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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 paper 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. [ABSTRACT FROM AUTHOR]

Multifractal analysis has become a standard signal processing tool, for which a promising new formulation, the \boldsymbol p-leader multifractal formalism, has recently been proposed. It relies on novel multiscale quantities, the \boldsymbol p-leaders, defined as local $\ell ^p$ norms of sets of wavelet coefficients located at infinitely many fine scales. Computing such infinite sums from actual finite-resolution data requires truncations to the finest available scale, which results in biased \boldsymbol p -leaders and thus in inaccurate estimates of multifractal properties. A systematic study of such finite-resolution effects leads to conjecture an explicit and universal closed-form correction that permits an accurate estimation of scaling exponents. This conjecture is formulated from the theoretical study of a particular class of models for multifractal processes, the wavelet-based cascades. The relevance and generality of the proposed conjecture is assessed by numerical simulations conducted over a large variety of multifractal processes. Finally, the relevance of the proposed corrected estimators is demonstrated on the analysis of heart rate variability data. [ABSTRACT FROM PUBLISHER]

Background: Multifractal analysis of human heartbeat dynamics has been demonstrated to provide promising markers of congestive heart failure (CHF). Yet, it crucially builds on the interpolation of RR interval series which has been generically performed with limited links to CHF pathophysiology. Objective: We devise a novel methodology estimating multifractal autonomic dynamics from heartbeat-derived series defined in the continuous time. We hypothesize that markers estimated from our novel framework are also effective for mortality prediction in severe CHF. Methods: We merge multifractal analysis within a methodological framework based on inhomogeneous point process models of heartbeat dynamics. Specifically, wavelet coefficients and wavelet leaders are computed over measures extracted from instantaneous statistics of probability density functions characterizing and predicting the time until the next heartbeat event occurs. The proposed approach is tested on data from 94 CHF patients aiming at predicting survivor and nonsurvivor individuals as determined after a four years follow up. Results and Discussion: Instantaneous markers of vagal and sympatho-vagal dynamics display power-law scaling for a large range of scales, from $\simeq 0.5$ to $\simeq 100$  s. Using standard support vector machine algorithms, the proposed inhomogeneous point-process representation-based multifractal analysis achieved the best CHF mortality prediction accuracy of 79.11% (sensitivity 90.48%, specificity 67.74%). Conclusion: Our results suggest that heartbeat scaling and multifractal properties in CHF patients are not generated at the sinus-node level, but rather by the intrinsic action of vagal short-term control and of sympatho-vagal fluctuations associated with circadian cardiovascular control especially within the very low frequency band. These markers might provide critical information in devising a clinical tool for individualized prediction of survivor and nonsurvivor CHF patients. [ABSTRACT FROM AUTHOR]