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Estimation of fractal dimension and fractal curvatures from digital images

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  • Spodarev, Evgeny
  • Straka, Peter
  • Winter, Steffen

Abstract

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e. volume, surface area, Euler characteristic, etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behavior, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.

Suggested Citation

  • Spodarev, Evgeny & Straka, Peter & Winter, Steffen, 2015. "Estimation of fractal dimension and fractal curvatures from digital images," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 134-152.
    • Handle: RePEc:eee:chsofr:v:75:y:2015:i:c:p:134-152
    DOI: 10.1016/j.chaos.2015.02.011
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    References listed on IDEAS

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    1. Martı́nez-López, F. & Cabrerizo-Vı́lchez, M.A. & Hidalgo-Álvarez, R., 2001. "An improved method to estimate the fractal dimension of physical fractals based on the Hausdorff definition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 298(3), pages 387-399.
    2. Cutler, C. D. & Dawson, D. A., 1989. "Estimation of dimension for spatially distributed data and related limit theorems," Journal of Multivariate Analysis, Elsevier, vol. 28(1), pages 115-148, January.
    3. DRYGAS, Hilmar, 1976. "Weak and strong consistency of the least squares estimators in regression models," LIDAM Reprints CORE 236, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Rodriguez-Romo, Suemi & Sosa-Herrera, Antonio, 2013. "Lacunarity and multifractal analysis of the large DLA mass distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(16), pages 3316-3328.
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    Cited by:

    1. Lahmiri, Salim, 2016. "Image characterization by fractal descriptors in variational mode decomposition domain: Application to brain magnetic resonance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 456(C), pages 235-243.
    2. Chamorro-Posada, Pedro, 2016. "A simple method for estimating the fractal dimension from digital images: The compression dimension," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 562-572.
    3. Zuo, Xue & Tang, Xiang & Zhou, Yuankai, 2020. "Influence of sampling length on estimated fractal dimension of surface profile," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).

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