Relevance of wavelets shape selection
Résumé
Wavelet analysis is a mathematical tool recently developed for signal processing. Its primary applications have been in the areas of signal analysis, image compression, sub band coding, vibration/motion analysis, and sound synthesis. Time-frequency analysis of signal and shape analysis have taken a very important part of signal processing, where the wavelet transform (WT) plays an essential role. The basic parameter of wavelet transform is the mother wavelet which has many defined shapes and fundamentally many more are possible. Various authors report that the influence of the function shape on the result of the wavelet transform is obvious and therefore an optimal function shape has to be defined for each application. In materials or mechanical sciences the surface topography plays a major role for the integrities and functionalities of materials (wear, corrosion, adhesion, brightness, acoustical noise…). For these reasons surface topography is often linearly recorded and gives signal (high amplitude, scanning length…) that can be treated with different processing tools – see figure 1.a.In this paper, discrete and continuous wavelet transform are used to analyse these signals. Tagushi’s experimental design is used to study the effects of the process conditions on the resulting roughness signal. A tactile profilometry was used to record 30 surface signals for each investigated configuration. These signals, considered at different scale lengths, were analysed by seven different wavelets (Daubechies, Symlets, Coifflets,Meyer, Bior, Mexican hat and Morlet) combined with three types of decompositions (low frequency, high frequency and "roughness decomposition"). A set of roughness parameters (Ra, Rq, Sm …) is computed for each signal at all scales. A methodology is built to assess the influence of wavelet choice towards finding the most relevant scale, and knowing the best type of wavelet for this analysis. This methodology is applicable in any application of signal processsing. By using it, the variance analysis is combined with the Bootstrap theory which takes into consideration the fact that a small variation in any score influences the value of the treatment index.The comparison between the different used wavelets allows us to conclude that at the same scale, the value of roughness parameters varies with the type of wavelet (figure 1.c), but the interaction between the wavelet and each experimental condition is very low (figure 1.b). The influence, quantified by the analysis of variance, of each process parameter has the same value regardless of the wavelet choice. The most relevant scale of each process parameter has the same spatial range regardless of the wavelet choice. Although the wavelet decompositions leads to different values for a given roughness parameter, statistical analysis shows that the influence and the spatial localisation of abrasion never depends on the wavelet choice. This conclusion somewhat counterbalances many previous assertions about the correct choice of a wavelet according to a specific application.