Optimal mass transport for shape matching and comparison.

Area of Science:

• Computer Vision
• Medical Imaging
• Computational Geometry

Background:

• Surface-based 3D shape analysis is crucial for computer vision and medical imaging.
• Existing methods like conformal mapping for surface registration can cause large area distortions, leading to instability.
• Optimal mass transport (OMT) offers a potential solution for shape matching and comparison.

Purpose of the Study:

• To propose an efficient optimal mass transport map for 3D shape analysis.
• To improve surface registration by creating a unique, area-preserving map.
• To introduce a novel Riemannian framework for shape space studies.

Main Methods:

• Utilizing Monge-Brenier theory for optimal mass transport map computation, reducing complexity from O(n^2) to O(n).
• Composing conformal maps with OMT maps to achieve unique, area-preserving, and diffeomorphic maps for surface registration.
• Developing the Conformal Wasserstein Shape Space framework by combining conformal geometry and OMT.

Main Results:

• The proposed OMT method significantly improves computational efficiency compared to conventional approaches.
• The combined conformal and OMT map addresses area distortion issues in surface registration, enhancing stability.
• The Conformal Wasserstein Shape Space provides a novel Riemannian metric for shape space analysis.

Conclusions:

• The proposed optimal mass transport approach offers an efficient and effective solution for 3D shape analysis.
• This work advances surface registration and shape space studies in computer vision and medical imaging.
• Experimental validation demonstrates the superiority of the proposed methods over existing techniques.