Ricerc@Sapienza

In this paper, we consider the problem of information transfer across shapes and propose an extension to the widely used functional map representation. Our main observation is that in addition to the vector space structure of the functional spaces, which has been heavily exploited in the functional map framework, the functional algebra (i.e., the ability to take pointwise products of functions) can significantly extend the power of this framework.

The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback of such bases is their inherently global nature, as the Laplacian eigenfunctions carry geometric and topological structure of the entire manifold. In this paper, we introduce a new framework for local spectral shape analysis.

Step functions are non-smooth and piecewise constant functions with a finite number of pieces. Each of these pieces indicates a local region contained in the entire domain. Several geometry processing applications involve step functions defined on non-Euclidean domains, such as shape segmentation, partial matching and self-similarity detection. Standard signal processing cannot handle this class of functions. The classical Fourier series, for instance, does not give a good representation of these non-smooth functions.

In this paper, we propose a new construction for the Mexican hat wavelets on shapes with applications to partial shape matching. Our approach takes its main inspiration from the well-established methodology of diffusion wavelets. This novel construction allows us to rapidly compute a multi-scale family of Mexican hat wavelet functions, by approximating the derivative of the heat kernel. We demonstrate that this leads to a family of functions that inherit many attractive properties of the heat kernel (e.g.

Shape correspondence is a fundamental problem in computer graphics and vision, with applications in various problems including animation, texture mapping, robotic vision, medical imaging, archaeology and many more. In settings where the shapes are allowed to undergo nonrigid deformations and only partial views are available, the problem becomes very challenging. In this chapter we describe recent techniques designed to tackle such problems. Specifically, we explain how the renown functional maps framework can be extended to tackle the partial setting.