Transport between RGB Images Motivated by Dynamic Optimal Transport

TL;DR

This paper introduces two novel models for interpolating RGB images using dynamic optimal transport, extending previous gray-scale methods to color images by considering RGB as periodic three-dimensional arrays and employing efficient numerical algorithms.

Contribution

It generalizes dynamic optimal transport to color images with new discrete variational models and demonstrates their existence, non-uniqueness, and efficient numerical solutions.

Findings

Models successfully interpolate RGB images.

Efficient primal-dual algorithm implemented.

Numerical examples validate the models.

Abstract

We propose two models for the interpolation between RGB images based on the dynamic optimal transport model of Benamou and Brenier [8]. While the application of dynamic optimal transport and its extensions to unbalanced transform were examined for gray-values images in various papers, this is the first attempt to generalize the idea to color images. The nontrivial task to incorporate color into the model is tackled by considering RGB images as three-dimensional arrays, where the transport in the RGB direction is performed in a periodic way. Following the approach of Papadakis et al. [35] for gray-value images we propose two discrete variational models, a constrained and a penalized one which can also handle unbalanced transport. We show that a minimizer of our discrete model exists, but it is not unique for some special initial/final images. For minimizing the resulting functionals we apply a primal-dual algorithm. One step of this algorithm requires the solution of a four-dimensional discretized Poisson equation with various boundary conditions in each dimension. For instance, for the penalized approach we have simultaneously zero, mirror and periodic boundary conditions. The solution can be computed efficiently using fast Sin-I, Cos-II and Fourier transforms. Numerical examples demonstrate the meaningfulness of our model.