Section: Application Domains

Image Processing and inverse problems

The Wasserstein distance between densities is the value function of the Optimal Mass Transportation problem. This distance may be considered to have "orthogonal" properties to the widely used least square distance. It is for instance quadratic with respect to dilations and translation. On the other hand it is not very sensitive to rigid transformations, [64] is an attempts at generalizing the CFD formulation in this context. The Wasserstein distance is an interesting tool for applications where distances between signals and in particular oscillatory signals need to to computed, this is assuming one understand how to transform the information into positive densities.

• Tannenbaum and co-authors have designed several variants of the CFD numerical method and applied it to warping, morphing and registration (using the Optimal Mass Transportation map) problems in medical imaging. [76][17]

• Gabriel Peyre and co-authors [73] have proposed an easier to compute relaxation of the Wasserstein distance (the sliced Wasserstein distance) and applied it to two image processing problems: color transfer and texture mixing.

• Froese Engquist [40] use a Monge-Ampère Solver to compute the Wasserstein distance between synthetic 2D Seismic signals (After some transformations). Applications to waveform inversion and registration are discussed and simple numerical examples are presented.