Transport between RGB Images Motivated by Dynamic Optimal Transport
For computer graphics and image processing, this work extends optimal transport interpolation from grayscale to color images, though the novelty is incremental as it adapts existing methods.
This paper proposes the first dynamic optimal transport models for interpolating between RGB images by treating them as 3D arrays with periodic transport in the color direction. The models produce meaningful interpolations, with efficient computation via fast transforms.
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.