An Efficient Algorithm for Matrix-Valued and Vector-Valued Optimal Mass Transport

TL;DR

This paper introduces a fast, efficient algorithm based on sequential quadratic programming for solving generalized optimal mass transport problems involving matrix-valued and vector-valued densities, with applications in imaging.

Contribution

The paper develops a novel SQP-based algorithm that efficiently handles matrix and vector optimal mass transport problems using approximate Hessians and Poisson equation solvers.

Findings

The algorithm converges rapidly while maintaining low per-iteration cost.

It effectively solves problems in diffusion tensor imaging and color image processing.

The method employs incomplete Cholesky preconditioning for efficiency.

Abstract

We present an efficient algorithm for recent generalizations of optimal mass transport theory to matrix-valued and vector-valued densities. These generalizations lead to several applications including diffusion tensor imaging, color images processing, and multi-modality imaging. The algorithm is based on sequential quadratic programming (SQP). By approximating the Hessian of the cost and solving each iteration in an inexact manner, we are able to solve each iteration with relatively low cost while still maintaining a fast convergent rate. The core of the algorithm is solving a weighted Poisson equation, where different efficient preconditioners may be employed. We utilize incomplete Cholesky factorization, which yields an efficient and straightforward solver for our problem. Several illustrative examples are presented for both the matrix and vector-valued cases.