A numerical study of a mean curvature denoising model using a novel augmented Lagrangian method
Wei Zhu
Inverse Problems & Imaging 2017, 11(6): 975-996 doi: 10.3934/ipi.2017045

In this paper, we propose a new augmented Lagrangian method for the mean curvature based image denoising model [33]. Different from the previous works in [21,35], this new method only involves two Lagrange multipliers, which significantly reduces the effort of choosing appropriate penalization parameters to ensure the convergence of the iterative process of finding the associated saddle points. With this new algorithm, we demonstrate the features of the model numerically, including the preservation of image contrasts and object corners, as well as its capability of generating smooth patches of image graphs. The data selection property and the role of the spatial mesh size for the model performance are also discussed.

keywords: Image denoising mean curvature augmented Lagrangian method high-order variational model
A fast modified Newton's method for curvature based denoising of 1D signals
Andy M. Yip Wei Zhu
Inverse Problems & Imaging 2013, 7(3): 1075-1097 doi: 10.3934/ipi.2013.7.1075
We propose a novel fast numerical method for denoising of 1D signals based on curvature minimization. Motivated by the primal-dual formulation for total variation minimization introduced by Chan, Golub, and Mulet, the proposed method makes use of some auxiliary variables to reformulate the stiff terms presented in the Euler-Lagrange equation which is a fourth-order differential equation. A direct application of Newton's method to the resulting system of equations often fails to converge. We propose a modified Newton's iteration which exhibits local superlinear convergence and global convergence in practical settings. The method is much faster than other existing methods for the model. Unlike all other existing methods, it also does not require tuning any additional parameter besides the model parameter. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.
keywords: total variation. variational method Newton's method mean curvature Denoising
Augmented Lagrangian method for a mean curvature based image denoising model
Wei Zhu Xue-Cheng Tai Tony Chan
Inverse Problems & Imaging 2013, 7(4): 1409-1432 doi: 10.3934/ipi.2013.7.1409
High order derivative information has been widely used in developing variational models in image processing to accomplish more advanced tasks. However, it is a nontrivial issue to construct efficient numerical algorithms to deal with the minimization of these variational models due to the associated high order Euler-Lagrange equations. In this paper, we propose an efficient numerical method for a mean curvature based image denoising model using the augmented Lagrangian method. A special technique is introduced to handle the mean curvature model for the augmented Lagrangian scheme. We detail the procedures of finding the related saddle-points of the functional. We present numerical experiments to illustrate the effectiveness and efficiency of the proposed numerical method, and show a few important features of the image denoising model such as keeping corners and image contrast. Moreover, a comparison with the gradient descent method further demonstrates the efficiency of the proposed augmented Lagrangian method.
keywords: mean curvature Augmented lagrangian method image denoising.
Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours
Egil Bae Xue-Cheng Tai Wei Zhu
Inverse Problems & Imaging 2017, 11(1): 1-23 doi: 10.3934/ipi.2017001

In this paper, we propose an image segmentation model where an $L^1$ variant of the Euler's elastica energy is used as boundary regularization. An interesting feature of this model lies in its preference for convex segmentation contours. However, due to the high order and non-differentiability of Euler's elastica energy, it is nontrivial to minimize the associated functional. As in recent work on the ordinary $L^2$-Euler's elastica model in imaging, we propose using an augmented Lagrangian method to tackle the minimization problem. Specifically, we design a novel augmented Lagrangian functional that deals with the mean curvature term differently than in previous works. The new treatment reduces the number of Lagrange multipliers employed, and more importantly, it helps represent the curvature more effectively and faithfully. Numerical experiments validate the efficiency of the proposed augmented Lagrangian method and also demonstrate new features of this particular segmentation model, such as shape driven and data driven properties.

keywords: Euler's elastica augmented Lagrangian method image segmentation convex contour variational model

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