lethal diseases. To overcome the limitations mentioned above, new approaches have been introduced to replace conventional CT imaging. For example, year 2007 a dental imaging technology company Palodex Group released an upgrade kit for standard panoramic X-ray device, called Volumetric Tomography (VT), which is based on limited angle tomography. In the first article, we demonstrated that limited angle tomography is able to give similar clinical information as CT devices in dental implant planning. Therefore, the implant planning could be executed more cost and dose effectively when suitable algorithms are applied throughout the reconstruction process. Since limited angle tomography system applies small number of X-ray images taken from a limited aperture, new image processing methods are required for clinically suitable image quality. For that reason, two novel imaging processing methods and one analyzing method were created and documented in this work. In the second article, a new image processing method based on modification of the constrained least-square filter for extremely sparse situations was introduced. In this method, called Wiener-filter based iterative reconstruction technique (WIRT), we considered the uncertainty of the interpolation as noise and utilized the regularization only in the regions where the uncertainty is the dominating factor. In the third article, a new sinogram estimation algorithm called sinogram interpolation technique (SINT) was created, where the missing sinogram columns were estimated based on the known columns. In the fourth article, a method named mutual information based technology (MINT) was developed to estimate the imaging geometry directly from the projection data. In this method, the imaging angles can be estimated based on the projection images without any external markers or additional constructions to the device. Therefore, this method simplifies workflow and the improves imaging angle accuracy significantly.Tämän väitöstyön tarkoituksena oli arvioida alhaisen sädeannoksen tuottavan tietokonetomografialaitteen käyttöä implanttisuunnittelussa sekä luoda uusia kuvanlaatua ja käytettävyyttä parantavia menetelmiä tähän konseptiin sekä yleisesti alhaisen säteilyn omaaviin tietokonetomografialaitteisiin. Tietokonetomografian tarkoitus on luoda kolmeulotteinen malli kohteesta perustuen useisiin röntgenkuviin, jotka ovat otettu eri suunnista kohdetta. Verrattuna tavalliseen röntgenkuvaukseen tietokonetomografia lisää röntgenkuvauksen kliinistä hyötyä, sillä se mahdollistaa kappaleiden muotojen ja niiden välisten etäisyyksien mittaamisen. Perinteinen tietokonetomografia ei kuitenkaan aina ole optimaalinen kustannuksen, käytettävyyden tai annoksen suhteen. Tämä johtuu siitä, että se perustuu mekaanisesti tarkkaan laitteistoon, joka suunnataan oikeassa ja ennalta määrätyssä kulmassa potilaaseen nähden ja kykenee ottamaan useita, jopa satoja, röntgenkuvia mahdollisimman lyhyessä ajassa. Tästä syystä tietokonetomografialaitteet ovat perinteisesti olleet hyvin raskaita ja kalliita laitteita, jotka tuottavat potilaalle korkean sädeannoksen. Edellä mainituista syistä on viimevuosina kehitelty uusia kolmeulotteisia kuvausmenetelmiä, joissa laitteisto on pyritty kehittämään kevyemmäksi ja sädeannokseltaan pienemmäksi kuin perinteiset tietokonetomografialaitteet. Pieni röntgenkuvamäärä asettaa kuitenkin suuria haasteita kolmeulotteisen tiedon muodostamiseen. Koska nykyiset tietokonetomografiassa käytetyt kuvankäsittelyoperaatiot on suunniteltu suurille säde-annoksille, ne eivät sellaisenaan sovellu pienannoksiseen tomografialaskentaan. Tässä työssä pyrittiin toteuttamaan, arvioimaan ja dokumentoimaan uusia kuvankäsittelyoperaatioita jotka soveltuvat pienemmän sädeannoksen kuvaukseen eli ns. harva- ja rajoitetun kulman tomografiaan. Väitöskirja sisältää kolme täysin uutta kuvankäsittelymenetelmää joista kaksi parantaa kliinistä kuvanlaatua ja kolmas laitteiston käytettävyyttä. Tässä väitöskirjassa osoitetaan myös että vaikka pienannoksinen tietokonetomografialaite ei tuota niin tarkkaa kuvaa kuin perinteinen tietokonetomografialaite, sillä saadaan riittävän tarkka tieto hammasimplanttisuunniteluun

Reconstruction for Time-Domain In Vivo EPR 3D Multigradient Oximetric Imaging—A Parallel Processing Perspective

Three-dimensional Oximetric Electron Paramagnetic Resonance Imaging using the Single Point Imaging modality generates unpaired spin density and oxygen images that can readily distinguish between normal and tumor tissues in small animals. It is also possible with fast imaging to track the changes in tissue oxygenation in response to the oxygen content in the breathing air. However, this involves dealing with gigabytes of data for each 3D oximetric imaging experiment involving digital band pass filtering and background noise subtraction, followed by 3D Fourier reconstruction. This process is rather slow in a conventional uniprocessor system. This paper presents a parallelization framework using OpenMP runtime support and parallel MATLAB to execute such computationally intensive programs. The Intel compiler is used to develop a parallel C++ code based on OpenMP. The code is executed on four Dual-Core AMD Opteron shared memory processors, to reduce the computational burden of the filtration task significantly. The results show that the parallel code for filtration has achieved a speed up factor of 46.66 as against the equivalent serial MATLAB code. In addition, a parallel MATLAB code has been developed to perform 3D Fourier reconstruction. Speedup factors of 4.57 and 4.25 have been achieved during the reconstruction process and oximetry computation, for a data set with 23 × 23 × 23 gradient steps. The execution time has been computed for both the serial and parallel implementations using different dimensions of the data and presented for comparison. The reported system has been designed to be easily accessible even from low-cost personal computers through local internet (NIHnet). The experimental results demonstrate that the parallel computing provides a source of high computational power to obtain biophysical parameters from 3D EPR oximetric imaging, almost in real-time

Scaled Projected-Directions Methods with Application to Transmission Tomography

Statistical image reconstruction in X-Ray computed tomography yields large-scale regularized linear least-squares problems with nonnegativity bounds, where the memory footprint of the operator is a concern. Discretizing images in cylindrical coordinates results in significant memory savings, and allows parallel operator-vector products without on-the-fly computation of the operator, without necessarily decreasing image quality. However, it deteriorates the conditioning of the operator. We improve the Hessian conditioning by way of a block-circulant scaling operator and we propose a strategy to handle nondiagonal scaling in the context of projected-directions methods for bound-constrained problems. We describe our implementation of the scaling strategy using two algorithms: TRON, a trust-region method with exact second derivatives, and L-BFGS-B, a linesearch method with a limited-memory quasi-Newton Hessian approximation. We compare our approach with one where a change of variable is made in the problem. On two reconstruction problems, our approach converges faster than the change of variable approach, and achieves much tighter accuracy in terms of optimality residual than a first-order method.Comment: 19 pages, 7 figure