Non-cyclic deconvolution using an augmented lagrangian method

Alternating direction methods have been recently employed for addressing the optimization problems resulting from the regularization approach to linear inverse problems. These methods have been found to achieve state-of-the-art speed in problems such as (frame-based or total-variation-based) cyclic deconvolution, MRI reconstruction, removal of multiplicative or Poisson noise, and inpainting. The speed of these algorithms has three main reasons: (a) they use a regularized version of the Hessian of part of the objective function; (b) they exploit the fact that products by the direct operator and its transpose can be computed efficiently (using fast implementations such as the FFT and/or fast wavelet transforms); (c) a matrix inversion required by the algorithm can be solved in closed form, again using properties of unitary transforms and tight frames. In this paper, we extend this class of algorithms to problems where condition (c) is not met; this is the case, for example, of non-cyclic deconvolution problems. Instead of the exact solution of the matrix inversion, we use an iterative method. Preliminary experiments reported show promising results in terms of computation speed.