Vector and Matrix Optimal Mass Transport: Theory, Algorithm, and Applications
For researchers in optimal transport and image processing, this work provides a principled extension to multi-channel data, though the novelty is incremental as it builds on established OMT theory.
This paper extends optimal mass transport to vector- and matrix-valued densities, providing a rigorous mathematical formulation, existence and duality results, and a scalable, parallelizable algorithm. GPU experiments demonstrate the method's effectiveness in applications like color image processing.
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.