Decoupled local energy and phase representation of a wavelet transform

The wavelet transform is increasingly popular for mathematical scale- space analysis in various aspects of signal processing. The squared power and full-wave rectification of the wavelet transform coefficients are the most frequently features used for further processing. However it is shown in this paper that, in general, these features are coupled with the local phase component that depends not only on the analyzed signal but also on the analyzing wavelet at the scale. This dependency causes two problems: 'spurious' spatial variations of features at each scale; and the difficulty of associating features meaningfully across scales. To overcome these problems, we present a decoupled local energy and local phase representation of a real-valued wavelet transform by applying the Hilbert transform at each scale. We show that although local energy is equivalent to the power of the wavelet transform coefficients in term of energy conservation, they differ in scale-space. The local energy representation not only provides a phase-independent local feature at each scale, but also facilitate the analysis of similarity in scale-space. Applications of this decoupled representation to signal segmentation and the analysis of fractal signals are presented. Examples are given through out, using both real infra-red line scan signals and simulated Fractional Brownian Motion data.