Wavelet Phase Harmonics

Alex Grossmann pointed out the importance of wavelet phase alignments across scales, to characterize local signal properties. This observation has drawn relatively little attention. Most wavelet research has concentrated on properties of wavelet coefficient amplitudes. This chapter shows that the phase is indeed crucial to characterize dependencies across scales, particularly to build non-Gaussian models of random processes. We introduce phase harmonic operators which capture the phase information with a windowed Fourier transform on the phase. We derive maximum entropy models of non-Gaussian stationary processes, conditioned by the covariance of wavelet phase harmonics. Relations with high-order moments and neural network coefficients are explained. It is shown that coherent structures of turbulent flows can be reproduced by wavelet phase harmonic models.