Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split B...

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Vydané v:	International journal of computer vision Ročník 92; číslo 3; s. 265 - 280
Hlavný autor:	Setzer, Simon
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Boston Springer US 01.05.2011 Springer Springer Nature B.V
Predmet:	Algorithms Applied sciences Artificial Intelligence Computer Imaging Computer Science Computer science; control theory; systems Equipment and supplies Exact sciences and technology Frames Hilbert space Image processing Image Processing and Computer Vision Image processing equipment industry Mathematical models Methods Operators Pattern Recognition Pattern Recognition and Graphics Pattern recognition. Digital image processing. Computational geometry Shrinkage Splitting Vision Wavelet transforms Forward-backward splitting Alternating split Bregman algorithm Augmented Lagrangian method Douglas-Rachford splitting Bregman methods Image denoising Lagrangian Image processing Total variation Image restoration Modeling Gradient descent Fix point Discretization Multistep method Operator splitting Variational calculus Lagrangian method
ISSN:	0920-5691, 1573-1405
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Abstract	We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view—as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods.
AbstractList	We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view--as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods. Keywords Douglas-Rachford splitting * Forward-backward splitting * Bregman methods * Augmented Lagrangian method * Alternating split Bregman algorithm * Image denoising We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view--as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods. We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view--as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods.[PUBLICATION ABSTRACT]
Audience	Academic
Author	Setzer, Simon
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Keywords	Forward-backward splitting Alternating split Bregman algorithm Augmented Lagrangian method Douglas-Rachford splitting Bregman methods Image denoising Lagrangian Image processing Total variation Image restoration Modeling Gradient descent Fix point Discretization Multistep method Operator splitting Variational calculus Lagrangian method
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Snippet	We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman...
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SubjectTerms	Algorithms Applied sciences Artificial Intelligence Computer Imaging Computer Science Computer science; control theory; systems Equipment and supplies Exact sciences and technology Frames Hilbert space Image processing Image Processing and Computer Vision Image processing equipment industry Mathematical models Methods Operators Pattern Recognition Pattern Recognition and Graphics Pattern recognition. Digital image processing. Computational geometry Shrinkage Splitting Vision Wavelet transforms
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Title	Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing
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