Analyzing of fringe patterns polluted by noise and nonlinearity using S-Transform

The S-transform, a time frequency representation proposed in 1996 by R.G. Stockwell, can be conceptually thought of either as a variable window short-time Fourier transform or a phase corrected Wavelet transform. Whose window size verifies with the frequency and here both the local amplitude and local phase spectrum of a time varying signal can be simultaneously estimated. As a reversible time-frequency analysis tool, it is well suited to analyzing of non-stationary signals and has many desirable characteristics. Distinct from the wavelet transformation, averaging the local S spectra over the direction of time can correctly form the Fourier transform spectra of the signal. Therefore S transform have direct relationship with the Fourier transform. In recent years, the S-transform has been introduced in three-dimensional optical measurement based on a fringe projection technique and attracted many researchers to work on the field. In this paper, for verifying the advantages of S transform, we compare the reconstruction of S transform, including S transform ridge method and S transform filtering method, with that of other methods, such as Fourier transform method, wavelet transform method for eliminating phase errors caused by the existences of both nonlinear factor and noise. In addition, we have a discussion and make a comparison on these methods by means of computer simulations appearing robust within different white Gaussian noise levels. It shows that S-transform profilometry based on the filtering way is helpful to the enhancement of measurement accuracy, which is verified by experiments for its better reconstruction results.