Aspects of Total Variation Regularized<i>L</i><sup>1</sup>Function Approximation

Chan, Tony F.; Esedoḡlu, Selim

doi:10.1137/040604297

Cited by 591 publications

(504 citation statements)

References 26 publications

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“…It has been extended recently to problems such as (1), in [10,8]. In image processing, the observation that (1) can be solved by nding the appropriate superlevel of the solution of (4) was also mentioned recently in [12,13].…”

Section: 4mentioning

confidence: 99%

Total Variation Minimization and a Class of Binary MRF Models

Chambolle

2005

Lecture Notes in Computer Science

261

277

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Section: 4mentioning

confidence: 99%

Total Variation Minimization and a Class of Binary MRF Models

Chambolle

2005

Lecture Notes in Computer Science

261

277

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“…Our computational method is designed for Θ as given in (1.3), or for any smooth and convex function Θ. Recently, data terms of the form Θ(v) = v 1 were shown to be useful if some data entries have to be preserved, which is appreciable, for instance, if n is impulse noise [39,14,27]. Our method is straightforward to extend to this situation, and we will indicate how this can be done.…”

mentioning

confidence: 99%

Efficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimization

Nikolova¹,

Ng²,

Zhang³

et al. 2008

SIAM J. Imaging Sci.

180

171

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Abstract. We consider the restoration of piecewise constant images where the number of the regions and their values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or a Radon transform). Thus we also address the generic problem of unsupervised segmentation in the context of linear inverse problems. The segmentation and the restoration tasks are solved jointly by minimizing an objective function (an energy) composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. The pertinence of such an energy is ensured by the analytical properties of its minimizers. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonsmooth nonconvex minimization. Indeed, the existing methods are unsatisfactory since they (implicitly or explicitly) involve a smooth approximation of the regularization term and often get stuck in shallow local minima. The goal of this paper is to design a method that efficiently handles the nonsmooth nonconvex minimization. More precisely, we propose a continuation method where one tracks the minimizers along a sequence of approximate nonsmooth energies {Jε}, the first of which being strictly convex and the last one the original energy to minimize. Knowing the importance of the nonsmoothness of the regularization term for the segmentation task, each Jε is nonsmooth and is expressed as the sum of an 1 regularization term and a smooth nonconvex function. Furthermore, the local minimization of each Jε is reformulated as the minimization of a smooth function subject to a set of linear constraints. The latter problem is solved by the modified primal-dual interior point method, which guarantees the descent direction at each step. Experimental results are presented and show the effectiveness and the efficiency of the proposed method. Comparison with simulated annealing methods further shows the advantage of our method.

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“…Thus, L 1 regularization on the function, or on its variations, have been developed to allow for more "compressed" reconstructions. These approaches are suitable for problems such as segmentation of images where boundaries are are sharp [16]. In these problems, the data is sparse in the sense that the boundaries in an image contain most of the information.…”

Section: Physical Constraints and Regularizationmentioning

confidence: 99%

“…This class of methods relies on optimization that uses "compressive" L 1 regularization terms in the objective function that favor solutions that are compactly supported [16,17]. This type of regularization term is not derived from any fundamental physical law, but represents prior knowledge that the function to be recovered is sparse in content.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of Cell Focal Adhesions using Physical Constraints and Compressive Regularization

Chang

Liu²,

Chou³

2017

Biophysical Journal

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We develop a method to reconstruct, from measured displacements of an underlying elastic substrate, the spatially dependent forces that cells or tissues impart on it. Given newly available high-resolution images of substrate displacements, it is desirable to be able to reconstruct small scale, compactly supported focal adhesions which are often localized and exist only within the footprint of a cell. In addition to the standard quadratic data mismatch terms that define least-squares fitting, we motivate a regularization term in the objective function that penalizes vectorial invariants of the reconstructed surface stress while preserving boundaries. We solve this inverse problem by providing a numerical method for setting up a discretized inverse problem that is solvable by standard convex optimization techniques. By minimizing the objective function subject to a number of important physically motivated constraints, we are able to efficiently reconstruct stress fields with localized structure from simulated and experimental substrate displacements. Our method incorporates the exact solution for the stress tensor accurate to first-order finite-differences and motivates the use of distance-based cut-offs for data inclusion and problem sparsification.

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Aspects of Total Variation RegularizedL¹Function Approximation

Cited by 591 publications

References 26 publications

Total Variation Minimization and a Class of Binary MRF Models

Total Variation Minimization and a Class of Binary MRF Models

Efficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimization

Reconstruction of Cell Focal Adhesions using Physical Constraints and Compressive Regularization

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Aspects of Total Variation RegularizedL1Function Approximation

Cited by 591 publications

References 26 publications

Total Variation Minimization and a Class of Binary MRF Models

Total Variation Minimization and a Class of Binary MRF Models

Efficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimization

Reconstruction of Cell Focal Adhesions using Physical Constraints and Compressive Regularization

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Product

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Aspects of Total Variation RegularizedL¹Function Approximation