Vector and Matrix Optimal Mass Transport: Theory, Algorithm, and Applications

TL;DR

This paper extends optimal mass transport theory to vector and matrix-valued data, providing rigorous formulations, scalable algorithms, and GPU implementations for applications like color image processing.

Contribution

The paper introduces a rigorous mathematical framework, scalable algorithms, and GPU-based implementations for vector and matrix optimal mass transport.

Findings

Proved existence of solutions and strong duality for vector/matrix-OMT.

Developed scalable, parallel algorithms for these problems.

Demonstrated applications in image processing with GPU acceleration.

Abstract

In many applications such as color image processing, data has more than one piece of information associated with each spatial coordinate, and in such cases the classical optimal mass transport (OMT) must be generalized to handle vector-valued or matrix-valued densities. In this paper, we discuss the vector and matrix optimal mass transport and present three contributions. We first present a rigorous mathematical formulation for these setups and provide analytical results including existence of solutions and strong duality. Next, we present a simple, scalable, and parallelizable methods to solve the vector and matrix-OMT problems. Finally, we implement the proposed methods on a CUDA GPU and present experiments and applications.