An Efficient Algorithm for Matrix-Valued and Vector-Valued Optimal Mass Transport
This work provides a practical computational tool for extending optimal transport to matrix and vector-valued densities, which is important for imaging applications but represents an incremental algorithmic improvement.
The authors developed an efficient algorithm for matrix-valued and vector-valued optimal mass transport, enabling applications in diffusion tensor imaging, color image processing, and multi-modality imaging. The algorithm uses sequential quadratic programming with inexact Hessian approximations and incomplete Cholesky preconditioning, achieving fast convergence at low per-iteration cost.
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.