Multifractal analysis has become a powerful signal processing tool that
characterizes signals or images via the fluctuations of their pointwise
regularity, quantified theoretically by the so-called multifractal spectrum.
The practical estimation of the multifractal spectrum fundamentally relies on
exploiting the scale dependence of statistical properties of appropriate
multiscale quantities, such as wavelet leaders, that can be robustly computed
from discrete data. Despite successes of multifractal analysis in various
real-world applications, current estimation procedures remain essentially
limited to providing concave upper-bound estimates, while there is a priori no
reason for the multifractal spectrum to be a concave function. This work
addresses this severe practical limitation and proposes a novel formalism for
multifractal analysis that enables nonconcave multifractal spectra to be
estimated in a stable way. The key contributions reside in the development and
theoretical study of a generalized multifractal formalism to assess the
multiscale statistics of wavelet leaders, and in devising a practical algorithm
that permits this formalism to be applied to real-world data, allowing for the
estimation of nonconcave multifractal spectra. Numerical experiments are
conducted on several synthetic multifractal processes as well as on a
real-world remote-sensing image and demonstrate the benefits of the proposed
multifractal formalism over the state of the art.