Spherical function regularization for parallel MRI reconstruction

From the optimization point of view, a difficulty with parallel MRI with simultaneous coil sensitivity estimation is the multiplicative nature of the non-linear forward operator: the image being reconstructed and the coil sensitivities compete against each other, causing the optimization process to be very sensitive to small perturbations. This can, to some extent, be avoided by regularizing the unknown in a suitably "orthogonal" fashion. In this paper, we introduce such a regularization based on spherical function bases. To perform this regularization, we represent efficient recurrence formulas for spherical Bessel functions and associated Legendre functions. Numerically, we study the solution of the model with non-linear ADMM. We perform various numerical simulations to demonstrate the efficacy of the proposed model in parallel MRI reconstruction.