Computing and processing correspondences with functional maps

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.