Metric-Induced Optimal Embedding for Intrinsic 3D Shape Analysis

Abstract

For various 3D shape analysis tasks, the Laplace-Beltrami(LB) embedding has become increasingly popular as it enables the efficient comparison of shapes based on intrinsic geometry. One fundamental difficulty in using the LB embedding, however, is the ambiguity in the eigen-system, and it is conventionally only handled in a heuristic way. In this work, we propose a novel and intrinsic metric, the spectral l$^{2}$-distance, to overcome this difficulty. We prove mathematically that this new distance satisfies the conditions of a rigorous metric. Using the resulting optimal embedding determined by the spectral l$^{2}$-distance, we can perform both local and global shape analysis intrinsically in the embedding space. We demonstrate this by developing a template matching approach in the optimal embedding space to solve the challenging problem of identifying major sulci on vervet cortical surfaces. In our experiments, we validate the robustness of our method by the successful identification of major sulcal lines on a large data set of 698 cortical surfaces and illustrate its potential in brain mapping studies.