Learned spherical ADMM for Rician MRI denoising

Abstract

Rician noise corruption in magnetic resonance imaging poses a challenging restoration problem due to its nonlinearity and signal dependence. In this paper, we propose a novel learned spherical alternating direction method of multipliers (LSADMMs) for effective Rician noise removal under spherical constraints. Grounded in a sphere-constrained variational formulation, the proposed architecture unfolds the iterations of a proximal linearized alternating direction method of multiplier solver into a deep neural network. Structurally, LSADMM alternates between lightweight, learnable gradient descent modules and fixed, physics-based operators. To enable blind denoising, we incorporate a noise-level estimation prefix that provides adaptive guidance across noise levels. Notably, a parameter-free spherical projection layer is incorporated to strictly enforce the geometric constraint of the Rician degradation model, ensuring that intermediate iterates remain physically consistent. Theoretically, we establish the boundedness of the unfolded iterates and prove the layer-wise stability of the unrolled dynamics. Extensive numerical experiments on synthetic and real-world datasets demonstrate that LSADMM achieves competitive restoration performance with a lightweight architecture, requiring substantially fewer parameters than conventional end-to-end deep learning methods.