A relative fractal dimension spectrum as a complexity measure

This paper presents a derivation of a new relative fractal dimension spectrum, D-Rq, to measure the dissimilarity between two finite probability distributions originating from various signals. This measure is an extension of the Kullback-Leibler (KL) distance and the Renyi fractal dimension spectrum, Dq. Like the KL distance, DRq determines the dissimilarity between two probability distibutions X and Y of the same size, but does it at different scales, while the scalar KL distance is a single-scale measure. Like the Renyi fractal dimension spectrum, the DRq is also a bounded vectorial measure obtained at different scales and for different moment orders, q. However, unlike the Dq, all the elements of the new DRq become zero when X and Y are the same. Experimental results show that this objective measure is consistent with the subjective mean-opinion-score (MOS) when evaluating the perceptual quality of images reconstructed after their compression. Thus, it could also be used in other areas of cognitive informatics.