Can the fractal dimension of images be measured?

Abstract

Fractal dimension is a popular parameter for explaining certain phenomena and for describing natural textures. The problem of estimating the fractal dimension of a profile or an image is more difficult and devious than theory suggests. This paper studies the accuracy and robustness of two common estimators of fractal dimension (box counting and the variation method) using two types of data (Brownian and Takagi). Poor results are demonstrated from applying theory directly, called naive estimation. Data is then interpreted in the most optimistic way possible by matching the estimator to the known fractal dimension. Experiments quantify the effects of resolution, or fineness of sampling, and quantization, or rounding of sampled values. Increasing resolution enhances the estimators when true dimension, D, is large, but may, possibly due to quantization effect, degrade estimators when D is small. Quantization simply causes shifts in estimates. The results suggest that one should not place much reliance in the absolute value of a fractal estimate, but that the estimates do vary monotonically with D and might be useful descriptors in tasks such as image segmentation and description.