Programmation mathématique en tomographie discrète

La tomographie est un ensemble de techniques visant à reconstruirel intérieur d un objet sans toucher l objet lui même comme dans le casd un scanner. Les principes théoriques de la tomographie ont été énoncéspar Radon en 1917. On peut assimiler l objet à reconstruire à une image,matrice, etc.Le problème de reconstruction tomographique consiste à estimer l objet àpartir d un ensemble de projections obtenues par mesures expérimentalesautour de l objet à reconstruire. La tomographie discrète étudie le cas où lenombre de projections est limité et l objet est défini de façon discrète. Leschamps d applications de la tomographie discrète sont nombreux et variés.Citons par exemple les applications de type non destructif comme l imageriemédicale. Il existe d autres applications de la tomographie discrète, commeles problèmes d emplois du temps.La tomographie discrète peut être considérée comme un problème d optimisationcombinatoire car le domaine de reconstruction est discret et le nombrede projections est fini. La programmation mathématique en nombres entiersconstitue un outil pour traiter les problèmes d optimisation combinatoire.L objectif de cette thèse est d étudier et d utiliser les techniques d optimisationcombinatoire pour résoudre les problèmes de tomographie.The tomographic imaging problem deals with reconstructing an objectfrom a data called a projections and collected by illuminating the objectfrom many different directions. A projection means the information derivedfrom the transmitted energies, when an object is illuminated from a particularangle. The solution to the problem of how to reconstruct an object fromits projections dates to 1917 by Radon. The tomographic reconstructingis applicable in many interesting contexts such as nondestructive testing,image processing, electron microscopy, data security, industrial tomographyand material sciences.Discete tomography (DT) deals with the reconstruction of discret objectfrom limited number of projections. The projections are the sums along fewangles of the object to be reconstruct. One of the main problems in DTis the reconstruction of binary matrices from two projections. In general,the reconstruction of binary matrices from a small number of projections isundetermined and the number of solutions can be very large. Moreover, theprojections data and the prior knowledge about the object to reconstructare not sufficient to determine a unique solution. So DT is usually reducedto an optimization problem to select the best solution in a certain sense.In this thesis, we deal with the tomographic reconstruction of binaryand colored images. In particular, research objectives are to derive thecombinatorial optimization techniques in discrete tomography problems.PARIS-CNAM (751032301) / SudocSudocFranceF

Network Flow Algorithms for Discrete Tomography

Tomography is a powerful technique to obtain images of the interior of an object in a nondestructive way. First, a series of projection images (e.g., X-ray images) is acquired and subsequently a reconstruction of the interior is computed from the available project data. The algorithms that are used to compute such reconstructions are known as tomographic reconstruction algorithms. Discrete tomography is concerned with the tomographic reconstruction of images that are known to contain only a few different gray levels. By using this knowledge in the reconstruction algorithm it is often possible to reduce the number of projections required to compute an accurate reconstruction, compared to algorithms that do not use prior knowledge. This thesis deals with new reconstruction algorithms for discrete tomography. In particular, the first five chapters are about reconstruction algorithms based on network flow methods. These algorithms make use of an elegant correspondence between certain types of tomography problems and network flow problems from the field of Operations Research. Chapter 6 deals with a problem that occurs in the application of discrete tomography to the reconstruction of nanocrystals from projections obtained by electron microscopy.The research for this thesis has been financially supported by the Netherlands Organisation for Scientific Research (NWO), project 613.000.112.UBL - phd migration 201

Selected topics in homogenization of transport processes in historical masonry structures

The paper reviews several topics associated with the homogenization of transport processed in historical masonry structures. Since these often experience an irregular or random pattern, we open the subject by summarizing essential steps in the formulation of a suitable computational model in the form of Statistically Equivalent Periodic Unit Cell (SEPUC). Accepting SEPUC as a reliable representative volume element is supported by application of the Fast Fourier Transform to both the SEPUC and large binary sample of real masonry in search for effective thermal conductivities limited here to a steady state heat conduction problem. Fully coupled non-stationary heat and moisture transport is addressed next in the framework of two-scale first-order homogenization approach with emphases on the application of boundary and initial conditions on the meso-scale.Comment: 19 pages, 13 figures, 2 table

Conditioning of and Algorithms for Image Reconstruction from Irregular Frequency Domain Samples.

The problem of reconstructing an image from irregular samples of its 2-D DTFT arises in synthetic aperture radar (SAR), magnetic resonance imaging (MRI), computed tomography (CT), limited angle tomography, and 2-D filter design. The problem of determining a configuration of a limited number of 2-D DTFT samples also arises in magnetic resonance spectroscopic imaging (MRSI) and 3-D MRI. This work first focuses on the selection of the measurement data. Since there is no 2-D Lagrange interpolation formula, sufficient conditions for the uniqueness and conditioning of the reconstruction problem are both not apparent. Kronecker substitutions, such as the Good-Thomas FFT, the helical scan FFT, and the 45-degree rotated support, unwrap the 2-D problem into a 1-D problem, resulting in uniqueness and insights into the problem conditioning. The variance of distances between the adjacent unwrapped 1-D DTFT samples was developed as a sensitivity measure to quickly and accurately estimate of the condition number of the system matrix. A well-conditioned configuration of DTFT samples, restricted to radial lines in CT or spirals in MRI, is found by simulated annealing with the variance sensitivity measure as the objective function. The preconditioned conjugate gradient method reconstructs the 1-D solution that is then rewrapped to a 2-D image. In unrestricted cases, 2-D DTFT configurations like a regular hexagonal pattern can be unwrapped to uniformly-spaced and perfectly conditioned 1-D configurations and quickly solved using an inverse 1-D DFT. The next focus is on developing fast reconstruction algorithms. A non-iterative DFT-based method of reconstructing an image is presented, by first masking the 2-D DTFT samples with the frequency response of a filter that is zeroed at the unknown 2-D DFT locations, and then quickly deconvolving the filtered image using three 2-D DFTs. The masking filter needs to be precomputed only once per DTFT configuration. A divide-and-conquer image reconstruction method is also presented using subband decomposition and Gabor filters to solve smaller subband problems, leading to a quick unaliased low-resolution image or later to be recombined into the full solution. All methods are applied to actual CT data resulting in faster reconstructions than POCS and FBP with equivalent errors.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77912/1/leebc_1.pd