Variational inference of the fiber orientation density using diffusion MR imaging

Diffusion MR imaging has enabled the in vivo exploration of the connectional architecture in human brain. This method particularly reveals the complex system of long-range nerve fibers that integrate the functionally distinct areas of the cerebral cortex. Since the fibers are not directly observed but the diffusion process of water molecules in the underlying material, a forward model is established that maps the microgeometry of nervous tissue onto the diffusion-weighted signals. This article proposes the spherical deconvolution of the fiber orientation density in a reproducing kernel Hilbert space, thereby generalizing previous approaches that perform a truncated Fourier analysis on the sphere. The specified inverse problem is solved within a smoothing spline framework which preserves the characteristic properties of a density function, namely its normalization and non-negativity. A Gaussian process model allows the specification of confidence bands for the estimated fiber orientation density and the rigorous selection of the hyperparameters, here the high-frequency content in the density function and the noise variance of the MR observations. In addition, we weaken the constant diffusivity assumption frequently made in the spherical convolution methodology. The novel approach, which uncovers the fiber orientation field of white matter, is demonstrated with diffusion-weighted data sets featuring high angular resolution.