Toward new multi-wavelets: associated filters and algorithms. Part I: theoretical framework and investigation of biomedical signals, ECG, and coronavirus cases

Abstract

Biosignals are nowadays important subjects for scientific researches from both theory, and applications, especially, with the appearance of new pandemics threatening the humanity such as the new coronavirus. One aim in the present work is to prove that wavelets may be a successful machinery to understand such phenomena by applying a step forward extension of wavelets to multi-wavelets. We proposed in a first step to improve multi-wavelet notion by constructing more general families using independent components for multi-scaling and multi-wavelet mother functions. A special multi-wavelet is then introduced, continuous, and discrete multi-wavelet transforms are associated, as well as new filters, and algorithms of decomposition, and reconstruction. Applied breakthroughs of the paper may be summarized in three aims. In a first direction, an approximation (reconstruction) of a classical (stationary, periodic) example dealing with Fourier modes has been conducted in order to confirm the efficiency of the HSch multi-wavelets in approximating such signals and in providing fast algorithms. The second experimentation is concerned with the decomposition and reconstruction application of the HSch multi-wavelet on an ECG signal. The last experimentation is concerned with a de-noising application on a strain of coronavirus signal permitting to localize approximately the transmembrane segments of such a series as neighborhoods of the local maxima of an numerized version of the strain. Accuracy of the method has been evaluated by means of error estimates and statistical tests.

Clifford wavelets for fetal ECG extraction

“…In the present research, the objective is to improve the signal processing methods used in fetal cardiographs, and to provide efficient solutions to this problem, by developing suitable techniques for extracting and filtering ECG signals from the fetuses recorded by an array of electrodes placed on the mother's womb. So for a better extraction of ECG wave-forms from the fetus in order to aid in the medical diagnosis of cardiac pathology, the approach envisaged consists in improving the estimation of the FECG signal using two wavelet/multiwavelet based methods such as the one developed in [27] and consisting of the simplest wavelet/multi-wavelet tollkit and the last recent one developed in [5,6] due to Clifford wavelets as the most recent forms in the field.…”

“…We proposed in a first step to improve wavelet processing by applying recent families of multi-wavelets issued from single ones where independent components for multi-scaling and multi-wavelet mother functions are used. We will consider as in [5,6,27,50] vector-valued mother multi-wavelet Ψ HF Sch = (ψ H , ψ F Sch ) for the case of Haar-Faber-Schauder multiwavelet essentially issued from [27], and Ψ Cl = (ψ 1 , ψ 2 ) for the case of Clifford multi-wavelets due to [5].…”

“…These advantages have been encouraging motivations and causes behind [27] where the authors have developed an entropy based procedure for approximating signals with such wavelets by considering a multi-wavelet case its components are exactly Haar and Faber-Schauder wavelets. In the present work, we continue to exploit such case and consider the Haar-Faber-Schauder multi-wavelet Ψ HF Sch = (ψ H ψ F Sch ) T , where the upper script T stands for the transpose.…”

“…In the present section we propose to apply multi-wavelets for the extraction of FECG signal. We will serve from the explicit HFSch multi-wavelets introduced in [27,50] as classical classes and the Clifford ones developed recently in [5,6], and recalled previously as explicit Clifford wavelets.…”

Computational Intelligence Approaches for Enhancing Biomedical Image Processing Applications Based on Breast Cancer

“…While, image processing is an area both in computer science and applied mathematics that concentrate on digital images and their metamorphosis, to enhance their quality or extract knowledge from them in application domain such as medicine, real life, geography, photography, etc. [3]. In addition, image processing is a subset of signal processing with a focus on images and other derived data such as video.…”