Multifractal Analysis and Wavelets

Abstract

Abstract. In this course, we give the basics of the part of multifractal theory that intersects wavelet theory. We start by characterizing the pointwise Hölder exponents by some decay rates of wavelet coefficients. Then, we give some examples of wavelet series having a multifractal behavior, and we explain how to build wavelet series with prescribed pointwise Hölder exponents. Next we develop the problematics of multifractal formalism, going from the intuitive formula by Frisch and Parisi to explicit and exploitable formulas. We prove that "multifractals are everywhere", in the sense that typical functions in Besov spaces or typical measures are multifractal in the sense of Baire's categories. We finish by some well-known examples of multifractal wavelet series, random and deterministic, focusing on the influence of certain adaptive threshold procedures to the multifractal properties of signals.

Counter-examples to multifractal formalisms

“…In this subsection, we propose a general construction which allows to construct many counter-examples to the formalism, amoung which homogeneous functions. The existence of such counter-examples was an open question of [27] and somehow homogeneity of the function is often seen as a way to ensure the validity of the multifractal formalism [38].…”

“…Note that this function has very particular properties on the distribution of wavelet coefficients: they are hierarchical and are decreasing at each scale in the translation index. An open question mentioned in [27] and in [38] is to know whether homogeneity of the signal could ensure the validity of the formalism. The same question arises with randomness.…”

Wavelet leader based formalism to compute multifractal features for classifying lung nodules in X-ray images

“…Multifractal analysis is a novel technique in medical image processing: it studies the local regularity and scale behavior of functions; it is an attempt to describe geometrical and statistical distributions of the singularities of functions [Seuret, 2016]; therefore, multifractal analysis is also useful to describe images and its results have been satisfactory in segmentation and classification tasks. The multifractal spectrum is a function that measures local regularity of a signal, from a global point of view.…”

Pointwise regularity of parameterized affine zipper fractal curves

“…to study the possible values, which occur as (regular) pointwise Hölder exponents, and determine the magnitude of the sets, where it appears. This property was studied for several types of singular functions, for example for wavelets by Barral and Seuret [4], Seuret [38], for Weierstrass-type functions Otani [32], for complex analogues of the Takagi function by Jaerisch and Sumi [22] or for different functional equations by Coiffard, Melot and Willer [10], by Okamura [31] and by Slimane [7] etc.…”

“…We call v as the linear parametrization of Γ. Such linear parameterizations occur in the study of Wavelet functions in a natural way, see for example Protasov [36], Protasov and Guglielmi [37], and Seuret [39]. A particular example for (1.3) is the de Rham's curve, see section 5 for details including an example of a graph of v generated by the de Rham's curve.…”