Toward new multi-wavelets: associated filters and algorithms. Part I: theoretical framework and investigation of...

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.