Collection Statistical Modeling for Shapes and Imaging
Organizer(s)
Date(s)
04/05/2024
Interaction between invariant structures for shape analysis
By Ron Kimmel
A classical approach for surface classification is to find a compact algebraic representation for each surface that would be similar for objects within the same class and preserve dissimilarities between classes. Self functional maps was suggested by Halimi and the lecturer as a surface representation that satisfies these properties, translating the geometric problem of surface classification into an algebraic form of classifying matrices. The proposed map transforms a given surface into a universal isometry invariant form defined by a unique matrix. The suggested representation is realized by applying the functional maps framework to map the surface into itself. The idea is to use two different metric spaces of the same surface for which the functional map serves as a signature. As an example we suggested the regular and the scale invariant surface laplacian operators to construct two families of eigenfunctions. The result is a matrix that encodes the interaction between the eigenfunctions resulted from two different Riemannian manifolds of the same surface. Using this representation, geometric shape similarity is converted into algebraic distances between matrices. If time permits, I will also comment on some of our efforts to migrate geometry into the arena of deep learning, in a sense learning to understand.
