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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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Lineární algebraické modelování úloh s nepřesnými daty
Vasilík, Kamil ; Hnětynková, Iveta (advisor) ; Janovský, Vladimír (referee)
In this thesis we consider problems Ax b arising from the discretization of ill-posed problems, where the right-hand side b is polluted by (unknown) noise. It was shown in [29] that under some natural assumptions, using the Golub-Kahan iterative bidiagonalization the noise level in the data can be estimated at a negligible cost. Such information can be further used in solving ill-posed problems. Here we suggest criteria for detecting the noise revealing iteration in the Golub-Kahan iterative bidiagonalization. We discuss the presence of noise of different colors. We study how the loss of orthogonality affects the noise revealing property of the bidiagonalization.
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Numerical computation with functions using Chebfun
Lébl, Matěj ; Tichý, Petr (advisor) ; Hnětynková, Iveta (referee)
Goal of this work is to introduce Chebfun software and show ideas behind it. In the first chapter we summarize the theory of polynomial interpolation with focus on the Chebyshev interpolants. In the second chapter we introduce Chebfun software, its basic commands and principles of constructing interpolants. The third chapter is devoted to demonstrate theorems from the first chapter and to show practical applications of Chebfun when finding roots of a function and solving differential equations. Powered by TCPDF (www.tcpdf.org)
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Efficient implementation of dimension reduction methods for high-dimensional statistics
Pekař, Vojtěch ; Duintjer Tebbens, Erik Jurjen (advisor) ; Hnětynková, Iveta (referee)
The main goal of our thesis is to make the implementation of a classification method called linear discriminant analysis more efficient. It is a model of multivariate statistics which, given samples and their membership to given groups, attempts to determine the group of a new sample. We focus especially on the high-dimensional case, meaning that the number of variables is higher than number of samples and the problem leads to a singular covariance matrix. If the number of variables is too high, it can be practically impossible to use the common methods because of the high computational cost. Therefore, we look at the topic from the perspective of numerical linear algebra and we rearrange the obtained tasks to their equivalent formulation with much lower dimension. We offer new ways of solution, provide examples of particular algorithms and discuss their efficiency. Powered by TCPDF (www.tcpdf.org)
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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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