Generalized Legendre transform multifractal formalism for nonconcave spectrum estimation

Despite widespread adoption of multifractal analysis as a signal processing tool, most practical multifractal formalisms suffer from a major drawback: since they are based on Legendre transforms, they can only yield concave estimates for multifractal spectrum that are, in most cases, only upper bounds on the (possibly nonconcave) true spectrum. Inspired by ideas borrowed from statistical physics, a procedure is devised for the estimation of not a priori concave spectra that retains the simple and efficient Legendre transform formalism structure. The potential and interest of the proposed procedure are illustrated and assessed on realizations of a synthetic multifractal process, with theoretically known nonconcave multifractal spectrum.