A comparison between wavelet fringe analysis algorithms
A Z Abid, M A Gdeisat, D R Burton and M J Lalor
General Engineering Research Institute (GERI), Liverpool John Moores University,
United Kingdom
[email protected], [email protected], [email protected] and
[email protected]
Abstract. Recently, there has been much interest in demodulating fringe patterns using
continuous wavelet transform. Different wavelet algorithms are used in the demodulation
process and they can be classified into two categories: phase estimation and frequency
estimation techniques. This paper compares these techniques in terms of their noise
performance, reliability and the required time to execute using digital computers. Real and
computer generated fringe patterns are used to perform this comparison.
1. Introduction
Today, non-contact measurement has been employed in many science and engineering applications to
compute the 3-D surface of an object. This computation could be done by the analysis of projected
fringe patterns using continuous wavelet transform (CWT) method. The analysis consists of
demodulating the deformed fringe patterns and extracting the phase information encoded into it and
hence the height profile of the object can be calculated. Different wavelet algorithms are used in the
demodulation process to extract the phase of the deformed fringe patterns and they can be classified
into two categories: phase estimation and frequency estimation techniques.
The phase estimation algorithm employs complex mother wavelets to estimate the phase of a fringe
pattern. The extracted phase suffers from 2 discontinuities and a phase unwrapping algorithm is
required to remove these 2 jumps. Zhong et al. [1] have used Gabor wavelet to extract the phase
distribution where phase unwrapping algorithm is required.
The frequency estimation technique estimates the instantaneous frequencies in a fringe pattern,
which are integrated to estimate the phase. The phase extracted using this technique is continuous;
consequently, phase unwrapping algorithms are not required. Complex or real mother wavelets can be
used to estimate the instantaneous frequencies in the fringe pattern. Dursun et al. [2] and Afifi et al.
[3] have used Morlet and Paul wavelets respectively to obtain the phase distribution of projected
fringes without using any unwrapping algorithms.
In this paper a comparison between these two approaches, in terms of their noise performance,
reliability and the required time to execute, using digital computer is done. The paper is organized as
follows. Section 2 introduces wavelet transform and explains the Morlet wavelet. Section 3 explains
the wavelet phase extraction algorithms. The computer simulation and the experimental work are
discussed in sections 4 and 5 respectively and finally the conclusion is stated in section 6.
2. Wavelet Transform
Wavelet transform is a suitable tool to analyze non-stationary signals and thus it has been developed as
an alternative approach to the current available transforms, such as Fourier transform, to analyze
fringe patterns. Moreover, it is worth mentioning that WT has a mutliresolution property in the time
and frequency domain which overcomes the resolution problem in other transforms.
The term wavelet means a small wave of limited duration and it can be real or complex. However,
two conditions must be satisfied in any wavelet which are: the wavelet must have an average value of
zero, called the ‘admissibility condition’; and it must have a finite energy. Many different types of
mother wavelets are available and for phase evaluation application, the most suitable mother wavelet
is probably the complex Morlet [2]. The Morlet wavelet is a plane wave modulated by a Gaussian
function, and is defined as
 ( x)   1/ 4 exp( icx ) exp(  x 2 / 2)
(1)
where c is a fixed spatial frequency, and chosen to be about 5 or 6 to satisfy the admissibility condition
[2]. Figure 1 shows the real part (dashed line) and the imaginary part (solid line) of the Morlet
wavelet.
Figure 1: Complex Morlet wavelet
The one-dimensional continuous wavelet transform (1D-CWT) of a row ƒ(x) of a fringe pattern is
obtained by translation on the x axis by b (with y fixed) and dilation by s of the mother wavelet ψ(x) as
given by
W ( s, b) 
1
s

 f ( x)

*
 x b 

 dx
 s 
(2)
where * denotes complex conjugation and W(s,b) is the calculated CWT coefficients which refers to
the closeness of the signal to the wavelet at a particular scale.
3. Wavelet phase extraction algorithms
Phase estimation and frequency estimation methods are two techniques that will be used in this work
to extract the phase distribution from two dimensional fringe patterns. In the phase estimation method,
complex Morlet wavelet will be applied to a row of the fringe pattern. The resultant wavelet transform
is a two dimensional complex array. Hence, the modulus and the phase arrays can be calculated by the
following equations
abs(s, b)  W (s, b)
(3)
 {W ( s, b)} 

 {W ( s, b)} 
 ( s, b)  tan 1 
(4)
To compute the phase of the row, the maximum value of each column of the modulus array is
determined and then its corresponding phase value is found from the phase array. By repeating this
process to all rows of the fringe pattern, a wrapped phase map is resulted and unwrapping algorithm is
needed to unwrap it.
In the frequency estimation method, complex Morlet wavelet is applied to a row of the fringe
pattern. The resultant wavelet transform is a two dimensional complex array. The modulus array can
be found using equation (3) and hence the maximum value for each column and its corresponding
scale value can be determined. Then the instantaneous frequencies are computed using the following
equation [2]

c  c2  2
f (b) 
 2 f o
2smax (b)
(5)
where ƒo is the spatial frequency. At the end, the phase distribution can be extracted by integrating the
estimated frequencies and no phase unwrapping algorithm is required.
4. Computer simulation
In our numerical simulation, we generate a fringe pattern of size 512 × 512 which is shown in figure 2.
Figure 2: Simulated fringe pattern.
The intensity distribution of this fringe pattern is given by
I ( x, y)  cos[ 2 f o x   ( x, y)]  NOISE
(6)
where ƒo is set to 1/16, NOISE represents a normally distributed noise and  ( x, y ) represents a
simulated object that phase modulates the fringes and is given by
 ( x, y)  0.15[( x  256) 2  ( y  256) 2 ]1/ 2 (7)
The computer-generated fringe pattern, having zero background illumination, is used to test the two
wavelet algorithms. The phase estimation technique has successfully demodulated the simulated fringe
pattern with the help of the unwrap procedure of Matlab to remove the 2π jumps.
On the other hand, the frequency estimation algorithm was not capable of demodulating the fringes
correctly. Figures 3.a and 3.b show the retrieved phase distribution using phase estimation and
frequency estimation methods respectively.
(a)
(b)
Figure 3: Demodulated simulated fringes using (a) phase estimation method; (b) frequency estimation method.
5. Experimental work
A real fringe pattern, shown in Figure 4, taken from the thorax of a female Mannequin was used to
test the wavelet algorithms. The image is of size 512 × 512 pixels. The phase estimation method has
successfully demodulated the real fringe pattern using the unwrap procedure of Matlab. On the other
hand, the frequency estimation method failed to retrieve the phase information. Figure 5 illustrates the
results of both methods.
6. Conclusion
Phase estimation and frequency estimation methods are two wavelet algorithms which are tested in
this work, using the complex Morlet wavelet, to retrieve the phase distribution of simulated and real
fringe patterns. The phase estimation method has successfully demodulated the fringes with the help
of the unwrap procedure of Matlab where the frequency estimation method was not capable of doing
so. By looking at the results of the demodulated real and simulated fringe patterns we conclude that
the phase estimation method behaves the same as the frequency estimation method in term of noise
performance. However, in term of execution time, frequency estimation algorithm has taken longer
execution time than the other method. And finally, in term of reliability, frequency estimation method
seems to be unreliable. Table 1 lists a comparison between the two algorithms. All the algorithms in
this work were programmed in Matlab with the help of YAWTB toolbox [4].
Figure 4: A real fringe pattern
(a)
(b)
Figure 5: Demodulated real fringe patterns using (a) phase estimation method; (b) frequency estimation method.
Table 1: A comparison between phase estimation and
frequency estimation methods.
Unwrapping algorithm
Noise performance
Reliability
Computation time
Phase
estimation
Required
Very good
Reliable
Short
Frequency
estimation
Not required
Good
Not reliable
Long
References
[1] Zohng, J. and Weng, J. Spatial carrier-fringe pattern analysis by means of wavelet transform:
wavelet transform profilometry. Appl. Opt. 2004; 34(26), pp. 4993-4998.
[2] Dursun, A., Ozder, S. and Ecevit, N. Continuous wavelet transform analysis of projected fringe
pattern analysis. Meas. Sci. Tech. 2004; 15, pp. 1768-1772.
[3] Afifi, M., Fassi-Fihri, M., Marjane, M., Nassim, K., Sidki, M. and Rachafi, S. Paul waveletbased algorithm for optical phase distribution evaluation. Opt. Comm. 2002; 211, pp. 47-51.
[4] Yet another wavelet toolbox (YAWTB) home page (accessed on Jan. 2006).
http://www.fyma.ul.ac.be/projects/yawtb/
[5]
Gdeisat, M., Burton, D. and Lalor, M. Spatial carrier fringe pattern demudulation using a twodimensional continuous wavelet transform. To be published in Appl. Opt. 2006.