Minimizing total variation flow

Andreu, F.; Ballester, Coloma; Caselles, Vicent; Mazón, José M.

doi:10.1016/s0764-4442(00)01729-8

Cited by 114 publications

(157 citation statements)

References 17 publications

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“…For p = 0, linear homogeneous diffusion is obtained, which is equivalent to Gaussian smoothing with standard deviation √ 2t, and forms the basis of Gaussian scale-space theory [22,42]. For p = 1 one obtains the total variation (TV) flow [2,14], the diffusion filter that corresponds to TV minimisation [40] with a penaliser Ψ(|∇u| 2 ) = 2|∇u|. TV flow offers a number of interesting properties such as finite extinction time [3], shape-preserving qualities [5], and equivalence to TV regularisation in 1-D [11].…”

Section: Diffusivity Functionsmentioning

confidence: 99%

“…In contrast to an earlier conference publication [10], the nonlinear structure tensor, as it is proposed here, applies the original matrix-valued diffusion techniques from [44] and [50], thus using all available information for steering the diffusion. Moreover, it employs diffusivity functions based on total variation (TV) flow [2,14], the diffusion filter corresponding to TV regularisation [40]. This flow offers a number of favourable properties, and it does not require additional contrast parameters such as most other diffusivity functions.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear structure tensors

Brox

Weickert

Burgeth

et al. 2006

Image and Vision Computing

219

139

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In this article we introduce nonlinear versions of the popular structure tensor, also known as second moment matrix. These nonlinear structure tensors replace the Gaussian smoothing of the classical structure tensor by discontinuity-preserving nonlinear diffusions. While nonlinear diffusion is a well-established tool for scalar and vectorvalued data, it has not often been used for tensor images so far. Two types of nonlinear diffusion processes for tensor data are studied: an isotropic one with a scalar-valued diffusivity, and its anisotropic counterpart with a diffusion tensor. We prove that these schemes preserve the positive semidefiniteness of a matrix field and are therefore appropriate for smoothing structure tensor fields. The use of diffusivity functions of total variation (TV) type allows us to construct nonlinear structure tensors without specifying additional parameters compared to the conventional structure tensor. The performance of nonlinear structure tensors is demonstrated in three fields where the classic structure tensor is frequently used: orientation estimation, optic flow computation, and corner detection. In all these cases the nonlinear structure tensors demonstrate their superiority over the classical linear one. Our experiments also show that for corner detection based on nonlinear structure tensors, anisotropic nonlinear tensors give the most precise localisation.

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Section: Diffusivity Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonlinear structure tensors

Brox

Weickert

Burgeth

et al. 2006

Image and Vision Computing

219

139

View full text Add to dashboard Cite

show abstract

“…Diffusion with this diffusivity is called total variation (TV) flow [1], which is the diffusion filter corresponding to TV regularization [32].…”

Section: Structure Tensors Based On Nonlinear Diffusionmentioning

confidence: 99%

Adaptive Structure Tensors and their Applications

Brox

Boomgaard

Lauze

et al. 2006

Mathematics and Visualization

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The structure tensor, also known as second moment matrix or Förstner interest operator, is a very popular tool in image processing. Its purpose is the estimation of orientation and the local analysis of structure in general. It is based on the integration of data from a local neighborhood. Normally, this neighborhood is defined by a Gaussian window function and the structure tensor is computed by the weighted sum within this window. Some recently proposed methods, however, adapt the computation of the structure tensor to the image data. There are several ways how to do that. This article wants to give an overview of the different approaches, whereas the focus lies on the methods based on robust statistics and nonlinear diffusion. Furthermore, the dataadaptive structure tensors are evaluated in some applications. Here the main focus lies on optic flow estimation, but also texture analysis and corner detection are considered.

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“…The total variation problem is however more "regular" than the inviscid Bingham problem, since it is also monotone in L 1 (Ω), as proved in [1]. The theory of monotone problems in Banach spaces is provided in [7,2,3].…”

Section: Introductionmentioning

confidence: 98%

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

2014

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We study approximations by conforming methods of the solution to the variational inequality ∂ t u, v − u + ψ(v) − ψ(u) ≥ f, v − u , which arises in the context of inviscid incompressible Bingham type fluid flows and of the total variation flow problem. In the general context of a convex lower semi-continuous functional ψ on a Hilbert space, we prove the convergence of time implicit space conforming approximations, without viscosity and for non-smooth data. Then we introduce a general class of total variation functionals ψ, for which we can apply the regularization method. We consider the time implicit regularized, linearized or not, algorithms, and prove their convergence for general total variation functionals. A comparison with an analytical solution concludes this study.

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Minimizing total variation flow

Cited by 114 publications

References 17 publications

Nonlinear structure tensors

Nonlinear structure tensors

Adaptive Structure Tensors and their Applications

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

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