Measuring the Fractal Dimensions of Surfaces

The fractal dimension of a surface is a measure of its geometric complexity and can take on any non-integer value between 2 and 3. Normally, the topological dimension of surfaces is 2 however, their fractal dimensions increase with greater amounts of complexity or roughness. For example, a fractal dimension of 2.3 is found to be a common value in describing the relief on the earth. This paper discusses and presents examples of an algorithm designed to measure the fracticality of surfaces. The algorithm was developed at The Ohio State University and is shown to be reliable and robust. It is placed in an interactive setting and is based on the premise that the complexity of isarithm lines may be used to approximate the complexity of a surface. The algorithm operates with the following scenario Starting with a matrix of Z-heights, an isarithm interval is selected and isarithm lines are constructed on the surface. A fractal dimension is computed for each isarithm line by calculating their lengths over a number of sampling intervals. The surfaces fractal dimension is the result of averaging the fractal dimensions of all the isarithm lines and adding 1. Potential applications for this technique include a new means for data compression, a quantitative measure of surface roughness, and be used for generalization and filtering.