## Restoration of turbulence-degraded extended object using the stochastic parallel gradient descent algorithm: numerical simulation

Optics Express, Vol. 17, Issue 5, pp. 3052-3062 (2009)

http://dx.doi.org/10.1364/OE.17.003052

Acrobat PDF (542 KB)

### Abstract

An adaptive optics (AO) system with Stochastic Parallel Gradient Descent (SPGD) algorithm and a 61-element deformable mirror is simulated to restore the image of a turbulence-degraded extended object. SPGD is used to search the optimum voltages for the actuators of the deformable mirror. We try to find a convenient image performance metric, which is needed by SPGD, merely from a gray level distorted image and without any additional optics elements. Simulation results show the gray level variance function acts more promising than other metrics, such as metrics based on the gray level gradient of each pixel. The restoration capability of the AO system is investigated with different images and different turbulence strength wave-front aberrations using SPGD with the above resultant image quality criterion. Numerical simulation results verify the performance metric is effective and the AO system can restore those images degraded by different turbulence strengths successfully.

© 2009 Optical Society of America

## 1. Introduction

2. R. A. Muller and A. Buffington. “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Am . A. **64**, 1200–1210 (1974). [CrossRef]

4. P. Yang, M. W. Ao, Y. Li, B. Xu, and W. H. Jiang, “Intracavity transverse modes controlled by a genetic algorithm based on Zernike mode coefficients,” Opt. Express **15**, 17051–17062 (2007). [CrossRef] [PubMed]

5. M. A. Vorontsov and G. W. Carhart. “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am . A. **17**, 1440–1453 (2000) [CrossRef]

6. M. S. Zakynthinaki and Y. G. Saridakis, “Stochastic optimization for a tip-tilt adaptive correcting system,” Comput. Phys. Commun . **150**, 274–292 (2003) [CrossRef]

## 2. Model of space object imaging

*x,y*) denotes coordinates of a point in the image plane and

*f*(

*x, y*) is gray level of the point (

*x, y*). For incoherent imaging, the image

*f*(

*x, y*) is given by the convolution of the object function

*I*(

*x, y*) and the intensity point spread function (PSF),

*h*(

*x, y*) , of the system[7]:

*λ*/

*D*) is supposed to be 5 pixels, so the full Field of View (FOV) is about 24 times the size of diffraction limited angle. The ideal imaging results of Image A and Image B are given in Figs. 1(c) and 1 (d) under the telescope parameters in this paper.

*D*/

*r*

_{0}is set at 5, 10 and 20 respectively.

## 3. Definition of several image sharpness functions

*F*

_{1}is defined according to the gray level variance of image, which expresses the discrete degree of gray levels distribution. Gray levels are distributed in a bigger range when the image is much clearer. Here

*M*×

*N*is the image size.

*F*

_{2}and the third

*F*

_{3}are defined according to the gray level gradient information of each pixel.

*F*

_{3}by using the Laplacian operator, which implements a second derivative operation on the image.

*F*

_{2}and

*F*

_{3}gives prominence to the contribution of some points, which have a large gray level gradient, to image quality evaluation function.

*F*

_{2}and

*F*

_{3}are also regarded as modification of sharpness function

*S*

_{1}in Ref. 2

2. R. A. Muller and A. Buffington. “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Am . A. **64**, 1200–1210 (1974). [CrossRef]

*P*(

*u,v*), with the weighting coefficient (

*u*

^{2}+

*v*

^{2}), so

*F*

_{4}can also be called as the mean square radius of the image spectrum.

## 4. Description of AO system

*f*(

*x,y*), an image quality analyzer that calculates the image quality metric and the SPGD algorithm that produces control signals

*u*= {

*u*

_{1},

*u*

_{2},…

*u*

_{61}} for a 61-element Deformable Mirror (DM) according to changes of metric. The phase compensation

*m*(

*x,y*) , introduced by the DM, can be combined linearly with influence functions of actuators:

*S*(

_{j}*x,y*) is the influence function and

*u*is the control signal of the

_{j}*j*th actuator. On the basis of experimental measurements, we know the actuator influence function of a 61-element DM actuators is approximately Gaussian [9

9. W. H. Jiang, N. Ling, X. J. Rao, and F. shi. “Fitting capability of deformable mirror,” SPIE **1542**130–137 (1991). [CrossRef]

*x*) is the location of the

_{j},y_{j}*j*th actuator,

*α*is the coupling value between actuators and is set to 0.08, and a is the Gaussian index and is set to 2. The distance between actuators is d which is about 0.1367 normalized in a unit circle. Fig. 3 gives actuators location distribution of DM. The circled line in the figure denotes the effective aperture and the layout of all actuators is hexagonal. We suppose the stroke of the DM is enough to correct the wave-front aberrations in the simulation.

5. M. A. Vorontsov and G. W. Carhart. “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am . A. **17**, 1440–1453 (2000) [CrossRef]

*u*= {∆

*u*

_{1},∆

*u*

_{2},…∆

*u*

_{61}} with fixed amplitude |∆

*u*|=σ and random signs with equal probabilities for Pr(∆

_{j}*u*= ±σ) = 0.5 [10

_{j}10. J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Autom. Control . **37**, 332–341 (1992). [CrossRef]

*u*, the control signals are updated with the rule:

*F*=

*F*(

*u*+ ∆

*u*) -

*F*(

*u*- ∆

*u*) is the corresponding perturbation of the image quality metric and

*γ*is a gain coefficient which scales the size of the control parameter corrections (positive for the case of metric maximization, negative for minimization and positive in this paper).

## 5. Numerical results and analysis

### 5.1 Adaptation process

*D*/

*r*

_{0}is 5, 10 and 20 respectively. Averaged evolution curves of four image-quality metrics are given in Fig. 4(a) to Fig. 7(a), in which the averaged evolution curves are normalized to be 1 in the optimal case. Corresponding standard deviation curves and Strehl Ratio curves during the control algorithm’s 1500 iterations are presented in Fig. 4(b) to Fig. 7(b) and Fig. 4(c) to Fig. 7(c).

*F*

_{1}and

*F*

_{4}have relatively smaller standard deviations than

*F*

_{2}and

*F*

_{3}, which shows that

*F*

_{1}and

*F*

_{4}have stronger adaptability to different turbulence realizations than

*F*

_{2}and

*F*

_{3}. Fig. 4(c) to Fig. 7(c) indicate that all the four image-quality metrics have strong correction ability for

*D*/

*r*

_{0}= 5 ,

*F*

_{1}and

*F*

_{4}are much better than

*F*

_{2}and

*F*

_{3}for

*D*/

*r*

_{0}= 10 , and

*F*

_{1}is the best for

*D*/

*r*

_{0}= 20 .

### 5.2 Zernike order and PSF of single frame phase screen

*D*/

*r*

_{0}= 20) when

*F*

_{1},

*F*

_{2},

*F*

_{3}and

*F*

_{4}are used as performance metrics optimized respectively. Corresponding PSFs are shown in Fig. 9. We also fit the DM figure to the phase screen using least squares to obtain the best correction achievable with the given 61-element DM. The fitting results are also shown in Fig. 8 and Fig. 9.

*F*

_{1}and

*F*

_{4}have much higher correction ability than

*F*

_{2}and

*F*

_{3}. Compared with the least squares fitting,

*F*

_{1}almost obtains the best correction achievable for the 61-element DM.

### 5.3 Cormparison of imaging results

*F*

_{1},

*F*

_{2},

*F*

_{3}and

*F*

_{4}respectively.

*F*

_{1},

*F*

_{2},

*F*

_{3}and

*F*

_{4}respectively.

*D*/

*r*

_{0}is 5,

*F*

_{1}and

*F*

_{4}are much better than

*F*

_{2}and

*F*

_{3}when

*D*/

*r*

_{0}is 10, and

*F*

_{1}is the best when

*D*/

*r*

_{0}is 20. These trends agree with the evolution curves in Fig. 4, Fig. 5, Fig. 6 and Fig. 7. The simulation results show that the AO system with the 61-element DM and SPGD algorithm can realize high resolution imaging for objects with different characteristics.

### 5.4 Correction ability analysis of performance metrics

*F*

_{1}is similar in correction ability to

*F*

_{4},

*F*

_{2}is similar to

*F*

_{3}, and

*F*

_{1}and

*F*

_{4}are better than

*F*

_{2}and

*F*

_{3}when

*D*/

*r*

_{0}is 10.

*F*

_{1}is the best when

*D*/

*r*

_{0}is 20. The correction capability of

*F*

_{1}is very close to the least squares fitting.

*F*

_{2}and

*F*

_{3}just use the information of point

*f*(

*x,y*) and its surrounding points, then sum all of points in the image plane. The variance function

*F*

_{1}in Eq. (2) relates each point to the mean of entire image and the frequency evaluation

*F*

_{4}in Eq. (5) makes use of mean frequency spectrum. From the above simulative results, we can conclude that

*F*

_{1}and

*F*

_{4}, which use global information of the image, have much stronger correction capability than

*F*

_{2}and

*F*

_{3}, which use only local information of the image. Just using local information may be the reason why

*F*

_{2}and

*F*

_{3}have relatively low correction ability. Contrasted to

*F*

_{4},

*F*

_{1}sustains performance to much bigger wave-front aberrations. The possible reason is that attenuation of high frequency content becomes serious, which makes the sensitivity of

*F*

_{4}to wave-front aberrations become weak gradually, as the turbulence strength increases. Fig. 13 gives the trends of four image-quality metrics versus the augment of turbulence strength, in which the curves are computed by averaging 50 different turbulence realizations.

*F*

_{2}and

*F*

_{3}based on gray level gradient are sensitive to the wave-front aberrations when

*D*/

*r*

_{0}is less than 10. The variance function

*F*

_{1}and the frequency evaluation function

*F*

_{4}have higher sensitivity than

*F*

_{2}and

*F*

_{3}when

*D*/

*r*

_{0}is bigger than 10. The sensitivity of

*F*

_{1}is a litter greater than

*F*

_{4}when

*D*/

*r*

_{0}is greater than 20. These trends verify our simulative results are reasonable

## 6. Conclusion

## Acknowledgments

## References and links

1. | R. K. Tyson, |

2. | R. A. Muller and A. Buffington. “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Am . A. |

3. | S. Zommer, E. N. Ribak, S. G. Lipson, and J. Adler. “Simulated annealing in ocular adaptive optics,” Opt. Lett . |

4. | P. Yang, M. W. Ao, Y. Li, B. Xu, and W. H. Jiang, “Intracavity transverse modes controlled by a genetic algorithm based on Zernike mode coefficients,” Opt. Express |

5. | M. A. Vorontsov and G. W. Carhart. “Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration,” J. Opt. Soc. Am . A. |

6. | M. S. Zakynthinaki and Y. G. Saridakis, “Stochastic optimization for a tip-tilt adaptive correcting system,” Comput. Phys. Commun . |

7. | J. W. Goodman. |

8. | N. Roddier. “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng . |

9. | W. H. Jiang, N. Ling, X. J. Rao, and F. shi. “Fitting capability of deformable mirror,” SPIE |

10. | J. C. Spall, “Multivariate stochastic approximation using a simultaneous perturbation gradient approximation,” IEEE Trans. Autom. Control . |

**OCIS Codes**

(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(110.0110) Imaging systems : Imaging systems

(350.4600) Other areas of optics : Optical engineering

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: November 3, 2008

Revised Manuscript: January 4, 2009

Manuscript Accepted: January 21, 2009

Published: February 17, 2009

**Citation**

Huizhen Yang, Xinyang Li, Chenglong Gong, and Wenhan Jiang, "Restoration of turbulence-degraded extended
object using the stochastic parallel gradient
descent algorithm: numerical simulation," Opt. Express **17**, 3052-3062 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3052

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### References

- R. K. Tyson, Principle of Adaptive Optics (Academic Press, 1991).
- R. A. Muller and A. Buffington, "Real-time correction of atmospherically degraded telescope images through image sharpening," J. Opt. Am. A. 64, 1200 -1210 (1974). [CrossRef]
- S. Zommer, E. N. Ribak, S. G. Lipson and J. Adler, "Simulated annealing in ocular adaptive optics," Opt. Lett. 31, 1-3 (2000).
- P. Yang, M. W. Ao, Y. Li, B. Xu, and W. H. Jiang, "Intracavity transverse modes controlled by a genetic algorithm based on Zernike mode coefficients," Opt. Express 15, 17051-17062 (2007). [CrossRef] [PubMed]
- M. A. Vorontsov and G. W. Carhart, "Adaptive optics based on analog parallel stochastic optimization: analysis and experimental demonstration," J. Opt. Soc. Am. A. 17, 1440 -1453 (2000) [CrossRef]
- M. S. Zakynthinaki and Y. G. Saridakis, "Stochastic optimization for a tip-tilt adaptive correcting system," Comput. Phys. Commun. 150, 274 -292 (2003) [CrossRef]
- J. W. Goodman. Introduction to Fourier Optics (Publishing House of Electronics Industry, 2006).
- N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174 -1180 (1990). [CrossRef]
- W. H. Jiang, N. Ling, X. J. Rao, and F. Shi. "Fitting capability of deformable mirror," Proc. SPIE 1542, 130 -137 (1991). [CrossRef]
- J. C. Spall, "Multivariate stochastic approximation using a simultaneous perturbation gradient approximation," IEEE Trans. Autom. Control. 37, 332 -341 (1992). [CrossRef]

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