On Semismooth Newton’s Methods for Total Variation Minimization

Ng, Michael K.; Qi, Liqun; Yang, Yu-Fei; Huang, Yu-Mei

doi:10.1007/s10851-007-0650-0

Cited by 78 publications

(45 citation statements)

References 7 publications

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“…For TV minimization without constraints, Carter [4], Chambolle [5], Hintermüller and Stadler [13] and Ng et al [21] studied methods for solving the dual problem directly. These methods work well for denoising problems where K = I.…”

Section: Propositionmentioning

confidence: 99%

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

2008

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In this paper, we propose a fast primal-dual algorithm for solving bilaterally constrained total variation minimization problems which subsume the bilaterally constrained total variation image deblurring model and the two-phase piecewise constant Mumford-Shah image segmentation model. The presence of the bilateral constraints makes the optimality conditions of the primal-dual problem semi-smooth which can be solved by a semi-smooth Newton's method superlinearly. But the linear system to solve at each iteration is very large and difficult to precondition. Using a primal-dual active-set strategy, we reduce the linear system to a much smaller and better structured one so that it can be solved efficiently by conjugate gradient with an approximating inverse preconditioner. Locally superlinear convergence results are derived for the proposed algorithm. Numerical experiments are also provided for both deblurring and segmentation problems. In particular, for the deblurring problem, we show that the addition of the bilateral constraints to the total variation model improves the quality of the solutions.

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Section: Propositionmentioning

confidence: 99%

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

2008

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“…Next, we plug (19) into (18) (27) Here, we identify the Lagrange multipliers for with . This is because of (23), which is the same as (19 Here, and is an arbitrary positive constant.…”

Section: A Primal-dual Programmentioning

confidence: 99%

“…and primal-dual relaxation methods. 3 Chambolle [26] presents a semi-implicit scheme and Ng et al [27] present a semi-smooth Newton's method 4 for solving the same dual problem. These algorithms have the advantage of not requiring an extra regularization of the TV norm.…”

Section: Introductionmentioning

confidence: 99%

A Primal-Dual Active-Set Method for Non-Negativity Constrained Total Variation Deblurring Problems

Krishnan

Lin

Yip

2007

IEEE Trans. on Image Process.

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Abstract-This paper studies image deblurring problems using a total variation-based model, with a non-negativity constraint. The addition of the non-negativity constraint improves the quality of the solutions, but makes the solution process a difficult one. The contribution of our work is a fast and robust numerical algorithm to solve the non-negatively constrained problem. To overcome the nondifferentiability of the total variation norm, we formulate the constrained deblurring problem as a primal-dual program which is a variant of the formulation proposed by Chan, Golub, and Mulet for unconstrained problems. Here, dual refers to a combination of the Lagrangian and Fenchel duals. To solve the constrained primal-dual program, we use a semi-smooth Newton's method. We exploit the relationship between the semi-smooth Newton's method and the primal-dual active set method to achieve considerable simplification of the computations. The main advantages of our proposed scheme are: no parameters need significant adjustment, a standard inverse preconditioner works very well, quadratic rate of local convergence (theoretical and numerical), numerical evidence of global convergence, and high accuracy of solving the optimality system. The scheme shows robustness of performance over a wide range of parameters. A comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme. 1

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“…In image restoration, the potential function plays a key role so that it is intensively studied in recent decades [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Two classes of regularization terms are well known.…”

Section: Introductionmentioning

confidence: 99%

Total Variation Image Restoration Method Based on Subspace Optimization

Liu

Gao

2018

Mathematical Problems in Engineering

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The alternating direction method is widely applied in total variation image restoration. However, the search directions of the method are not accurate enough. In this paper, one method based on the subspace optimization is proposed to improve its optimization performance. This method corrects the search directions of primal alternating direction method by using the energy function and a linear combination of the previous search directions. In addition, the convergence of the primal alternating direction method is proven under some weaker conditions. Thus the convergence of the corrected method could be easily obtained since it has same convergence with the primal alternating direction method. Numerical examples are given to show the performance of proposed method finally.

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On Semismooth Newton’s Methods for Total Variation Minimization

Cited by 78 publications

References 7 publications

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

A Primal-Dual Active-Set Method for Non-Negativity Constrained Total Variation Deblurring Problems

Total Variation Image Restoration Method Based on Subspace Optimization

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