Fractal Dimension Analysis

Application Scenarios

The CFAImageFD class is used to calculate the fractal dimension of 2D images, serving as an important tool for image complexity analysis and texture feature extraction. Main application scenarios include:

Image Texture Analysis: Quantify image roughness and complexity
Medical Image Analysis: Analyze tissue structure complexity
Materials Science: Study fractal features of material surfaces
Computer Vision: Image feature extraction and classification
Geology: Terrain and landform complexity analysis

Usage Examples

Basic Usage

import cv2
from FreeAeonFractal.FAImageFD import CFAImageFD
from FreeAeonFractal.FAImage import CFAImage

# Read image
rgb_image = cv2.imread('./images/fractal.png')
gray_image = cv2.cvtColor(rgb_image, cv2.COLOR_BGR2GRAY)

# For BC method, need binary image
bin_image, threshold = CFAImage.otsu_binarize(gray_image)

# Calculate three fractal dimensions
fd_bc = CFAImageFD(bin_image).get_bc_fd(corp_type=-1)
fd_dbc = CFAImageFD(gray_image).get_dbc_fd(corp_type=-1)
fd_sdbc = CFAImageFD(gray_image).get_sdbc_fd(corp_type=-1)

# Output results
print("BC Fractal Dimension:", fd_bc['fd'])
print("DBC Fractal Dimension:", fd_dbc['fd'])
print("SDBC Fractal Dimension:", fd_sdbc['fd'])

# Visualize results
CFAImageFD.plot(rgb_image, gray_image, bin_image, fd_bc, fd_dbc, fd_sdbc)

GPU Accelerated Version

from FreeAeonFractal.FAImageFDGPU import CFAImageFDGPU

fd_bc = CFAImageFDGPU(bin_image, device='cuda').get_bc_fd()
fd_dbc = CFAImageFDGPU(gray_image, device='cuda').get_dbc_fd()
fd_sdbc = CFAImageFDGPU(gray_image, device='cuda').get_sdbc_fd()
# Note: p_value is None in GPU version (not computed)

Batch Processing

import cv2, glob
import numpy as np
from FreeAeonFractal.FAImageFD import CFAImageFD

images = [cv2.imread(f, cv2.IMREAD_GRAYSCALE) for f in glob.glob('./images/*.png')]

# Batch box-counting
results = CFAImageFD.get_batch_bc(images)
for r in results:
    print("BC FD:", r['fd'])

Installation

pip install FreeAeon-Fractal

Class Description

CFAImageFD

Description: Class for calculating 2D image fractal dimensions, supporting three different calculation methods: Box-Counting (BC), Differential Box-Counting (DBC), and Shifted DBC (SDBC).

Initialization Parameters

Parameter	Type	Default	Description
`image`	numpy.ndarray	Required	Input image (single channel)
`max_size`	int	None	Maximum box size (default: minimum image dimension)
`max_scales`	int	30	Maximum number of scales
`with_progress`	bool	True	Whether to show progress bar
`min_size`	int	2	Minimum box size

Main Methods

1. get_bc_fd(corp_type=-1, fit_range=None)

Description: Calculate fractal dimension using Box-Counting (BC) method.

Parameters:

corp_type (int): Image cropping method
- -1: Auto crop to multiples of box size
- 0: No processing (requires image dimensions to be multiples of box size)
- 1: Padding
fit_range (tuple or None): Optional (min_scale, max_scale) to restrict log-log fit range

Return Value (dict):

{
    'fd': float,              # Fractal dimension value
    'scales': list,           # Scale list
    'counts': list,           # Box count list
    'log_scales': list,       # Logarithmic scales
    'log_counts': list,       # Logarithmic counts
    'intercept': float,       # Fit intercept
    'r_value': float,         # Correlation coefficient
    'p_value': float,         # p-value
    'std_err': float          # Standard error
}

Use Case: Suitable for binary images, calculating the dimension of occupied space.

2. get_dbc_fd(corp_type=-1, fit_range=None)

Description: Calculate fractal dimension using Differential Box-Counting (DBC) method (Sarkar & Chaudhuri 1994). Uses n_r = ceil(I_max/h) - ceil(I_min/h) + 1.

Parameters: Same as get_bc_fd

Return Value: Same as get_bc_fd

Use Case: Suitable for grayscale images, considering grayscale information in fractal dimension calculation.

3. get_sdbc_fd(corp_type=-1, fit_range=None)

Description: Calculate fractal dimension using Shifted DBC (SDBC) method (Chen 1995). Uses n_r = floor((I_max - I_min)/h) + 1.

Parameters: Same as get_bc_fd

Return Value: Same as get_bc_fd

Use Case: SDBC is a simplified version of DBC, faster computation, suitable for grayscale images.

4. get_fd(scale_list, box_count_list)

Description: Utility method to perform log-log fit on custom scale and count data.

Parameters:

scale_list: List of scale values
box_count_list: Corresponding box counts

Return Value: Same dict structure as get_bc_fd

5. plot(raw_img, gray_img, bin_img, fd_bc, fd_dbc, fd_sdbc)

Description: Static method to visualize original image, grayscale image, binary image, and fitting results of three fractal dimensions.

Parameters:

raw_img: Original RGB image
gray_img: Grayscale image
bin_img: Binary image
fd_bc: BC method result dictionary
fd_dbc: DBC method result dictionary
fd_sdbc: SDBC method result dictionary

6. get_batch_bc / get_batch_dbc / get_batch_sdbc(images, ...)

Description: Static methods for batch processing of multiple images.

Parameters:

images (list): List of input images
Additional keyword arguments passed to the corresponding single-image method

Return Value: List of result dictionaries

Algorithm Description

BC (Box-Counting) Method

The box-counting method calculates fractal dimension by counting the number of non-empty boxes needed to cover the image at different scales. Formula:

D = lim(ε→0) log(N(ε)) / log(1/ε)

Where N(ε) is the number of boxes at scale ε.

DBC (Differential Box-Counting) Method

DBC considers grayscale height information, treating the image as a 3D surface. The height of each box is:

n_r = ceil(I_max / h) - ceil(I_min / h) + 1

Where h = max_val / (image_size / r) is the height of a grayscale unit.

SDBC (Simplified DBC) Method

Shifted DBC uses a simplified height calculation:

n_r = floor((I_max - I_min) / h) + 1

This avoids the rounding artefact in the original DBC formula.

Important Notes

Image Preprocessing:
- BC method requires binary images, recommend using Otsu auto-thresholding
- DBC and SDBC methods are suitable for grayscale images
Scale Selection:
- max_size affects the scale range of analysis
- max_scales affects sampling density, recommend default value 30
- Use fit_range to exclude unreliable small/large scales
Result Interpretation:
- Fractal dimension range typically 1-2 (for 2D images)
- Larger values indicate more complex images
- r_value close to 1 indicates good power-law fit
Performance Optimization:
- For large images, can downsample first
- Set with_progress=False to improve computation speed
- Use GPU version (CFAImageFDGPU) for batch processing
- Note: GPU version sets p_value=None (not computed)

References

Mandelbrot, B. B. (1982). The Fractal Geometry of Nature.
Sarkar, N., & Chaudhuri, B. B. (1994). An efficient differential box-counting approach to compute fractal dimension of image. IEEE Transactions on Systems, Man, and Cybernetics.
Chen, W. S., et al. (1995). Efficient fractal coding of images based on differential box counting. Pattern Recognition.