Singularity spectrum of fractal signals from wavelet analysis: Exact results

Journal Article · Mon Feb 01 00:00:00 EST 1993 · Journal of Statistical Physics; (United States)

DOI:https://doi.org/10.1007/BF01053588· OSTI ID:6654865

Bacry, E; Muzy, J F; Arneodo, A ^[1]

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1. Centre de Recherche Paul Pascal, Pessac (France)

The multifractal formalism for singular measures is revisited using the wave transform. For Bernoulli invariant measures of some expanding Markov maps, the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima. The generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures. It is demonstrated that the Hausdorff dimension D(h) of the set of singularities of Hoelder exponent h can be directly determined from the wavelet transform modulus maxima. The singularity spectrum so obtained is shown to be not disturbed by the presence, in the signal, of a superimposed polynomial behavior of order n, provided one uses an analyzing wavelet that possesses at least N > n vanishing moments. However, it is shown that a C[infinity] behavior generally induces a phase transition in the D(h) singularity spectrum that somewhat masks the weakest singularities. This phase transition actually depends on the number N of vanishing moments of the analyzing wavelet; its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal. These theoretical results are illustrated with numerical examples. They are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional Brownian motions and turbulent velocity signals.

• OSTI ID: 6654865
• Journal Information: Journal of Statistical Physics; (United States), Vol. 70:3-4; ISSN 0022-4715
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
FRACTALS
DIAGNOSTIC USES
WAVE PROPAGATION
DYNAMICS
BERNOULLI LAW
BROWNIAN MOVEMENT
DISTRIBUTION FUNCTIONS
MARKOV PROCESS
PARTITION FUNCTIONS
SINGULARITY
TURBULENT FLOW
FLUID FLOW
FUNCTIONS
MECHANICS
STOCHASTIC PROCESSES
USES
661300* - Other Aspects of Physical Science- (1992-)
990200 - Mathematics & Computers