Dynamic optimal transport with mixed boundary condition for color image processing

Recently, Papadakis et al. [11] 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 [1] 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.

[1]  Gabriele Steidl,et al.  First order algorithms in variational image processing , 2014, ArXiv.

[2]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[3]  Anil Kokaram,et al.  The linear Monge-Kantorovitch linear colour mapping for example-based colour transfer , 2007 .

[4]  Xavier Bresson,et al.  Local Histogram Based Segmentation Using the Wasserstein Distance , 2009, International Journal of Computer Vision.

[5]  Christoph Schnörr,et al.  Convex Variational Image Restoration with Histogram Priors , 2013, SIAM J. Imaging Sci..

[6]  Gui-Song Xia,et al.  Synthesizing and Mixing Stationary Gaussian Texture Models , 2014, SIAM J. Imaging Sci..

[7]  Mila Nikolova,et al.  Fast Ordering Algorithm for Exact Histogram Specification , 2014, IEEE Transactions on Image Processing.

[8]  Julien Rabin,et al.  Wasserstein Barycenter and Its Application to Texture Mixing , 2011, SSVM.

[9]  Pierre Maréchal,et al.  The role of perspective functions in convexity, polyconvexity, rank-one convexity and separate convexity , 2008 .

[10]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[11]  Leonidas J. Guibas,et al.  The Earth Mover's Distance as a Metric for Image Retrieval , 2000, International Journal of Computer Vision.

[12]  Mohamed-Jalal Fadili,et al.  Wasserstein active contours , 2012, 2012 19th IEEE International Conference on Image Processing.

[13]  Gabriel Peyré,et al.  Optimal Transport with Proximal Splitting , 2013, SIAM J. Imaging Sci..

[14]  Katja Bachmeier,et al.  Numerical Heat Transfer And Fluid Flow , 2016 .

[15]  Julien Rabin,et al.  Wasserstein regularization of imaging problem , 2011, 2011 18th IEEE International Conference on Image Processing.

[16]  M. Rumpf,et al.  A generalized model for optimal transport of images including dissipation and density modulation , 2015, 1504.01988.

[17]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[18]  C. Villani.  Topics in Optimal Transportation , 2003 .

[19]  Julien Rabin,et al.  Transportation Distances on the Circle , 2009, Journal of Mathematical Imaging and Vision.

[20]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.