Sparse representation of step functions on manifolds

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 define a new frame for the sparse approximation and transfer of the step functions defined on manifolds. The definition of our frame is completely spectral and provides a concise representation through an efficient computation. We exploit the sparse representation that takes full advantage of the proposed frame. Furthermore, our frame is built specifically to enhance its use in combination with the functional maps, a powerful tool for transferring signals between manifolds. This functional approach makes the proposed framework stable to isometric and non-isometric deformations. A large set of experiments confirms that the proposed frame improves the sparse approximation and transfer of step functions.