Fast Linearized Augmented Lagrangian Method for Euler's Elastica Model

2017 ◽  
Vol 10 (1) ◽  
pp. 98-115 ◽  
Author(s):  
Jun Zhang ◽  
Rongliang Chen ◽  
Chengzhi Deng ◽  
Shengqian Wang
Keyword(s):  
Augmented Lagrangian ◽  
Complexity Analysis ◽  
Augmented Lagrangian Method ◽  
Saddle Points ◽  
Computational Cost ◽  
Lagrangian Method ◽  
Previous Method ◽  
High Order ◽  
Noise Elimination ◽  
Euler’S Elastica

AbstractRecently, many variational models involving high order derivatives have been widely used in image processing, because they can reduce staircase effects during noise elimination. However, it is very challenging to construct efficient algorithms to obtain the minimizers of original high order functionals. In this paper, we propose a new linearized augmented Lagrangian method for Euler's elastica image denoising model. We detail the procedures of finding the saddle-points of the augmented Lagrangian functional. Instead of solving associated linear systems by FFT or linear iterative methods (e.g., the Gauss-Seidel method), we adopt a linearized strategy to get an iteration sequence so as to reduce computational cost. In addition, we give some simple complexity analysis for the proposed method. Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method, and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.

Data Driven Multivariate Adaptive Regression Splines Based Simulation Optimization

2010 ◽  
Vol 44-47 ◽  
pp. 3800-3806
Author(s):  
Hu Ping Mao ◽  
Yi Zhong Wu ◽  
Li Ping Chen
Keyword(s):  
Data Processing ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Computational Cost ◽  
Lagrangian Method ◽  
Multivariate Adaptive Regression Splines ◽  
Data Driven ◽  
Optimization Approach ◽  
Global Test ◽  
Adaptive Regression

This paper proposes a data driven based optimization approach which combines augmented Lagrangian method, MARS with effective data processing. In the approach, an expensive simulation run is required if and only if a nearby data point does not exist in the cumulatively growing database. Over time the database matures and is enriched as more and more optimizations have been performed. MARS is a self-adaptive regression process, which fits in with the multidimensional problems, and uses a modified recursive partitioning strategy to simplify high-dimensional problems into smaller yet highly accurate models. Combining the local response surface of MARS and augmented Lagrangian method improve sequential approximation optimization and reduce simulation times by effective data processing, yet maintain a low computational cost. The approach is applied to a six dimensional test function, a ten dimensional engineering problem and a two dimensional global test functions to demonstrate its feasibility and convergence, and yet some limiting properties.

scholarly journals An augmented Lagrangian method for solving total variation (TV)-based image registration model

2020 ◽  
Vol 14 ◽  
pp. 174830262097353
Author(s):  
Noppadol Chumchob ◽  
Ke Chen
Keyword(s):  
Image Registration ◽  
Augmented Lagrangian ◽  
Numerical Solutions ◽  
Augmented Lagrangian Method ◽  
Closed Form Solution ◽  
Numerical Algorithms ◽  
Lagrangian Method ◽  
Form Solution ◽  
The Other ◽  
Other Hand

Variational methods for image registration basically involve a regularizer to ensure that the resulting well-posed problem admits a solution. Different choices of regularizers lead to different deformations. On one hand, the conventional regularizers, such as the elastic, diffusion and curvature regularizers, are able to generate globally smooth deformations and generally useful for many applications. On the other hand, these regularizers become poor in some applications where discontinuities or steep gradients in the deformations are required. As is well-known, the total (TV) variation regularizer is more appropriate to preserve discontinuities of the deformations. However, it is difficult in developing an efficient numerical method to ensure that numerical solutions satisfy this requirement because of the non-differentiability and non-linearity of the TV regularizer. In this work we focus on computational challenges arising in approximately solving TV-based image registration model. Motivated by many efficient numerical algorithms in image restoration, we propose to use augmented Lagrangian method (ALM). At each iteration, the computation of our ALM requires to solve two subproblems. On one hand for the first subproblem, it is impossible to obtain exact solution. On the other hand for the second subproblem, it has a closed-form solution. To this end, we propose an efficient nonlinear multigrid (NMG) method to obtain an approximate solution to the first subproblem. Numerical results on real medical images not only confirm that our proposed ALM is more computationally efficient than some existing methods, but also that the proposed ALM delivers the accurate registration results with the desired property of the constructed deformations in a reasonable number of iterations.

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