Morphological Amoebas and Partial Differential Equations

Welk, Martin; Breuß, Michael

doi:10.1016/b978-0-12-800144-8.00003-3

Cited by 9 publications

(30 citation statements)

References 44 publications

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“…Asymptotic analysis results are available for Oja and transformation-retransformation L 1 median filtering. A PDE limit stated in [80] for standard L 1 median filtering of n-variate planar images that could be applied to trivariate planar images was incomplete; a corrected result still needs to be published. Also in the case of trivariate planar images both Oja and transformation-retransformation L 1 median filtering approximate the same PDE as stated in the following results.…”

Section: Trivariate Planar Imagesmentioning

confidence: 99%

Multivariate Medians for Image and Shape Analysis

Welk¹

2019

Preprint

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Having been studied since long by statisticians, multivariate median concepts found their way into the image processing literature in the course of the last decades, being used to construct robust and efficient denoising filters for multivariate images such as colour images but also matrix-valued images. Based on the similarities between image and geometric data as results of the sampling of continuous physical quantities, it can be expected that the understanding of multivariate median filters for images provides a starting point for the development of shape processing techniques. This paper presents an overview of multivariate median concepts relevant for image and shape processing. It focusses on their mathematical principles and discusses important properties especially in the context of image processing.

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Section: Trivariate Planar Imagesmentioning

confidence: 99%

Multivariate Medians for Image and Shape Analysis

Welk¹

2019

Preprint

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“…The proof of the proposition is analogous to the proof of Theorem 3, using the special case u x = v y = 1 of the following lemma. The lemma itself is corrected from [33] and rewritten for the three-channel case. (14), (15), and…”

Section: Affine Equivariant Transformed L 1 Medianmentioning

confidence: 99%

“…Multivariate median filters and PDE. While the above-mentioned relationship between univariate median filtering and the mean curvature motion PDE could be extended to relate also adaptive median filtering procedures [34] and further discrete filters [33] to well-understood PDEs of image processing, the picture changes when turning to multivariate median filtering. As demonstrated in [33], it is possible to derive some PDE for median filtering based on the spatial median as in [25].…”

Section: Introductionmentioning

confidence: 99%

Multivariate Median Filters and Partial Differential Equations

Welk

2016

J Math Imaging Vis

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Multivariate median filters have been proposed as generalisations of the well-established median filter for grey-value images to multi-channel images. As multivariate median, most of the recent approaches use the $L^1$ median, i.e.\ the minimiser of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalise to such a filter. However, the famous result by Guichard and Morel about approximation of the mean curvature motion PDE by median filtering does not have a comparably simple counterpart for $L^1$ multivariate median filtering. We discuss the affine equivariant Oja median and the affine equivariant transformation--retransformation $L^1$ median as alternatives to $L^1$ median filtering. We analyse multivariate median filters in a space-continuous setting, including the formulation of a space-continuous version of the transformation--retransformation $L^1$ median, and derive PDEs approximated by these filters in the cases of bivariate planar images, three-channel volume images and three-channel planar images. The PDEs for the affine equivariant filters can be interpreted geometrically as combinations of a diffusion and a principal-component-wise curvature motion contribution with a cross-effect term based on torsions of principal components. Numerical experiments are presented that demonstrate the validity of the approximation results.Comment: v2: Minor revision; a few equations, some text, and one reference added; typos correcte

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“…With various settings for these norms, it has been used e.g. in [45,46,77,78] to construct larger spatially adaptive neighbourhoods in images, so-called morphological amoebas. See also [75] for a more detailed description of the amoeba framework in a graph-based terminology.…”

Section: Graph Constructionmentioning

confidence: 99%

“…Second, we define the adaptive patch graph G A w (p, ) as the subgraph of G w which includes all nodes q for which G w contains a path from p to q with total weight less or equal . In the terminology of [45,46,77,78], the node set of G A w (p, ) is a morphological amoeba of amoeba radius around p, which we will denote by A (p). Note that the graph G A w (p, ) encodes image information not only in its edge weights, but also in its node set A (p).…”

Section: Graph Constructionmentioning

confidence: 99%

Graph Entropies in Texture Segmentation of Images

Welk¹

2016

Mathematical Foundations and Applications of Graph Entropy

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We study the applicability of a set of texture descriptors introduced in recent work by the author to texture-based segmentation of images. The texture descriptors under investigation result from applying graph indices from quantitative graph theory to graphs encoding the local structure of images. The underlying graphs arise from the computation of morphological amoebas as structuring elements for adaptive morphology, either as weighted or unweighted Dijkstra search trees or as edge-weighted pixel graphs within structuring elements. In the present paper we focus on texture descriptors in which the graph indices are entropy-based, and use them in a geodesic active contour framework for image segmentation. Experiments on several synthetic and one real-world image are shown to demonstrate texture segmentation by this approach. Forthermore, we undertake an attempt to analyse selected entropy-based texture descriptors with regard to what information about texture they actually encode. Whereas this analysis uses some heuristic assumptions, it indicates that the graph-based texture descriptors are related to fractal dimension measures that have been proven useful in texture analysis.

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Morphological Amoebas and Partial Differential Equations

Cited by 9 publications

References 44 publications

Multivariate Medians for Image and Shape Analysis

Multivariate Medians for Image and Shape Analysis

Multivariate Median Filters and Partial Differential Equations

Graph Entropies in Texture Segmentation of Images

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