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A numerical method to differentiate between “pure” and “impure” fractals

Tom 109 / 2016

Martin Obert, Mailin Tiefenbach, Gunhild Martels, Barbara Ahlemeyer, Gabriele A. Krombach, Marian Kampschulte Banach Center Publications 109 (2016), 143-153 MSC: Primary 92B99; Secondary 92-08. DOI: 10.4064/bc109-0-9

Streszczenie

We introduce a numerical method that enables the differentiation between natural fractal image representations of objects that have only fractal properties (“pure” fractals) from those that have partial Euclidean and partial fractal properties (“impure” fractals). We evaluate our classification method on a numerically constructed fractal that serves as “pure” fractal and compare it with an “impure” digital image representation of the blood vessel system of a mouse-liver. Our classification method is based on the study of the invariance of a data set under different levels of image-data-point reductions — comparing a dissection and a random erosion method. The various image-sets were characterized by the fractal dimension $D$, evaluated by the mass radius method. The differentiation between “pure” and “impure” fractals is performed by their different and opposite behavior when the reduction is performed at average reduction levels: We find for the “pure” fractal that dissections lead to no $D$-difference between complete and reduced sets; contrary, the erosion method leads to a noticeable $D$-difference. These findings swap for the “impure” fractal. Here we find that dissections lead to distinct $D$-differences between complete and reduced sets; contrary, the erosion method leads to no change in these $D$-differences. We assume that this differentiation can be applied successfully when automated image classifications are desired or necessary.

Autorzy

  • Martin ObertDepartment of Diagnostic and Interventional Radiology
    University Hospital Giessen
    Justus-Liebig-University Giessen
    Klinikstrasse 33
    35392 Giessen, Germany
    e-mail
  • Mailin TiefenbachDepartment of Diagnostic and Interventional Radiology
    University Hospital Giessen
    Justus-Liebig-University Giessen
    Klinikstrasse 33
    35392 Giessen, Germany
  • Gunhild MartelsDepartment of Diagnostic and Interventional Radiology
    University Hospital Giessen
    Justus-Liebig-University Giessen
    Klinikstrasse 33
    35392 Giessen, Germany
  • Barbara AhlemeyerInstitute for Anatomy and Cell Biology
    Division of Medical Cell Biology II
    Justus-Liebig-University Giessen
    Aulweg 123
    35392 Giessen, Germany
  • Gabriele A. KrombachDepartment of Diagnostic and Interventional Radiology
    University Hospital Giessen
    Justus-Liebig-University Giessen
    Klinikstrasse 33
    35392 Giessen, Germany
  • Marian KampschulteDepartment of Diagnostic and Interventional Radiology
    University Hospital Giessen
    Justus-Liebig-University Giessen
    Klinikstrasse 33
    35392 Giessen, Germany

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