Fast optimal transport averaging of neuroimaging data

Knowing how the Human brain is anatomically and functionally organized at the level of a group of healthy individuals or patients is the primary goal of neuroimaging research. Yet computing an average of brain imaging data defined over a voxel grid or a triangulation remains a challenge. Data are large, the geometry of the brain is complex and the between subjects variability leads to spatially or temporally nonoverlapping effects of interest. To address the problem of variability, data are commonly smoothed before performing a linear group averaging. In this work we build on ideas originally introduced by Kantorovich [18] to propose a new algorithm that can average efficiently non-normalized data defined over arbitrary discrete domains using transportation metrics. We show how Kantorovich means can be linked toWasserstein barycenters in order to take advantage of the entropic smoothing approach used by [7]. It leads to a smooth convex optimization problem and an algorithm with strong convergence guarantees. We illustrate the versatility of this tool and its empirical behavior on functional neuroimaging data, functional MRI and magnetoencephalography (MEG) source estimates, defined on voxel grids and triangulations of the folded cortical surface.