Wavelet Transform Profilometry Using the Phase Gadient Method

To extract the phase of a fringe pattern, the intensity data from the fringe pattern is applied to the 1D-CWT algorithm on a row by row basis. An example of a fringe pattern is shown in figure 1(a). A row of the fringe pattern  is indicated by the black line in figure 1(a)  and is plotted as a graph of intensity vs  pixel position in figure 1(b). The 1D-CWT of the row is calculated for each row using the equation

The wavelet transform for a row is a two-dimensional complex array. Figure 1(c) indicates the modulus of the complex array for the row shown in figure 1(b). The horizontal axis in figure 1(c) is the translation parameter b; whereas the vertical axis is the scale factor s. The scale is discretized to a vector that contains 640 elements and varies from 0.1 to 64 with a step size of 0.1.

The phase of the row is extracted as follows. The maximum value for each column in figure 1(c) is determined, indicated by a dotted curve in the figure, along with its corresponding scale value. The scale values s_max(b) are shown in figure 1(d). When using the Morlet wavelet, the estimated instantaneous frequencies  and the phase gradient values are calculated using

Where f_c is the frequency of the mother wavelet and it is set here to 2π. The term f_o is the spatial carrier frequency of the fringe pattern. The phase gradient values are integrated in order to extract the phase of the row, which is shown in figure 1(e). The demodulated phase of the entire fringe pattern is shown in figure 1(f). The resultant phase is continuous and an unwrapping algorithm is therefore not required in this case.

A Matlab program that demonstrates the operation of this algorithm can be downloaded here.

share was this page helpful?