Fractal dimension from image

Suppose that we want to estimate the fractal dimension from an object represented by an image:

Image depicting the Sierpinski  triangle.

The image depicts the Sierpinski triangle, a well known fractal object. Computing the fractal dimension of an object can be useful, for example, in image feature extraction.

In our case, the fractal object is represented by a binary image I, where non-zero pixels belong to the object and zero pixels constitute the background. We can estimate the Haussdorf fractal dimension of the object using the following algorithm:

1. Pad the image with background pixels so that its dimensions are a power of 2.
2. Set the box size 'e' to the size of the image.
3. Compute N(e), which corresponds to the number of boxes of size 'e' which contains at least one object pixel.
4. If e > 1 then e = e / 2 and repeat step 3.
5. Compute the points log(N(e)) x log(1/e) and use the least squares method to fit a line to the points.
6. The returned Haussdorf fractal dimension D is the slope of the line.

 

Matlab code

This link provides my implementation of the previous algorithm. The hausDim() function can be used to estimate the fractal dimension of the depicted fractal. Note that first it is necessary to convert the image to a binary matrix using the im2bw function:

% Read the image.
>> I = imread('https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQHyGYhHQzfTBDzSCm9fH5atXyrVLG5E4mAGc-Ikn2EJWovGkhi_PpGfhrVEax21b9c4Afzqlt2vUQJP_dtXZ52RK5HJZ08rEhkVkrOLIMK9FdBNZ0yWVaxt_O-S7QuAl7M9udzGjsjhyphenhyphen-/s1600/sierpinski.png');

% Convert to a binary image and negate pixel values.
>> I = ~im2bw(I);
>> hausDim(I)

ans = 1.5496

The estimated value is 1.5496. This is quite close to the exact value which is log(3) / log(2) = 1.58496...