Fringe Pattern Analysis Using Wavelet Transforms and Phase Demodulation Using Wavelet Transform
The phase information in fringe patterns can be extracted by means of many different methods. For example, by Fourier transform profilometry, phase shifting profilometry, windowed Fourier transform profilometry, frequency tracking methods or wavelet transform profilometry (WTP).
In the WTP method, the continuous wavelet transform (CWT) is used to retrieve the phase of a fringe pattern. The fringe pattern image can either be processed on a row-by-row basis using the one-dimensional continuous wavelet transform (1D-CWT), or alternatively the entire image can be processed as a single unit using the two-dimensional continuous wavelet transform (2D-CWT). Our experience with both methods has revealed that for many practical applications the 1D-CWT is more suitable than the 2D-CWT, and is also much faster in terms of excecution times when implemented in digital computers.
The process of demodulating the phase of a fringe pattern using the WTP technique consists of three stages. In the first stage the continuous wavelet transform of the fringe pattern is calculated. In this step, a suitable mother wavelet should be determined. For example, the following mother wavelets may be used; the Paul, the derivative of Gaussian, Morlet, Shannon, or the frequency b-spline. The choice of an appropriate mother wavelet depends primarily on the properties of the fringe pattern that is being analysed. For example, the Paul wavelet should be used to analyse fringe patterns that have high signal-to-noise ratios and contain rapid phase changes. On the other hand, it is advisable to use the Morlet wavelet to analyse noisy fringe patterns.
In the second stage of the phase demodulation process, the phase, or the derivative of the phase, of the continuous wavelet transform coefficients is computed (which was computed in the first stage). We refer to both methods as the phase estimation and phase gradient techniques respectively. In the first method, the phase of the CWT coefficients is calculated directly using the arctangent mathematical function. In the second method, the gradient of the phase is calculated by estimating the instantaneous frequencies that exist in the fringe pattern. Then, the phase of the CWT coefficients is computed by integrating the values produced by the phase gradient method.
The direct calculation of the phase of the CWT coefficients using the phase estimation method produces more accurate results than the phase gradient technique. Additionally, the phase estimation method performs better against noise and it is faster to execute using digital computers. On the other hand, the phase gradient method could be useful in applications where the derivative of the phase is required, such as the non-contact measurement of strain.
In the third stage, the phase of the fringe pattern is extracted from the phase values that were calculated in stage 2. To perform this task, an algorithm should be used such as the maximum ridge extraction algorithm, phase ridge extraction algorithm, or the cost function ridge extraction algorithm. Probably, the latter method has the best performance amongst the other algorithms.
The WTP method introduces errors at the edges of the phase maps that are constructed. These distortions can be alleviated by artificially extending the borders of a fringe pattern and then analysing it using the WTP technique.
In summary, in order to analyse a fringe pattern using the WTP method, you should choose the parameters that are most suitable for your application. For example, should either the 1D-CWT or the 2D-CWT should be used? What is the most appropriate mother wavelet for the fringe pattern in hand? Are we interested in the phase information of a fringe pattern or the gradient of the phase? What is the most suitable method for extracting the ridge? Is the presence of distortion at the borders of the produced phase map critical for your application? In this webpage, we will provide you with the required information and software to determine whether wavelet transform profilometry could be suitable for your application.
Great effort has been put into programming this WTP software and publishing it on this website. We provide our software free of charge. But we request that if you wish to use our software, please inform us by emailing m.a.gdeisat@ljmu.ac.uk, cite our paper below, and acknowledge the use of the WTP software in any published work.
This not only benefits us, but may be a source of future collaboration and helps us to customise our algorithms to perform in an increasing number of application areas. If our algorithms do not exactly meet the requirments of your application, contact us and we may be willing to customise a solution for you.
References:
[1] Gdeisat. M. A, Burton. D. R and Lalor. M. J, "Spatial Carrier Fringe Pattern Demodulation Using a Two-Dimensional Continuous Wavelet Transform," Applied Optics, doi:10.1364/AO.45.008722, Vol. 45, No. 34, pp 8722-8732, 2006.
[2] Gdeisat, M A., Abid, A., Burton, D R., Lalor, M J., Lilley, F., Moore, C J., Qudeisat, M., "Spatial and Temporal Carrier Fringe Pattern Demodulation using the One-Dimensional Continuous Wavelet Transform: Recent Progress, Challenges and Suggested Developments", Optics and Lasers in Engineering, doi:10.1016/j.optlaseng.2009.07.009 , Vol. 47, pp 1348-1361, 2009.
