Wavelet Localisation and Local Modulation Freezing in MRW Unwrapping

2026-06-14

AI Summary

This work addresses scale aliasing and covariance distortion in the disentanglement of multifractal random walks (MRW) caused by local variations in the multiplicative modulation field. The authors propose a novel framework based on wavelet localization and local modulation freezing, which leverages compactly supported wavelets to approximately freeze the modulation field within localized regions. This transformation converts log-wavelet magnitudes into an additive structure, reformulating covariance disentanglement as a local multiscale operator problem. Innovatively treating wavelet localization itself as a mechanism for probing local stochastic structure, the study elucidates how support geometry, scale overlap, and interactions with the modulation field influence covariance, and establishes a theoretical link between local regularity and compactly supported multiscale operators. Numerical experiments confirm the scale-dependent nature of modulation freezing, residual aliasing, and scale collapse, with results aligning closely with theoretical predictions and validating the proposed framework.

Abstract

We develop a localised wavelet formulation of multifractal random walk unwrapping based on the local multiplicative modulation freezing. The framework is motivated by the observation that finite-support wavelet localisation may induce approximate local factorisation of multiplicatively modulated stochastic fields, allowing the modulation component to become effectively frozen within sufficiently localised probing domains. Within this regime, logarithmic wavelet amplitudes admit an approximate additive decomposition linking local wavelet statistics directly to the underlying modulation field. This viewpoint reformulates covariance-based MRW unwrapping as a localised multiscale operator problem in which wavelet coefficients act as finite-support probes of multiplicative organisation. The validity of the approximation depends explicitly on support geometry, scale-dependent overlap, and residual multiscale mixing generated by internal modulation variability. We show that these effects naturally produce finite-scale deviations from ideal logarithmic covariance scaling and lead to structured covariance distortions whose form depends on the interaction between the modulation field and the geometry of the wavelet representation. In the resulting framework, localisation itself becomes the operational mechanism enabling multiscale probing of local stochastic organisation. Numerical investigations using orthonormal wavelet decompositions support the proposed interpretation and demonstrate the emergence of scale-dependent freezing regimes, residual covariance mixing, and finite-support breakdown effects consistent with the theory. The proposed framework suggests a broader connection between wavelet localisation, local regularity organisation, and finite-support multiscale stochastic operators. Wavelet localisation becomes an operational mechanism for probing localised multiscale structure.

Problem

• multifractal random walk
• wavelet localisation
• multiplicative modulation
• covariance distortion
• finite-support probing

Innovation

• wavelet localisation
• multifractal random walk
• local modulation freezing
• multiplicative stochastic fields
• multiscale probing