Numerical aspects of TV flow

Breuß, Michael; Bürgel, Andrea; Brox, Thomas; Sonar, Thomas; Weickert, Joachim

doi:10.1007/s11075-005-9006-2

Cited by 9 publications

(6 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The experimental orders of convergence show some variations. In most cases, they are in the range [4,6] for p = 5 and in [4,7] for p = 6. There are only small deviations (up to approximately 10%) between the errors of the split forms and the corresponding unsplit forms.…”

Section: Convergence Studiesmentioning

confidence: 99%

On conservation and stability properties for summation-by-parts schemes

Nordström

Ruggiu

2017

Journal of Computational Physics

View full text Add to dashboard Cite

High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework. However, if generalised SBP operators are used, the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely -interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain. Here, new formulations are proposed, allowing the creation of discretisations using general SBP operators that are both conservative and stable. Thus, several shortcomings that might be attributed to generalised SBP operators are overcome (cf.There are classical finite difference operators that can be interpreted in an analytical setting similar to the one described above [19]. However, in general -to the authors knowledge -it is not known whether finite difference SBP operators correspond to an analytical basis. Nevertheless, the basic requirement of the SBP property (4) can be enhanced by accuracy conditions in order to get a useful definition of (numerical) SBP operators, cf. [11,59].Definition 1. Using a numerical representation u = (u 0 , . . . , u p ) T of a function u in a nodal basis, the operators D , R , M , and B described above form a qth order SBP discretisation, if 1. the derivative matrix D is exact for polynomials of degree q, 2. the mass / norm matrix M is symmetric and positive definite, 3. the SBP property (4) is fulfilled, and

show abstract

Section: Convergence Studiesmentioning

confidence: 99%

On conservation and stability properties for summation-by-parts schemes

Nordström

Ruggiu

2017

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…[38,40,41,194]). It is obtained if the ROF-model (2) is considered as an implicit time discretization of a flow with time step D 1 and iterated to compute u.t C / as a solution of…”

Section: Total Variation Flowmentioning

confidence: 99%

A Guide to the TV Zoo

Burger

Osher²

2013

Level Set and PDE Based Reconstruction Methods in Imaging

118

View full text Add to dashboard Cite

Total variation methods and similar approaches based on regularizations with`1-type norms (and seminorms) have become a very popular tool in image processing and inverse problems due to peculiar features that cannot be realized with smooth regularizations. In particular total variation techniques had particular success due to their ability to realize cartoon-type reconstructions with sharp edges. Due to an explosion of new developments in this field within the last decade it is a difficult task to keep an overview of the major results in analysis, the computational schemes, and the application fields. With these lectures we attempt to provide such an overview, of course biased by our major lines of research.We are focusing on the basic analysis of total variation methods and the extension of the original ROF-denoising model due various application fields. Furthermore we provide a brief discussion of state-of-the art computational methods and give an outlook to applications in different disciplines.

show abstract

“…For singular diffusion equations, our LAS approaches can also be regarded as alternatives to the finite difference schemes in [8], the finite element scheme in [29], and the level set approach considered in [21].…”

Section: Related Workmentioning

confidence: 99%

Locally analytic schemes: A link between diffusion filtering and wavelet shrinkage

Welk

Steidl

Weickert

2008

Applied and Computational Harmonic Analysis

Self Cite

View full text Add to dashboard Cite

We study a class of numerical schemes for nonlinear diffusion filtering that offers insights on the design of novel wavelet shrinkage rules for isotropic and anisotropic image enhancement. These schemes utilise analytical or semi-analytical solutions to dynamical systems that result from space-discrete nonlinear diffusion filtering on minimalistic images with 2 × 2 pixels. We call them locally analytic schemes (LAS) and locally semi-analytic schemes (LSAS), respectively. They can be motivated from discrete energy functionals, offer sharp edges due to their locality, are very simple to implement because of their explicit nature, and enjoy unconditional absolute stability. They are applicable to singular nonlinear diffusion filters such as TV flow, to bounded nonlinear diffusion filters of Perona-Malik type, and to tensor-driven anisotropic methods such as edge-enhancing or coherence-enhancing diffusion filtering. The fact that these schemes use processes within 2 × 2-pixel blocks allows to connect them to shift-invariant Haar wavelet shrinkage on a single scale. This interpretation leads to novel shrinkage rules for two-and higher-dimensional images that are scalar-, vector-or tensor-valued. Unlike classical shrinkage strategies they employ a diffusion-inspired coupling of the wavelet channels that guarantees an approximation with an excellent degree of rotation invariance. By extending these schemes from a single scale to a multi-scale setting, we end up at hybrid methods that demonstrate the possibility to realise the effects of the most sophisticated diffusion filters within a fairly simplistic wavelet setting that requires only Haar wavelets in conjunction with coupled shrinkage rules.

show abstract

Numerical aspects of TV flow

Cited by 9 publications

References 20 publications

On conservation and stability properties for summation-by-parts schemes

On conservation and stability properties for summation-by-parts schemes

A Guide to the TV Zoo

Locally analytic schemes: A link between diffusion filtering and wavelet shrinkage

Contact Info

Product

Resources

About