Convolutional wasserstein distances

Abstract

Figure 1: Shape interpolation from a cow to a duck to a torus via convolutional Wasserstein barycenters on a 100×100×100 grid, using the method at the beginning of §7. AbstractThis paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains. Our main contribution is to show that optimal transportation can be made tractable over large domains used in graphics, such as images and triangle meshes, improving performance by orders of magnitude compared to previous work. To this end, we approximate optimal transportation distances using entropic regularization. The resulting objective contains a geodesic distance-based kernel that can be approximated with the heat kernel. This approach leads to simple iterative numerical schemes with linear convergence, in which each iteration only requires Gaussian convolution or the solution of a sparse, pre-factored linear system. We demonstrate the versatility and efficiency of our method on tasks including reflectance interpolation, color transfer, and geometry processing.

Sample‐Based Optimal Transport and Barycenter Problems

“…Optimal transport and barycenter problems have been used to solve shape transform problems [42]. The task is to find maps or barycenters for different shapes in two and three-dimensional spaces.…”

“…In image processing applications, several approaches have been proposed to regularize the discrete linear optimization problem of the earth mover's distance. The entropy regularization approach [42] adds an entropy term that leads to efficient algorithms to derive new solutions. It was proposed in [14] to add a graph regularization term to generate more regular solutions.…”

Discovery and visualization of structural biomarkers from MRI using transport-based morphometry

“…Computational cost also becomes a challenge in approaches utilizing a system of linear equations that arise from the finite-difference implementation of the linearized Monge-Ampere equation [41, 42]. Another family of solvers [43, 44, 29] are based on Kantorovich’s formulation of the problem. In short, Kantorovich’s formulation searches for the optimal transport plan π defined on Ω × Ω with marginals μ and σ that minimizes the following, $$\underset{\pi \in true\prod false(\mu ,\sigma false)}{\text{min}}true{\int }_{\mathrm{normal\Omega }\times \mathrm{normal\Omega }}c(x,y)d\pi (x,y)$$Here Π( μ, σ ) is the set of all transport plans with marginals μ and σ .…”

Restricting Voronoi diagrams to meshes using corner validation

“…Such a distance can be used to evaluate methods that modify meshes (remeshing, simplification or compression), by giving a measure of the difference between input and output. Optimal transport algorithms and the related Wasserstein distance are currently gaining popularity due to recent advances making the computation of optimal transport maps tractable [FCRT14, SdGP∗15, BPC16]. Our contribution is one more step in making such computations affordable.…”