MATLAB Implementation for Calculating Differential Box Counting Dimension

The differential box counting dimension is a widely used method for calculating fractal dimensions of images, particularly suitable for analyzing texture patterns or surface roughness. This approach computes the dimension based on the relationship between the number of boxes required to cover the image and the box size.

The computational procedure for differential box counting dimension primarily involves the following steps: Image Preprocessing: First convert the image to grayscale format using MATLAB's rgb2gray() function for subsequent processing. Box Covering: Partition the image into boxes of different sizes (e.g., 2×2, 4×4, ..., up to half the image dimensions) using nested loops and meshgrid operations. Difference Calculation: Within each box, identify the maximum and minimum pixel values using max() and min() functions, then compute their difference to determine the required number of "layers" in the height direction to cover the box. Box Counting: Based on box size and calculated differences, count the total number of boxes needed to cover the entire image using accumulation algorithms. Regression Analysis: Take logarithms of both box sizes and corresponding box counts, then perform linear regression using polyfit() function - the slope of the regression line represents the differential box counting dimension.

This method is particularly applicable to natural textures, medical images, or geological data analysis, effectively reflecting image complexity and self-similarity. Since fractal dimension itself is a statistical measure, it's crucial to select appropriate box size ranges and regression intervals during computation to ensure result accuracy. MATLAB implementation typically involves optimizing box size selection through vectorization and preallocation for efficient large-scale image processing.