We formulate a novel approach to infer conditional independence models or Markov structure of a multivariate distribution. Specifically, our objective is to place informative prior distributions over graphs (decomposable and unrestricted) and sample efficiently from the induced posterior distribution. We also explore the idea of factorizing according to complete sets of a graph; which implies working with a hypergraph that cannot be retrieved from the graph alone. The key idea we develop in this paper is a parametrization of hypergraphs using the geometry of points in $R^m$. This induces informative priors on graphs from specified priors on finite sets of points. Constructing hypergraphs from finite point sets has been well studied in the fields of computational topology and random geometric graphs. We develop the framework underlying this idea and illustrate its efficacy using simulations. / Dissertation

With perfect channel state information at all the transmission terminals, the asymptotic capacity of multi-hop multiple-input multiple-output(MIMO) amplify-andforward(AF) relay channels is derived. Although the derivation is based on the assumption of a large number of antennas, simulation results show that the derived expression is surprisingly accurate for even a small number of antennas, and may even be superior to existing results. In addition, based on the asymptotic result, we present different power allocation schemes to (i) minimization the transmit power; (ii) maximization the network throughput; (iii) minimization the transmit power over all source. Fortunately, the proposed power allocation problems can be formulated using geometric programming(GP). Therefore, the optimal power distribution among the multi-hop relay can be obtained efficiently. For multiuser scenarios, since it is possible that the QoS of each user cannot be satisfied simultaneously, we study jointly admission control and power allocation optimization problem. This joint problem is NP-hard. Therefor, we propose an iterative algorithm to reduced the complexity.

In this thesis, we introduce a new class of problems, which we call Optimal Areal Positioning (OAP), and study a special form of these problems. OAPs have important applications in earth observation satellite management, tele-robotics, multi-camera control, and surveillance. In OAP, we would like to find the optimal position of a set of floating geometric objects (targets) on a two-dimensional plane to (partially) cover another set of fixed geometric objects (requests) in order to maximize the total reward obtained from covered parts of requests. In this thesis, we consider the special form of OAP in which targets and requests are parallel axes rectangles and targets are of equal size. A predetermined reward is associated with covering an area unit of each request. Based on the number of target rectangles, we classify rectangular OAP into two categories: Single Target Problem (STP) and Multi-Target Problem (MTP). The structure of MTP can be compared to the planar p-center which is NP-complete, if p is part of the input. In fact, we conjecture that MTP is NP-complete. The existing literature does not contain any work on MTP. The research contributions of this thesis are as follows: We develop new theoretical properties for the solution of STP and devised a new solution approach for it. This approach is based on a novel branch-and-bound (BB) algorithm devised over a reduced solution space. Branching is done using a clustering scheme. Our computational results show that in many cases our approach significantly outperforms the existing Plateau Vertex Traversal and brute force algorithms, especially for problems with many requests appearing in clusters over a large region. We perform a theoretical study of MTP for the first time and prove several theoretical properties for its solution. We have introduced a reduced solution space using these properties. We present the first exact algorithm to solve MTP. This algorithm has a branch-and-bound framework. The reduced solution space calls for a novel branching strategy for MTP. The algorithm has a main branch-and-bound tree with a special structure along with two trees (one for each axis) to store the information required for branching in the main tree in an efficient format. Branching is done using a clustering scheme. We perform computational experiments to evaluate the performance of our algorithm. Our algorithm solves relatively large instances of MTP in a short time.

Scales are a visually striking feature that grows on many animals. These small, rigid plates embedded in the skin form an integral part of our description of fish and reptiles, some plants, and many extinct animals. Scales exist in many shapes and sizes, and serve as protection, camouflage, and plumage for animals. The variety of scales and the animals they grow from pose an interesting problem in the field of Computer Graphics. This dissertation presents a method for generating scales and scale-like structures on a polygonal mesh through surface replacement. A triangular mesh was covered with scales and one or more proxy-models were used as the scales shape. A user began scale generation by drawing a lateral line on the model to control the distribution and orientation of scales on the surface. Next, a vector field was created over the surface to control an anisotropic Voronoi tessellation, which represents the region occupied by each scale. Then these regions were replaced by cutting the proxy model to match the boundary of the Voronoi region and deform the cut model onto the surface. The final result is a fully connected 2-manifold that is suitable for subsequent post-processing applications, like surface subdivision.

This thesis presents an investigation into Paleoindian projectile-point morphology. A goal of this research is to determine if evidence of a normative cultural manufacturing protocol can be identified on Clovis projectile points which can then be used to address research questions concerning Clovis point variability, and ultimately, the spread of this tool-form across North America. This paper addresses obstacles to behavioral investigations of stone tool morphology such as the effects of resharpening and raw material type on tool shape. I argue that a culturally normative process of manufacture was maintained throughout the life-history of Clovis projectile points which translated into a specific shape maintained to the time of exhaustion and discard. As an analytical tool, this study utilizes the geometric morphometric method to retain the geometry of each artifact throughout analysis by focusing on spatial covariation among landmarks uniformly found on each tool. This thesis investigates variability in 123 fluted projectile points from 23 archaeological sites in North America which met criteria meant to control for security of context in the archaeological record. Principle components describing the shape-variability inherent in this data-set were generated using geometric morphometrics and multivariate statistical analyses were employed to identify major factors of variability. This research concluded that Clovis projectile-point shape was determined by normative cultural behavior maintained throughout the life of the artifact and not the result of raw material type or resharpening processes. Therefore, the projectile-point variability found to be geographically patterned provided evidence of Paleoindian movement and the spread of tool form. Multivariate analysis of variance determined that a regional trend in variability was present. The distribution of within-site variance suggested that artifacts from sites in the West were very homogeneous while artifacts from Eastern sites were more variable. The multivariate cluster and discriminant function analyses also demonstrated a closer affinity between artifacts in the Southwest and Northwest than either has with the Northeast. The similarities in projectile point morphology between the Southwest and Northwest regions suggest movement beginning with a Southwest point of origin from which Pleistocene peoples may have carried their fluted point technology north and east.

As the popularity of the digital Web-cam¡Athese devices are more and more cheaper and powerful. We can apply computer vision techniques with camera and projector to build a more convenient presentation system. In presentation, sometimes due to the position of projector, the images appear the perspective distortion (keystone distortion). The user should manually adjust the position of projector or use the keystone corrections of the projector. But when the distortion is not trapezium, the built-in keystone corrections are not suitable in this situation. We present a computer-vision based method that uses a Web-cam to calibrate the keystone distortion. The Web-cam takes the images that the projector projected on the wall. If the Web-cam observes keystone distortions of the projected images, we use a geometric transform that pre-warps the images in the projector frame, such that these images appears rectangle with known aspect ratio after being projected on the wall. Besides, we implement the virtual buttons that allow users to interact with the computer. The virtual buttons means that when the camera detect the laser point is on the virtual buttons, computer triggers the event as the virtual button being pushed. This paper uses point-matching pairs to obtain the homography between camera image frame and source image frame. The homography, that is the fundamental of calibrating perspective distortions also help us to search the position of the laser point.

The aim of this study is on the development of young children's arithmetic and geometric concepts. To achieve this aim, teachers motivated young children's curiosity through the use of interesting illustrated books. In particular, the target is the advance in young children's knowing and understanding of geometric figures and numbers within 10. There were altogether three phases. In the first phase, the investigator designed a mathematical activity (a pre-test) for the teacher to capture young children's entry behavior. The second phase referred to the use of illustrated books in ordinary instruction. During the third phase, the mathematical activity (identical to that of phase 1) was carried out again, in order to find out if advancement in young children's mathematical ability was present. Data collection and administration included investigator's observation notes; video analyses; notes on teacher interviews; and investigator's self reflections record. Data analyses and results were two: geometric concepts and arithmetic concepts. Geometric concepts. Using illustrated books could advance in understanding and increase in curiosity of young children; also to promote young children's recognition in variation in polygon; and arouse young children's imagination and creativity in individual figures and combination of patterns. Arithmetic concepts. Using illustrated books could cause the interests in change of numbers; cultivate young children's sensitivity in the addition and subtraction of numbers; and promote the ability of combination and decomposition of numbers within 10. The findings of this study suggested that illustrated books are important and useful in assisting young children to explore mathematical concepts. Illustrated books encourage young children to think and allow room for oral presentation, clarifying and reflecting one self. Finally, teachers are suggested to integrate the use illustrated books in teaching mathematics, and follow up with related mathematics activities, to upgrade young children's development in various mathematics concepts.

The purpose of the study is to investigate the problem-solving strategies of children in fourth-grade, fifth-grade and sixth-grade. The investigators tried to guide children through a variety of problem-solving strategies that were in written, figural, or symbolic forms. Then the investigators examined the variation of children¡¦s spatial concepts by analyzing the problem-solving strategies used in calculating the volumes of various solids. In order to improve the teaching materials, the investigator identified the performance of spatial concepts by referring to QCAI (QUASAR Cognitive Assessment Instrument). For the purpose of designing future lessons, the investigator examined the relationship between children¡¦s problem-solving strategies level and the performance in the seven content areas in the curriculum. There are two results of this study: The first result is on the problem-solving strategies found within the same grade (4th; 5th; and 6th) and there are 3 findings. First, strategies of fourth-graders are focused on low-levels and middle-levels; strategies of fifth-graders focused on middle-level geometric space concept, and finally, strategies of sixth-grades students focused on middle-levels and high-levels. Second, there are no significant differences relating to gender. Third, the higher the grade of children is the higher the levels of their spatial concepts. In addition, the second result is the relationship between strategy levels and performance in seven content areas: revealing position corrections, and having mathematics as the most significant, and the rest in order: social studies, language arts, science and technology, arts and humanities, integrative activities, health and physical education.

The multi-speed internal gear hub of a bicycle is a well-closed gear shifting system that works perfectly under any challenging riding environment, and is developed specifically to improve fragile chain-drive derailleur using planetary gear trains. In recent years, bicycle internal hub gears have been developed toward multi-speed transmission, and in the development process of internal gear hubs, the structure of speed changing mechanism has changed from a single planetary gear train to multiple planetary gear trains, and is becoming more complex. The primary purpose of this Study aims to establish a systemized and efficient design process, and develop an effective theory and method for designing multiple-speed internal gear hubs with the design concept and common features of multiple-speed internal gear hub products developed in recent years, to favor the innovation and development of internal gear hubs. First, existing multiple speed internal hub products are analyzed and summarized for their basic features, limitations, and demands as the reference for the design of multiple speed internal gear hubs, and a catalog of usable planetary structures is systematically established with the coupling and connection of basic high and low ratio speed changing modules. Secondly, usable planetary structures that offer best performance of geometric progression speed ratio distribution of gear hub are matched with gear positions, and a table of gear sequence is confirmed; third, based on the maximum external diameter required by design, tooth numbers for every gear in a hub are defined according to the relationship between tooth number and speed ratio, as well as the gear sequences; and finally, the systematic design process above is applied to the development of an easy-to-use computer aided design software with the lowest possible number of variables using Visual Basic 6.0 for designers, in order to favor the innovation and development of internal gear hubs.

The goal of this thesis is to introduce new methods to create intricate perforated shapes in a computing environment. Modeling shapes with a large number of holes and handles, while requiring minimal human interaction, is an unsolved research problem in computer graphics. In this thesis, we have developed two methods for interactively modeling such shapes. Both methods developed create perforated shapes by building a framework of tube like elements, such that each edge of a given mesh is replaced by a pipe. The first method called Wire modeling replaces each edge with a pipe that has a square cross-section. The result looks like a shape that is created by a framework of matchsticks. The second method, called Column modeling allows more rounded cross-sections for the pipes. The cross-sections can be any uniform polygon, and the users are able to control the number of the segments in the cross-section. These methods are implemented as an extension to an existing modeling system guaranteeing that the pipes are connected and the resulting shape can be physically constructed. Our methods require an initial input mesh that can either be imported from a commercially available software package, or created directly in this modeling system. The system also allows the users to export the models in obj file format, so that the models can be animated and rendered in other software packages.