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

In this paper, we propose a fast fixed point algorithm and apply it to total variation (TV) deblurring and segmentation. The TV-based models can be written in the form of a general minimization problem. The novel method is derived from the idea of establishing the relation between solutions of the general minimization problem and new variables, which can be obtained by a fixed point algorithm efficiently. Under gentle conditions it provides a platform to develop efficient numerical algorithms for various image processing tasks. We then specialize this fixed point methodology to the TV-based image deblurring and segmentation models, and the resulting algorithms are compared with the split Bregman method, which is a strong contender for the state-of-the-art algorithms. Numerical experiments demonstrate that the algorithm proposed here performs favorably.

show abstract

“…Such a model was proposed in [15,16] and shown to be able to improve the quality of the deblurred image.…”

Section: Total Variation Image Deblurring Modelmentioning

confidence: 97%

A Fast Fixed Point Algorithm for Total Variation Deblurring and Segmentation

2011

View full text Add to dashboard Cite

show abstract

“…The active-set reduced-space method consists of two major steps: (a) in the first phase, an index set with respect to the computational domain is decomposed into active and inactive parts, based on a criterion specifying a certain active set method; and (b) in the second phase, a reduced linear system associated with the inactive set is solved. We would like to point out that the class of active-set reduced-space method has proven to be very efficient in a variety of applications [23,28,34,35,56], but very little work has been done in the two-phase flow problem as far as we know.…”

Section: B595mentioning

confidence: 99%

Active-Set Reduced-Space Methods with Nonlinear Elimination for Two-Phase Flow Problems in Porous Media

Yang¹,

Yang²,

Sun³

2016

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. Fully implicit methods are drawing more attention in scientific and engineering applications due to the allowance of large time steps in extreme-scale simulations. When using a fully implicit method to solve two-phase flow problems in porous media, one major challenge is the solution of the resultant nonlinear system at each time step. To solve such nonlinear systems, traditional nonlinear iterative methods, such as the class of the Newton methods, often fail to achieve the desired convergent rate due to the high nonlinearity of the system and/or the violation of the boundedness requirement of the saturation. In the paper, we reformulate the two-phase model as a variational inequality that naturally ensures the physical feasibility of the saturation variable. The variational inequality is then solved by an active-set reduced-space method with a nonlinear elimination preconditioner to remove the high nonlinear components that often causes the failure of the nonlinear iteration for convergence. To validate the effectiveness of the proposed method, we compare it with the classical implicit pressure-explicit saturation method for two-phase flow problems with strong heterogeneity. The numerical results show that our nonlinear solver overcomes the often severe limits on the time step associated with existing methods, results in superior convergence performance, and achieves reduction in the total computing time by more than one order of magnitude.

show abstract

“…Routh et al (2007) solved the optimization with non-smooth regularization and physical bounds using an interior point method. Krishnan et al (2009) developed a primaldual active-set algorithm for bound constrained TV deblurring problems. Chartrand and Wohlberg (2010) solved a TV regularization with bound constraints by a splitting approach, thus allowing existing TV solvers be employed with minimal alteration.…”

Section: Constrained Tv Regularizationmentioning

confidence: 99%

Rayleigh wave dispersion curve inversion: Occam versus the L1‐norm

Haney

2010

SEG Technical Program Expanded Abstracts 2010

View full text Add to dashboard Cite

SUMMARYWe compare inversions of Rayleigh wave dispersion curves for shear wave velocity depth profiles based on the L2-norm (Occam's Inversion) and L1-norm (TV Regularization). We forward model Rayleigh waves using a finite-element method instead of the conventional technique based on a recursion formula and root-finding. The forward modeling naturally leads to an inverse problem that is overparameterized in depth. Solving the inverse problem with Occam's Inversion gives the smoothest subsurface model that satisfies the data. However, the subsurface need not be smooth and we therefore also solve the inverse problem with TV Regularization, a procedure that does not penalize discontinuities. The use of such a regularization scheme for such an overparameterized inverse problem means blocky subsurface models can be obtained without fixing the layer boundaries in advance. This represents an entirely new philosophy for surface wave inversion.

show abstract

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

Cited by 19 publications

References 29 publications

A Fast Fixed Point Algorithm for Total Variation Deblurring and Segmentation

A Fast Fixed Point Algorithm for Total Variation Deblurring and Segmentation

Active-Set Reduced-Space Methods with Nonlinear Elimination for Two-Phase Flow Problems in Porous Media

Rayleigh wave dispersion curve inversion: Occam versus the L1‐norm

Contact Info

Product

Resources

About