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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Veröffentlicht in:International journal of computer vision Jg. 92; H. 3; S. 265 - 280
1. Verfasser: Setzer, Simon
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Boston Springer US 01.05.2011
Springer
Springer Nature B.V
Schlagworte:
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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Beschreibung
Zusammenfassung: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.
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ISSN:0920-5691
1573-1405
DOI:10.1007/s11263-010-0357-3