Dynamic Optimal Transport with Mixed Boundary Condition for Color Image Processing

Abstract: Recently, Papadakis et al. proposed an efficient primal-dual algorithm for solving the dynamic optimal transport problem with quadratic ground cost and measures having densities with respect to the Lebesgue measure. It is based on the fluid mechanics formulation by Benamou and Brenier and proximal splitting schemes. In this paper we extend the framework to color image processing. We show how the transportation problem for RGB color images can be tackled by prescribing periodic boundary conditions in the color dimension. This requires the solution of a 4D Poisson equation with mixed Neumann and periodic boundary conditions in each iteration step of the algorithm. This 4D Poisson equation can be efficiently handled by fast Fourier and Cosine transforms. Furthermore, we sketch how the same idea can be used in a modified way to transport periodic 1D data such as the histogram of cyclic hue components of images. We discuss the existence and uniqueness of a minimizer of the associated energy functional. Numerical examples illustrate the meaningfulness of our approach.