This dissertation presents results for two separate problems, both in the context of variational fracture models. The first problem involved developing and analyzing models of fracture in which we modeled the energy dissipated by crack growth as concentrated on the front of the crack. While many engineering models of fracture are based on a notion of crack front, there had not been a rigorous definition. We present the first work in this area, which includes a natural weak definition of crack front and front speed, a model of fracture whose evolution is described at the crack front, and a relaxation result that shows that these front based dissipations are all effectively equivalent to a Griffith-type dissipation. The second problem involved the computation of stationary points for Mumford-Shah and fracture using a level set method. Our method improves on existing techniques in that it can handle tips in the singular set and can find minimizers that previous techniques are unable to resolve.

Our method can be naturally applied in the application of manga stylization rendering. We propose a novel colorization technique that propagates color over manga regions exhibiting pattern-continuity as well as intensity-continuity. The proposed method works effectively on colorizing black-and-white manga which contains intensive amount of strokes, hatching, halftoning and screening. Once the user scribbles on the drawing, a local, statistical based pattern feature obtained with Gabor wavelet filters is applied to measure the pattern-continuity. The boundary is then propagated by the level set method that monitors the pattern-continuity. Regions with open boundaries or multiple disjointed regions with similar patterns can be sensibly segmented by a single scribble. With the segmented regions, various colorization techniques can be applied to replace colors, colorize with stroke preservation, or even convert pattern to shading. Based on the observation of the manga features, we then propose a framework to generate manga-style backgrounds from real photographs. It frees manga artists from the tedious and time-consuming background production. To mimic how manga artists produce the tidy background, it composes of two major steps, the line drawing and the screen laying. A line importance model is proposed to simplify and embolden lines in a stylish way. During the screen laying, texture analysis is utilized to automatically match the regions in photographs with screens in the database. Two types of screening mechanisms are proposed for matching shading & texture as well as the high-level structures. / Segmentation has always been a crucial and challenging topic in various computer vision and graphics applications. Information from image features can be incorporated into image segmentation techniques to make the process more accurate and robust. This dissertation focuses on encoding these feature information into the image processing procedures, including image segmentation and manga stylization rendering. / The fundamental part of this dissertation includes the discussion on the construction of the speed function, which is important in applying the curve-evolution based image segmentation. We firstly investigate the curvature term in the speed function, and then show how to transform the image segmentation problem into an interface propagating problem. We propose two formulations to enhance the speed function in level set methods, in order to tackle the segmentation problem of tagged MR images. First, a relaxation factor is introduced, aiming at relaxing the boundary condition when the boundary is unclear or blurry. Second, in order to incorporate human visual sensitive information from the image, a simple and general model is introduced to incorporate shape, texture and color features. By further extending this model, we present a unified approach for segmenting and tracking of the high-resolution color anatomical Chinese Visible Human (CVH) data. The underlying relationship of these two applications relies on the proposed variational framework for the speed function. Our proposed method can be used to segment the first slice of the volume data. Then based on the extracted boundary on the first slice, our method can also be adapted to track the boundary of the homogeneous organs among the subsequent serial images. In addition to the promising segmentation results, the tracking procedure requires only a small amount of user intervention. / Qu Yingge. / "July 2007." / Advisers: Heng Pheng Ann; Wong Tien Tsin. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0433. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 107-120). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract in English and Chinese. / School code: 1307.

Zheng Xi Yin. / "December 2002." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (p. 117-121). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

by Li Jian Lin. / "November 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 85-90) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Lai, Chun Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 83-87). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Spectral sets in Rd --- p.11 / Chapter 2.1 --- Preliminaries --- p.11 / Chapter 2.2 --- Fundamental domains and convex sets in Rd --- p.15 / Chapter 2.3 --- Finite union of cubes --- p.20 / Chapter 3 --- Spectral theory on discrete groups --- p.27 / Chapter 3.1 --- Finite groups and Zd --- p.28 / Chapter 3.2 --- Rational spectrums and tiling sets --- p.32 / Chapter 3.3 --- Fuglede´ةs Problem in R1 --- p.37 / Chapter 3.4 --- "Failure of Fuglede´ةs Conjecture in Rd, d >3" --- p.42 / Chapter 4 --- Self-similar tiles in R1 --- p.49 / Chapter 4.1 --- Basics of self-similar tiles --- p.49 / Chapter 4.2 --- Self-similar tile digit sets and spectral problem --- p.52 / Chapter 4.3 --- Kenyon criterion --- p.55 / Chapter 5 --- Spectral self-similar measures --- p.66 / Chapter 5.1 --- Spectral self-similar measures --- p.66 / Chapter 5.2 --- One-dimensional self-similar measures --- p.72 / Chapter 5.3 --- General properties of spectral measures --- p.80 / Bibliography --- p.83

Wong, Wing Chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 66-67). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Background Study --- p.4 / Chapter 2.1 --- Basic Definitions and Notations --- p.5 / Chapter 2.2 --- Tarjan and Trojanowski's algorithm --- p.6 / Chapter 2.2.1 --- Techniques --- p.6 / Chapter 2.2.2 --- Algorithm --- p.8 / Chapter 2.3 --- "Fomin, Grandoni and Kratsch's Algorithm" --- p.9 / Chapter 2.3.1 --- Techniques --- p.9 / Chapter 2.3.2 --- Algorithm --- p.14 / Chapter 3 --- Improvements --- p.18 / Chapter 3.1 --- Tarjan and Trojanowski´ةs Algorithm --- p.18 / Chapter 3.1.1 --- Correctness and Running Time Analysis --- p.28 / Chapter 3.1.2 --- Improvement --- p.30 / Chapter 3.1.3 --- Using more weights --- p.35 / Chapter 3.2 --- The First Algorithm --- p.37 / Chapter 3.2.1 --- Standard Analysis --- p.37 / Chapter 3.2.2 --- Measure and Conquer --- p.38 / Chapter 3.2.3 --- Using more weights --- p.42 / Chapter 3.3 --- The Second Algorithm --- p.43 / Chapter 3.3.1 --- Running Time Analysis --- p.44 / Chapter 3.3.2 --- Using More Weights --- p.45 / Chapter 3.4 --- The Third Algorithm --- p.46 / Chapter 4 --- Lower Bounds --- p.52 / Chapter 4.1 --- Tarjan and Trojanowski's Algorithm --- p.52 / Chapter 4.2 --- The First Algorithm --- p.55 / Chapter 4.3 --- The Second Algorithm --- p.58 / Chapter 4.4 --- The Third Algorithm --- p.60 / Chapter 5 --- Conclusion --- p.63 / Bibliography --- p.66

The nature and assessment of executive function are areas of active research. Many current assessments of executive function are complex, have limited reliability and validity, and suffer from task impurity, meaning other cognitive processes may indirectly influence task performance. Additionally, measures may be culture, language, or education bound limiting their use in certain populations (Miyake, Emerson, & Friedman, 2000; Miyake, Friedman, et al., 2000; Strauss, Sherman, & Spreen, 2006; Stuss, 2007). The purpose of this project was to develop a novel set of executive function measures to address issues with current clinical measures. The new measures 1) can be used in an ERP environment, 2) use the same stimulus set to address task impurity and 3) use simpler cognitive operations of inhibition, set-shifting, and updating, identified in previous research by Miyake et al., (2000). Twenty-nine undergraduate participants at the University of South Florida were administered currently used clinical measures of executive function theorized to engage in inhibition, set-shifting, and updating and the set of the novel tasks. ERP data was collected during the administration of the novel tasks. Behaviorally, conditions theorized to engage executive function resulted in slower response reaction time than control conditions. Additionally, behavioral results indicated that performance on novel tasks were differentially related to different clinical EF tasks. ERP differences were observed between both Go/No-Go conditions (inhibition) and among N-back conditions (updating). Results suggest the novel executive function tasks are tapping into different cognitive processes and may be a viable tool for studying executive function in the future.

This study investigated the distribution of bibliographic records amongst online bibliographic databases. The topic of Fuzzy Set Theory over the period of 1965 to 1993 was chosen to provide the case study for this investigation. From the DIALOG database host, searches were conducted on 114 databases to determine the number of journal article records relating to the topic of Fuzzy Sets. Both the number of records in each database, as well as the overlap of coverage between the databases were calculated. Six counting techniques were developed to allocate records to databases based on different methods for handling records that were duplicated between databases. When duplicate records are included, the top database accounts for 19% of the records; when duplicates are removed, the top database was found to account for 37% of the records. The distribution of records in databases was found to conform to the Bradford-Zipf hyperbolic distribution. Various other analyses were undertaken including: the duplicate records themselves, the total size of the DIALOG database system over time and the density of Fuzzy Set records in databases over time. A secondary aim of this study was to perform an informetric study on the literature of Fuzzy Set Theory itself. Results obtained include an analysis of the growth of the Fuzzy Set literature, an analysis of the journals covering the topic of Fuzzy Sets, an analysis of the terminology used in describing topics related to Fuzzy Sets. Also, the Ulrich's database was used to provide a subject classification of the journals to analyse the diffusion of the topic of Fuzzy Sets into other disciplines. Apart from the discipline of mathematics, the top disciplines into which Fuzzy Sets have diffused were found to be applied physics, systems and computing. The third aim of the thesis was to refine and develop the methodology used to perform large scale informetric studies using data from a variety of online bibliographic databases. Commercially available software was used wherever possible, but where this was not possible or infeasible, custom written programs were developed to perform various steps in the methodology.

A random variable that is defined as the absorption time of an evanescent finite-state continuous-time Markov chain is said to have a phase-type distribution. A phase-type distribution is said to have a representation (α,T ) where α is the initial state probability distribution and T is the infinitesimal generator of the Markov chain. The distribution function of a phase-type distribution can be expressed in terms of this representation. The wider class of matrix-exponential distributions have distribution functions of the same form as phase-type distributions, but their representations do not need to have a simple probabilistic interpretation. This class can be equivalently defined as the class of all distributions that have rational Laplace-Stieltjes transform. There exists a one-to-one correspondence between the Laplace-Stieltjes transform of a matrix- exponential distribution and a representation (β,S) for it where S is a companion matrix. In order to use matrix-exponential distributions to fit data or approximate probability distributions the following question needs to be answered: “Given a rational Laplace-Stieltjes transform, or a pair (β,S) where S is a companion matrix, when do they correspond to a matrix-exponential distribution?” In this thesis we address this problem and demonstrate how its solution can be applied to the abovementioned fitting or approximation problem. / Thesis (Ph.D.)--School of Applied Mathematics, 2003.

The purpose of the thesis was to explore the possibilities of using a Level-Set method to design a time-optimal path planar of a subject to direction-dependent maximum velocities. A promising application for such a planning approach lies in sailboat navigation planning, because of the dynamic ocean waves, current, wind and the characteristics of a sailboat. In the thesis, we developed an IOS application to simulate such scenario as environment properties with wind, static obstacles and the sailboat mapped into direction-dependent velocities in different locations of the environment.  Considering the wind is the main power for the sailboat, a wind speed generation function was created, based on different locations. The Level-Set method is widely used in image processing because of its various advantages, for instance, the ability to deal with topology change and stability. It also can be applied in path planning, in which the process of the Level-Set method can be considered as a continuous wave front propagating with a speed from the start location. A grid-based map was used to represent the environment. While the wave front was crossing the cell on the grid, a time was recorded for every cell, following the negative gradient direction of such crossing time, and then an optimal path could be found. In addition, we used the Narrow Band method to speed up the calculation of processing the level set equation. Finally, this report gives the results of the experiments of static obstacle avoidance, wind effects and smooth path planning.