Variational Rician Noise Removal via Splitting on Spheres

We propose a novel variational method for Rician noise removal in magnitude-based magnetic resonance (MR) imaging. We first explore the link between the Gaussian noise removal for complex images and the Rician noise removal for magnitude images. Then we establish the constraint optimization model via signal-noise splitting, consisting of a total variation regularizer, two quadratic terms, and a constraint on the field of spheres. Specifically, this constraint represents the forward model of calculating the magnitude of complex images corrupted by Gaussian noises. Namely, the proposed model is completely different from the existing maximum a posteriori based methods, which inevitably involved the sophisticated Bessel function causing high computation costs. It is further efficiently solved by the alternating direction method of multipliers with convergence guarantee. Numerical comparisons with existing variational methods show that the proposed method produces comparable results in terms of image quality, but saves about 50% of overall computational cost on average.