Lagrangian multipliers and split Bregman methods for minimization problems constrained on S ^n-1

The numerical methods of total variation (TV) model for image denoising, especially Rudin-Osher-Fatemi (ROF) model, is widely studied in the literature. However, the S ^n-1 constrained counterpart is less addressed. The classical gradient descent method for the constrained problem is limited in two aspects: one is the small time step size to ensure stability; the other is that the data must be projected onto S ^n-1 during evolution since the unit norm constraint is poorly satisfied. In order to avoid these drawbacks, in this paper, we propose two alternative numerical methods based on the Lagrangian multipliers and split Bregman methods. Both algorithms are efficient and easy to implement. A number of experiments demonstrate that the proposed algorithms are quite effective in denoising of data constrained on S ¹ or S ², including general direction data diffusion and chromaticity denoising.