A Fast Relaxed Normal Two Split Method and an Effective Weighted TV Approach for Euler's Elastica Image Inpainting

Yashtini, Maryam; Kang, Sung Ha

doi:10.1137/16m1063757

Cited by 45 publications

(33 citation statements)

References 43 publications

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“…In Fig. 4, we also monitor the energy changes of the subproblems and of the original problem (9). From this figure, it is clear that the energies of the p n+1/3 subproblem (27), of the λ n+1/3 subproblem (29), of (n + 2/3) subproblem (31) (including p n+2/3 and λ n+2/3 subproblems) all decrease as n increases, while the energy of the p n+1 subproblem (53) increases to a stable value.…”

Section: Application Of the Proposed Methods To Image Smoothingmentioning

confidence: 99%

“…and suppose that (p, λ) is a minimizer of the functional in (9). We have then u = v p and the following system of (necessary) optimality conditions holds:…”

Section: Optimality Conditions and Associated Dynamical Flow Problemmentioning

confidence: 99%

“…Recently, the Euler elastica energy defined by (2) found applications in image processing, such as: denoising [1,2], segmentation [3,4,5,6], inpainting [7,8,1,9], zooming [1], illusory contour [10,11,12], segmentation with depth [13,14,15]. In [7], Chan et al discuss the mathematical foundation of the Euler elastica model and its mathematical properties, motivated by applications to image inpainting.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, Zhang et al proposed in [2] a linearized strategy to speed up ALM, and applied it to the solution of image denoising problems. In [9], two numerical algorithms were proposed to solve inpainting related problems involving the Euler elastica energy (2): The first algorithm is an improved variant of the ALM based algorithm discussed in [1]. The second algorithm is obtained by applying a split-Bregman method to a linearized elastica model proposed in [16].…”

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: Application Of the Proposed Methods To Image Smoothingmentioning

confidence: 99%

“…and suppose that (p, λ) is a minimizer of the functional in (9). We have then u = v p and the following system of (necessary) optimality conditions holds:…”

Section: Optimality Conditions and Associated Dynamical Flow Problemmentioning

confidence: 99%

Section: Introductionmentioning

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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“…Besides, the curvature-related terms will bring extra computational complexity due to the existence of nonlinear higherorder derivatives. This issue also appears in other variational models such as the nontexture image inpainting [20] and image denoising [21] with features (edge, corner, smoothness, contrast, etc.) preservation.…”

Section: Introductionmentioning

confidence: 91%

A Novel Euler’s Elastica-Based Segmentation Approach for Noisy Images Using the Progressive Hedging Algorithm

Tan

Liu

et al. 2019

J Math Imaging Vis

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Eulers elastica based unsupervised segmentation models have strong capability of completing the missing boundaries for existing objects in a clean image, but they are not working well for noisy images. This paper aims to establish a Eulers elastica based approach that properly deals with the random noises to improve the segmentation performance for noisy images. We solve the corresponding optimization problem via using the progressive hedging algorithm (PHA) with a step length suggested by the alternating direction method of multipliers (ADMM). Technically, all the simplified convex versions of the sub problems derived from the major framework of PHA can be obtained by using the curvature weighted approach and the convex relaxation method. Then an alternating optimization strategy is applied with the merits of using some powerful accelerating techniques including the fast Fourier transform (FFT) and generalized soft threshold formulas. Extensive experiments have been conducted on both synthetic and real images, which validated some significant gains of the proposed segmentation models and demonstrated the advantages of the developed algorithm.

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On Variable Splitting and Augmented Lagrangian Method for Total Variation-Related Image Restoration Models

Liu¹,

Duan²,

Wu³

et al. 2021

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

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A Fast Relaxed Normal Two Split Method and an Effective Weighted TV Approach for Euler's Elastica Image Inpainting

Cited by 45 publications

References 43 publications

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

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

A Novel Euler’s Elastica-Based Segmentation Approach for Noisy Images Using the Progressive Hedging Algorithm

On Variable Splitting and Augmented Lagrangian Method for Total Variation-Related Image Restoration Models

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