ACCURACY OF GENERALIZED DIMENSIONS ESTIMATED FROM GRAYSCALE IMAGES USING THE METHOD OF MOMENTS

Abstract

The moment-based box counting method of multifractal analysis is widely used for estimating generalized dimensions, Dq, from two-dimensional grayscale images. An evaluation of the accuracy of this method is needed to establish confidence in the resulting estimates of Dq. We estimated Dq from q = -10 to +10 for 23 random geometrical multifractal fields with different grid sizes, and known analytical Dq versus q functions. The fields were transformed to give normalized grayscale values between zero and one. Comparison of the estimated and analytical functions indicated the moment-based box counting method overestimates Dq by as much as 6.9% when q ≪ 0. The root mean square error, RMSE, for the entire range of q values examined ranged from 7.81 × 10-6 to 1.35 × 10-1, with a geometric mean of 6.50 × 10-3. The RMSE decreased with decreasing grid size and increasing heterogeneity. These trends appear to be largely due to the presence of zeros in the normalized grayscale fields. Variations in the slope of the log-transformed partition function, ln [χ(q,δ)], with box size resulted in the overestimation of Dq when q ≪ 0. An alternative procedure for estimating Dq was developed based on the numerical first derivatives of ln [χ(q,δ)]. Using this approach the maximum deviation in Dq values was only 1.2%, while the RMSE varied from 3.11 × 10-6 to 2.72 × 10-2, with a geometric mean of 2.57 × 10-4. When analyzing normalized grayscale fields, moment-based estimates of Dq should be interpreted with care. An order of magnitude increase in the accuracy of Dq can be achieved for such fields if the numerical first derivatives of ln [χ(q,δ)] are used in the analysis instead of standard linear regression.