Wavelet Shrinkage With Consistent Cycle Spinning Generalizes Total Variation Denoising

We introduce a new wavelet-based method for the implementation of Total-Variation-type denoising. The data term is least-squares, while the regularization term is gradient-based. The particularity of our method is to exploit a link between the discrete gradient and wavelet shrinkage with cycle spinning, which we express by using redundant wavelets. The redundancy of the representation gives us the freedom to enforce additional constraints (e.g., normalization) on the solution to the denoising problem. We perform optimization in an augmented-Lagrangian framework, which decouples the difficult
n
-dimensional constrained-optimization problem into a sequence of
n
easier scalar unconstrained problems that we solve efficiently via traditional wavelet shrinkage. Our method can handle arbitrary gradient-based regularizers. In particular, it can be made to adhere to the popular principle of least total variation. It can also be used as a maximum a posteriori estimator for a variety of priors. We illustrate the performance of our method for image denoising and for the statistical estimation of sparse stochastic processes.