Wavelet-based multifractal analysis of 1-D and 2-D signals: New results

Abstract

During the past decade, new tools stemming from fractal geometry and wavelet analysis are meeting with great success in signal image processing. This paper will focus on these two topics: Wavelets and Multifractal. Both themes evolved towards self contained theories, and yet, a host of reasons justify for coupling them in same applications. It is well known that both analyses share the same conceptual backbone of ''scale'': it is the ''mathematical zoom'' commonly associated to wavelet analysis and it is the ''scaling laws'' that underlie multifractal structures. Very naturally then, wavelets stood as a privileged tool for analyzing and characterizing multifractal signals and images. Hence, the purpose of this paper is to illuminate some of the issue involved in taking advantage of the current advances in wavelets and multifractals analysis. We discuss continuous, discrete, orthogonal wavelets and present applications to fracture processes and medical ultrasound imaging.

Study of the Transition to Instability in Second-Harmonic Generation with Scale Index Method

“…Non-stationary signals make wavelets more suitable for detecting such periodicity. Moreover, wavelets have been successfully used in the analysis of many chaotic systems [31][32][33]. Non-periodicity as a hallmark of a chaotic behavior could be quantified through scale index analysis [33,34].…”

Optical imaging of high-frequency focused ultrasonic field using a Nomarski interferometer coupled with multichannel lock-in detection

“…[21][22][23][24][25][26][27][28][29] Subsequently, we can also show that it will yield the measurement of the refractive index variation in the fluid, locally in a distance of a half period of the acoustic signal, which corresponds to the distance between the two beams. [21][22][23][24][25][26][27][28][29] Subsequently, we can also show that it will yield the measurement of the refractive index variation in the fluid, locally in a distance of a half period of the acoustic signal, which corresponds to the distance between the two beams.…”

Local multifractal detrended fluctuation analysis for non-stationary image's texture segmentation

“…One is local singularity exponent, namely Hölder coefficients [3,4], based on capacity measure, denoted as ˛c, the other is multifractal dimensions D q based on differential box-counting (MDBC). The local singularity coefficients ˛c features, defined as ˛c ≡ lim ε→0 ln (D r )/ ln(ε), can well describe the characteristic of a texture image and be used in lots of fields [13,23,27].…”

“…Hence, it can bring us a more efficacious way to process various texture recognition problems by the MFA [2][3][4]22,23,27]. Such as, Xu et al [22] proposed a robust texture descriptor combining the MFA and Gabor filter.…”