Estimation of fractal dimension and fractal curvatures from digital images

Spodarev, Evgeny; Straka, Peter; Winter, Steffen

doi:10.1016/j.chaos.2015.02.011

Cited by 11 publications

(4 citation statements)

References 52 publications

(96 reference statements)

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“…We briefly recall their definitions. in connection with the modified Weyl-Berry conjecture, see, for example, [19] and the relevant references therein, and the Minkowski content is also a popular texture index (lacunarity) in applications characterizing the geometry of a given fractal structure beyond its fractal dimensions, see, for example, [2,20,25].…”

Section: Introductionmentioning

confidence: 99%

“…The numbers

{\underset{̲}{dim}}_{M} A : = inf false{t ⩾ 0 : {\underset{̲}{scriptM}}^{t} (A) = 0 false} = sup false{t ⩾ 0 : {\underset{̲}{scriptM}}^{t} (A) = \infty false}

and

{\bar{dim}}_{M} A = inf false{t ⩾ 0 : {\bar{scriptM}}^{t} (A) = 0 false} = sup false{t ⩾ 0 : {\bar{scriptM}}^{t} (A) = \infty false}

are usually called the lower and upper Minkowski dimension of

A

and lower and upper S‐dimension

{\underset{̲}{dim}}_{S} A

and

{\bar{dim}}_{S} A

A

are defined analogously with

{\underset{̲}{M}}^{t} false(A false)

and

{\bar{M}}^{t} false(A false)

replaced by

{\underset{̲}{S}}^{t} false(A false)

and

{\bar{S}}^{t} false(A false)

, respectively. Minkowski measurability plays an important role for instance in connection with the modified Weyl–Berry conjecture, see, for example, and the relevant references therein, and the Minkowski content is also a popular texture index (lacunarity) in applications characterizing the geometry of a given fractal structure beyond its fractal dimensions, see, for example, .…”

Section: Introductionmentioning

confidence: 99%

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Localization results for Minkowski contents

Winter

2018

J. London Math. Soc.

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It was shown recently that the Minkowski content of a bounded set A in double-struckRd with volume zero can be characterized in terms of the asymptotic behaviour of the boundary surface area of its parallel sets Ar as the parallel radius r tends to 0. Here we discuss localizations of such results. The asymptotic behaviour of the local parallel volume of A relative to a suitable second set normalΩ can be understood in terms of the suitably defined local surface area relative to normalΩ. Also a measure version of this relation is shown: Viewing the Minkowski content as a locally determined measure, this local Minkowski content can be obtained as a weak limit of suitably rescaled surface measures of close parallel sets (local S‐content). Such measure relations had been observed before for self‐similar and some self‐conformal sets. They are now established for arbitrary closed sets, including even the case of unbounded sets. The results are based on a localization of Stachó's famous formula relating the boundary surface area of Ar to the derivative of the volume function at r.

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Section: Introductionmentioning

confidence: 99%

“…The numbers

{\underset{̲}{dim}}_{M} A : = inf false{t ⩾ 0 : {\underset{̲}{scriptM}}^{t} (A) = 0 false} = sup false{t ⩾ 0 : {\underset{̲}{scriptM}}^{t} (A) = \infty false}

and

{\bar{dim}}_{M} A = inf false{t ⩾ 0 : {\bar{scriptM}}^{t} (A) = 0 false} = sup false{t ⩾ 0 : {\bar{scriptM}}^{t} (A) = \infty false}

are usually called the lower and upper Minkowski dimension of

A

and lower and upper S‐dimension

{\underset{̲}{dim}}_{S} A

and

{\bar{dim}}_{S} A

A

are defined analogously with

{\underset{̲}{M}}^{t} false(A false)

and

{\bar{M}}^{t} false(A false)

replaced by

{\underset{̲}{S}}^{t} false(A false)

and

{\bar{S}}^{t} false(A false)

Section: Introductionmentioning

confidence: 99%

Localization results for Minkowski contents

Winter

2018

J. London Math. Soc.

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“…Minkowski functionals and Minkowski tensors [41] with the aim of extracting new fractal features from fundus images [42]. In addition, fractal analysis will be used to detect red dark lesions as microaneurysms and haemorrhages.…”

Section: Discussionmentioning

confidence: 99%

Evaluation of fractal dimension effectiveness for damage detection in retinal background

Colomer

Naranjo

Janvier

et al. 2018

Journal of Computational and Applied Mathematics

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This work investigates the characterization of bright lesions in retinal fundus images using texture analysis techniques. Exudates and drusen are evidences of retinal damage in diabetic retinopathy (DR) and age-related macular degeneration (AMD) respectively. An automatic detection of pathological tissues could make possible an early detection of these diseases. In this work, fractal analysis is explored in order to discriminate between pathological and healthy retinal texture. After a deep preprocessing step, in which a spatial and colour normal

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“…The algorithms normally used for calculating the fractal dimension of images [9][10][11] are rather involved and the absence of simple to use and freely distributed software tools can limit the widespread use of fractal methods in digital image analysis. Furthermore, they are typically based on the processing of the image data that should be extracted from the picture file when this is the available source format.…”

Section: Introductionmentioning

confidence: 99%

A simple method for estimating the fractal dimension from digital images: The compression dimension

Chamorro‐Posada

2016

Chaos, Solitons & Fractals

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The fractal structure of real world objects is often analyzed using digital images.In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored as digital image files. The computational scheme can be implemented using readily available free software. Its simplicity also makes it very interesting for introductory elaborations of basic concepts of fractal geometry, complexity, and information theory. A test of the computational scheme using limited-quality images of well-defined fractal sets obtained from the Internet and free software has been performed. Also, a systematic evaluation of the proposed method using computer generated images of the Weierstrass cosine function shows an accuracy comparable to those of the methods most commonly used to estimate the dimension of fractal data sequences applied to the same test problem. * pedcha@tel.uva.es

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Estimation of fractal dimension and fractal curvatures from digital images

Cited by 11 publications

References 52 publications

Localization results for Minkowski contents

Localization results for Minkowski contents

Evaluation of fractal dimension effectiveness for damage detection in retinal background

A simple method for estimating the fractal dimension from digital images: The compression dimension

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