Notions of similarity and correspondence between geometric shapes and images are central to many tasks in geometry processing, computer vision, and computer graphics. The goal of this course is to familiarize the audience with a set of recent techniques that greatly facilitate the computation of mappings or correspondences between geometric datasets, such as 3D shapes or 2D images by formulating them as mappings between functions rather than points or triangles. Methods based on the functional map framework have recently led to state-of-the-art results in problems as diverse as non-rigid shape matching, image co-segmentation and even some aspects of tangent vector field design. One challenge in adopting these methods in practice, however, is that their exposition often assumes a significant amount of background in geometry processing, spectral methods and functional analysis, which can make it difficult to gain an intuition about their performance or about their applicability to real-life problems. In this course, we try to provide all the tools necessary to appreciate and use these techniques, while assuming very little background knowledge. We also give a unifying treatment of these techniques, which may be difficult to extract from the individual publications and, at the same time, hint at the generality of this point of view, which can help tackle many problems in the analysis and creation of visual content. This course is structured as a half day course. We will assume that the participants have knowledge of basic linear algebra and some knowledge of differential geometry, to the extent of being familiar with the concepts of a manifold and a tangent vector space. We will discuss in detail the functional approach to finding correspondences between non-rigid shapes, the design and analysis of tangent vector fields on surfaces, consistent map estimation in networks of shapes and applications to shape and image segmentation, shape variability analysis, and other areas. i 1. INTRODUCTION
We propose a generalized proximal point algorithm (PPA), in the generic setting of finding a root of a maximal monotone operator. In addition to the classical PPA, a number of benchmark operator splitting methods in the PDE and optimization literatures can be retrieved by this generalized PPA scheme. We establish the convergence rate of this generalized PPA scheme under different conditions, including estimating its worst-case convergence rate measured by the iteration complexity under mild assumptions and deriving its linear convergence rate under certain stronger conditions. Throughout our discussion, we pay particular attention to the special case where the operator is the sum of two maximal monotone operators, and specify our theoretical results in generic setting to this special case. Our result turns out to be a general and unified study on the convergence rate of a number of existing methods, and subsumes some existing results in the literature.
Abstract. We present a novel method for computing correspondences between pairs of non-rigid shapes. Unlike the majority of existing techniques that assume a deformation model, such as intrinsic isometries, a priori and use a pre-defined set of point or part descriptors, we consider the problem of learning a correspondence model given a collection of reference pairs with known mappings between them. Our formulation is purely intrinsic and does not rely on a consistent parametrization or spatial positions of vertices on the shapes. Instead, we consider the problem of finding the optimal set of descriptors that can be jointly used to reproduce the given reference maps. We show how this problem can be formalized and solved for efficiently by using the recently proposed functional maps framework. Moreover, we demonstrate how to extract the functional subspaces that can be mapped reliably across shapes. This gives us a way to not only obtain better functional correspondences, but also to associate a confidence value to the different parts of the mappings. We demonstrate the efficiency and usefulness of the proposed approach on a variety of challenging shape matching tasks.
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