Wavelet Filtering with the Mellin Transform

TL;DR

This paper demonstrates that convolution operators can be exactly represented as multiplications in the wavelet domain via the Mellin transform, offering improved denoising of signals affected by atmospheric turbulence.

Contribution

It introduces a novel approach linking frequency and scale filters through the Mellin transform, extending wavelet theory and enhancing denoising techniques for turbulent signals.

Findings

Exact representation of convolution as multiplication in wavelet domain

Mellin transform establishes a correspondence between frequency and scale filters

Proposed method better handles spectral power laws in atmospheric turbulence

Abstract

It is shown that any convolution operator in the time domain can be represented exactly as a multiplication operator in the time-scale (wavelet) domain. The Mellin transform gives a one-to-one correspondence between frequency filters (multiplications in the frequency domain) and scale filters (multiplications in the scale domain), subject to the convergence of the defining integrals. The usual wavelet reconstruction theorem is a special case. Applications to the denoising of random signals are proposed. It is argued that the present method is more suitable for removing the effects of atmospheric turbulence than the conventional procedures because it is ideally suited for resolving spectral power laws.