TV denoising algorithm implementations in MATLAB using Bregman bias correction techniques for different noise models
This case is solved using Bregman iterations as proposed in [1] by repeatedly solving modified versions of the ROF model with a corrected noise term. In a subroutine, the ROF model [2] is solved using Chambolle's algorithm [3] that reduces the problem to the computation of a nonlinear projection in the dual space. Code is provided for the 1D and the 2D problem.
In the case of Poisson noise (and in particular for low counts), the data fidelity term must be the Kullback-Leibler divergence, hence the ROF model is not a good basis for a denoising or reconstruction algorithm. Instead, the signal/image is denoised using the Bregman FB-EM-TV algorithm. Reconstruction is carried out by a forward-backward scheme, splitting the iteration into an EM step and a TV correction step with a weight function in the TV functional. Bias correction is carried out by employing a few Bregman iterations.
The denoising case is a special case of the image reconstruction case above, only differing from the latter by the vanishing EM step.
[1] Osher, S.; Burger, M.; Goldfarb, D.; Xu, J.; Yin, W. An Iterative Regularization Method for Total Variation-Based Image Restoration
Multiscale Modeling; Simulation, Society for Industrial and Applied Mathematics, 2005, 4, 460-489
[2] Rudin, L. I.; Osher, S.; Fatemi, E. Nonlinear total variation based noise removal algorithms
Physica D: Nonlinear Phenomena, 1992, 60, 259-268
[3] Chambolle, A. An Algorithm for Total Variation Minimization and Applications
Journal of Mathematical Imaging and Vision, 2004, 20, 89-97
[4] Sawatzky, A.; Brune, C.; Kösters, T.; Wübbeling, F.; Burger, M. EM-TV Methods for Inverse Problems with Poisson Noise
In: Burger, M.; Mennucci, A. C.; Osher, S.; Rumpf, M. (Eds.). Level Set and PDE Based Reconstruction Methods in Imaging
Springer International Publishing, 2013, 71-142