Shape Correspondence and 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 chapter is to give an overview of 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 nonrigid shape matching, image cosegmentation and even some aspects of tangent vector field design. In this chapter, we try to provide all the necessary foundation to appreciate and use these techniques, while assuming very little background knowledge. We also aim to provide practical implementation details for the methods within this domain and, at the same time, hint at the generality of the “functional” point of view, which can help tackle many problems in the analysis and creation of visual content.