Hilbert assisted wavelet transform method of optical fringe pattern phase reconstruction for optical profilometry and interferometry

A Hilbert assisted wavelet transform (HWT) method is presented for phase reconstruction of 3D profilometry and interferometry. A rigorous mathematical demonstration about this HWT method is given in this paper. An important conclusion that the phase of the optical fringe pattern is equal to the phase of its HWT coefficients on the ridge is theoretically clarified. The strict relation between the scale parameter and the phase gradient (also called the angular frequency) of the fringe pattern is also given. Computer simulations and experiments reveal the validity of the method and the correctness of the mathematical demonstration. Since the filtering process is avoided in this method, it can deal with the frequency overlapping problem to a certain extent. Applications of this method in both optical profilometry and interferometry are shown and discussed in the experiment section.