On 1-Laplacian Elliptic Equations Modeling Magnetic Resonance Image Rician Denoising

TL;DR

This paper introduces a novel approach for Rician denoising of MRI images using a 1-Laplacian elliptic equation framework, providing theoretical guarantees and demonstrating superior numerical performance over existing TV-based methods.

Contribution

It develops a new mathematical model involving the 1-Laplacian for MRI Rician denoising, with an existence theory and a convergent algorithm for solving the non-smooth, non-convex problem.

Findings

The proposed method outperforms previous TV-based models in denoising quality.

Theoretical existence of solutions is established for the nonlinear elliptic equations.

Numerical results on synthetic and real MRI data validate the effectiveness of the approach.

Abstract

Modeling magnitude Magnetic Resonance Images (MRI) rician denoising in a Bayesian or generalized Tikhonov framework using Total Variation (TV) leads naturally to the consideration of nonlinear elliptic equations. These involve the so called $1$-Laplacian operator and special care is needed to properly formulate the problem. The rician statistics of the data are introduced through a singular equation with a reaction term defined in terms of modified first order Bessel functions. An existence theory is provided here together with other qualitative properties of the solutions. Remarkably, each positive global minimum of the associated functional is one of such solutions. Moreover, we directly solve this non--smooth non--convex minimization problem using a convergent Proximal Point Algorithm. Numerical results based on synthetic and real MRI demonstrate a better performance of the proposed method when compared to previous TV based models for rician denoising which regularize or convexify the problem. Finally, an application on real Diffusion Tensor Images, a strongly affected by rician noise MRI modality, is presented and discussed.