&lt;title&gt;Shocks and other nonlinear filtering applied to image processing&lt;/title&gt;

Osher, Stanley; Rudin, Leonid

doi:10.1117/12.50835

Cited by 26 publications

(28 citation statements)

References 2 publications

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“…For the stabilised inverse linear diffusion process introduced by Osher and Rudin, it was not possible to establish a continuous well-posedness theory, but a stable minmod discretisation proved to work well in practice [9]. Later on, Breuß and Welk [2] showed that staircasing cannot be avoided by suitable space discretisations.…”

Section: Introductionmentioning

confidence: 99%

Theoretical Foundations for Discrete Forward-and-Backward Diffusion Filtering

Welk

Gilboa

Weickert

2009

Lecture Notes in Computer Science

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Abstract. Forward-and-backward (FAB) diffusion is a method for sharpening blurry images (Gilboa et al. 2002). It combines forward diffusion with a positive diffusivity and backward diffusion where negative diffusivities are used. The well-posedness properties of FAB diffusion are unknown, and it has been observed that standard discretisations can violate a maximum-minimum principle. We show that for a novel nonstandard space discretisation which pays specific attention to image extrema, one can apply a modification of the space-discrete well-posedness and scale-space framework of Weickert (1998). This allows to establish well-posedness and a maximum-minimum principle for the resulting dynamical system. In the fully discrete 1-D case with an explicit time discretisation, a maximum-minimum principle and total variation reduction are proved in spite of the fact that negative diffusivities may appear. This provides a theoretical justification for applying FAB diffusion to digital images.

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Section: Introductionmentioning

confidence: 99%

Theoretical Foundations for Discrete Forward-and-Backward Diffusion Filtering

Welk

Gilboa

Weickert

2009

Lecture Notes in Computer Science

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“…A considerable number of variants of shock filters have been considered in the literature [1,20,36,41,45,56]. When they are applied iteratively, experiments show that their steady state is given by a piecewise constant image with discontinuities ("shocks") between adjacent segments of constant grey value.…”

Section: Morphological Operationsmentioning

confidence: 99%

PDE-based Morphology for Matrix Fields: Numerical Solution Schemes

Burgeth

Breuß

Didas

et al. 2009

Tensors in Image Processing and Computer Vision

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Tensor fields are important in digital imaging and computer vision. Hence there is a demand for morphological operations to perform e.g. shape analysis, segmentation or enhancement procedures. Recently, fundamental morphological concepts have been transferred to the setting of fields of symmetric positive definite matrices, which are symmetric rank two tensors. This has been achieved by a matrixvalued extension of the nonlinear morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images. Having these two basic operations at our disposal, more advanced morphological operators such as top hats or morphological derivatives for matrix fields with symmetric, positive semidefinite matrices can be constructed. The approach realises a proper coupling of the matrix channels rather than treating them independently. However, from the algorithmic side the usual scalar morphological PDEs are transport equations that require special upwind-schemes or novel high-accuracy predictor-corrector approaches for their adequate numerical treatment. In this chapter we propose the non-trivial extension of these schemes to the matrix-valued setting by exploiting the special algebraic structure available for symmetric matrices. Furthermore we compare the performance and juxtapose the results of these novel matrix-valued high-resolution-type (HRT) numerical schemes by considering top hats and morphological derivatives applied to artificial and real world data sets.

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“…A considerable number of variants of shock filters have been considered in the literature [2,24,32,36,39,48]. When they are applied iteratively, experiments show that their steady state is given by a piecewise constant image with discontinuities ("shocks") between adjacent segments of constant grey value.…”

Section: Morphological Operationsmentioning

confidence: 99%

Adaptive Continuous-Scale Morphology for Matrix Fields

et al. 2009

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In this article we consider adaptive, PDE-driven morphological operations for 3D matrix fields arising e.g. in diffusion tensor magnetic resonance imaging (DT-MRI). The anisotropic evolution is steered by a matrix constructed from a structure tensor for matrix valued data. An important novelty is an intrinsically one-dimensional directional variant of the matrix-valued upwind schemes such as the Rouy-Tourin scheme. It enables our method to complete or enhance anisotropic structures effectively. A special advantage of our approach is that upwind schemes are utilised only in their basic one-dimensional version. No higher dimensional variants of the schemes themselves are required. Experiments with synthetic and real-world data substantiate the gap-closing and line-completing properties of the proposed method.

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<title>Shocks and other nonlinear filtering applied to image processing</title>

Cited by 26 publications

References 2 publications

Theoretical Foundations for Discrete Forward-and-Backward Diffusion Filtering

Theoretical Foundations for Discrete Forward-and-Backward Diffusion Filtering

PDE-based Morphology for Matrix Fields: Numerical Solution Schemes

Adaptive Continuous-Scale Morphology for Matrix Fields

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