Non-rigid shapes are ubiquitous in the world surrounding us, at all levels from micro to macro. The need to study such shapes and model their behavior arises in a wide spectrum of applications. In recent years, non-rigid shapes have attracted a growing interest, which has led to rapid development of the field, where state-of-the-art results from very different sciences - theoretical and numerical geometry, optimization, linear algebra, graph theory, machine learning and computer graphics, to mention a few - are applied to find solutions. In this talk, we will approach the non-rigid world from the perspective of metric geometry. The discussion will focus on two archetype problems in non-rigid shape analysis: similarity and correspondence. We will start with the construction of similarity criteria for deformation-invariant shape comparison, based on intrinsic geometric properties of the shapes, and show that such criteria are related to the Gromov-Hausdorff distance. Next, we will extend the problem of similarity to partially similar shapes, and present a construction of set-valued distances, based on the notion of Pareto optimality. Next, we will consider a particular setting of self-similarity of shapes, which will allow us to determine intrinsic symmetries of the shapes. Finally, we will show that the correspondence between non-rigid shapes can be obtained as a byproduct of the non-rigid similarity problem. Part of the talk will be dedicated to the numerical core of our method, the generalized multidimensional scaling. As examples of applications, we will use problems from the fields of computer vision and computer graphics.
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