Simultaneous denoising and enhancement of signals by a fractal conservation law

TL;DR

This paper introduces a novel fractal conservation law-based filter for simultaneous noise reduction and signal enhancement, demonstrating superior performance over traditional methods through numerical tests and multi-scale analysis.

Contribution

The paper presents a new filtering approach using a fractal scalar conservation law with a fractional anti-diffusive term, offering improved denoising and enhancement capabilities.

Findings

Outperforms Savitzky-Golay filter in denoising

Exhibits multi-scale properties for frequency control

Enables simultaneous noise reduction and contrast enhancement

Abstract

In this paper, a new filtering method is presented for simultaneous noise reduction and enhancement of signals using a fractal scalar conservation law which is simply the forward heat equation modified by a fractional anti-diffusive term of lower order. This kind of equation has been first introduced by physicists to describe morphodynamics of sand dunes. To evaluate the performance of this new filter, we perform a number of numerical tests on various signals. Numerical simulations are based on finite difference schemes or Fast and Fourier Transform. We used two well-known measuring metrics in signal processing for the comparison. The results indicate that the proposed method outperforms the well-known Savitzky-Golay filter in signal denoising. Interesting multi-scale properties w.r.t. signal frequencies are exhibited allowing to control both denoising and contrast enhancement.