Název práce: Segmentace mikroskopických snímků pomocí level-set metod Autor: Zuzana Bílková Katedra: Katedra numerické matematiky Vedoucí diplomové práce: RNDr. Václav Kučera, Ph.D., KNM, MFF UK Konzultant: RNDr. Jindřich Soukup, ÚTIA, AV ČR Abstrakt: Tato diplomová práce představuje novou metodu pro segmentaci snímků pořízených mikroskopem s fázovým konrastem. Cílem je oddělit buňky od pozadí. Algoritmus je založen na variační formulaci level set metod, tedy na minimalizaci funkcionálu popisujícího level set funkci. Funkcionál je minimalizován gradientním tokem popsaným evoluční parciální diferenciální rovnicí. Nejdůležitější nové myšlenky jsou inicializace pomocí prahování a nové členy ve funkcionálu, které zrychlují konvergenci a zpřesňují výsledky. Také jsme použili nové funkce napsané v jazyce C k počítání gradientu a Laplaceova operátoru. Tato implementace je třikrát rychlejší než standardní funkce v MATLABu. Dosáhli jsme lepších výsledků než algoritmy, se kterými jsme metodu porovnávali. Klíčová slova: Segmentace, level set metody, aktivní kontury Title: Segmentation of microscopic images using level set methods Author: Zuzana Bílková Department: Department of Numerical Mathematics Supervisor: RNDr....

Recently, solutions to the problem of image segmentation and denoising are developed based on the Mumford-Shah model. The model provides an energy functional, called the Mumford-Shah functional, which should be minimized. Since the minimization of the functional has some difficulties, approximate approaches are proposed. Two such methods are the gradient flows method and the Chan-Vese active contour method. The performance evolution in terms of speed shows that the gradient flows method converges to the boundaries of the smooth parts faster / but for the hierarchical four-phase segmentation, it is observed that this method sometimes gives unsatisfactory results. In this work, a fast hierarchical four-phase segmentation method is proposed where the Chan-Vese active contour method is applied following the gradient flows method. After the segmentation process, the segmented regions are denoised using diffusion filters. Additionally, for the low signal-to-noise ratio applications, the prefiltering scheme using nonlinear diffusion filters is included in the proposed method. Simulations have shown that the proposed method provides an effective solution to the image segmentation and denoising problem.

Ce travail porte sur la résolution de problèmes faisant intervenir des mouvements d'interfaces. Dans les différentes parties de cette thèse, on cherche à déterminer ces mouvements d'interfaces en résolvant des modèles approchés consistant en des équations ou des systèmes d'équations sur des champs. Les problèmes obtenus sont des équations paraboliques et des systèmes hyperboliques. Dans la première partie (chapitre 2), on étudie un modèle simplifié pour la propagation d'une onde de souffle en dynamique des fluides compressibles. Ce modèle peut s'écrire sous la forme d'un système hyperbolique, et on construit un algorithme résolvant numériquement ce système par une méthode de type Fast-Marching. On mène également une étude théorique de ce système pour déterminer des solutions de référence et tester la validité de l'algorithme. Dans la deuxième partie (chapitres 3 à 5), les équations approchées sont de type parabolique, et on cherche à montrer l'existence de solutions de type régime permanent à ces équations. Dans les chapitres 3 et 4, on étudie une équation générique en une dimension associée à des phénomènes de réaction-diffusion. Dans le chapitre 3, on montre l'existence de solutions quasi-planes pour un terme de réaction (terme non-linéaire) assez général, et dans le chapitre 4 on utilise ces résultats pour montrer l'existence d'ondes pulsatoires progressives dans le cas spécifique d'une non-linéarité bistable. Le modèle étudié dans le chapitre 5 est un modèle de champ de phase approchant un modèle de dynamique des dislocations dans un cristal, dans un domaine correspondant physiquement à une source de Frank-Read / This work is about the resolution of problems associated with the motion of interfaces. In each part of this thesis, the goal is to determine the motion of interfaces by the use of approached models consisting of equations or systems of equation on fields. The problems we get are parabolic equations and hyperbolic systems. In the first part (Chapter 2), we study a simplified model for the propagation of a shock wave in compressible fluid dynamics. This model can be written as a hyperbolic system, and we construct an algorithm to solve it numerically by a Fast-Marching like method. We also conduct a theoretical study of this system to determine reference solutions and test the algorithm. In the second part (Chapters 3 to 5), the approached models yield parabolic equations, and our goal is to show the existence of permanent regime solutions for these equations. Chapter 3 and 4 are dedicated to the study of a generic one-dimensional equation modelling reaction-diffusion phenomena. In Chapter 3, we show the existence of plane-like solutions for a general reaction term, and in Chapter 4 we use this result to show the existence of pulsating travelling waves in the specific case of a bistable nonlinearity. In Chapter 5, we study a phase-field model approaching a model for the dynamics of dislocations in a crystal, in a domain corresponding to a Frank-Read source

Nesta dissertação de Mestrado será apresentada uma poderosa técnica numérica, conhecida como método Level Set, capaz de simular e analisar movimentos de curvas em diferentes cenários físicos. Tal método - formulado por Osher e Sethian [1] - está sedimentado na seguinte idéia: representar uma determinada curva (ou superfície) Γ como a curva de nível zero (zero level set) de uma função Φ de maior dimensão (denominada função Level Set). A equação diferencial do tipo Hamilton-Jacobi que descreve a evolução da função Level Set é discretizada através da utilização de acurados esquemas hiperbólicos e, como resultado de tal acurácia, obtém-se uma formulação numérica capaz de tratar eficazmente mudanças topológicas e/ou descontinuidades que, eventualmente, podem surgir no decorrer da propagação da curva (ou superfície) de nível zero. Em virtude da eficácia e versatilidade do método Level Set, esta técnica numérica está sendo amplamente aplicada à diversas áreas científicas, incluindo mecânica dos fluidos, processamento de imagens e visão computacional, crescimento de cristais, geometria computacional e ciência dos materiais. Particularmente, o propósito deste trabalho equivale ao estudo dos fundamentos do método Level Set e, por fim, visa-se aplicar tal modelo numérico à problemas existentes na área de crescimento de cristais. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988. / In this dissertation, we present a powerful numerical technique known as Level Set Method for computing and analyzing moving fronts in different physical settings. The method -formulated by Osher and Sethian [1] - is based on the following idea: a curve (or surface) is embedded as the zero level set of a higher-dimensional function Φ (called level set function). Then, we can link the evolution of this function Φ to the propagation of the curve itself through a time-dependent initial value problem. At any time, the curve is given by the zero level set of the time-dependent level set function Φ. The evolution of the level set function Φ is described by a Hamilton-Jacobi type partial differential equation, which can be discretised by the use of accurate methods for hyperbolic equations. As a result, the Level Set Method is able to track complex curves that can develop large spikes, sharp corners or change its topology as they evolve. Because of its versatility and efficacy, this numerical technique has found applications in a large number of areas, including fluid mechanics, image processing and computer vision, crystal growth, computational geometry and materials science. Particularly, the aim of this dissertation has been to understand the fundamentals of Level Set Method and its final goal is compute the motion of bondaries in crystal growth using this numerical model. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988.

Nesta dissertação de Mestrado será apresentada uma poderosa técnica numérica, conhecida como método Level Set, capaz de simular e analisar movimentos de curvas em diferentes cenários físicos. Tal método - formulado por Osher e Sethian [1] - está sedimentado na seguinte idéia: representar uma determinada curva (ou superfície) Γ como a curva de nível zero (zero level set) de uma função Φ de maior dimensão (denominada função Level Set). A equação diferencial do tipo Hamilton-Jacobi que descreve a evolução da função Level Set é discretizada através da utilização de acurados esquemas hiperbólicos e, como resultado de tal acurácia, obtém-se uma formulação numérica capaz de tratar eficazmente mudanças topológicas e/ou descontinuidades que, eventualmente, podem surgir no decorrer da propagação da curva (ou superfície) de nível zero. Em virtude da eficácia e versatilidade do método Level Set, esta técnica numérica está sendo amplamente aplicada à diversas áreas científicas, incluindo mecânica dos fluidos, processamento de imagens e visão computacional, crescimento de cristais, geometria computacional e ciência dos materiais. Particularmente, o propósito deste trabalho equivale ao estudo dos fundamentos do método Level Set e, por fim, visa-se aplicar tal modelo numérico à problemas existentes na área de crescimento de cristais. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988. / In this dissertation, we present a powerful numerical technique known as Level Set Method for computing and analyzing moving fronts in different physical settings. The method -formulated by Osher and Sethian [1] - is based on the following idea: a curve (or surface) is embedded as the zero level set of a higher-dimensional function Φ (called level set function). Then, we can link the evolution of this function Φ to the propagation of the curve itself through a time-dependent initial value problem. At any time, the curve is given by the zero level set of the time-dependent level set function Φ. The evolution of the level set function Φ is described by a Hamilton-Jacobi type partial differential equation, which can be discretised by the use of accurate methods for hyperbolic equations. As a result, the Level Set Method is able to track complex curves that can develop large spikes, sharp corners or change its topology as they evolve. Because of its versatility and efficacy, this numerical technique has found applications in a large number of areas, including fluid mechanics, image processing and computer vision, crystal growth, computational geometry and materials science. Particularly, the aim of this dissertation has been to understand the fundamentals of Level Set Method and its final goal is compute the motion of bondaries in crystal growth using this numerical model. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988.

The level set method offers a simple and robust approach to modeling liquid-vapor interfaces that arise in boiling and condensing flows. The current liquid-vapor phase-transition techniques used with the level set method are not able to account for different thermal conductivities and specific heats in each respective phase, nor are they able to accurately account for latent heat absorption and release. This paper presents a new level set based technique for liquid-vapor phase-transition that accounts for different material properties in each respective phase, such as thermal conductivity and specific heat, while maintaining the interface at the saturation temperature. The phase-transition technique is built on the ghost fluid framework coupled with the standard level set method. A new technique is presented for constructing ghost nodes that implicitly captures the immersed boundary conditions and is second order accurate. The method is tested against analytical solutions, and it is used to model film boiling. The new phase-transition technique will greatly assist efforts to accurately capture the physics of boiling and condensing flows. In addition to presenting a new phase transition technique, a coupled level set volume of fluid advection scheme is developed for phase transition flows. The new scheme resolves the mass loss problem associated with the level set method, and the method provides an easy way to accurately calculate the curvature of an interface, which can be difficult with the volume of fluid method. A film boiling simulation is performed to illustrate the superior performance of the coupled level set volume of fluid approach over the level set method and the volume of fluid method.

In this thesis, the image segmentation methods based on the Mumford&amp / #8211 / Shah variational approach have been studied. By obtaining an optimum point of the Mumford-Shah functional which is a piecewise smooth approximate image and a set of edge curves, an image can be decomposed into regions. This piecewise smooth approximate image is smooth inside of regions, but it is allowed to be discontinuous region wise. Unfortunately, because of the irregularity of the Mumford Shah functional, it cannot be directly used for image segmentation. On the other hand, there are several approaches to approximate the Mumford-Shah functional. In the first approach, suggested by Ambrosio-Tortorelli, it is regularized in a special way. The regularized functional (Ambrosio-Tortorelli functional) is supposed to be gamma-convergent to the Mumford-Shah functional. In the second approach, the Mumford-Shah functional is minimized in two steps. In the first minimization step, the edge set is held constant and the resultant functional is minimized. The second minimization step is about updating the edge set by using level set methods. The second approximation to the Mumford-Shah functional is known as the Chan-Vese method. In both approaches, resultant PDE equations (Euler-Lagrange equations of associated functionals) are solved by finite difference methods. In this study, both approaches are implemented in a MATLAB environment. The overall performance of the algorithms has been investigated based on computer simulations over a series of images from simple to complicated.

This thesis explores two possible ways of detecting soft plaque present in the coronary arteries, using CTA imagery. The coronary arteries are vessels that supply oxidized blood to the cardiac muscle and are thus important for the proper functioning of heart. Cholesterol or reactive oxygen species from cigarette smoke and other toxins may get adhered to the walls of coronary arteries and trigger chronic inflammation that leads to formation of the soft plaque. When the soft plaque grows bigger in volume, it occludes the blood flow to the cardiac muscle and finally results in ischemic heart attack. Moreover, smaller plaque can easily rupture due to the blood flow in arteries and can result in complications such as stroke. Hence there is a need to detect the soft plaque using non-invasive or minimally invasive techniques. In CTA imagery, the cardiac muscle appears as a dark gray color, while the blood appears as dull white color and the the calcified plaque appears as bright white. The soft plaque has an intensity which falls between the intensity level of the blood and cardiac muscle, making it difficult to directly segment the soft plaque using standard segmentation methods. Soft plaque in its advanced stages forms a concavity in the blood lumen. A watershed based segmentation method was used to detect the presence of this concavity which in turn identifies the location of the soft plaque. For segmenting the soft plaque at its earlier stages, a novel segmentation technique was used. In this technique the surface is evolved based on a region-based energy calculated in the local neighborhood around each point on the evolving surface. This method seems to be superior to the watershed based segmentation method in detecting smaller plaque deposits.

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After a review of the notions of Hausdorff and Fourier dimensions from fractal geometry and Fourier analysis and the properties of local times of Brownian motion, we study the Fourier structure of Brownian level sets. We show that if δa(X) is the Dirac measure of one-dimensional Brownian motion X at the level a, that is the measure defined by the Brownian local time La at level a, and μ is its restriction to the random interval [0, L−1 a (1)], then the Fourier transform of μ is such that, with positive probability, for all 0 ≤ β < 1/2, the function u → |u|β|μ(u)|2, (u ∈ R), is bounded. This growth rate is the best possible. Consequently, each Brownian level set, reduced to a compact interval, is with positive probability, a Salem set of dimension 1/2. We also show that the zero set of X reduced to the interval [0, L−1 0 (1)] is, almost surely, a Salem set. Finally, we show that the restriction μ of δ0(X) to the deterministic interval [0, 1] is such that its Fourier transform satisfies E (|ˆμ(u)|2) ≤ C|u|−1/2, u 6= 0 and C > 0. Key words: Hausdorff dimension, Fourier dimension, Salem sets, Brownian motion, local times, level sets, Fourier transform, inverse local times. / Decision Sciences / PhD. (Operations Research)