Error caused by sampling in S-Transform Profilometry

S-Transform, combining the advantages of both the windowed Fourier transform and the wavelet transform, is a suitable approach for the phase demodulation of the deformed fringe pattern to obtain the reconstruction of the tested object in optical three-dimensional (3D) surface measurement. In the S-Transform algorithm, the translated spectra are needed to calculate the S-Transform coefficients. When the signal is discrete, its Fourier spectrum is periodically repeated in the frequency domain. That is to say, not only the large height variation rate of the tested object can cause spectrum aliasing in the same frequency “island”, but also the spectrum repetition caused by sampling operation may lead to the spectrum overlapping between adjacent “islands” if the sampling rate is not high enough to satisfy the sampling theory. The paper researches the sampling problem of the S-Transform for guaranteeing its accuracy in fringe analysis. The frequency-domain description of the one-dimensional S-Transform and the two-dimensional S-Transform coefficients of the discrete fringe pattern are deduced. In addition, the criteria for the selection of sampling frequency in the S-Transform Profilometry are derived. Computer simulations and processing results of practical experiments demonstrate our analysis.