Rotationally Invariant Wavelet Shrinkage

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

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“…SΛ denotes the coupled shrinkage functioñ (Chambolle 2005;Mrázek and Weickert 2003;Welk et al 2008). Similarly, we obtain…”

Section: Application To Image Denoisingmentioning

confidence: 99%

“…The corresponding circulant matrices are denoted by H 0 ∈ R n,n and H 1 ∈ R n,n . Then, the following matrix fulfills W * W = I N but WW * = I 4N : Mrázek and Weickert (2003) and Welk et al (2008) showed that…”

Section: Total Variation Regularizationmentioning

confidence: 99%

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

2010

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“…Mrazek and Weickert [52] presented a translation and rotation invariant method that combines wavelet transforms with nonlinear diffusion filters. The iterated filter is not completely rotation invariant, but the asymmetry decreases with a higher number of iterations.…”

Section: Invariance; Review Of Previous Workmentioning

confidence: 99%

Three-dimensional object registration using wavelet features

Chalfant

Patrikalakis

2009

Engineering with Computers

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Recent developments in shape-based modeling and data acquisition have brought three-dimensional models to the forefront of computer graphics and visualization research. New data acquisition methods are producing large numbers of models in a variety of fields. Three-dimensional shape-based matching and registration (alignment) are key to the useful application of such models in areas from automated surface inspection to cancer detection and surgery. The three-dimensional models in these applications are typically huge. State-of-the-art simulations in computational fluid dynamics produce upward of four terabytes of data per second of flow. Research-level magnetic resonance imaging (MRI) resolutions can reach 1 cubic micro-meter. As a result, object registration and matching algorithms must handle very large amounts of data.The algorithms developed in this thesis accomplish automatic registration and matching of three-dimensional voxelized models. We employ features in a wavelet transform domain to accomplish registration. The features are extracted in a multiresolutional format, thus delineating features at various scales for robust and rapid matching. Registration is achieved through seeking peaks in sets of rotation quaternions using a voting scheme, then separately identifying translation. The method is robust to occlusion, clutter and noise. The efficacy of the algorithm is demonstrated through examples from solid modeling and medical imaging applications.

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“…In [13], the authors proposed to choose a diffusion inspired shrinkage function that simultaneously depends on c 1,2 , c 2,1 and c 2,2 . In contrast to the classical wavelet shrinkage, this results in an improved rotation invariance of the resulting image.…”

Section: Haar Wavelet Shrinkagementioning

confidence: 99%

“…It does not require regularised diffusivities of type (5). Interestingly, it can also be related to a recently introduced family of shift invariant wavelet shrinkage methods with coupled shrinkage functions [13].…”

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confidence: 99%

A Four-Pixel Scheme for Singular Differential Equations

Welk

Weickert

Steidl

2005

Lecture Notes in Computer Science

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Abstract. Singular diffusion equations such as total variation (TV) and balanced forward-backward (BFB) diffusion are appealing: They have a finite extinction time, and experiments show that piecewise constant structures evolve. Unfortunately, their implementation is awkward. The goal of this paper is to introduce a novel class of numerical methods for these equations in the 2D case. They are simple to implement, absolutely stable and do not require any regularisation in order to make the diffusivity bounded. Our schemes are based on analytical solutions for 2×2-pixel images which are combined by means of an additive operator splitting (AOS). We show that they may also be regarded as iterated 2D Haar wavelet shrinkage. Experiments demonstrate the favourable performance of our numerical algorithm.

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Rotationally Invariant Wavelet Shrinkage

Cited by 31 publications

References 14 publications

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

Three-dimensional object registration using wavelet features

A Four-Pixel Scheme for Singular Differential Equations

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