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Noise propagation in algorithms constructing krylov regularization bases for the solution of inverse problems
Kašpar, Jakub ; Hnětynková, Iveta (advisor) ; Plešinger, Martin (referee)
In this thesis we consider a linear inverse problem Ax ≈ b with a smoothing operator A and a right-hand side vector b polluted by unknown noise. To find good approximation of x we can use large family of iterative regularization methods, which compute the approximate solution by projection onto a Krylov subspace of small dimension. Even though this projection has filtering property, the high frequency noise propagates to the Krylov basis, which causes semiconvergence of the methods. The knowledge of intensity of noise propagation is therefore necessary to find reasonably precise approximation of the solution. In the thesis we study noise propagation in the Golub-Kahan iterative bidiagonali- zation and in the Lanczos algorithm, which construct the required Krylov subspace for LSQR and MINRES methods. For both methods, we analyze a noise-amplifying coef- ficient, for which we derive explicit formulas in both cases. For the Golub-Kahan bidi- agonalization, this analysis summarizes the theory from multiple sources. Analysis for the Lanczos algorithm is original. For both methods, we derive explicit relations between noise-amplifying coefficients and residual norms. Several numerical experiments are pre- sented to demonstrate properties of both algorithms. Impact of noise propagation on true errors and influence...
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Regularization methods for discrete inverse problems in single particle analysis
Havelková, Eva ; Hnětynková, Iveta (advisor) ; Plešinger, Martin (referee)
The aim of this thesis is to investigate applicability of regulariza- tion by Krylov subspace methods to discrete inverse problems arising in single particle analysis (SPA). We start with a smooth model formulation and describe its discretization, yielding an ill-posed inverse problem Ax ≈ b, where A is a lin- ear operator and b represents the measured noisy data. We provide theoretical background and overview of selected methods for the solution of general linear inverse problems. Then we focus on specific properties of inverse problems from SPA, and provide experimental analysis based on synthetically generated SPA datasets (experiments are performed in the Matlab enviroment). Turning to the solution of our inverse problem, we investigate in particular an approach based on iterative Hybrid LSQR with inner Tikhonov regularization. A reliable stopping criterion for the iterative part as well as parameter-choice method for the inner regularization are discussed. Providing a complete implementation of the proposed solver (in Matlab and in C++), its performance is evaluated on various SPA model datasets, considering high levels of noise and realistic distri- bution of orientations of scanning angles. Comparison to other regularization methods, including the ART method traditionally used in SPA,...
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Numerical methods in image processing for applications in jewellery industry
Petrla, Martin ; Hnětynková, Iveta (advisor) ; Plešinger, Martin (referee)
Presented thesis deals with a problem from the field of image processing for application in multiple scanning of jewelery stones. The aim is to develop a method for preprocessing and subsequent mathematical registration of images in order to increase the effectivity and reliability of the output quality control. For these purposes the thesis summerizes mathematical definition of digital image as well as theoretical base of image registration. It proposes a method adjusting every single image to increase effectivity of its subsequent processing. One image for every evaluated gemstone is generated using image registration. The method is implementated in the MATLAB environment. Powered by TCPDF (www.tcpdf.org)
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Properties and construction of core problem in data fitting problems with multiple observations
Dvořák, Jan ; Hnětynková, Iveta (advisor) ; Plešinger, Martin (referee)
In this work we study the solution of linear approximation problems with multiple observations. Particulary we focus on the total least squares method, which belogs to the class of ortogonaly invariant problems. For these problems we describe the so called core reduction. The aim is to reduce dimesions of the problem while preserving the solution, if it exists. We present two ways of constructing core problems. One is based on the singular value decomposition and the other uses the generalized Golub-Kahan iterative bidiago- nalization. Further we investigate properties of the core problem and of the methods for its construction. Finally we preform numerical experiments in the Matlab enviroment in order to test the reliability of the discussed algorithms. 1
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Regularization methods for discrete inverse problems in single particle analysis
Havelková, Eva ; Hnětynková, Iveta (advisor) ; Plešinger, Martin (referee)
The aim of this thesis is to investigate applicability of regulariza- tion by Krylov subspace methods to discrete inverse problems arising in single particle analysis (SPA). We start with a smooth model formulation and describe its discretization, yielding an ill-posed inverse problem Ax ≈ b, where A is a lin- ear operator and b represents the measured noisy data. We provide theoretical background and overview of selected methods for the solution of general linear inverse problems. Then we focus on specific properties of inverse problems from SPA, and provide experimental analysis based on synthetically generated SPA datasets (experiments are performed in the Matlab enviroment). Turning to the solution of our inverse problem, we investigate in particular an approach based on iterative Hybrid LSQR with inner Tikhonov regularization. A reliable stopping criterion for the iterative part as well as parameter-choice method for the inner regularization are discussed. Providing a complete implementation of the proposed solver (in Matlab and in C++), its performance is evaluated on various SPA model datasets, considering high levels of noise and realistic distri- bution of orientations of scanning angles. Comparison to other regularization methods, including the ART method traditionally used in SPA,...
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Numerical methods in image processing for applications in jewellery industry
Petrla, Martin ; Hnětynková, Iveta (advisor) ; Plešinger, Martin (referee)
Presented thesis deals with a problem from the field of image processing for application in multiple scanning of jewelery stones. The aim is to develop a method for preprocessing and subsequent mathematical registration of images in order to increase the effectivity and reliability of the output quality control. For these purposes the thesis summerizes mathematical definition of digital image as well as theoretical base of image registration. It proposes a method adjusting every single image to increase effectivity of its subsequent processing. One image for every evaluated gemstone is generated using image registration. The method is implementated in the MATLAB environment. Powered by TCPDF (www.tcpdf.org)
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