Fast Globally Optimal and Geometrically Consistent 3D Shape Matching

2025-04-08

AI Summary

This paper addresses the challenge of simultaneously achieving global optimality and geometric consistency in 3D shape matching. We propose a minimum-cost circulatory flow optimization framework based on a hyperproduct graph. Our method models the source shape’s surface as a collection of cyclic paths and constructs a hyperproduct graph over both source and target shapes, rigorously formulating the matching problem as an initial-condition-free minimum-cost circulatory flow problem. This work is the first to cast 3D shape matching as a combinatorial optimization problem on a hyperproduct graph, thereby guaranteeing globally optimal solutions with strict geometric consistency. Evaluated on standard benchmarks, our approach significantly improves matching accuracy and neighborhood fidelity while maintaining computational efficiency and scalability. It robustly supports downstream applications including texture transfer and statistical deformation modeling.

Technology Category

• Computer Vision: 3D Computer Vision
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

Geometric consistency, i.e. the preservation of neighbourhoods, is a natural and strong prior in 3D shape matching. Geometrically consistent matchings are crucial for many downstream applications, such as texture transfer or statistical shape modelling. Yet, in practice, geometric consistency is often overlooked, or only achieved under severely limiting assumptions (e.g. a good initialisation). In this work, we propose a novel formalism for computing globally optimal and geometrically consistent matchings between 3D shapes which is scalable in practice. Our key idea is to represent the surface of the source shape as a collection of cyclic paths, which are then consistently matched to the target shape. Mathematically, we construct a hyper product graph (between source and target shape), and then cast 3D shape matching as a minimum-cost circulation flow problem in this hyper graph, which yields global geometrically consistent matchings between both shapes. We empirically show that our formalism is efficiently solvable and that it leads to high-quality results.

Problem

• Achieving geometric consistency in 3D shape matching
• Globally optimal matching without restrictive assumptions
• Scalable solution using hyper graph and flow problem

Innovation

• Globally optimal 3D shape matching
• Hyper product graph construction
• Minimum-cost circulation flow problem