In this paper we introduce a highly adaptive 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 number of numerical experiments including biomedical applications, and the detection of abnormal road surface based on tire sensor signals.

In this work, we consider the problem of detecting the phenomenon of brake disc runout, which is usually caused by deformations of the brake disc in passenger vehicles. We introduce a novel measurement system that collects and processes information on the vibration profile of the vehicle, as well as the pressure of the brake fluid. Our novel processing method for the collected signals relies on model-driven adaptive wavelet neural networks. We demonstrate that the proposed signal processing methodology outperforms previous approaches to detect aberrations of the brake disc. In addition, the proposed neural network models are defined by interpretable parameters and have reduced complexity when compared to traditional and similar approaches. These characteristics make the proposed construction suitable for automotive applications.