Polarization phase relationships via multiple Morse wavelets. I. Fundamentals
Sofia C. Olhede, Andrew T. Walden · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2003
The purpose of this paper is to provide a detailed analysis of the use of multiple generalized Morse wavelets for polarization analysis of multi–component recordings where phase relationships between components of the signal can be transient. Special attention will be given to the case of three components. The use of complex and analytic wavelets enables detection of coherent motion with elliptical polarization because of sensitivity to phase shifts among components. We adopt a singular–valuedecomposition approach. The principal polarization is given by the first eigenvector of the multiple–wavelet multivariate scalogram at scale a and time b. Having described in detail a deterministic signal plus random noise model for the recorded data, we show that for (a,b) in a domain Ω, we can approximate the phase between components of the original multi–component signal via the estimated phase of the components of the first right–singular vector; detailed results on the accuracy of the approximation are given. The exact form of the domain Ω, which depends on the transient nature of the phase relationship, is described.