Wavelet Signs: A New Tool for Signal Analysis

We propose a new analysis tool for signals, called signature, that is based on complex wavelet signs. The complexvalued signature of a signal at some spatial location is defined as the fine-scale limit of the signs of its complex wavelet coefficients. We show that the signature equals zero at sufficiently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. We establish that signature is invariant under fractional differentiation and rotates in the complex plane under fractional Hilbert transforms. We derive an appropriate discretization, which shows that wavelet signatures can be computed explicitly. This allows an immediate application to signal analysis.