Bures-Wasserstein Barycentric Coordinates with Application to Diffusion Tensor Image Smoothing

This article considers the Wasserstein barycentric coordinates problem for Gaussian distributions which is the inverse problem of the Wasserstein barycenter problem. These coordinates take into account the underlying geometry of the measure space of Gaussian distributions and are thus meaningful for applications such as diffusion analysis and distributed information fusion. When the probability supports are discrete and identical, the theory of Wasserstein barycentric coordinates is well developed. However, for general probability distributions, the computation of Wasserstein barycentric coordinates is intractable since the technical hurdles involve solving a non-convex and non-concave optimization problem. For Gaussian distributions, we derive the closed-form expression of the derivatives for the objective function and propose a projected gradient descent method to solve the problem. Finally, we illustrate its application in diffusion tensor image (DTI) denoising including simulated DTI with different noise levels and DTI of the human brain.