Hyperbolic Wasserstein Distance for Shape Indexing

Abstract

Shape space is an active research topic in computer vision and medical imaging fields. The distance defined in a shape space may provide a simple and refined index to represent a unique shape. This work studies the Wasserstein space and proposes a novel framework to compute the Wasserstein distance between general topological surfaces by integrating hyperbolic Ricci flow, hyperbolic harmonic map, and hyperbolic power Voronoi diagram algorithms. The resulting hyperbolic Wasserstein distance can intrinsically measure the similarity between general topological surfaces. Our proposed algorithms are theoretically rigorous and practically efficient. It has the potential to be a powerful tool for 3D shape indexing research. We tested our algorithm with human face classification and Alzheimer's disease (AD) progression tracking studies. Experimental results demonstrated that our work may provide a succinct and effective shape index.

Fig. 1.

(a) A multiply connected human face surface; (b) the fundamental group of surface (a); (c) the fundamental domain of surface (a); (d) another fundamental domain of surface (a), which is deformed from (c) with a Fuchsian transformation of surface (a).

Fig. 2.

Illustrations of hyperbolic geometry: (a) Poincaré disk model; (b) geodesics in Poincaré disk; (c) Klein model.

Fig. 3.

The overall computational pipeline of the proposed hyperbolic Wasserstein distance.

Fig. 4.

(a) A hyperbolic triangle; (b) the circle packing metric on a hyperbolic triangle.

Fig. 5.

Front (a) and back (b) views of a left cortical surface with six landmark curves and the fundamental group of paths; (c) embedding of the cortical surface onto the Poincaré disk.

Fig. 6.

Geodesic curve lifting: (a) Fuchsian transformations that map $T_i^{+}$ and $T_i^{-}$ to the u axis; (b) Fuchsian transformation that maps $T_i^{+}$ to $T_i^{-}$; (c) a finite portion of the universal covering space; (d) recomputation of the fundamental group of paths as geodesics in the Poincaré disk.

Fig. 7.

When the recomputed fundamental group of paths are lifted to 3D surfaces, they are consistent across different surfaces (a) and (b).

Fig. 8.

Canonical fundamental domain decomposition for the initial map construction between surfaces.

Fig. 9.

Illustration of the power Voronoi diagram on the Poincaré disk, where each point (black dot) is associated with a Voronoi cell (green boundary) and the radius of each circle (blue) centered at each point.

Fig. 10.

Experimental results of human facial expression analysis with hyperbolic Wasserstein distance.

Fig. 11.

Multiply connected surfaces of the two persons (a) and (b) from Texas 3DFRD.

Fig. 12.

Landmark curves on a left cortical surface, which are automatically labeled by Caret, showing in two different views (a) and (b), on both original and inflated surfaces. The inflated surface is obtained with FreeSurfer [68].

Fig. 13.

Hyperbolic Wasserstein distance computation results of a pair of brain left cortical surfaces of an AD patient (a, c and e) and a healthy control subject (b, d, and f).

Fig. 14.

Robustness of the hyperbolic Wasserstein distance to imaging noise: (a) is the original MR image and (b) is the image with manually added Racian noise.