## Optical image processing using an optoelectronic feedback system with electronic distortion correction

Optics Express, Vol. 13, Issue 12, pp. 4657-4665 (2005)

http://dx.doi.org/10.1364/OPEX.13.004657

Acrobat PDF (889 KB)

### Abstract

Spontaneous pattern formation in an optoelectronic system with an optical diffractive feedback loop exhibits a contrast enhancement effect, a spatial filtering effect, and filling-up of vacant space while maintaining surrounding structures. These effects allow image processing with defect tolerance. Aberrations and slight misalignments that inevitably exist in optical systems distort the spatial structures of the formed patterns. Distortion also increases due to a small aspect ratio difference between a display device and an image sensor. We experimentally demonstrate that the spatial distortion of the optoelectronic feedback system is reduced by electronic distortion correction and the initial structure of a seed optical pattern is preserved for a long time. We also demonstrate image processing of a fingerprint pattern based on seeded spontaneous optical pattern formation with electronic distortion correction.

© 2005 Optical Society of America

## 1. Introduction

1. M. Kreuzer, W. Balzer, and T. Tschudi, “Formation of spatial structures in bistable optical elements containing nematic liquid crystals,” Appl. Opt. **29**, 579–582 (1990). [CrossRef] [PubMed]

12. E. V. Degtiarev and M. A. Vorontsov, “Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression,” J. Opt. Soc. Am. B **12**, 1238–1248 (1995). [CrossRef]

18. R. Neubecker, E. Benkler, R. Martin, and G. -L. Oppo, “Manipulation and removal of defects in spontaneous optical patterns,” Phys. Rev. Lett. **91**, 113903 (2003). [CrossRef] [PubMed]

19. R. Neubecker and A. Zimmermann, “Spatial forcing of spontaneous optical patterns,” Phys. Rev. E **65**, 035205(R) (2002). [CrossRef]

20. R. Neubecker, A. Zimmermann, and O. Jakoby, “Utilizing nonlinear optical pattern formation for a simple image-processing task,” Appl. Phys. B **76**, 383–392 (2003). [CrossRef]

21. M. A. Vorontsov, G. W. Carhart, and R. Dou, “Spontaneous optical pattern formation in a large array of optoelectronic feedback circuits,” J. Opt. Soc. Am. B **17**, 266–274 (2000). [CrossRef]

23. Y. Hayasaki, E. Hikosaka, H. Yamamoto, and N. Nishida, “Image processing based on seeded spontaneous optical pattern formation using optoelectronic feedback,” Appl. Opt. **44**, 236–240 (2005). [CrossRef] [PubMed]

23. Y. Hayasaki, E. Hikosaka, H. Yamamoto, and N. Nishida, “Image processing based on seeded spontaneous optical pattern formation using optoelectronic feedback,” Appl. Opt. **44**, 236–240 (2005). [CrossRef] [PubMed]

23. Y. Hayasaki, E. Hikosaka, H. Yamamoto, and N. Nishida, “Image processing based on seeded spontaneous optical pattern formation using optoelectronic feedback,” Appl. Opt. **44**, 236–240 (2005). [CrossRef] [PubMed]

## 2. Optoelectronic feedback system

*u*(

*x*,

*y*,

*t*) of the LCD at a position (

*x*,

*y*) and time

*t*to a control signal

*p*(

*x*,

*y*,

*t*), the space-time evolution of the internal state is

*τ*is the time constant of the LCD and

*x*and

*y*directions that describes transverse diffusion with a diffusion length

*l*[21

21. M. A. Vorontsov, G. W. Carhart, and R. Dou, “Spontaneous optical pattern formation in a large array of optoelectronic feedback circuits,” J. Opt. Soc. Am. B **17**, 266–274 (2000). [CrossRef]

*F*on the internal state

*u*, and the output light wave

*A*and its intensity

*I*is

*F*is approximated as

*F*(

*u*)=

*I*

_{0}[1-cos(

*u*)]/2, where

*I*

_{0}is the incident light intensity. The control signal

*p*(

*x*,

*y*,

*t*) for a diffractive feedback light intensity

*I*

_{d}(

*x*,

*y*,

*t*) on the CCD image sensor is described as

*G*includes the input/output characteristic of the CCD image sensor, the electric conversion performed by a computer, which in this experiment is an inversion characteristic, and the conversion from an electric signal given to the LCD to the internal state

*u*. The characteristic between the light intensities

*I*

_{1}and

*I*

_{2}as described by

*I*

_{2}=

*G*[

*F*(

*I*

_{1})] is a sigmoid function. The characteristic is presented in Ref. 22, when the CCD image sensor and the LCD are directly connected without the electric conversion by a computer. The feedback light is formulated by considering free-space propagation over a length Z. By use of the stationary, scalar, paraxial wave equation, the amplitude of the diffracted feedback wave

*A*

_{d}(

*x*,

*y*,

*Z*,

*t*) satisfies

*k*=2

*π*/

*λ*is the wave number and

*A*

_{d}(

*x*,

*y*, 0,

*t*)=

*A*(

*x*,

*y*,

*t*). The model equations from (1) to (4) are equivalent to those for an OFS [10

10. R. Neubecker, G.-L. Oppo, B. Thuering, and T. Tschudi, “Pattern formation in a liquid-crystal light valve with feedback including polarization, saturation, and internal threshold effect,” Phys. Rev. A **52**, 791–808 (1995). [CrossRef] [PubMed]

22. Y. Hayasaki, Y. Tamura, H. Yamamoto, and N. Nishida, “Spatial property of formed patterns depending on focus condition in a two-dimensional optoelectronic feedback system,” Jpn. J. Appl. Phys. **40**, 165–169 (2001). [CrossRef]

*I*

_{d}(

*x*,

*y*,

*Z*,

*t*) is distorted. A personal computer and fram-grabber are used not only to give an initial seed pattern, to capture the image sequences, and to analyze the temporal evolution of the patterns, but also to calculate the correction of these distortions.

## 3. Electronic distortion correction

*x*,

*y*) and (

*u*,

*v*), respectively. The geometrical transformation from the LCD plane to the CCD image sensor plane involves a lateral shift between the devices, lateral magnifications, including the aspect ratio mismatch between the devices, device rotation, and nonlinear distortions. The output point (

*x*

_{1},

*y*

_{1}) is transformed by lateral deviation and lateral magnification from the input point (

*x*,

*y*) as:

*x*

_{1},

*y*

_{1}) is transformed to the point (

*x*

_{2},

*y*

_{2}) by a rotation of the CCD image sensor by an angle

*Θ*, and is denoted as:

*x*

_{2},

*y*

_{2}) is converted to polar coordinates (

*r*

_{2},

*θ*

_{2}), the nonlinear distortion is approximated with a 3rd order polynomial as:

*C*is a coefficient, and

*r*

_{2}=(

^{1/2}. Then, (

*r*

_{2},

*θ*

_{2}) is converted back to the rectangular coordinates (

*u*,

*v*) on the CCD image sensor by:

*θ*

_{2}=arctan(

*y*

_{2}/

*x*

_{2}). Therefore, the transformation from (

*x*,

*y*) to (

*u*,

*v*) that represents the geometry of the OEFS is described by Eqs. (5–8).

*A*

_{0},

*A*

_{1},

*B*

_{0},

*B*

_{1},

*C*, and

*Θ*are set based on the following calculation. At first, a 27×19 array of

*N*small bright points is displayed on the LCD as sampling points at the same time, and the image is detected by the CCD image sensor through the optical system. Next, the coordinates of

*n*-th detected point, [

*u*

_{0}(

*n*),

*v*

_{0}(

*n*)] (

*n*=1, 2, …,

*N*), are obtained by the threshold operation of the image and the centroid detection of each bright area. From the relation between a point at (

*x*,

*y*) on the LCD and the corresponding point at (

*u*,

*v*) on the CCD image sensor, the coefficients in Eqs. (5–8) are determined by the downhill simplex method [24] for the estimation function

*u*-

*u*

_{0}(

*n*)]

^{2}+[

*v*-

*v*

_{0}(

*n*)]

^{2}. Finally, the coordinates (

*u*,

*v*) on the CCD image sensor corresponding to all pixels on the LCD are calculated. The coefficients calculated with the method described above were

*A*

_{0}=0.977,

*A*

_{1}=1.00,

*B*

_{0}=-0.267,

*B*

_{1}=-0.537,

*C*=0.000176, and

*Θ*=0.00125. In this case, since the optical system was set such that the deviation at its center was small, the distortion became bigger at the periphery of the operation region. The system has not only linear distortions, such as a lateral displacement, magnification mismatch, rotation, and aspect ratio difference, but also the nonlinear distortion caused by aberrations of the optical system.

*u*,

*v*) calculated from a point (

*x*,

*y*) by Eqs. (5–8) do not match the pixel position of the CCD image sensor. Therefore,

*I*(

*u*,

*v*) is calculated using linear interpolation of the four pixel values on the CCD image sensor,

*I*(

*i*,

*j*),

*I*(

*i*,

*j*+1),

*I*(

*i*+1,

*j*),

*I*(

*i*+1,

*j*+1), where

*i*and

*j*are integer, that are neared to (

*u*,

*v*) as follows:

*p*=

*u*/

*W-i*and

*q*=

*v*/

*W-j*, and

*W*is the defined size of a pixel; in our experiments,

*W*=1.

## 4. Experimental results

*K*

_{z}=2

*π*(

*λ*|

*Z*|)

^{-0.5}on the plane

*P*, for the free-propagation length

*Z*[8

8. T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO_{3},” Opt. Lett. **18**, 598–600 (1993). [CrossRef] [PubMed]

16. G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A **58**, 2577–2585 (1998). [CrossRef]

17. Y. Hayasaki, H. Yamamoto, and N. Nishida, “Selection of optical patterns using direct modulation method of spatial frequency in a nonlinear optical feedback system,” Opt. Commun. **187**, 49–55 (2001). [CrossRef]

*K*

_{z}are initially supplied, the circular fringes should be maintained for a long time. In a practical OEFS, however, the initially-given circular fringes with

*K*

_{z}became distorted gradually, as shown in Fig. 3, because the OEFS has aberrations, slight misalignments, and an aspect ratio difference, as mentioned above. Figure 3(a) shows the temporal evolution of patterns whose center coincides with the optical axis of the system (on-axis region). The size of the images is 100×100 pixels on the LCD. Figure 3(b) shows patterns whose center is shifted from the optical axis (off-axis region) by 100 pixels in the

*x*-axis and 100 pixels in the

*y*-axis on the LCD. These patterns were observed at 1/30 s, 5/30 s, 10/30 s, 15/30 s, and 20/30 s, respectively. The distortion of the formed patterns becomes larger at the periphery. The patterns in Figs. 3(a) and 3(b) were obtained without electronic distortion correction.

*I*

_{1}(

*i*,

*j*,

*t*) and the pattern at the 1st frame. The SSD for measuring the difference between a temporally evolving pattern

*I*

_{1}(

*i*,

*j*,

*t*) with a mean

*µ*

_{1}(

*t*) and a variance

*σ*

_{1}(

*t*)

^{2}and a temporally evolving pattern

*I*

_{2}(

*i*,

*j*, t) with a mean

*µ*

_{2}(

*t*) and a variance

*σ*

_{2}(

*t*)

^{2}is defined as

*N*is the number of pixels. In this experiment,

*I*

_{2}(

*i*,

*j*,

*t*)=

*I*

_{1}(

*i*,

*j*, 1),

*µ*

_{2}(

*t*)=

*µ*

_{1}(1), and

*σ*

_{2}(

*t*)=

*σ*

_{1}(1). For example, the SSD between two random patterns is about 0.015. In the on-axis region, because the distortion of the system is small, the effect of the distortion correction is not noticeable. On the other hand, in the off-axis region, the reduction of the SSD is remarkable and the distortion correction worked very well. By introducing the electronic distortion correction, the OEFS became to preserve the initial structures for long time in the pattern evolution process.

**A**with a natural defect and the same fingerprint pattern

**A′**with an artificial defect, respectively. The fingerprint pattern was obtained by pushing it against a glass plate, illuminating it with a red light emitting diode, and detecting the reflected light intensity distribution with a CCD image sensor. When the fingerprint patterns without and with the artificial defect,

**A**and

**A′**, were initially supplied to the OEFS with electronic distortion correction, the respective temporal evolutions at 1/30 s, 7/30 s, and 22/30 s are shown in Figs. 7(a) to 7(c) and Figs. 7(d) to 7(f), respectively. These patterns were in the off-axis region. Both temporal evolutions proceed while increasing the contrast and removing both the natural and the artificial defects, when the system is configured so that the principal wave number of the fingerprint pattern matches

*K*

_{z}, because the optical pattern in the vacant space (defect) is formed while maintaining and connecting surrounding structures with

*K*

_{z}, when the shorter side of the defect has the length shorter than or approximately equal to the fringe spacing (2

*π*/

*K*

_{z}), as the artificial defect given to the fingerprint pattern shown in Fig. 6(b). The bifurcation, which is a typical minutia, which initially existed in the fingerprint pattern, was well preserved, and its position was almost fixed. Furthermore, the pattern transformed from the fingerprint pattern

**A′**with an artificial defect, shown in Fig. 7(c), almost agreed with the pattern transformed from the original fingerprint pattern according to the temporal evolution of patterns, as shown in Fig. 7(f). Therefore, the OEFS performs image conversion while exhibiting defect tolerance. The defect tolerance is originated from filling-up effect of vacant space while maintaining surrounding structures in spontaneous optical pattern formation. This characteristic is most important and novel in the use of the optical pattern evolutions in OEFS for optical image processing.

**A**and those of the fingerprint pattern

**A′**with the defect. The bold dashed curve and the bold solid curve indicate the SSDs when the OEFS was used without and with the electronic distortion correction, respectively. The SSDs in the OEFS gradually decayed with oscillation. Introducing the electronic distortion correction resulted in a smaller SSD. The effect of the correction was particularly noticeable in the off-axis region. The oscillation is caused by a drop in the frame rate of the OEFS [22

22. Y. Hayasaki, Y. Tamura, H. Yamamoto, and N. Nishida, “Spatial property of formed patterns depending on focus condition in a two-dimensional optoelectronic feedback system,” Jpn. J. Appl. Phys. **40**, 165–169 (2001). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and Links

1. | M. Kreuzer, W. Balzer, and T. Tschudi, “Formation of spatial structures in bistable optical elements containing nematic liquid crystals,” Appl. Opt. |

2. | F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. |

3. | G. D’Alessandro and W. J. Firth, “Spontaneous hexagon formation in a nonlinear optical medium with feedback mirror,” Phys. Rev. Lett. |

4. | S. A. Akhmanov, M. A. Vorontsov, V. Yu. Ivanov, A. V. Larichev, and N. I. Zheleznykh, “Controlling transverse-wave interactions in nonlinear optics: generation and interaction of spatiotemporal structures,” J. Opt. Soc. Am. B |

5. | R. Macdonald and H. J. Eichler, “Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror,” Opt. Commun. |

6. | B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in liquid crystal light valve feedback system,” Opt. Commun. |

7. | E. Ciaramella, M. Tamburrini, and E. Santamoto, “Talbot assisted hexagonal beam patterning in a thin liquid crystal film with a single feedback mirror at negative distance,” Appl. Phys. Lett. |

8. | T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO |

9. | P. P. Banerjee, H. L. Yu, D. A. Gregory, N. Kukhtarev, and H. J. Caufield, “Self-organization of scattering in photorefractive KNbO |

10. | R. Neubecker, G.-L. Oppo, B. Thuering, and T. Tschudi, “Pattern formation in a liquid-crystal light valve with feedback including polarization, saturation, and internal threshold effect,” Phys. Rev. A |

11. | M. A. Vorontsov and W. B. Miller (Eds.), “Self-organization in Optical systems and applications in information technology,” Chapter 2 (Berlin, Springer-Verlag, 1995). |

12. | E. V. Degtiarev and M. A. Vorontsov, “Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression,” J. Opt. Soc. Am. B |

13. | Y. Hayasaki, H. Yamamoto, and N. Nishida, “Optical dependence of spatial frequency of formed patterns on focusing condition in a nonlinear optical ring resonator,” Opt. Commun. |

14. | V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. |

15. | S. J. Jensen, M. Schwab, and C. Denz, “Manipulation, stabilization, and control of pattern formation using Fourier space filtering,” Phys, Rev. Lett. |

16. | G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A |

17. | Y. Hayasaki, H. Yamamoto, and N. Nishida, “Selection of optical patterns using direct modulation method of spatial frequency in a nonlinear optical feedback system,” Opt. Commun. |

18. | R. Neubecker, E. Benkler, R. Martin, and G. -L. Oppo, “Manipulation and removal of defects in spontaneous optical patterns,” Phys. Rev. Lett. |

19. | R. Neubecker and A. Zimmermann, “Spatial forcing of spontaneous optical patterns,” Phys. Rev. E |

20. | R. Neubecker, A. Zimmermann, and O. Jakoby, “Utilizing nonlinear optical pattern formation for a simple image-processing task,” Appl. Phys. B |

21. | M. A. Vorontsov, G. W. Carhart, and R. Dou, “Spontaneous optical pattern formation in a large array of optoelectronic feedback circuits,” J. Opt. Soc. Am. B |

22. | Y. Hayasaki, Y. Tamura, H. Yamamoto, and N. Nishida, “Spatial property of formed patterns depending on focus condition in a two-dimensional optoelectronic feedback system,” Jpn. J. Appl. Phys. |

23. | Y. Hayasaki, E. Hikosaka, H. Yamamoto, and N. Nishida, “Image processing based on seeded spontaneous optical pattern formation using optoelectronic feedback,” Appl. Opt. |

24. | J. A. Nelder and R. Mead, “Simplex method for function minimization,” Computer J. |

**OCIS Codes**

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(200.3050) Optics in computing : Information processing

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 13, 2005

Revised Manuscript: June 4, 2005

Published: June 13, 2005

**Citation**

Yoshio Hayasaki, Ei-ichiro Hikosaka, Hirotsugu Yamamoto, and Nobuo Nishida, "Optical image processing using an optoelectronic feedback system with electronic distortion correction," Opt. Express **13**, 4657-4665 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4657

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### References

- M. Kreuzer, W. Balzer, and T. Tschudi, �??Formation of spatial structures in bistable optical elements containing nematic liquid crystals,�?? Appl. Opt. 29, 579-582 (1990). [CrossRef] [PubMed]
- F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, �??Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,�?? Phys. Rev. Lett. 65, 2531-2534 (1990). [CrossRef] [PubMed]
- G. D'Alessandro and W. J. Firth, �??Spontaneous hexagon formation in a nonlinear optical medium with feedback mirror,�?? Phys. Rev. Lett. 66, 2597-2600 (1991). [CrossRef] [PubMed]
- S. A. Akhmanov, M. A. Vorontsov, V. Yu. Ivanov, A. V. Larichev, and N. I. Zheleznykh, �??Controlling transverse-wave interactions in nonlinear optics: generation and interaction of spatiotemporal structures,�?? J. Opt. Soc. Am. B 9, 78-90 (1992). [CrossRef]
- R. Macdonald and H. J. Eichler, �??Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror,�?? Opt. Commun. 89, 289-295 (1992). [CrossRef]
- B. Thüring, R. Neubecker, and T. Tschudi, �??Transverse pattern formation in liquid crystal light valve feedback system,�?? Opt. Commun. 102, 111-115 (1993). [CrossRef]
- E. Ciaramella and M. Tamburrini, and E. Santamoto, �??Talbot assisted hexagonal beam patterning in a thin liquid crystal film with a single feedback mirror at negative distance,�?? Appl. Phys. Lett. 63, 1604-1606 (1993). [CrossRef]
- T. Honda, �??Hexagonal pattern formation due to counterpropagation in KNbO3,�?? Opt. Lett. 18, 598-600 (1993). [CrossRef] [PubMed]
- P. P. Banerjee, H. L. Yu, D. A. Gregory, N. Kukhtarev, and H. J. Caufield, �??Self-organization of scattering in photorefractive KNbO3 into reconfigurable hexagonal spot array,�?? Opt. Lett. 20, 10-12 (1995). [CrossRef] [PubMed]
- R. Neubecker, G.-L. Oppo, B. Thuering, and T. Tschudi, �??Pattern formation in a liquid-crystal light valve with feedback including polarization, saturation, and internal threshold effect,�?? Phys. Rev. A 52, 791-808 (1995). [CrossRef] [PubMed]
- M. A. Vorontsov and W. B. Miller (Eds.), �??Self-organization in Optical systems and applications in information technology,�?? Chapter 2 (Berlin, Springer-Verlag, 1995).
- E. V. Degtiarev and M. A. Vorontsov, �??Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression,�?? J. Opt. Soc. Am. B 12, 1238-1248 (1995). [CrossRef]
- Y. Hayasaki, H. Yamamoto, and N. Nishida, �??Optical dependence of spatial frequency of formed patterns on focusing condition in a nonlinear optical ring resonator,�?? 151, 263-267 (1998). [CrossRef]
- V. Mamaev and M. Saffman, �??Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,�?? Phys. Rev. Lett. 80, 3499-3502 (1998). [CrossRef]
- S. J. Jensen, M. Schwab, and C. Denz, �??Manipulation, stabilization, and control of pattern formation using Fourier space filtering,�?? Phys, Rev. Lett. 81, 1614-1617 (1998). [CrossRef]
- G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, �??Elimination of spatiotemporal disorder by Fourier space techniques,�?? Phys. Rev. A 58, 2577-2585 (1998). [CrossRef]
- Y. Hayasaki, H. Yamamoto, and N. Nishida, �??Selection of optical patterns using direct modulation method of spatial frequency in a nonlinear optical feedback system,�?? Opt. Commun. 187, 49-55 (2001). [CrossRef]
- R. Neubecker , E. Benkler, R. Martin, and G. -L. Oppo, "Manipulation and removal of defects in spontaneous optical patterns," Phys. Rev. Lett. 91, 113903 (2003). [CrossRef] [PubMed]
- R. Neubecker and A. Zimmermann, �??Spatial forcing of spontaneous optical patterns,�?? Phys. Rev. E 65, 035205(R) (2002). [CrossRef]
- R. Neubecker, A. Zimmermann, and O. Jakoby, �??Utilizing nonlinear optical pattern formation for a simple image-processing task,�?? Appl. Phys. B 76, 383-392 (2003). [CrossRef]
- M. A. Vorontsov, G. W. Carhart, and R. Dou, �??Spontaneous optical pattern formation in a large array of optoelectronic feedback circuits,�?? J. Opt. Soc. Am. B 17, 266-274 (2000). [CrossRef]
- Y. Hayasaki, Y. Tamura, H. Yamamoto, and N. Nishida, �??Spatial property of formed patterns depending on focus condition in a two-dimensional optoelectronic feedback system,�?? Jpn. J. Appl. Phys. 40, 165-169 (2001). [CrossRef]
- Y. Hayasaki, E. Hikosaka, H. Yamamoto, and N. Nishida, �??Image processing based on seeded spontaneous optical pattern formation using optoelectronic feedback,�?? Appl. Opt. 44, 236-240 (2005). [CrossRef] [PubMed]
- J. A. Nelder and R. Mead, �??Simplex method for function minimization,�?? Computer J. 7, 308-313 (1965).

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