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Texture Discrimination via Hilbert Curve Path Based Information Quantifiers

Aurelio F. Bariviera, Roberta Hansen, Verónica E. Pastor

TL;DR

The paper addresses robust texture classification under rotation, noise, and color variations by transforming 2D textures into 1D sequences using a Hilbert curve, then applying Bandt-Pompe symbolization to obtain a probability distribution over ordinal patterns. It computes three information-theoretic quantifiers, namely the normalized Shannon entropy ${\cal H}$, the permutation-based statistical complexity ${\cal C}_{JS}$, and the Fisher information measure ${\mathcal F}$, and analyzes them via the Complexity-Entropy and Entropy-Fisher planes (CECP/FECP). The approach is validated on self-similar multifractal and Brownian surfaces, and on Normalized and Colored Brodatz textures, showing invariance to rotations and, in the color case, robustness to color variations, with clear discrimination among texture classes. The method offers a simple, locality-preserving, and scalable descriptor that can be extended to higher dimensions and has potential applications in domains such as medical imaging and industrial texture analysis.

Abstract

The analysis of the spatial arrangement of colors and roughness/smoothness of figures is relevant due to its wide range of applications. This paper proposes a texture classification method that extracts data from images using the Hilbert curve. Three information theory quantifiers are then computed: permutation entropy, permutation complexity, and Fisher information measure. The proposal exhibits some important properties: (i) it allows to discriminate figures according to varying degrees of correlations (as measured by the Hurst exponent), (ii) it is invariant to rotation and symmetry transformations, (iii) it can be used either in black and white or color images. Validations have been made not only using synthetic images but also using the well-known Brodatz image database.

Texture Discrimination via Hilbert Curve Path Based Information Quantifiers

TL;DR

The paper addresses robust texture classification under rotation, noise, and color variations by transforming 2D textures into 1D sequences using a Hilbert curve, then applying Bandt-Pompe symbolization to obtain a probability distribution over ordinal patterns. It computes three information-theoretic quantifiers, namely the normalized Shannon entropy , the permutation-based statistical complexity , and the Fisher information measure , and analyzes them via the Complexity-Entropy and Entropy-Fisher planes (CECP/FECP). The approach is validated on self-similar multifractal and Brownian surfaces, and on Normalized and Colored Brodatz textures, showing invariance to rotations and, in the color case, robustness to color variations, with clear discrimination among texture classes. The method offers a simple, locality-preserving, and scalable descriptor that can be extended to higher dimensions and has potential applications in domains such as medical imaging and industrial texture analysis.

Abstract

The analysis of the spatial arrangement of colors and roughness/smoothness of figures is relevant due to its wide range of applications. This paper proposes a texture classification method that extracts data from images using the Hilbert curve. Three information theory quantifiers are then computed: permutation entropy, permutation complexity, and Fisher information measure. The proposal exhibits some important properties: (i) it allows to discriminate figures according to varying degrees of correlations (as measured by the Hurst exponent), (ii) it is invariant to rotation and symmetry transformations, (iii) it can be used either in black and white or color images. Validations have been made not only using synthetic images but also using the well-known Brodatz image database.
Paper Structure (18 sections, 8 equations, 19 figures)

This paper contains 18 sections, 8 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Statistical Information Quantifiers
  3. Shannon Entropy
  4. Statistical complexity
  5. Fisher Information Measure
  6. Bandt and Pompe symbolization method
  7. Complexity Entropy and Entropy Fisher Causality Planes
  8. Proposed method
  9. The Hilbert curve
  10. Results
  11. Self-similar Multifractal Measures as Complex Images
  12. Multinomial multiplicative cascades
  13. Worked example
  14. Brownian surfaces
  15. Normalized Brodatz
  16. ...and 3 more sections

Figures (19)

  • Figure 1: Levels of planar Hilbert curve. (a) level 1, (b) level 2, (c) level 3, (d) level 6.
  • Figure 2: Construction of a SMM $\mu\!=\![p_1,p_2]$ (binomial cascade) on the unit interval with $p_1+p_2\!=\!1$, $p_i\!>\!0,i\!=\!1,2$, starting from the uniform distribution. Scheme of the first three steps and an advanced step of the recursive process are shown (see the text for the explanation).
  • Figure 3: Three steps of SMM construction, $\mu\!=\![p_1,p_2,p_3,p_4]$ on the unit square, $p_1\!=\!0.2434,p_2\!=\!0.2522,p_3\!=\!0.2566$ and $p_4\!=\!0.2478$. (a) Measure values attached to sub-squares at step 1, (b) The same but at step 2, (c) The same but at step 3 (only a few). The colours are in accordance to the measure values.
  • Figure 4: Multifractal-type image. (a) Measure values attached to sub-squares at step 10 (only the colours), (b) 3-dimensional rendering of (a) showing the SMM $\mu$. The colours are in accordance to the measure values. ($p_1\!<\!p_4\!<\!p_2\!<\!p_3$).
  • Figure 5: (a) Regular image obtained by ordering the $\mu$-values of the multifractal surface in Fig. \ref{['cuatrinom-2']} from lowest to highest, (b) 3-dimensional rendering of (a).
  • ...and 14 more figures