APPLIED MATH SEMINAR

Speaker: Matt Feiszli, Mathematics, Yale University

Title:
Scale-invariant metrics on spaces of curves and surfaces, with applications to matching and denoising.

When/where: Tuesday, September 21st, 4:15 PM, AKW 200

Abstract:

The study of metrics and distances on spaces of curves and surfaces is a very active area of research with important applications in vision and imaging.  We present several multiscale families of distances based on geometric variants of ``square functions'' from the harmonic analysis literature; these methods enable the use of essentially linear methods on highly nonlinear spaces of curves and surfaces.  As applications,  we demonstrate scale-invariant approaches to curve matching and denoising.  Our fast, greedy matching algorithm exhibits state-of-the-art performance on the MPEG7 CE-Shape-1b test dataset, generally regarded as the most challenging shape matching task commonly used in the literature.  Our denoising method is essentially non-parametric and adapts to local noise.  In particular, our noise-adaptivity eliminates the arbitrarily-chosen ``diffusion time'' parameter common in many other methods.  In cases where the distinction between signal and noise is unclear, we show how statistics gathered from the curve can be used to identify a finite number of ``good'' choices for the denoising.  Finally, our theoretical results provide a unifying point of view on some of the most successful curve-matching methods in the literature; in particular we can understand and relate the diffeomorphic warping models of Grenander, Trouve, Younes, Mumford et al, the shape contexts of Belongie and Malik, and the shape-tree of Felzenszwalb.  In each case we find the method uses a square function of angle and distance to measure deformations.