3D shape analysis is an extremely active research topic in both computer graphics and computer vision. In computer vision, 3D shape acquisition and modeling are generally the result of complex data processing and data analysis methods. There are many practical situations where a visual shape is modeled by a point cloud observed with a variety of 2D and 3D sensors. Unlike the graphical data, the sensory data are not, in the general case, uniformly distributed across the surfaces of the observed objects and they are often corrupted by sensor noise, outliers, surface properties (scattering, specularities, color, etc.), self occlusions, varying lighting conditions. Moreover, the same object that is observed by different sensors, from slightly different viewpoints, or at different time instances may yield completely different point distributions, noise levels and, most notably, topological differences, e.g., merging of hands. In this thesis we outline single and multi-scale representation of articulated 3D shapes and devise new shape analysis methods, keeping in mind the challenges posed by visual shape data. In particular, we discuss in detail the heat diffusion framework for multi-scale shape representation and propose solutions for shape segmentation and dense shape registration using the spectral graph methods and various other machine learning algorithms, namely, the Gaussian Mixture Model (GMM) and the Expectation Maximization (EM). We first introduce the mathematical background on differential geometry and graph isomorphism followed by the introduction of pose-invariant spectral embedding representation of 3D articulated shapes. Next we present a novel unsupervised method for visual shape segmentation by analyzing the Laplacian eigenvectors. We then outline a semi-supervised solution for shape segmentation based upon a new learn, align and transfer paradigm. Next we extend the shape representation to a multi-scale setup by outlining the heat-kernel framework. Finally, we present a topologically-robust dense shape matching method using the multi-scale heat kernel representation and conclude with a detailed discussion and future direction of work.
