Equivariant Spherical Deconvolution: Learning Sparse Orientation Distribution Functions from Spherical Data

We present a rotation-equivariant unsupervised learning framework for the\nsparse deconvolution of non-negative scalar fields defined on the unit sphere.\nSpherical signals with multiple peaks naturally arise in Diffusion MRI (dMRI),\nwhere each voxel consists of one or more signal sources corresponding to\nanisotropic tissue structure such as white matter. Due to spatial and spectral\npartial voluming, clinically-feasible dMRI struggles to resolve crossing-fiber\nwhite matter configurations, leading to extensive development in spherical\ndeconvolution methodology to recover underlying fiber directions. However,\nthese methods are typically linear and struggle with small crossing-angles and\npartial volume fraction estimation. In this work, we improve on current\nmethodologies by nonlinearly estimating fiber structures via unsupervised\nspherical convolutional networks with guaranteed equivariance to spherical\nrotation. Experimentally, we first validate our proposition via extensive\nsingle and multi-shell synthetic benchmarks demonstrating competitive\nperformance against common baselines. We then show improved downstream\nperformance on fiber tractography measures on the Tractometer benchmark\ndataset. Finally, we show downstream improvements in terms of tractography and\npartial volume estimation on a multi-shell dataset of human subjects.\n