**Fringe Pattern Demodulation Using Evolutionary Algorithms**

L. E. Toledo1, F. J. Cuevas1, J.F. Jimenez Vielma2 and J. H. Sossa2 *1Centro de Investigaciones en Optica A.C., Dept. of Computer Vision and Artificial Intelligence, Optical Division, Leon, 2Center for Computing Research, National Polytehnical Institute, Artificial Inteligence Laboratory, Mexico* 

#### **1. Introduction**

78 Advanced Topics in Measurements

A particular attention has to be devoted to the error budget on fluence determination, more precisely on the measurement of beam equivalent area. CCD cameras have to be carefully qualified. Thus the error on calculation could be estimated. It is vital to ensure that this equivalent area determined on the reference path is equal to that on the sample controlling the CCD position and by verifying that the optics in front of the camera do not alter the beam profile. This measurement has to be recorded at the frequency laser to monitor shotto-shot laser fluctuations. The calculations on error bars not only allow comparing results from several samples tested on several facilities but also give an upper limit of damage density, particularly useful when a small area is scanned or in a low damage density range. Depending on the available area on the sample to be tested and the level of damage density, it appears that the 1/1 and rasterscan procedures are comparable and complementary with the use of an appropriate data reduction. More, these procedures give access to representative measurements when compared to results and behaviours observed on high

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**8. References** 

Interferometers are used in metrology to measure temperature, displacement, stress and other physical variables. A typical interferometer split a laser beam using a beam divisor. Beam A is called reference, and is projected directly over a film or a CCD camera using mirrors or fiber optic. Beam B interact with the physical phenomenon to be measured. The interaction modifies the optic path of beam B; then it is projected over the same film or CCD camera that beam A. The total irradiance is modelled on eq. 1.

$$I(\mathbf{x}, y) = a(\mathbf{x}, y) + b(\mathbf{x}, y) \cos(\varphi(\mathbf{x}, y)) \tag{1}$$

The information about the measure is embodied on an interferogram, that is, a fringe pattern image. In optical metrology, a fringe pattern carries information embedded in its phase, that represents the difference in optical path between beam A and beam B. *x y*, are integer values representing coordinates of the pixel location in the fringe image, *a x*(,) *y* is the background illumination, *b x*(,) *y* is the amplitude modulation, and (,) *x y* is the phase term related to the physical quantity being measured. Figure 1 shows an interferogram and its associated phase (,) *x y* .

Fig. 1. Fringe pattern(a) and its phase map (b).

Fringe Pattern Demodulation Using Evolutionary Algorithms 81

(a) (b) (c)

(a) (b)

Methods like the Phase Tracker (Servin et al, 2001a) and the Two-dimensional Hilbert Transform (Larkin et al, 2001) are used for closed fringes, normalized images. These methods are robust again a large amount of noise, but a subjacent condition is to fulfil Nyquist condition. Phase tracker gives an unwrapped phase so there is not necessary to use an unwrapping method. The phase tracker and Hilbert transform proposed a cost function that depends of some measure of the difference between the real phase and the estimated phase. The real phase is unknown so the original interferogram is used and compared to the

fringe pattern of the proposed phase. More terms are added to introduce restrictions.

A problem with minimize a cost function is the danger of fall in a local minimum, far away from the optimal point. It is also possible to use soft computing algorithms, such as neural networks and evolutionary algorithms (EA). In the neural network technique, a multilayer neural network (MLNN) is trained by using fringe patterns, and the phase gradients associated with them, from calibrated objects (Cuevas et al, 2000); after the training, the MLNN can estimate the phase gradient when the fringe pattern is presented in the MLNN input. A genetic algorithm (GA) is a particular type of EAs. GA´s are optimization algorithms that simulate natural evolution (Holland, 1975), and whereas GAs do not search for the best solution to a given problem, they can discover highly precise functional solutions and are very useful for nonlinear optimization problems or in the presence of

Fig. 4. (a) Original fringe pattern I; (b) Adding a constant phase of 120 degrees, I2;

(c) Adding a phase of -120 degrees, I3.

Fig. 5. (a) Wrapped phase; (b) Unwrapped phase.

The problem is to recover the phase map from the fringe pattern image. The demodulation process can be achieved by different methods, depending on the characteristics of the fringe pattern. If the fringe pattern, or interferogram has open fringes (see fig. 2c) by adding a carrier or tilt onto the phase, phase is obtained using Takeda's Fourier Transform method (Takeda & Kobayashi, 1982) or the Phase Locked Loop (Servin & Rodriguez Vera, 1993). A carrier is a phase that increases or decreases linearly with *x* and or *y* .

PLL and Fourier methods can not be used if the interferogram has closed fringes or is not normalized. A normalized fringe pattern means *axy* (,) 0 and *bxy* (,) 1 (see fig 3). Many methods can be used to normalize a fringe pattern (Quiroga et al, 2001). An interferogram can be normalized due to a tilt in the plane waves of beams, defocus, speckle noise, etc.

Fig. 3. (a)Non-normalized fringe pattern. *a x*( , ) 0.001 *y y* , *b x*(,) 2 *y x* ; (b) Normalized fringe pattern.

If it is possible to take three or more images and add a constant on the phase, a constant shift that is different for each fringe pattern (see fig. 4), phase shifting techniques are suitable (Malacara et al, 1998). It is not necessary to normalize the fringe pattern, but it is assumed that 123 *a x*(,) (,) (,) *y a x y a x y* and 123 *b x*(,) (,) (,) *y b x y b x y .*

A drawback to phase shifted method is the real phase are not obtained, but a mod 2 of the phase (fig. 5a). An unwrapping method (Ghiglia & Romero, 1994) is necessary to obtain the real phase (fig. 4b). The fringe patterns used on phase shifting should be normalized.

The problem is to recover the phase map from the fringe pattern image. The demodulation process can be achieved by different methods, depending on the characteristics of the fringe pattern. If the fringe pattern, or interferogram has open fringes (see fig. 2c) by adding a carrier or tilt onto the phase, phase is obtained using Takeda's Fourier Transform method (Takeda & Kobayashi, 1982) or the Phase Locked Loop (Servin & Rodriguez Vera, 1993). A

PLL and Fourier methods can not be used if the interferogram has closed fringes or is not normalized. A normalized fringe pattern means *axy* (,) 0 and *bxy* (,) 1 (see fig 3). Many methods can be used to normalize a fringe pattern (Quiroga et al, 2001). An interferogram can be normalized due to a tilt in the plane waves of beams, defocus, speckle noise, etc.

(a) (b) (c)

Fig. 2. Original phase (a), fringe pattern without carrier; (b), fringe pattern by adding a carrier (c).

(a) (b)

If it is possible to take three or more images and add a constant on the phase, a constant shift that is different for each fringe pattern (see fig. 4), phase shifting techniques are suitable (Malacara et al, 1998). It is not necessary to normalize the fringe pattern, but it is assumed

phase (fig. 5a). An unwrapping method (Ghiglia & Romero, 1994) is necessary to obtain the

of the

Fig. 3. (a)Non-normalized fringe pattern. *a x*( , ) 0.001 *y y* , *b x*(,) 2 *y x* ; (b) Normalized

A drawback to phase shifted method is the real phase are not obtained, but a mod 2

real phase (fig. 4b). The fringe patterns used on phase shifting should be normalized.

that 123 *a x*(,) (,) (,) *y a x y a x y* and 123 *b x*(,) (,) (,) *y b x y b x y .*

fringe pattern.

carrier is a phase that increases or decreases linearly with *x* and or *y* .

Fig. 4. (a) Original fringe pattern I; (b) Adding a constant phase of 120 degrees, I2; (c) Adding a phase of -120 degrees, I3.

Fig. 5. (a) Wrapped phase; (b) Unwrapped phase.

Methods like the Phase Tracker (Servin et al, 2001a) and the Two-dimensional Hilbert Transform (Larkin et al, 2001) are used for closed fringes, normalized images. These methods are robust again a large amount of noise, but a subjacent condition is to fulfil Nyquist condition. Phase tracker gives an unwrapped phase so there is not necessary to use an unwrapping method. The phase tracker and Hilbert transform proposed a cost function that depends of some measure of the difference between the real phase and the estimated phase. The real phase is unknown so the original interferogram is used and compared to the fringe pattern of the proposed phase. More terms are added to introduce restrictions.

A problem with minimize a cost function is the danger of fall in a local minimum, far away from the optimal point. It is also possible to use soft computing algorithms, such as neural networks and evolutionary algorithms (EA). In the neural network technique, a multilayer neural network (MLNN) is trained by using fringe patterns, and the phase gradients associated with them, from calibrated objects (Cuevas et al, 2000); after the training, the MLNN can estimate the phase gradient when the fringe pattern is presented in the MLNN input. A genetic algorithm (GA) is a particular type of EAs. GA´s are optimization algorithms that simulate natural evolution (Holland, 1975), and whereas GAs do not search for the best solution to a given problem, they can discover highly precise functional solutions and are very useful for nonlinear optimization problems or in the presence of

Fringe Pattern Demodulation Using Evolutionary Algorithms 83

Genetic algorithms find application in bioinformatics, computational science, engineering,

(a) (b) Fig. 7. (a) Representation of the solution domain; (b) Each gene is codified with a bit string.

A standard representation of the solution is as an array of bits. Arrays of other types and structures can be used in essentially the same way. The main property that makes these genetic representations convenient is that their parts are easily aligned due to their fixed size, which facilitates simple crossover operations. Variable length representations may also

The fitness function is defined over the genetic representation and measures the quality of the represented solution. The fitness function is always problem dependent. In some problems, it is hard or even impossible to define the fitness expression; in these cases,

Once we have the genetic representation and the fitness function defined, GA proceeds to initialize a population of solutions randomly. Improve it through repetitive application of

At the first iteration many individual solutions are randomly generated to form the population. The population size depends on the nature of the problem, but typically contains several hundreds or thousands of possible solutions. Traditionally, the population is generated randomly, covering the entire range of possible solutions (the search space).

During each successive generation, a proportion of the existing population is selected to breed a new generation. Individual solutions are selected through a fitness-based process, where fitter solutions (as measured by a fitness function) are typically more likely to be selected. Certain selection methods rate the fitness of each solution and preferentially select the best solutions. Other methods rate only a random sample of the population, as this

economics, chemistry, manufacturing, mathematics, physics and other fields.

1. A genetic representation of the solution domain, see fig. 7. 2. A fitness function to evaluate the solution domain.

be used, but crossover implementation is more complex in this case.

A typical genetic algorithm requires:

interactive genetic algorithms are used.

process may be very time-consuming.

**2.1 Initialization** 

**2.2 Selection** 

mutation, crossover, inversion and selection operators.

multiple minimums (Goldberg, 1989), where classic techniques like gradient descent, deterministic hill climbing or random search (with no heredity) fail.

Methods using GA (Cuevas et al, 2002), approximate the phase through the estimation of parametric functions. The chosen functions could be Bessel in the case of having fringes from a vibrating plate experiment, or Zernike polynomials, in the case of an optical testing experiment, and when not much information is known about the experiment, a set of low degree polynomials *p*(,,) **a** *x y* can be used. A complicated pattern is demodulated dividing it into a set of partially overlapping windows fitting a low dimensional polynomial function in each window, so that no further unwrapping is needed (Cuevas et al, 2006).
