A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.

Thesis (Ph. D.)--York University, 1998. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 73-77). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004 & res_dat=xri:pqdiss & rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation & rft_dat=xri:pqdiss:NQ39271.

Shape processing defines a set of theoretical and algorithmic tools for creating, measuring and modifying digital representations of shapes.  Such tools are of paramount importance to many disciplines of computer graphics, including modeling, animation, visualization, and image processing.  Many applications of shape processing can be found in the entertainment and medical industries. In an attempt to improve upon many previous shape processing techniques, the present thesis explores the theoretical and algorithmic aspects of a difference measure, which involves fitting a ball (disk in 2D and sphere in 3D) so that it has at least one tangential contact with each shape and the ball interior is disjoint from both shapes. We propose a set of ball-based operators and discuss their properties, implementations, and applications.  We divide the group of ball-based operations into unary and binary as follows: Unary operators include: * Identifying details (sharp, salient features, constrictions) * Smoothing shapes by removing such details, replacing them by fillets and roundings * Segmentation (recognition, abstract modelization via centerline and radius variation) of tubular structures Binary operators include: * Measuring the local discrepancy between two shapes * Computing the average of two shapes * Computing point-to-point correspondence between two shapes * Computing circular trajectories between corresponding points that meet both shapes at right angles * Using these trajectories to support smooth morphing (inbetweening) * Using a curve morph to construct surfaces that interpolate between contours on consecutive slices The technical contributions of this thesis focus on the implementation of these tangent-ball operators and their usefulness in applications of shape processing. We show specific applications in the areas of animation and computer-aided medical diagnosis.  These algorithms are simple to implement, mathematically elegant, and fast to execute.

The research in the thesis investigates the use of minimal path techniques to track and detect cracks, modeled as curves, in critical infrastructure like pavements and bridges. We developed a novel minimal path algorithm to detect curves with complex topology that may have both closed cycles and open sections using an arbitrary point on the curve as the sole input. Specically, we applied the novel algorithm to three problems: semi-automatic crack detection, detection of continuous cracks for crack sealing applications and detection of crack growth in structures like bridges. The current state of the art minimal path techniques only work with prior knowledge of either both terminal points or one terminal point plus total length of the curve. For curves with multiple branches, all terminal points need to be known. Therefore, we developed a new algorithm that detects curves and relaxes the necessary user input to one arbitrary point on the curve. The document presents the systematic development of this algorithm in three stages. First, an algorithm that can detect open curves with branches was formulated. Then this algorithm was modied to detect curves that also have closed cycles. Finally, a robust curve detection algorithm was devised that can increase the accuracy of curve detection. The algorithm was applied to crack images and the results of crack detection were validated against the ground truth. In addition, the algorithm was also used to detect features like catheter tube and optical nerves in medical images. The results demonstrate that the algorithm is able to accurately detect objects that can be modeled as open curves.

With technological advances digital video and imaging are becoming more and more relevant. Medical, remote-learning, surveillance, conferencing and home monitoring are just a few applications of these technologies. Along with compression, there is now a need for analysis and extraction of data. During the days of film and early digital cameras the processing and manipulation of data from such cameras was transparent to the end user. This transparency has been decreasing and the industry is moving towards `smart users' - people who will be enabled to program and manipulate their video and imaging systems. Smart cameras can currently zoom, refocus and adjust lighting by sourcing out current from the camera itself to the headlight. Such cameras are used in the industry for inspection, quality control and even counting objects in jewelry stores and museums, but could eventually allow user defined programmability. However, all this will not happen without interactive software as well as capabilities in the hardware to allow programmability. In this research, compression, expansion and detail extraction from videos in the surveillance arena are addressed. Here, a video codec is defined that can embed contextual details of a video stream depending on user defined requirements creating a video summary. This codec also carries out motion based segmentation that helps in object detection. Once an object is segmented it is matched against a database using its shape and color information. If the object is not a good match, the user can either add it to the database or consider it an anomaly. RGB vector angle information is used to generate object descriptors to match objects to a database. This descriptor implicitly incorporates the shape and color information while keeping the size of the database manageable. Color images of objects that are considered `safe' are taken from various angles and distances (with the same background as that covered by the camera is question) and their RGB vector angle based descriptors constitute the information contained in the database. This research is a first step towards building a compression and detection system for specific surveillance applications. While the user has to build and maintain a database, there are no restrictions on the size of the images, zoom and angle requirements, thus, reducing the burden on the end user in creating such a database. This also allows use of different types of cameras and doesn't need a lot of up-front planning on camera location, etc.

Thesis (Ph.D.)--York University, [2004]. Graduate Programme in [Mathematics]. / Typescript. Includes bibliographical references (leaves 163-166). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ99158

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In this thesis, Hausdorff, packing and capacity dimensions are studied by evaluating sets in the Euclidean space R^. Also the lower entropy dimension is calculated for some Cantor sets. By incorporating technics of Munroe and of Saint Raymond and Tricot, outer measures are created. A Vitali covering theorem for packings is proved. Methods (by Taylor and Tricot, Kahane and Salem, and Schweiger) for determining the Hausdorff and capacity dimensions of sets using probability measures are discussed and extended. The packing pre-measure and measure are shown to be scaled after an affine transformation. A Cantor set constructed by L.D. Pitt is shown to be dimensionless using methods developed in this thesis. A Cantor set is constructed for which all four dimensions are different. Graph directed constructions (compositions of similitudes follow a path in a directed graph) used by Mauldin and Willjams are presented. Mauldin and Williams calculate the Hausdorff dimension, or, of the object of a graph directed construction and show that if the graph is strongly connected, then the a—Hausdorff measure is positive and finite. Similar results will be shown for the packing dimension and the packing measure. When the graph is strongly connected, there is a constant so that the constant times the Hausdorff measure is greater than or equal to the packing measure when a subset of the realization is evaluated. Self—affine Sierpinski carpets, which have been analyzed by McMullen with respect to their Hausdorff dimension and capacity dimension, are analyzed with respect to their packing dimension. Conditions under which the Hausdorff measure of the construction object is positive and finite are given.

In the context of fractal geometry, the natural extension of volume in Euclidean space is given by Hausdorff and packing measures. These measures arise naturally in the context of iterated function systems (IFS). For example, if the IFS is finite and conformal, then the Hausdorff and packing dimensions of the limit sets agree and the corresponding Hausdorff and packing measures are positive and finite. Moreover, the map which takes the IFS to its dimension is continuous. Developing on previous work, we show that the map which takes a finite conformal IFS to the numerical value of its packing measure is continuous. In the context of self-similar sets, we introduce the super separation condition. We then combine this condition with known density theorems to get a better handle on finding balls of maximum density. This allows us to extend the work of others and give exact formulas for the numerical value of packing measure for classes of Cantor sets, Sierpinski N-gons, and Sierpinski simplexes.

For every positive, decreasing, summable sequence $a=(a_i)$, we can construct a Cantor set $C_a$ associated with $a$. These Cantor sets are not necessarily self-similar. Their dimensional properties and measures have been studied in terms of the sequence $a$. In this thesis, we extend these results to a more general collection of Cantor sets. We study their Hausdorff and packing measures, and compare the size of Cantor sets with the more refined notion of dimension partitions. The properties of these Cantor sets in relation to the collection of cut-out sets are then considered. The multifractal spectrum of $\mathbf{p}$-Cantor measures on these Cantor sets are also computed. We then focus on the special case of homogeneous Cantor sets and obtain a more accurate estimate of their exact measures. Finally, we prove the $L^p$-improving property of the $\mathbf{p}$-Cantor measure on a homogeneous Cantor set as a convolution operator.