Journal of Mathematical Imaging and Vision

, Volume 56, Issue 3, pp 409–429 | Cite as

Transport Between RGB Images Motivated by Dynamic Optimal Transport

  • Jan Henrik Fitschen
  • Friederike Laus
  • Gabriele Steidl
Article

Abstract

We propose two models for the interpolation between RGB images based on the dynamic optimal transport model of Benamou and Brenier (Numer Math 84:375–393, 2000). While the application of dynamic optimal transport and its extensions to unbalanced transform were examined for gray-value images in various papers, this is the first attempt to generalize the idea to color images. The non-trivial 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. (SIAM J Imaging Sci 7:212–238, 2014) 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.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

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