Advances in Fringe Processing
using Genetic Algorithms
L.E.TOLEDOa
–
F. J. CUEVASb
Centro de
Investigaciones en Óptica, A.C, Department
of Optical Metrology, Loma del bosque 115, Col. Lomas
del Campestre, León 37150, Guanajuato, MÉXICO.
ltoledo@cio.mx
,
fjcuevas@cio.mx
Introduction.
What is a Genetic Algorithm(GA)?
GA operators:
Selection.
Crossover.
Mutation.
GA’s for phase recovering.
Tests and experiments.
Conclusion.
Aknowledgments.
References.
Introduction
A fringe pattern carries information embedded in its
phase. The fringe pattern would have open or closed
fringes
Methods commonly used:
Phase shifting.
Requires three or more images. Not useful
for transient events
Fourier method, Syncronous method, Phase
losked loop method.
Used only for open fringes images.
Regularization techniques: Regularized phase
tracker (RPT), Two dimensional Hilbert
transform method.
They establish a cost function and optimize
it, sensitive to noise, fall on local
minimums.
Fig. 1: Fringe pattern.
I(x,y) = a(x,y) + b(x,y)
cos(
φ
[x,y]
)
What is a genetic algorithm?
GAs are optimization algorithms that simulate natural evolution. They can discover highly
precise functional solutions, and are very useful for nonlinear optimization problems, or in
the presence of multiple minimums.
Genetic Algorithms were first proposed and analyzed by John Holland. A GA is a special case
of evolutionary algorithms (EA), which involve the reproduction, random variation,
competition and selection of contending individuals in a population.
We present a variation in the WFPD method by Cuevas et. al. [7] to demodulate complicated
fringe patterns using a GA to fit a polynomial on sub

sampled images. The Fringe Processing
on Independent Windows method (FPIW) is applied on a set of partially overlapping
windows extracted from the original fringe pattern. The independent phases obtained by the
GAs are used to reconstruct the whole phase field, adding splicing phases from adjacent
windows. Results can be refined using gradient descent, and we present two different ways
to do the splicing process.
Chromosomes.
The individual structures to be evolved are called chromosomes. Chromosomes are the
genotype that is manipulated by the GA. A chromosome is formed by genes; the value in
each gene is called an allele. A given allele could be a bit or a real value number.
Chromosomes codify a given solution in the domain of the function to be optimized, and this
solution is decoded in the evaluation routine.
Genetic Algorithms operators
Selection.
Some of these individuals are randomly selected to generated the new population, with a
probability that is a function of their fitness value. Fitness values for the demodulation
process are given by the fitness function. The function to be optimized is the fitness function
U(a
p
).
There are many methods to calculate the selection probability of a individual. Ruleta method
give probabilities calculated by the eq. 1:
But ruleta is not very useful, because it don’t avoid a premature convergence (the search
space is not completely explored and fall in local minimums).Boltzmann Method used a
“temperature” to control the convergence. Temperature is decreased on succesive
generations.
Crossover operator.
It simulates the sexual reproduction, and increases the variety of chromosomes inside the
population.
The information in some segments of the chromosomes is exchanged with the information in the
same segment of another selected chromosome, with a given probability
Pc
.
Pc
determines what percentage of the population will be mixed, and typical values
are around 0.9. A random number is generated, and if it is above
Pc
, chromosomes
are added to the new population without changes. If the number is below
Pc
, the
information is exchanged.
Mutation operator.
To avoid premature convergence and explore new regions in the function domain, a mutation
operator is applied on the new chromosomes.
A gene is altered with a probability
Pm
, given by Equation (6) for three hundred generations. If
the allele is binary, the bit is exchanged by its complement. If the allele is a real number, a
random quantity, which could be positive or negative, is added to the actual value.
Decreasing the mutation rate helps to explore all the search space and avoid premature
convergence.
GA’s for Phase Recovering
Codification.
The GA’s can be used in many optimization problems; the algoritmh is always the same.
GA’s is adapted to a problem by defining the fitness function, and the codification of the
chromosomes.
In the problem of demodulation, a chromosome codified a given function:
The coefficientes of these polynomial are the genes of the chromosome. The function p(x,y)
represents the phase
φ
(x,y)
of a fringe pattern.
GA look for the best chromosome that represents better the phase of a given fringe pattern.
Fitness Function.
It evaluates the chromosome aptitude to solve the given problem. In fringe pattern analysis, it
includes a similarity term that compares the original fringe pattern and the fringe pattern
generated by a given chromosome.
Others terms reduce search space to smooth solutions (first and second derivatives). Two
coefficients
λ
1 and
λ
2 are used to to control the effect of the smooth terms over the fitness
function
GA’s search for a maximum, minus sign transform the fitness function optimal to a maximum.
This function is interpolated over a sub

image from the original fringe pattern. More overlapped
sub

images are taken so the entire image is covered.
After all windows has been demodulated, all phases are spliced to reconstruct the whole phase
field.
1.

The demodulated phase from the first window is used as the initial reference.
2.

From the GA current fitted phase window
f(x,y)
, it is calculated a second phase field
f’
(
x,y
)
(with a negative concavity).
3.

Two DC bias are calculated, one for
f
(
x,y
) and one for
f’
(
x,y
) using:
where
N
is the overlapped neighbourhood region and
A
is the overlapped area (pixel2) of
N
.
3.

The RMS error for the two alternative phase window field (
f
(
x,y
) and

f’
(
x,y
))
compared
against the reconstructed phase field
F(x,y)
is calculated as:
4.

The phase described by the function with the minimum
RMS
error value (
f+DC1
or
f’+DC2
) is
spliced with the demodulated phase field
F
.
5.

If there are more windows to splice, the next window in the sequence is labelled as the
current window and go to step 2. Otherwise the splicing process is finished.
Reconstructing the phase field
Tests and Experiments
Test 1.
The GA was tested using a computer generated interferograms. It is necessary to
apply the segmented window approximation, to follow the complexity of the phase
map. The fringe image is a complicated one with closed fringes and under

sampled
fringes.
The phase and fringe pattern for this equation are shown below. The resolution of
this image is 50x20. A 9x9 window was moved over the fringes with an overlapping
region between 40% and 60% of the window.
(a)
Original phase.
(b)
Original Fringe image.
The population has 500 chromosomes. Boltzmann selection and a two point crossover operator
were used. The parameters were set up in the values given: λ1=0.025, λ2=0.001, 300
generations,
CP =
90%
,
The population was evolved during 300 generations and then the best chromosome was chosen
like the vector
ap
that best estimates the phase on these windows. Then the phase was spliced
with the previous estimated phase stored in the phase map.
The phase estimated by the GA is shown below. A media filter was applied over the phase map to
smooth the patch’s edges. The RMS error is 0.265 rad.
(a)
Phase map demodulated from 7(b).
(b)
Fringe pattern from (a).
Test 2.
A computer fringe image was generated with a resolution of 168x168 pixels.
The fringe pattern
was sampled to produce an image of 42x42 pixels. A window size of 9x9 was chosen.
λ
1
and
λ
2
were initialized with the values:
l1=
0.05,
l2=
0.005. The GA was applied on each window until
RMS error was less than 0.4.
a)
Original
computer
generated
phase
168
x
168
.
(b)
Original
fringe
image
168
x
168
.
(a)
Sampled Fringe pattern 42x42.
(b)
Fringe pattern associated with demodulated phase.
(c)
Error graph between original and demodulated phase.
RMS error between both phases is 0.154 rads. Demodulation processes can be done now in
parallel and the phase map behaviour is correctly reconstructed. See that the fringes lost in the
sub

sampled regions were recovered.
FPIW was applied on a 42x42 size image. WFPD and FPIW method are less restricted by
smoothness assumptions because these methods include upper grade terms.
Finally, methods using GA’s works near Nyquist (FPIW) or in sub

Nyquist(WFPD).
(a)
Demodulated
phase
in
a
resolution
of
168
x
168
.
(b)
(b)
Fringe
pattern
associated
with
(a)
.
Test 3. Interferogram from optical testing.
The binarized fringe pattern was demodulated using a window of 15x15.
Smopth coeficients
were set up in the values :
λ
1=
0.05
,
λ
2=
0.007. A media filter implemented with a window
of 7x7 was used to smooth the phase.
Conclusions
A new technique to estimate the phase in a complicated fringe image is presented. It is based on
avoiding the overlapping similarity criterion from the fitness function in the WFPD method. An
algorithm to splice the independent phases from all windows is presented.
The new technique shown in this paper is based on the assumption that it is not necessary to
know the phase on the neighbours to estimate the phase in a given window. This made it
possible to eliminate the overlapping similarity criterion in the fitness function, and instead only
take into account the smoothness of a given solution, given a fitness function that is easy to
evaluate and that is independent from the phase on other windows. This condition makes the
algorithm presented in this paper robust to demodulation errors from other windows. The
demodulation process can be done in parallel.
This algorithm was tested with computer generated interferograms with wide frequency content,
closed fringes and under

sampled fringes, and it was able to demodulate the phase on these
cases.
The proposed algorithm to splice the different phases is able to reconstruct the phase map, with
the sole condition that the phases to be joined do not present oscillations in their overlapping
areas. But even in cases where these oscillations appear, the algorithm can correct the phase in
the regions by adjusting
λ1, λ2
.
ACKNOWLEDGMENTS
We acknowledge the support of the Consejo Nacional de Ciencia y Tecnología de México, Consejo
de Ciencia y Tecnología del Estado de Guanajuato and Centro de Investigaciones en Óptica, A.C.
To Guillermo Garnica for its invaluable technical support.
References:
[1] Malacara D., Servin M., Malacara Z. (1998) Interferogram Analysis for Optical testing, Marcel
Dekker, New York.
[2] Takeda M., Ina H., Kobayashi S. (1982) Fourier

transform method of fringe

pattern analysis
for computer based topography and interferometry. J. Opt. Soc. Am., 72:156.
[3] Ichioka, Y., Inuiya, M. (1972) Direct phase detecting system. Appl. Opt., 11:1507

1514.
[4] Servin, M., Rodriguez

Vera, R (1993) Two dimensional phase locked loop demodulation of
interferogram. J. Mod. Opt., 40:2087

2094,.
[5] Servin M., Marroquín J.L., Cuevas F.J. (2001) Fringe

follower regularized phase tracker for
demodulation of closed

fringe interferograms. J. Opt. Soc. Am. A, 18:689

695.
[6] Larkin K.G., Bone D.J., Oldfield M.A. (2001) Natural demodulation of two

dimensional fringe
patterns in general background of the spiral phase quadrature transform. J. Opt. Soc. Am. A,
18:1862

1870.
[7] Cuevas F.J., Mendoza F., Servin M., Sossa

Azuela J.H. (2006) Window fringe pattern
demodulation by multi

functional fitting using a genetic algorithm. Opt. Commun. 261:231

239.
[8] Cuevas F.J., Sossa

Azuela J.H., Servin M. (2006) A parametric method applied to phase
recovery from a fringe pattern based on a genetic algorithm. Opt. Commun. 261:231

239.
[9] Goldberg D. (1989) Genetic Algorithms: Search and Optimization Algorithms, Addison

Wesley
Publishing, MA.
[10] Holland J.H.(1975) Adaptation in natural and Artificial Systems. University of Michigan Press.
Michigan.
[11] Bäck, Fogel, Michalewicz (2000) Evolutionary computation. Institute of Physics publishing,
Bristol and Philadelfia.
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