Dimension spectra of fractal measures from uniform partitions and correlation integrals

The spectra of generalized dimensions D-q and of local exponents f(alpha) f or fractal measures are evaluated by using the uniform partitions to comput e the free energy. The numerical results obtained from optimal algorithms a re compared with the analytical results obtained from the free energy evalu ated with dynamical partitions, in the case of IFS measures. It is proved t hat the spectra D-q obtained from correlation integrals and dynamical parti tions are the same even for q < 1. The spectra obtained from the uniform pa rtitions agree with the analytical result of dynamical partitions for any q > 1 and for q < 1 only if the support of the measure is not fractal or if the dynamical partitions are a subset of the uniform partitions. The spectr a obtained from a numerical approximation of the correlation integrals prov ide the correct result for any value of q. The algorithms based on the unif orm partitions are fast and can be used for real-time analysis of digitized images.