2016/12/20 by Gabriel Peyré, Peyré, Gabriel, Lénaïc Chizat +5 · 1 citation
Medicine · Engineering · Computer Science · #Advanced Neuroimaging Techniques and Applications #Sparse and Compressive Sensing Techniques #Medical Image Segmentation Techniques
paper · pdf · doi:10.48550/arxiv.1612.08731
This article introduces a new notion of optimal transport (OT) between tensor fields, which are measures whose values are positive semidefinite (PSD) matrices. This "quantum" formulation of OT (Q-OT) corresponds to a relaxed version of the classical Kantorovich transport problem, where the fidelity between the input PSD-valued measures is captured using the geometry of the Von-Neumann quantum entropy. We propose a quantum-entropic regularization of the resulting convex optimization problem, which can be solved efficiently using an iterative scaling algorithm. This method is a generalization of the celebrated Sinkhorn algorithm to the quantum setting of PSD matrices. We extend this formulation and the quantum Sinkhorn algorithm to compute barycenters within a collection of input tensor fields. We illustrate the usefulness of the proposed approach on applications to procedural noise generation, anisotropic meshing, diffusion tensor imaging and spectral texture synthesis.