An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications to Surface Smoothing and Imaging

Líu, Hao; Tai, Xue–Cheng; Glowinski, Roland

doi:10.48550/arxiv.2108.01914

Cited by 3 publications

(3 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The expression of ( 24) and ( 25) are well suited to be time discretized by operator-splitting methods, as what has been done in [9,28,27]. We refer the readers to [18] for a complete discussion of operator-splitting methods.…”

Section: Operator-splitting Schemesmentioning

confidence: 99%

Elastica Models for Color Image Regularization

Líu¹,

Tai²,

Kimmel³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

The choice of a proper regularization measure plays an important role in the field of image processing. One classical approach treats color images as two dimensional surfaces embedded in a five dimensional spatial-chromatic space. In this case, a natural regularization term arises as the image surface area. Choosing the chromatic coordinates as dominating over the spatial ones, the image spatial coordinates could be thought of as a paramterization of the image surface manifold in a three dimensional color space. Minimizing the area of the image manifold leads to the Beltrami flow or mean curvature flow of the image surface in the 3D color space, while minimizing the elastica of the image surface yields an additional interesting regularization. Recently, the authors proposed a color elastica model, which minimizes both the surface area and elastica of the image manifold. In this paper, we propose to modify the color elastica and introduce two new models for color image regularization. The revised measures are motivated by the relations between the color elastica model, Euler's elastica model and the total variation model for gray level images. Compared to our previous color elastica model, the new models are direct extensions of Euler's elastica model to color images. The proposed models are nonlinear and challenging to minimize. To overcome this difficulty, two operator-splitting methods are suggested. Specifically, nonlinearities are decoupled by introducing new vector-and matrix-valued variables. Then, the minimization problems are converted to solving initial value problems which are time-discretized by operator splitting. Each subproblem, after splitting either, has a closed-form solution or can be solved efficiently. The effectiveness and advantages of the proposed models are demonstrated by comprehensive experiments. The benefits of incorporating the elastica of the image surface as regularization terms compared to common alternatives are empirically validated.

show abstract

Section: Operator-splitting Schemesmentioning

confidence: 99%

Elastica Models for Color Image Regularization

Líu¹,

Tai²,

Kimmel³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…All variables will be updated in an alternative fashion, where each subproblem either has an explicit solution or can be solved efficiently. The operator-splitting method has been applied to numerically solving PDEs [28,37], image processing [39,15,38,40], surface reconstruction [32], inverse problems [27], obstacle problems [41], and computational fluid dynamics [8,7]. We refer readers to monographs [29,30] for detailed discussions on operatorsplitting methods.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Operator-Splitting Method for the Eigenvalue Problem of the Monge-Ampère Equation

Líu¹,

Leung²,

Qian³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Ampère operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by Abedin and Kitagawa (Inverse iteration for the Monge-Ampère eigenvalue problem, Proceedings of the American Mathematical Society, 148 (2020), no. 11, 4975-4886). Modifying the theoretical formulation, we develop an efficient algorithm for computing the eigenvalue and eigenfunction of the Monge-Ampère operator by solving a constrained Monge-Ampère equation during each iteration. Our method consists of four essential steps: (i) Formulate the Monge-Ampère eigenvalue problem as an optimization problem with a constraint; (ii) Adopt an indicator function to treat the constraint; (iii) Introduce an auxiliary variable to decouple the original constrained optimization problem into simpler optimization subproblems and associate the resulting new optimization problem with an initial value problem; and (iv) Discretize the resulting initial-value problem by an operator-splitting method in time and a mixed finite element method in space. The performance of our method is demonstrated by several experiments. Compared to existing methods, the new method is more efficient in terms of computational cost and has a comparable rate of convergence in terms of accuracy.

show abstract

“…All variables will be updated in an alternative fashion, where each subproblem either has an explicit solution or can be solved efficiently. The operator-splitting method has been applied to numerically solving PDEs [30,39], image processing [16,40,41,42], surface reconstruction [34], inverse problems [29], obstacle problems [43], and computational fluid dynamics [7,8]. We refer readers to monographs [31,32] for detailed discussions on operator-splitting methods.…”

Section: Introductionmentioning

confidence: 99%

An efficient operator-splitting method for the eigenvalue problem of the Monge-Ampère equation

Líu¹,

Leung²,

Qian³

2022

Commu. Optim. Theory

View full text Add to dashboard Cite

We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Ampère operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by Abedin and Kitagawa (Inverse iteration for the Monge-Ampère eigenvalue problem, Proceedings of the American Mathematical Society, 148 (2020) 4975-4886). Modifying the theoretical formulation, we develop an efficient algorithm for computing the eigenvalue and eigenfunction of the Monge-Ampère operator by solving a constrained Monge-Ampère equation during each iteration. Our method consists of four essential steps: (i) Formulate the Monge-Ampère eigenvalue problem as an optimization problem with a constraint; (ii) Adopt an indicator function to treat the constraint; (iii) Introduce an auxiliary variable to decouple the original constrained optimization problem into simpler optimization subproblems and associate the resulting new optimization problem with an initial value problem; and (iv) Discretize the resulting initial-value problem by an operator-splitting method in time and a mixed finite element method in space. The performance of our method is demonstrated by several experiments. Compared to existing methods, the new method is more efficient in terms of computational cost and has a comparable rate of convergence in terms of accuracy.

show abstract

An Operator-Splitting Method for the Gaussian Curvature Regularization Model with Applications to Surface Smoothing and Imaging

Cited by 3 publications

References 40 publications

Elastica Models for Color Image Regularization

Elastica Models for Color Image Regularization

An Efficient Operator-Splitting Method for the Eigenvalue Problem of the Monge-Ampère Equation

An efficient operator-splitting method for the eigenvalue problem of the Monge-Ampère equation

Contact Info

Product

Resources

About