Gauss-Map Variation for Image Denoising: Geometric Analysis and an Anderson--Accelerated Majorization--Minimization Method

We propose a Gauss-map variation (GMV) model for image denoising that measures the spatial variation of the tangent-plane projectors of the scaled image graph. We establish an equivalent representation of the regularizer in terms of the corresponding Gauss map and, using differential geometric tools including tubular coordinates and the Frenet frame, analyze its behavior across general $C^2$ and piecewise $C^2$ boundaries. The resulting estimates provide edge- and corner-contrast preservation properties. To solve the proposed model, we introduce a bilinear decomposition involving a unit normal field and a scalar magnitude field and develop an Anderson-accelerated majorization--minimization algorithm. The normal field subproblem admits an explicit pointwise majorization--minimization update, which is combined with an Anderson acceleration. For both $L^1$ and $L^2$ data fidelity terms, we establish sufficient decrease and boundedness of the iterates and prove that the generated sequence converges to a critical point of the penalized model. Numerical experiments on synthetic and natural images demonstrate the boundary preserving capability of the proposed model and its competitive performance in removing Gaussian and impulsive noise.