A Fast Augmented Lagrangian Method for Euler's Elastica Models

Duan, Yuping; Wang, Yu; Hahn, Jooyoung

doi:10.4208/nmtma.2013.mssvm03

Cited by 25 publications

(29 citation statements)

References 36 publications

(49 reference statements)

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“…Remark 5.1: In some earlier works, c.f. [1,58,2], the Euler elastica model was applied to image denoising. We found, however, that "edge-preserving smoothing" describes better than "denoising" the properties of the proposed method.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…As shown by the results reported in [1], the THC method can solve the Euler elastica problem hundredfold times faster than the Chan-Kang-Shen (CKS) method in [7]. After the THC method, some promising approaches, see e.g., [58,4,2,6], were proposed for the Euler elastica problem. In [58], Duan et al proposed another fast augmented Lagrangian method to solve the Euler elastica problem based on the framework of the THC method.…”

Section: Advantages Of the Proposed Methodsmentioning

confidence: 99%

“…After the THC method, some promising approaches, see e.g., [58,4,2,6], were proposed for the Euler elastica problem. In [58], Duan et al proposed another fast augmented Lagrangian method to solve the Euler elastica problem based on the framework of the THC method. Afterwards, Duan et al in [4] applied the THC based method in to solve the Euler elastica regularized Mumford-Shah problem, aiming to deal with two- Noisy Image TV Elastica ReErr(Log) Figure 4: We take the image "square" which is the last one in Fig.…”

Section: Advantages Of the Proposed Methodsmentioning

confidence: 99%

“…ADMM is a primal-dual method, closely related to the Douglas-Rachford alternating direction method (a well-known operator-splitting method). Following [1], several extensions were proposed for solving, via the Euler elastica energy functional, a large variety of imaging problems (see [3,23,24,6]). Actually, readers can find further curvature based methods in [25,26,27,28,29].…”

Section: Introductionmentioning

confidence: 99%

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A New Operator Splitting Method for the Euler Elastica Model for Image Smoothing

Deng

Glowinski

Tai³

2019

SIAM J. Imaging Sci.

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Euler's elastica model has a wide range of applications in Image Processing and Computer Vision. However, the non-convexity, the non-smoothness and the nonlinearity of the associated energy functional make its minimization a challenging task, further complicated by the presence of high order derivatives in the model. In this article we propose a new operator-splitting algorithm to minimize the Euler elastica functional. This algorithm is obtained by applying an operator-splitting based time discretization scheme to an initial value problem (dynamical flow) associated with the optimality system (a system of multivalued equations). The sub-problems associated with the three fractional steps of the splitting scheme have either closed form solutions or can be handled by fast dedicated solvers. Compared with earlier approaches relying on ADMM (Alternating Direction Method of Multipliers), the new method has, essentially, only the time discretization step as free parameter to choose, resulting in a very robust and stable algorithm. The simplicity of the sub-problems and its modularity make this algorithm quite efficient. Applications to the numerical solution of smoothing test problems demonstrate the efficiency and robustness of the proposed methodology.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Advantages Of the Proposed Methodsmentioning

confidence: 99%

Section: Advantages Of the Proposed Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A New Operator Splitting Method for the Euler Elastica Model for Image Smoothing

Deng

Glowinski

Tai³

2019

SIAM J. Imaging Sci.

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“…In [43], Tai, Hahn and Chung propose a fast and efficient algorithm for Euler's elastica model via augmented Lagrangian method. The algorithms for curvature based variational model, one can see [18,42,50,51] for details.…”

Section: Introductionmentioning

confidence: 99%

Proximal ADMM for Euler's Elastica Based Image Decomposition Model

Liu

Wali

Duan

et al. 2019

NMTMA

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This paper studies image decomposition models which involve functional related to total variation and Euler's elastica energy. Such kind of variational models with first order and higher order derivatives have been widely used in image processing to accomplish advanced tasks. However, these non-linear partial differential equations usually take high computational cost by the gradient descent method. In this paper, we propose a proximal alternating direction method of multipliers (ADM-M) for total variation (TV) based Vese-Osher's decomposition model [L.A. Vese and S.J. Osher, J. Sci. Comput., 19.1 (2003), pp. 553-572] and its extension with Euler's elastica regularization. We demonstrate that efficient and effective solutions to these minimization problems can be obtained by proximal based numerical algorithms. In numerical experiments, we present numerous results on image decomposition and image denoising, which conforms significant improvement of the proposed models over standard models.

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