ACQUIRE: an inexact iteratively reweighted norm approach for TV-based Poisson image restoration

Abstract

We propose a method, called ACQUIRE, for the solution of constrained optimization problems modeling the restoration of images corrupted by Poisson noise. The objective function is the sum of a generalized Kullback-Leibler divergence term and a TV regularizer, subject to nonnegativity and possibly other constraints, such as flux conservation. ACQUIRE is a line-search method that considers a smoothed version of TV, based on a Huber-like function, and computes the search directions by minimizing quadratic approximations of the problem, built by exploiting some second-order information. A classical second-order Taylor approximation is used for the Kullback-Leibler term and an iteratively reweighted norm approach for the smoothed TV term. We prove that the sequence generated by the method has a subsequence converging to a minimizer of the smoothed problem and any limit point is a minimizer. Furthermore, if the problem is strictly convex, the whole sequence is convergent. We note that convergence is achieved without requiring the exact minimization of the quadratic subproblems; low accuracy in this minimization can be used in practice, as shown by numerical results. Experiments on reference test problems show that our method is competitive with well-established methods for TV-based Poisson image restoration, in terms of both computational efficiency and image quality.

Directional TGV-Based Image Restoration under Poisson Noise

“…We note that problem ( 6 ) is a nonsmooth convex optimization problem because of the properties of the KL divergence (see, e.g., [ 11 ]) and the DTGV operator (see, e.g., [ 10 ]).…”

Split Bregman iteration for multi-period mean variance portfolio optimization

“…The nonsmoothness of the l 1 -type regularization terms precludes the use of standard descent methods for smooth objective functions. Problems of this kind can be solved either by smoothing the l 1 terms, e.g., [5] , [6] , and applying optimization solvers for differentiable problems such as gradient methods [7] , [8] , [9] or by using directly optimization solvers for nondifferentiable problems, such as Bregman, proximal and ADMM methods [2] , [10] , [11] , [12] . Due to the additive structure in (1) , splitting methods have became popular because they yield algorithms which consist at each iteration of subproblems that are easier to solve [13] , [14] .…”

Spatially Adaptive Regularization in Image Segmentation

“…Our techniques are based on the so-called cartoon-texture decomposition of the given image, on the mean and median filters, and on a thresholding technique. The resulting locally adaptive segmentation model can be solved either by smoothing the discrete TV term-see, e.g., [16,17]-and applying optimization solvers for differentiable problems, such as spectral gradient methods [18][19][20][21], or by using directly optimization solvers for nondifferentiable problems, such as Bregman, proximal and ADMM methods [22][23][24][25][26][27][28]. In this work, we use an alternating minimization procedure exploiting the split Bregman (SB) method proposed in [24].…”