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Gradient entropy (GradEn): The two dimensional version of slope entropy for image analysis

Runze Jiang, Pengjian Shang

TL;DR

This work addresses the challenge of quantifying irregularity in 2D image data by introducing GradEn, a two-dimensional extension of Slope Entropy that combines gradient-based symbolic patterns with amplitude information. GradEn computes horizontal, vertical, and diagonal gradients on 2×2 blocks, standardizes them, maps to a five-symbol alphabet, and evaluates the resulting 125 symbolic patterns via a normalized entropy $GradEn(X) = - \sum_{k=1}^{5^3} p(\pi_k) \log p(\pi_k) / \log(5^3)$. Extensive experiments on simulated data (colored noises, mixed processes, and logistic-map–derived distance matrices) and real-world datasets (texture images, gear faults, railway corrugation) show that GradEn offers superior discriminative power and lower computational cost than existing 2D entropy measures ($SampEn_{2d}$, $DistrEn_{2d}$, $PerEn_{2d}$). The results demonstrate GradEn’s potential as a practical tool for image characterization, texture analysis, and fault diagnostics, with avenues for generalization to broader parameter spaces and higher-dimensional data.

Abstract

Information theory and Shannon entropy are essential for quantifying irregularity in complex systems or signals. Recently, two-dimensional entropy methods, such as two-dimensional sample entropy, distribution entropy, and permutation entropy, have been proposed for analyzing 2D texture or image data. This paper introduces Gradient entropy (GradEn), an extension of slope entropy to 2D, which considers both symbolic patterns and amplitude information, enabling better feature extraction from image data. We evaluate GradEn with simulated data, including 2D colored noise, 2D mixed processes, and the logistic map. Results show the ability of GradEn to distinguish images with various characteristics while maintaining low computational cost. Real-world datasets, consist of texture, fault gear, and railway corrugation signals, demonstrate the superior performance of GradEn in classification tasks compared to other 2D entropy methods. In conclusion, GradEn is an effective tool for image characterization, offering a novel approach for image processing and recognition.

Gradient entropy (GradEn): The two dimensional version of slope entropy for image analysis

TL;DR

This work addresses the challenge of quantifying irregularity in 2D image data by introducing GradEn, a two-dimensional extension of Slope Entropy that combines gradient-based symbolic patterns with amplitude information. GradEn computes horizontal, vertical, and diagonal gradients on 2×2 blocks, standardizes them, maps to a five-symbol alphabet, and evaluates the resulting 125 symbolic patterns via a normalized entropy . Extensive experiments on simulated data (colored noises, mixed processes, and logistic-map–derived distance matrices) and real-world datasets (texture images, gear faults, railway corrugation) show that GradEn offers superior discriminative power and lower computational cost than existing 2D entropy measures (, , ). The results demonstrate GradEn’s potential as a practical tool for image characterization, texture analysis, and fault diagnostics, with avenues for generalization to broader parameter spaces and higher-dimensional data.

Abstract

Information theory and Shannon entropy are essential for quantifying irregularity in complex systems or signals. Recently, two-dimensional entropy methods, such as two-dimensional sample entropy, distribution entropy, and permutation entropy, have been proposed for analyzing 2D texture or image data. This paper introduces Gradient entropy (GradEn), an extension of slope entropy to 2D, which considers both symbolic patterns and amplitude information, enabling better feature extraction from image data. We evaluate GradEn with simulated data, including 2D colored noise, 2D mixed processes, and the logistic map. Results show the ability of GradEn to distinguish images with various characteristics while maintaining low computational cost. Real-world datasets, consist of texture, fault gear, and railway corrugation signals, demonstrate the superior performance of GradEn in classification tasks compared to other 2D entropy methods. In conclusion, GradEn is an effective tool for image characterization, offering a novel approach for image processing and recognition.

Paper Structure

This paper contains 14 sections, 4 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Gradient Entropy
  3. Experiments in simuated data
  4. Parameter Analysis
  5. Colored noise classification
  6. Robust Analysis
  7. Logistic map
  8. Computation cost
  9. Experiments in real-world data
  10. Texture Classification
  11. Gear fault diagnosis
  12. Railway Corrugation Diagnosis
  13. Conclusion
  14. Statement

Figures (15)

  • Figure 1: The sketch map of GradEn symbolization algorithm.
  • Figure 2: Three types of pixel size of GradEn algorithm.
  • Figure 3: The GradEn value of 2D Whitenoise with the change of $a$ and $b$.
  • Figure 4: The example of four types of 2D noises with the size equals to 100*100.
  • Figure 5: The boxplot results of $DistrEn_{2d}$ (left) and $GradEn$ (right) of four type noises.
  • ...and 10 more figures