Shortest Paths in Graphs with Matrix-Valued Edges: Concepts, Algorithm and Application to 3D Multi-Shape Analysis

TL;DR

This paper introduces a novel concept of shortest paths in graphs with matrix-valued edges, enabling more expressive modeling for complex relations, and demonstrates its application in 3D multi-shape analysis.

Contribution

It proposes a new graph-theoretic framework for shortest paths with matrix-valued edges and an algorithm to compute them, extending traditional scalar edge models.

Findings

Effective algorithm for shortest paths with matrix-valued edges

Application demonstrated in 3D multi-shape analysis

Enhanced expressivity over scalar edge models

Abstract

Finding shortest paths in a graph is relevant for numerous problems in computer vision and graphics, including image segmentation, shape matching, or the computation of geodesic distances on discrete surfaces. Traditionally, the concept of a shortest path is considered for graphs with scalar edge weights, which makes it possible to compute the length of a path by adding up the individual edge weights. Yet, graphs with scalar edge weights are severely limited in their expressivity, since oftentimes edges are used to encode significantly more complex interrelations. In this work we compensate for this modelling limitation and introduce the novel graph-theoretic concept of a shortest path in a graph with matrix-valued edges. To this end, we define a meaningful way for quantifying the path length for matrix-valued edges, and we propose a simple yet effective algorithm to compute the respective shortest path. While our formalism is universal and thus applicable to a wide range of settings in vision, graphics and beyond, we focus on demonstrating its merits in the context of 3D multi-shape analysis.