Two-dimensional wavelet transform profilometry (2D-WTP)
The two-dimensional wavelet transform profilometry (2D-WTP) can be used to extract the phase of fringe patterns deeply immersed in noise and have slow phase varaitions. Generally, I advise you to use the 1D-WTP algorithm to analyse fringe patterns unless they are very noisy and have slow phase variations. In this case only you should use the 2D-WTP algorithm.
In the 2D-WTP, the fringe pattern g(x,y) is projected onto the daughter wavelet ψa,b,s,θ by translation on x and y-axes by a and b respectively, dilation by s, and rotation by the angle θ of the mother wavelet ψ(x,y). The resulting wavelet transform is four-dimensional. The wavelet transform of the fringe pattern g(x,y) is given by:
Where x and y are pixel indices.
Suppose that a fringe pattern with the size of N Χ M pixels is analyzed using the 2D-WTP algorithm that employs ns scales and nθ angles. The resulting wavelet transform is a four-dimensional complex array with the dimensions of N Χ M Χ ns Χ nθ. The four-dimensional array can be visualized as a collection of N Χ M sub-arrays each with the dimensions of ns Χ nθ associated with each pixel in the fringe pattern. These sub-arrays are known as ridge arrays.
The phase of a pixel (m,n) in a fringe pattern is extracted as follows. Initially, the modulus of the complex ridge array associated with this pixel (m,n) is calculated. Then the element p with the maximum modulus in the ridge array is determined and its argument is calculated using the equation below. The extracted argument value is the phase of the pixel
Where I{p}and R{p}are the imaginary and real parts of the element p respectively. This procedure is repeated until the phase of all the pixels in the fringe pattern is determined. The resulted phase map may contain 2π jumps and a phase unwrapping algorithm may be required to remove these 2π phase steps.
The Morlet, modified Morlet, fan or Gabor mother wavelets can be used in conjunction of the 2D-WTP method to extract the phase of the fringe pattern.
1. 2D-Morlet Wavelet
The two-dimensional complex Morlet mother wavelet is essentially a plane wave within a Gaussian window and is given by [1]
Where θ is the rotation angle and ko is set to 5.336. The Fourier transform of this wavelet is
A Matlab program that demonstrates fringe pattern demodulation using the two-dimensional Morlet wavelet can be downloaded here.
2. 2D-Fan Wavelet
The two-dimensional fan mother wavelet is formed by superposing a number of complex Morlet wavelets. The superposition is constructed by averaging a series of 2D Morlet wavelets over a finite number of directions, Nθ [1].
The Fourier transform of this wavelet is
where θj=θk=kδθ, and δθ is the azimuth increment between successive Morlet wavelets
A Matlab program that demonstrates fringe pattern demodulation using the two-dimensional fan wavelet can be downloaded here. To run this Matlab program, you need also to download the fan2D function.
3. 2D-Gabor Wavelet
The two-dimensional Gabor mother wavelet is given by the equation [2].
The Fourier transform of this wavelet is
A Matlab program that demonstrates fringe pattern demodulation using the two-dimensional Gabor wavelet can be downloaded here.
Note that the 2D-Morlet wavelet is a special case of Gabor wavelet with σ set to 1 [1].
The advanced 2D-Morlet wavelet is also a special case of the Gabor wavelet with σ set to 0.5 in the Gabor wavelet equation.
4. 2D-Paul Wavelet
The time-domain two-dimensional Paul mother wavelet is given by the equation.
The Fourier transform of this wavelet is given by the equation.
A Matlab program that demonstrates fringe pattern demodulation using the two-dimensional Paul wavelet can be downloaded here. To run this Matlab program, you need also to download the Paul2D function.
The 2D Morlet wavelet has the best noise performance among the other wavelets but it lacks the ability of demodulatin fringe patterns that contain large phase variations. One the other hand, the 2D-Paul wavelet has the best ability to demodulate fringe patterns with large phase variations but it has the worst noise performance among all other 2D wavelets.
It is worth mentioning that the Matlab programs here use the YAWTB toolbox which employs the frequency domain equations to calculate the wavelet transorm.
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] Niu H., Quan C., Tay C., "Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform," Optics and Lasers in Engineering, 2009, In press.
[3] Wang Z., "Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing," Optical Engineering, Vol. 45, No. 4, pp 045601-1-045601-5, 2006.
