PhD Disssertation Defense

Thursday, February 11, 2016 - 10:00 to 11:30
Room 220, New Computer Science

PhD Candidate: Ming Zhong
Novel Spectral Representations and Sparsity-Driven Algorithms for Shape Modeling and Analysis

In this dissertation, we focus on extending classical spectral shape analysis by incorporating spectral graph wavelets and sparsity-seeking
algorithms. Defined with the graph Laplacian eigenbasis, the spectral graph
wavelets are localized both in the vertex domain and graph spectral
domain, and thus are very effective in describing local geometry.
With a rich dictionary of elementary vectors and forcing certain sparsity
constraints, a real life signal can often be well approximated by a very sparse coefficient representation. The many successful applications of sparse signal representation in computer vision and image processing inspire us to explore the idea of employing sparse modeling techniques with dictionary of spectral basis to solve various shape modeling problems.

Conventional spectral mesh compression uses the eigenfunctions of mesh
Laplacian as shape bases, which are highly inefficient in representing local
geometry. To ameliorate, we advocate an innovative approach to 3D mesh
compression using spectral graph wavelets as dictionary to encode mesh geometry. The spectral graph wavelets are locally defined at individual vertices and can better capture local shape information than Laplacian eigenbasis. The multi-scale SGWs form a redundant dictionary as shape basis, so we formulate the compression of 3D shape as a sparse approximation problem that can be readily handled by greedy pursuit algorithms.

Surface inpainting refers to the completion or recovery of missing shape
geometry based on the shape information that is currently available. We
devise a new surface inpainting algorithm founded upon the theory and
techniques of sparse signal recovery. Instead of estimating the missing
geometry directly, our novel method is to find this low-dimensional
representation which describes the entire original shape. More specifically,
we find that, for many shapes, the vertex coordinate function can be well
approximated by a very sparse coefficient representation with respect to the dictionary comprising its Laplacian eigenbasis, and it is then possible to
recover this sparse representation from partial measurements of the original shape. Taking advantage of the sparsity cue, we advocate a novel variational approach for surface inpainting, integrating data fidelity constraints on the shape domain with coefficient sparsity constraints on the transformed domain. Because of the powerful properties of Laplacian eigenbasis, the inpainting results of our method tend to be globally coherent with the remaining shape.

Informative and discriminative feature descriptors are vital in qualitative and
quantitative shape analysis for a large variety of graphics applications. We
advocate novel strategies to define generalized, user-specified features on
shapes. Our new region descriptors are primarily built upon the coefficients of spectral graph wavelets that are both multi-scale and multi-level in nature, consisting of both local and global information. Based on our novel spectral feature descriptor, we developed a user-specified feature
detection framework and a tensor-based shape matching algorithm.

Through various experiments, we demonstrate the competitive performance of our proposed methods and the great potential of spectral basis and sparsity-driven methods for shape modeling.

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PhD Disssertation Defense