Advances in Fringe Processing using Genetic Algorithms

1. Advances in Fringe Processing using Genetic Algorithms L.E.TOLEDOa – F. J. CUEVASb Centro deInvestigaciones 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

2. Introduction. • What is a Genetic Algorithm(GA)? • GA operators: • Selection. • Crossover. • Mutation. • GA’s for phase recovering. • Tests and experiments. • Conclusion. • Aknowledgments. • References.

3. 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])

4. 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.

5. 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(ap). • 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.

6. 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.

7. 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.

8. 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.

9. 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 λ1 and λ2 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.

10. 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.

11. Reconstructing the phase field

12. 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.

13. 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 apthat 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).

14. 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. λ1 andλ2 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 168x168. (b) Original fringe image 168x168.

15. Sampled Fringe pattern 42x42. (b) Fringe pattern associated with demodulated phase. • (c) Error graph between original and demodulated phase.

16. Demodulated phase in a resolution of 168x168. • (b) Fringe pattern associated with (a). • 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).

17. 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 : λ1=0.05, λ2=0.007. A media filter implemented with a window of 7x7 was used to smooth the phase.

18. 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.

19. 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.