Mother Wavelets
A number of analytical mother wavelets could potentially be used in fringe pattern phase demodulation. The analytical wavelets are represented in complex form and their Fourier transformation produces zero values for the negative frequencies. Examples of analytical wavelets are the Paul, Gaussian, Morlet, b-spline, and Shannon mother wavelets, which are defined by the following equations respectively.
Where n is the order of the Paul mother wavelet and is here set to a value of 4. The complex Gaussian wavelet is calculated by taking the pth derivative of the function
Cpexp(-ix)exp(-x2)
Where Cp is a constant factor which depends on the derivative order p of
Cpexp(-ix)exp(-x2)
function and is computed to normalise the Gaussian wavelet.function and is computed to normalise the Gaussian wavelet. Here, p is set to 4. fb is the variance of the window and this is set here to a value of 1. The term fc is the mother wavelet centre frequency and it is set here to a value of 2π. m is an integer order parameter (where m > 1). For more infromation, please see the Mathworks website.
The mother wavelets menioned above are ordered according to their time localization properties. The Paul wavelet has the best time localization capability amongst the five different mother wavelets, but at the same time it has the worst frequency localization. This makes the Paul mother wavelet more suitable for demodulating fringe patterns that exhibit high signal to noise ratio and rapid phase variations. On the other hand, the Morlet wavelet has better localization in the frequency domain than the Paul wavelet and it is more suitable for demodulating fringe patterns with slow phase variations and low signal to noise ratios.
The WTP techniques implemented using each of the five mother wavelets mentioned above, utilising phase estimation and the maximim ridge extraction algorithm and programmed in Matlab can be downloaded from the list below.
- Paul wavelet. Also you need to download the Paul wavelet definition.
- Gaussian wavelet.
- Morlet wavelet.
- Frequency b-spline wavelet.
- Shannon wavelet.
Note: Matlab accepts files with a '.m' extension and the files downloaded from this webpage have a '.txt' extension. So you will need to change the file extensions to '.m' before running them using Matlab.
If you do not have the Matlab wavelet toolbox, you can still experiment with the wavelets and download the required files. These programs implement the WTP in the frequecny domain. The time domain implementation of the WTP using Matlab can be downloaded here.
Figure 1(a) shows a fringe pattern and a row of the fringe pattern (indicated by the red line) is shown in Figure 1(b). The complex continuous wavelet transform has been used to analyse this row. The modulus of the wavelet transform for the row is shown in Figure 1(c) in grey scale. The white colour indicates large values in the transform, whereas the black colour indicates small values. The argument of the wavelet transform is shown in Figure 1(d) in grey scale. The phase of the signal shown in Figure 1(b) can be computed as follows. The maximum value of the modulus for each column in Figure 1(c) is found, and then the corresponding phase is chosen from Figure 1(d). The resulting wrapped phase is shown in Figure 1(e). The unwrapped phase is shown in Figure 1(f). The wrapped phase map of the fringe pattern is shown in Figure 1(g) and the unwrapped phase is shown in Figure 1(h).
