Constrained Total Variation Deblurring Models and Fast Algorithms Based on Alternating Direction Method of Multipliers

Abstract

Abstract. The total variation (TV) model is attractive for being able to preserve sharp attributes in images. However, the restored images from TV-based methods do not usually stay in a given dynamic range, and hence projection is required to bring them back into the dynamic range for visual presentation or for storage in digital media. This will affect the accuracy of the restoration as the projected image will no longer be the minimizer of the given TV model. In this paper, we show that one can get much more accurate solutions by imposing box constraints on the TV models and solving the resulting constrained models. Our numerical results show that for some images where there are many pixels with values lying on the boundary of the dynamic range, the gain can be as great as 10.28dB in peak signal-to-noise ratio. One traditional hinderance of using the constrained model is that it is difficult to solve. However, in this paper, we propose to use the alternating direction method of multipliers (ADMM) to solve the constrained models. This leads to a fast and convergent algorithm that is applicable for both Gaussian and impulse noise. Numerical results show that our ADMM algorithm is better than some state-of-the-art algorithms for unconstrained models both in terms of accuracy and robustness with respect to the regularization parameter.

Investigating the Influence of Box-Constraints on the Solution of a Total Variation Model via an Efficient Primal-Dual Method

“…An implementation of the ADMM for solving problem (1) is presented in [51], which we refer to as the ADMM in the sequel. For solving problem (3) a possible implementation is suggested in [27]. However, for comparison purposes we use a slightly different version, which uses the same succession of updates as the ADMM in [51], see Appendix B for a description of this version.…”

“…2.1) it follows that there exists a constant α ≥ 0 such that problem (3) is equivalent to problem (4). For image restoration box-constraints have been considered for example in [5,[27][28][29]. In [29] a functional consisting of an L 2 -data term and a Tikhonov-like regularization term (i.e., L 2 -norm of some derivative of u) in connection with box-constrained is presented together with a Newton-like numerical scheme.…”

“…For box-constrained total variation minimization in [5] a fast gradient-based algorithm, called monoton fast iterative shrinkage/thresholding algorithm (MFISTA), is proposed and a rate of convergence is proven. Based on the alternating direction method of multipliers (ADMM) [30] in [27] a solver for the box-constrained L 2 -TV and L 1 -TV model is derived and shown to be faster than MFISTA. In [28] a primal-dual algorithm for the box-constrained L 1 -TV model and for box-constrained non-local total variation is presented.…”

Primal-Dual Decomposition by Operator Splitting and Applications to Image Deblurring

“…As pointed out in [Vog02,BT09,CTY13] the explicit addition of box or nonnegativity constraints on x can improve the quality of the restoration substantially.…”

A New Study of Blind Deconvolution with Implicit Incorporation of Nonnegativity Constraints

“…Such tasks have been widely studied in the last few decades; see [6] for decoupling noise and blur modeling, [7] for imposing box constraints, [8] for a fast iterative solver for noise and blur modeling, and [9,10] for general surveys. However there are still many outstanding issues to be addressed, especially when both the noise type and the blur type are unknown.…”