Denoising of diffusion magnetic resonance images using a modified and differentiable Monge–Kantorovich distance for measure-valued functions

We report on the implementation of a novel total-variation denoising method for diffusion spectrum images (DSI). Our method works on the raw signal obtained from dMRI. From the Stejskal-Tanner equation [6], the signals S(x, sk), 1 ≤ k ≤ K, at a given voxel location x may be considered as samplings of a measure supported on the unit sphere S2∈R3 at locations sk=(θk,ϕk)∈S2 which quantify the ease/difficulty of diffusion in these directions. We consider the entire signal S as a measure-valued function in a complete metric space which employs the Monge–Kantorovich (MK) metric. A total variation (TV) for measures and measure-valued functions is also defined. A major advance in this paper is the use of a modification of the standard MK distance which permits rapid computation in higher dimensions. An added bonus is that this modified metric is differentiable. The resulting TV-based denoising problem is a convex optimization problem.