Rational Gaussian wavelets and corresponding model driven neural networks

TL;DR

This paper addresses the limitation of fixed mother wavelets by introducing a parametric family of rational Gaussian wavelets (RGW) whose morphology is governed by poles and zeros in a rational term, e.g., $\psi^{\boldsymbol{\eta}}(t) = C(\boldsymbol{\eta}) P^{\boldsymbol{\eta}}(t) v^{\boldsymbol{\eta}}(t) e^{-t^2/2}$. RGWs are shown to be admissible and differentiable with respect to their parameters, enabling gradient-based optimization and seamless integration into variable projection networks (RGW-VP). The RGW-VP layer jointly learns wavelet morphology and translation/scale parameters to produce sparse, interpretable representations, applicable to biomedical signals. A case study on ventricular ectopic beat detection in ECG demonstrates competitive accuracy with fewer parameters and clear physiological interpretation of learned features. This work provides a transparent, trainable wavelet-based feature extractor suitable for safety-critical biomedical applications and beyond.

Abstract

In this paper we consider the continuous wavelet transform using Gaussian wavelets multiplied by an appropriate rational term. The zeros and poles of this rational modifier act as free parameters and their choice highly influences the shape of the mother wavelet. This allows the proposed construction to approximate signals with complex morphology using only a few wavelet coefficients. We show that the proposed rational Gaussian wavelets are admissible and provide numerical approximations of the wavelet coefficients using variable projection operators. In addition, we show how the proposed variable projection based rational Gaussian wavelet transform can be used in neural networks to obtain a highly interpretable feature learning layer. We demonstrate the effectiveness of the proposed scheme through a biomedical application, namely, the detection of ventricular ectopic beats (VEBs) in real ECG measurements.

Figures (7)

• Figure 1: The frequency components of the signal change as a function of time. Scalograms returned by MatLab's CWT routine using different analyzing wavelets. The scalogram in the bottom row captures the frequency profile of the signal much more clearly, demonstrating that the quality of the information provided by wavelet coefficients heavily depends on the choice of the analyzing wavelet.
• Figure 2: The analyzing wavelets used to construct Fig. \ref{['fig:wavcomp']}.
• Figure 3: LEFT: A rational Gaussian motherwavelet. Here $n=3$ poles and $p=10$ zeros were used. RIGHT: Reconstruction of ECG signal using $m=8$ wavelet coefficients.
• Figure 4: Schematic of the proposed, rational Gaussian wavelet based neural network architecture. The first layer acts as an automatic feature learning block passing the learned wavelet coefficients to an underlying fully connected network.
• Figure 5: CWT coefficients of a VEB signal represented by their scalogram and a VEB signal plot, respectively. Trained coefficients marked with green dots on the scalogram.

Theorems & Definitions (8)

• Theorem 1: Admissibility property
• Definition 1: Rational Gaussian wavelet
• Theorem 2: Admissibility of rational Gaussian wavelets
• Theorem 3: Error of variable projection based continuous wavelet coefficients
• Theorem : Admissibility of rational Gaussian wavelets (Theorem \ref{['thm:admisrgw']})
• proof
• Theorem : Error of variable projection based continuous wavelet coefficients (Theorem \ref{['them:errorest']})
• proof