Geometric Polynomial Constraints in Higher-Order Graph Matching

TL;DR

This paper introduces a novel higher-order graph matching method that applies geometric polynomial constraints directly as affinity weights, enabling correspondence matching without unary features or explicit polynomial system solving.

Contribution

It proposes a new approach to geometric constraints in graph matching using polynomial equations as affinity weights, bypassing traditional polynomial system solutions.

Findings

Supports correspondence matching without unary features.

Handles multiple geometric transformations like articulated motions.

Avoids solving polynomial systems by deriving affinity weights directly.

Abstract

Correspondence is a ubiquitous problem in computer vision and graph matching has been a natural way to formalize correspondence as an optimization problem. Recently, graph matching solvers have included higher-order terms representing affinities beyond the unary and pairwise level. Such higher-order terms have a particular appeal for geometric constraints that include three or more correspondences like the PnP 2D-3D pose problems. In this paper, we address the problem of finding correspondences in the absence of unary or pairwise constraints as it emerges in problems where unary appearance similarity like SIFT matches is not available. Current higher order matching approaches have targeted problems where higher order affinity can simply be formulated as a difference of invariances such as lengths, angles, or cross-ratios. In this paper, we present a method of how to apply geometric constraints modeled as polynomial equation systems. As opposed to RANSAC where such systems have to be solved and then tested for inlier hypotheses, our constraints are derived as a single affinity weight based on $n>2$ hypothesized correspondences without solving the polynomial system. Since the result is directly a correspondence without a transformation model, our approach supports correspondence matching in the presence of multiple geometric transforms like articulated motions.