Fractional Dimension of Images

This activity allows you to analyze photographs and determine their fractional dimension. You may use the photographs provided below or use your own images.

Introduction

All images and objects have a dimension. For example, a straight line has a dimension of 1, a sheet of paper has a dimension of 2, and a solid cube has a dimension of 3. But what about objects or pictures that are somewhere in between? A squiggly line can be drawn on a two dimensional piece of paper, however it does not fill the paper so it is not two dimensional and it is not straight so it is not one dimensional. A sponge or piece of bread appears to be 3 dimensional but they are not solid because of their many holes. In fact on a very small scale a sponge may appear as a two dimensional surface that curves much like the squiggly line. Such objects or images have fractional dimensions. That is, an object with a fractional dimension has a dimension that is not an integer but rather a fraction.

There are two ways to find the fractional dimension of an image - the Hausdorff Dimension and the Correlation Dimension.

The Hausdorff Dimension
Consider a squiggly line on a sheet of paper. Cover this line with 2 dimensional cubes of side E and let N(E) be the smallest number of E-sided cubes that can cover this line. The dimension of the line is then

or in simpler terms N(E) ~ E^-D

If you take the logarithm of both sides you will find an equation that matches the form of the equation for a line: logN = -D logE so that if you use several different size boxes and plot logN versus logE the slope of the line will tell you the dimension.

Try this example:

What is the fractional dimension of this squiggly line?
You should get 1.3. Does this make sense?

The Correlation Dimension
To determine the Correlation dimension you apply the same idea as the Hausdorff dimension except use circles instead of squares. Let R be the radius of the circle and let N(R) be the number of pixels that lie within the radius of the circle. The Correlation dimension is then defined by: N(R) ~ R^C where C is the Correlation dimension.

If you take the logarithm of both sides you will find an equation that matches the form of the equation for a line: logN = C logR so that if you use several different size circles and plot logN versus logR the slope of the line will tell you the dimension.

Try this example:

What is the Correlation dimension of this squiggly line? How does it compare to the Hausdorff dimension?

Materials

• computer
• Fractal Dimension software from the Polymer Center at Boston University
• images

Procedure

Since it is very time consuming to draw boxes and count pixels, we will use Fractal Dimension, a computer program developed by the Polymer Center at Boston University, to determine the fractional dimension of our images.

Choose an image that you believe to have be interesting and determine its fractal dimension. Then answer the following questions.

• What is this a picture of?
• Why would you think of it as a fractal?
• What is the Hausdorff dimension (box method)?
• What is the correlation dimension (circle method)?
• How do the two dimensions compare? Are they similar or different? Why do you think that is?
• Explain why the fractional dimension does (not) make sense for your image (Is it what you expected?)
• Does form follow function? (Is the fractional dimension apporpriate for what your image depicts?)

The following images are used with permission from the DHD Photo Gallery




BUBBLES


CLOUDS



FROST



AERIAL VEIW OF MOUNTAINS

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