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Regularization methods for discrete inverse problems in single particle analysis
Havelková, Eva ; Hnětynková, Iveta (advisor)
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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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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Methods for enforcing non-negativity of solution in Krylov regularization
Hoang, Phuong Thao ; Hnětynková, Iveta (advisor) ; Pozza, Stefano (referee)
The purpose of this thesis is to study how to overcome difficulties one typically encounters when solving non-negative inverse problems by standard Krylov subspace methods. We first give a theoretical background to the non-negative inverse problems. Then we concentrate on selected modifications of Krylov subspace methods known to improve the solution significantly. We describe their properties, provide their implementation and propose an improvement for one of them. After that, numerical experiments are presented giving a comparison of the methods and analyzing the influence of the present parameters on the behavior of the solvers. It is clearly demonstrated, that the methods imposing nonnegativity perform better than the unconstrained methods. Moreover, our improvement leads in some cases to a certain reduction of the number of iterations and consequently to savings of the computational time while preserving a good quality of the approximation.
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Application of computational methods in classification of glass stones
Lébl, Matěj ; Hnětynková, Iveta (advisor)
Application of computational methods in classification of glass stones Bc. Matěj Lébl Abstrakt: The goal of this thesis is to employ mathematical image processing methods in automatic quality control of glass jewellery stones. The main math- ematical subject is a matrix of specific attributes representing digital image of the studied products. First, the thesis summarizes mathematical definition of digital image and some standard image processing methods. Then, a complete solution to the considered problem is presented. The solution consists of stone localization within the image followed by analysis of the localized area. Two lo- calization approaches are presented. The first is based on the matrix convolution and optimized through the Fourier transform. The second uses mathematical methods of thresholding and median filtering, and data projection into one di- mension. The localized area is analyzed based on statistical distribution of the stone brightness. All methods are implemented in the MATLAB environment. 1
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Regularization techniques based on the least squares method
Kubínová, Marie ; Hnětynková, Iveta (advisor)
Title: Regularization Techniques Based on the Least Squares Method Author: Marie Michenková Department: Department of Numerical Mathematics Supervisor: RNDr. Iveta Hnětynková, Ph.D. Abstract: In this thesis we consider a linear inverse problem Ax ≈ b, where A is a linear operator with smoothing property and b represents an observation vector polluted by unknown noise. It was shown in [Hnětynková, Plešinger, Strakoš, 2009] that high-frequency noise reveals during the Golub-Kahan iterative bidiagonalization in the left bidiagonalization vectors. We propose a method that identifies the iteration with maximal noise revealing and reduces a portion of high-frequency noise in the data by subtracting the corresponding (properly scaled) left bidiagonalization vector from b. This method is tested for different types of noise. Further, Hnětynková, Plešinger, and Strakoš provided an estimator of the noise level in the data. We propose a modification of this estimator based on the knowledge of the point of noise revealing. Keywords: ill-posed problems, regularization, Golub-Kahan iterative bidiagonalization, noise revealing, noise estimate, denoising 1
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Behavior of total least squares method for models with multiple observations
Slavenko, Matvei ; Hnětynková, Iveta (advisor) ; Tůma, Miroslav (referee)
Linear approximation problems arise in various applications and can be solved by a large variety of methods. One of such methods is total least squares (TLS), an approach that allows to correct errors both in the linear model and available set of observations. In this work we collect and compare the main theoretical results related to TLS with multiple right-hand side. Particularly we describe the classification of TLS problems and summarise the solvability analysis that has currently been spread over various sources. The second part of the work is dedicated to an approach called core data reduction (CDR) and proof-of-concept programme demonstrating the CDR numerical behaviour. 1
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Multicriteria graph partitioning
Houška, Ondřej ; Tůma, Miroslav (advisor) ; Hnětynková, Iveta (referee)
The thesis is about graph partitioning and applications of graph partitioning in paral- lel algorithms for solving big sparse linear equations. The problem of graph partitioning is thorougly described and standard graph partitioning algorithms are explained. The appli- cation part is focusing on the Conjugate Gradient method preconditioned by a variant of incomplete Cholesky factorization based on drop tolerance. The role of graph partitioning in the problem decomposition is described and a load balancing problem is studied. 1
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Demosaicing as an ill-posed inverse problem
Mariničová, Veronika ; Šroubek, Filip (advisor) ; Hnětynková, Iveta (referee)
Color information of a scene is only recorded partially by a digital camera.Specifically, only one of the red, green, and blue color components is sampled at each pixel.The missing color values must be estimated － a process called demosaicing. Demosaicing can be solved as an individual step in the image processing pipeline. In this case, any errors and artefacts produced by this step are carried over into further steps in the image processing pipeline and are possibly magnified. Alternatively, we can try to resolve several degradations at once in a joint solution, which eliminates this effect. We present one such solution, that in addition to demosaicing, also jointly solves denoising, deconvolution, and super-resolution in the form of a convex optimization problem. We provide an overview of demosaicing methods and evaluate the results from our solution against selected existing methods.
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Regularization properties of Krylov subspace methods
Kučerová, Andrea ; Hnětynková, Iveta (advisor) ; Kučera, Václav (referee)
The aim of this thesis is to study and describe regularizing properties of iterative Krylov subspace methods for finding a solution of linear algebraic ill- posed problems contaminated by white noise. First we explain properties of this kind of problems, especially their sensitivity to small perturbations in data. It is shown that classical methods for solving approximation problems (such as the least squares method) fail here. Thus we turn to explanation of regularizing pro- perties of projections onto Krylov subspaces. Basic Krylov regularizing methods are considered, namely RRGMRES, CGLS, and LSQR. The results are illustrated on model problems from Regularization toolbox in MATLAB. 1
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Comparison of least squares methods for problems with errors in the model
Dvořák, Jan ; Hnětynková, Iveta (advisor) ; Kopal, Jiří (referee)
In this work we study the least squares and the total least squares problem for the solution of linear aproximation problems. We introduce both formulations and we discuss the existence and uniqueness of their solutions. We show selected computational methods based on the singular value decomposition. Further we focus on algebraic relations between both solutions and associated residuals. We present several estimates for the norm of the difference between the least squares and total least squares solution. We discuss their dependance on singular values of A and ( A b ) . Finally we illustrate the methods on selected test problems in the Matlab enviroment. 1
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