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影像風格化是一種改變輸入影像的重要技術，其用來加強圖像的特徵並傳達視覺上的資訊，其中圖案的形狀與分布構成的風格化的基本要素。然而對於初學者而言，設計一個新的風格化圖案以及適當的分佈配置是不容易的。在本文中，提出了一個使用異向性反應擴散的圖像風格化以及圖案生成的方法，並從已知的反應擴散圖案為基礎做延伸，同時保有反應擴散的自我組織圖案的特性。為了能夠有效的控制圖案的生成，利用調整後的異向性擴散用來控制圖案的形狀並結合流場調整圖案的排列。圖案的大小、密度、方向以及風格化樣式可藉由閥值的調整以及顏色映射加以控制。最後本系統用來生成剪紙、風格化半色調影像以及流場的視覺化的結果以突顯本系統之特色。 / Image stylization is an essential technique to create the style of input images, enhance image features, and express visual cues. The shape and distribution of the primitive are essential elements in stylization. However, designing a new pattern or creating an appropriate distribution can be challenging for novice users. In this paper, an anisotropic reaction diffusion system for image stylization and pattern generation is proposed. This system starts from modify existed reaction diffusion formula, but keeps the behaviors of reaction diffusion: self-organized patterns, stable pattern generation and multiple styled pattern. To enable more effective control over pattern generation, the proposed method utilizes a set of modifications on anisotropic diffusion to control shape and introduces a flow field to guide pattern arrangement. The size, density, orientation, and pattern style can be controlled by thresholding and toon mapping. The proposed system was used to generate images in the paper-cut, stylized halftone, and flow visualization, and the results are presented to highlight the control factors of the proposed system.

We study diffusion-reaction-advection models describing population dynamics of aquatic organisms subject to a constant drift, with reflecting upstream and outflow downstream boundary conditions. We consider three different models: single logistically growing species, two and three competing species. In the case of a single population, we determine conditions for existence, uniqueness and stability of non-trivial steady-state solutions. We analyze the dependence of such solutions on advection speed, growth rate and length of the habitat. Such analysis offers a possible explanation of the "drift paradox" in our context. We also introduce a spatially implicit ODE (nonspatial approximation) model which captures the essential behavior of the original PDE model. In the case of two competing species, we use a diffusion-advection version of the Lotka-Volterra competition model. Combining numerical and analytical techniques, in both the spatial and nonspatial approximation settings, we describe the effect of advection on competitive outcomes. Finally, in the case of three species, we use the nonspatial approximation approach to analyze and classify the possible scenarios as we change the flow speed in the habitat.

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

Competitive-Consecutive and Competitive-Parallel reactions are both mixing sensitive reactions; the yield of desired product from these reactions depends on how fast the reactants are brought together. Recent experimental results have suggested that the mixing effect may depend strongly on the stoichiometry of the reactions. To investigate this, a 1-D, non-dimensional, reaction-diffusion model at the micro-mixing scale has been developed. Assuming constant mass concentration and diffusivities, systems of PDEs have been derived on a mass fraction basis for both types of reactions. A single general Damkhler number and specific dimensionless reaction rate ratios were derived for both reaction schemes. The resulting dimensionless equations were simulated to investigate the effects of mixing, reaction rate ratio and stoichiometry of the reactions. It was found that decreasing the striation thickness and the dimensionless rate ratio maximizes yield for both types of reactions and that the stoichiometry has a considerable effect on yield. All three variables were found to interact strongly. Phase plots showing the interactions between the three variables were developed.

Doctor of Philosophy / A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

Thesis (Ph. D.)--Florida State University, 2005. / Advisor: Oliver Steinbock, Florida State University, College of Arts and Sciences, Dept. of Chemistry and Biochemistry. Title and description from dissertation home page (viewed Jan. 24, 2006). Document formatted into pages; contains xii, 123 pages. Includes bibliographical references.

Diploma thesis is about stationary solutions to reaction-diffusion system of the activator-inhibitor type on a one-dimensional domain. Three homogeneous boudary value problems are studied---with pure Neumann boudary conditions, with mixed Neumann-Dirichlet boudary conditions and with Neumann conditions on the boundary where simultaneously an additional homogeneous condition is prescribed in a given point in the interior of the domain. For all three boudary value problems the existence of so-called critical points (diffusion parameters, for which a non-trivial solution exists) is proved.

L'objet de cette thèse est l'étude de deux exemples de propagation pour des équations de réaction-diffusion hétérogènes.Le but de la première partie est de déterminer quels sont les effets d'échanges non locaux entre une ligne de diffusion rapide et un environnement bidimensionnel dans lequel a lieu un phénomène de réaction-diffusion de type KPP usuel. Dans le premier chapitre nous étudions comment ce couplage non local entre la ligne et le plan accélère la propagation dans la direction de la ligne ; on détermine aussi comment différentes fonctions d'échanges maximisent ou non la vitesse d'invasion. Le deuxième chapitre est consacré à la limite singulière de termes d'échanges qui convergent vers des masses de Dirac. On montre alors que la dynamique converge avec une certaine uniformité. Dans le troisième chapitre nous étudions la limite d'échanges étalés à l'infini. Ils permettent de donner un infimum sur la vitesse de propagation pour ce type de modèle qui peut cependant être supérieure à la vitesse KPP usuelle.La seconde partie de cette thèse est consacrée à l'étude de solutions entières (ou éternelles) pour des équations bistables hétérogènes. On considère un domaine bidimensionnel infini dans une direction, borné dans l'autre, qui converge vers un cylindre quand x tend vers moins l'infini. On montre alors l'existence d'une solution entière dans un tel domaine qui est égal à l'onde bistable en t tend vers moins l'infini. Cela nous conduit à étudier un modèle unidimensionnel avec un terme de réaction hétérogène, pour lequel on obtient le même résultat. / The aim of this thesis is to study two examples of propagation phenomena in heterogeneous reaction-diffusion equations.The purpose of the first part is to understand the effect of nonlocal exchanges between a line of fast diffusion and a two dimensional environment in which reaction-diffusion of KPP type occurs. The initial model was introduced in 2013 by Berestycki, Roquejoffre, and Rossi. In the first chapter we investigate how the nonlocal coupling between the line and the plane enhances the spreading in the direction of the line; we also investigate how different exchange functions may maximize or not the spreading speed.The second chapter is concerned with the singular limit of nonlocal exchanges that tend to Dirac masses. We show the convergence of the dynamics in a rather strong sense. In the third chapter we study the limit of long range exchanges with constant mass. It gives an infimum for the asymptotic speed of spreading for these models that still could be bigger than the usual KPP spreading speed.The second part of this thesis is concerned with entire solutions for heterogeneous bistable equations.We consider a two dimensional domain infinite in one direction, bounded in the other, that converges to a cylinder as x goes to minus infinity. We prove the existence of an entire solution in such a domain which is the bistable wave for t tends to minus infinity. It also lead us to investigate a one dimensional model with a non-homogeneous reaction term,for which we prove the same property.

Esta tese de doutorado descreve um novo modelo para o acoplamento de difusão química contínua e eventos celulares discretos dentro de um ambiente de simulação biologicamente inspirado. Nosso objetivo é definir e explorar um conjunto minimalista de recursos que também são expressivos, permitindo a criação de padrões 2D complexos usando apenas poucas regras. Por não nos restringirmos a uma grade estática ou regular, mostramos que muitos fenômenos diferentes podem ser simulados, como sistemas tradicionais de reação-difusão, autômatos celulares e padrões de pigmentação de seres vivos. Em particular, demonstramos que a adição de saturação química aumenta significativamente a gama de padrões simulados usando reação-difusão, incluindo padrões que não eram possíveis anteriormente. Nossos resultados sugerem um possível modelo universal que pode integrar abordagens de formação de padrões anteriores, fornecendo nova base para experimentação e texturas de aparência realista para uso geral em Computação Gráfica. / This doctoral thesis describes a novel model for coupling continuous chemical diffusion and discrete cellular events inside a biologically inspired simulation environment. Our goal is to define and explore a minimalist set of features that are also expressive, enabling the creation of complex 2D patterns using just a few rules. By not being constrained into a static or regular grid, we show that many different phenomena can be simulated, such as traditional reaction-diffusion systems, cellular automata, and pigmentation patterns from living beings. In particular, we demonstrate that adding chemical saturation increases significantly the range of simulated patterns using reaction-diffusion, including patterns not possible before. Our results suggest a possible universal model that can integrate previous pattern formation approaches, providing new ground for experimentation and realistic-looking textures for general use in Computer Graphics.