Signal analysis based on complex wavelet signs
L. Demaret, P. Massopust, M. Storath · 2016
We propose a new analysis tool for signals that is based on complex wavelet signs, called a signature. The complex-valued 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. At such feature points, the orientation of the signature in the complex plane can be interpreted as an indicator of local symmetry and antisymmetry. We establish that signature is invariant under fractional differentiation and rotates in the complex plane under fractional Hilbert transforms. Thus, the signature may be regarded as a complementary object to the local Sobolev regularity index. We derive an appropriate discretization, which shows that wavelet signatures can be computed explicitly. This allows an immediate application to signal analysis.