Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaf 36.

Zhao, Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 111-117) and index. / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation --- p.1 / Chapter 1.2 --- Objective of the study --- p.2 / Chapter 1.3 --- Challenges and our approach --- p.4 / Chapter 1.4 --- Original contributions --- p.6 / Chapter 1.5 --- Organization of this dissertation --- p.6 / Chapter 2 --- Related Work --- p.9 / Chapter 2.1 --- Critical configuration for motion estimation and projective reconstruction --- p.9 / Chapter 2.1.1 --- Point feature --- p.9 / Chapter 2.1.2 --- Line feature --- p.12 / Chapter 2.2 --- Camera motion estimation --- p.14 / Chapter 2.2.1 --- Line tracking --- p.15 / Chapter 2.2.2 --- Determining camera motion --- p.19 / Chapter 3 --- Preliminaries on Three-View Geometry and Trifocal Tensor --- p.23 / Chapter 3.1 --- Projective spaces P3 and transformations --- p.23 / Chapter 3.2 --- The trifocal tensor --- p.24 / Chapter 3.3 --- Computation of the trifocal tensor-Normalized linear algorithm --- p.31 / Chapter 4 --- Linear Line Structures --- p.33 / Chapter 4.1 --- Models of line space --- p.33 / Chapter 4.2 --- Line structures --- p.35 / Chapter 4.2.1 --- Linear line space --- p.37 / Chapter 4.2.2 --- Ruled surface --- p.37 / Chapter 4.2.3 --- Line congruence --- p.38 / Chapter 4.2.4 --- Line complex --- p.38 / Chapter 5 --- Critical Configurations of Three Views Revealed by Line Correspondences --- p.41 / Chapter 5.1 --- Two-view degeneracy --- p.41 / Chapter 5.2 --- Three-view degeneracy --- p.42 / Chapter 5.2.1 --- Introduction --- p.42 / Chapter 5.2.2 --- Linear line space --- p.44 / Chapter 5.2.3 --- Linear ruled surface --- p.54 / Chapter 5.2.4 --- Linear line congruence --- p.55 / Chapter 5.2.5 --- Linear line complex --- p.57 / Chapter 5.3 --- Retrieving tensor in critical configurations --- p.60 / Chapter 5.4 --- Rank classification of non-linear line structures --- p.61 / Chapter 6 --- Camera Motion Estimation Framework --- p.63 / Chapter 6.1 --- Line extraction --- p.64 / Chapter 6.2 --- Line tracking --- p.65 / Chapter 6.2.1 --- Preliminary geometric tracking --- p.65 / Chapter 6.2.2 --- Experimental results --- p.69 / Chapter 6.3 --- Camera motion estimation framework using EKF --- p.71 / Chapter 7 --- Experimental Results --- p.75 / Chapter 7.1 --- Simulated data experiments --- p.75 / Chapter 7.2 --- Real data experiments --- p.76 / Chapter 7.2.1 --- Linear line space --- p.80 / Chapter 7.2.2 --- Linear ruled surface --- p.84 / Chapter 7.2.3 --- Linear line congruence --- p.84 / Chapter 7.2.4 --- Linear line complex --- p.91 / Chapter 7.3 --- Empirical observation: ruled plane for line transfer --- p.93 / Chapter 7.4 --- Simulation for non-linear line structures --- p.94 / Chapter 8 --- Conclusions and Future Work --- p.97 / Chapter 8.1 --- Summary --- p.97 / Chapter 8.2 --- Future work --- p.99 / Chapter A --- Notations --- p.101 / Chapter B --- Tensor --- p.103 / Chapter C --- Matrix Decomposition and Estimation Techniques --- p.104 / Chapter D --- MATLAB Files --- p.107 / Chapter D.1 --- Estimation matrix --- p.107 / Chapter D.2 --- Line transfer --- p.109 / Chapter D.3 --- Simulation --- p.109

Tensor scale (t-scale) is a parametric representation of local structure morphology that simultaneously describes its orientation, shape and isotropic scale. At any image location, t-scale is the parametric representation of the largest ellipse (an ellipsoid in 3D) centered at that location and contained in the same homogeneous region. Recently, we have improved the t-scale computation algorithm by: (1) optimizing digital representations for LoG and DoG kernels for edge detection and (2) ellipse fitting by using minimization of both algebraic and geometric distance errors. Also, t-scale has been applied to computing the deformation vector field with applications to medical image registration. Currently, the method is implemented in two-dimension (2D) and the deformation vector field is directly computed from t-scale-derived normal vectors at matching locations in two images to be registered. Also, the method has been used to develop a simple algorithm for computing 2D warping from one shape onto another. Meanwhile, t-scale has been applied to generating interpolation lines with applications to medical image interpolation using normal vector. Normal vector yields local structure orientation pointing to the closest edge. However, this information is less reliable along the medial axis of a shape as it may be associated with either of the two opposite edges of the local shape. This problem is overcome using a shape-linearity measure estimating relative changes in scale along the orthogonal direction. Preliminary results demonstrate the method's potential in estimating deformation between two images and interpolating between neighboring slices in a grey scale image.

<p>The question about how to determine the rank of a tensor has been widely studied in the literature. However the analytical methods to compute the decomposition of tensors have not been so much developed even for low-rank tensors.</p><p>In this report we present analytical methods for finding real and complex PARAFAC decompositions of 2 x 2 x 2 tensors before computing the actual rank of the tensor.</p><p>These methods are also implemented in MATLAB.</p><p>We also consider the question of how best lower-rank approximation gives rise to problems of degeneracy, and give some analytical explanations for these issues.</p>

We present a new method for rendering novel images of flexible 3D objects from a small number of example images in correspondence. The strength of the method is the ability to synthesize images whose viewing position is significantly far away from the viewing cone of the example images ("view extrapolation"), yet without ever modeling the 3D structure of the scene. The method relies on synthesizing a chain of "trilinear tensors" that governs the warping function from the example images to the novel image, together with a multi-dimensional interpolation function that synthesizes the non-rigid motions of the viewed object from the virtual camera position. We show that two closely spaced example images alone are sufficient in practice to synthesize a significant viewing cone, thus demonstrating the ability of representing an object by a relatively small number of model images --- for the purpose of cheap and fast viewers that can run on standard hardware.

This paper presents an image-based rendering system using algebraic relations between different views of an object. The system uses pictures of an object taken from known positions. Given three such images it can generate "virtual'' ones as the object would look from any position near the ones that the two input images were taken from. The extrapolation from the example images can be up to about 60 degrees of rotation. The system is based on the trilinear constraints that bind any three view so fan object. As a side result, we propose two new methods for camera calibration. We developed and used one of them. We implemented the system and tested it on real images of objects and faces. We also show experimentally that even when only two images taken from unknown positions are given, the system can be used to render the object from other view points as long as we have a good estimate of the internal parameters of the camera used and we are able to find good correspondence between the example images. In addition, we present the relation between these algebraic constraints and a factorization method for shape and motion estimation. As a result we propose a method for motion estimation in the special case of orthographic projection.

The question about how to determine the rank of a tensor has been widely studied in the literature. However the analytical methods to compute the decomposition of tensors have not been so much developed even for low-rank tensors. In this report we present analytical methods for finding real and complex PARAFAC decompositions of 2 x 2 x 2 tensors before computing the actual rank of the tensor. These methods are also implemented in MATLAB. We also consider the question of how best lower-rank approximation gives rise to problems of degeneracy, and give some analytical explanations for these issues.

Meaning based search retrieves objects from search index repository based on user's search Meanings and meaning of objects rather than keyword matching. It requires techniques to capture user's search Meanings and meanings of objects, transform them to a representation that can be stored and compared efficiently on computers. Meaning of objects can be adequately captured in terms of a hierarchical composition structure called concept tree. This thesis describes the design and development of an algorithm that transforms the hierarchical concept tree to a tensor representation using tensor algebra theory. These tensor representations can capture the information need of a user in a better way and can be used for similarity comparisons in meaning based search. A preliminary evaluation showed that the proposed framework outperforms the TF-IDF vector model in 95% of the cases and vector based conceptual search model in 92% of the cases in adequately comparing meaning of objects. The tensor conversion tool also was used to verify the salient properties of the meaning comparison framework. The results show that the salient properties are consistent with the tensor similarity values of the meaning comparison framework.

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 131-141). Also available in print.

Classical and quantum Chern-Simons with gauge group U(1)N were classified by Belov and Moore in [BM05]. They studied both ordinary topological quantum field theories as well as spin theories. On the other hand a correspondence is well known between ordinary (2 + 1)-dimensional TQFTs and modular tensor categories. We study group categories and extend them slightly to produce modular tensor categories that correspond to toral Chern-Simons. Group categories have been widely studied in other contexts in the literature [FK93],[Qui99],[JS93],[ENO05],[DGNO07]. The main result is a proof that the associated projective representation of the mapping class group is isomorphic to the one from toral Chern-Simons. We also remark on an algebraic theorem of Nikulin that is used in this paper. / text