## 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

Adaptive optics (AO) techniques can be used to compensate static or dynamic aberrations of a light beam after propagation through a distorting medium. The conventional AO systems [1] are implemented with a deformable mirror to correct the distorted wave-front, a wave-front sensor to measure the aberrations present in the incoming beam, and a feedback control algorithm to link these two elements in real time. This kind of wave-front correction needs the information about a reference wave-front that corresponds to the undistorted image when it is applied to extended object imaging correction. Model-free optimization techniques provide an alternative approach to the control problem that does not require the use of any priori knowledge of a system model. Generally, this area of research is based on a “sharpness” criterion which is used as an image metric to measure the degree of correction of the wave-front phase [2]. The system considers the criterion as a function of the control parameters and then uses some optimization algorithm to improve image quality.

An appropriate image metric, also called a sharpness function and a suitable control algorithm are two keys to the success of this technique. The relatively popular image sharpness metrics include image gray variation function in the time domain, frequency spectrum evaluation function in the frequency domain, entropy evaluation function in informatics area and statistics function based on statistical information, etc. Several optimization algorithms, such as Simulated Annealing [3], Genetic Algorithm [4], Stochastic Parallel Gradient Descent (SPGD) [5], and Algorithm Of Pattern Extraction (Alopex) [6], are the common stochastic parallel search algorithms in AO system. In order to find a convenient image performance metric only from the distorted gray level image, we compare several image quality metrics in time domain or frequency domain. Based on SPGD algorithm and the above resultant image quality criterion, the high resolution imaging capability of the AO system is investigated through different images and different turbulence strength wave-front aberrations.

## 2. Model of space object imaging

The symbol (*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]:

For clarity, magnification factors are omitted. All aberration effects are manifested in the PSF which can be obtained through Fast Fourier Transform of the wave-front in simulation. The original images used are given in Figs. 1(a) and 1(b), and are called Image A and Image B respectively.

Image A has a clear background and the object has clear edges, while the object in Image B almost fills the image plane. The image size is 128 × 128 pixels and the gray level is from 0 to 255. The diffraction limit angle (*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.

We use the method proposed by N. Roddier, which makes use of a Zernike expansion of randomly weighted Karhunen-Loeve functions, to simulate atmospherically distorted wavefronts [8]. Considering that the low-order aberrations (tilts, defocus, astigmatism, etc) have the most significant impact on image quality, we use the first 104 Zernike polynomial orders. Different phase screens generated according to this method are not correlated to each other and represent the Kolmogorov spectrum. The phase screens *ϕ*(*r*) are defined over
128 × 128 pixels which is also the grid of the wave-front corrector and don’t include the tip/tilt aberrations. The tip/tilt aberrations are usually controlled by another control loop and are considered as being removed completely in our simulation. Atmospheric turbulence strength for a receiver system with aperture size *D* can be characterized by the following two parameters: the ratio *D*/*r*
_{0} and the averaged Strehl Ratio (SR) of phase fluctuations. Phase screens of different atmospheric turbulence strength can be obtained through changing *r*
_{0} in the simulation program.

The correction capability of the AO imaging system is analyzed through three different turbulence strength wave-front aberrations where *D*/*r*
_{0} is set at 5, 10 and 20 respectively.

## 3. Definition of several image sharpness functions

Four candidate image-quality metrics are examined in this paper. The first metric *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

and $A=\left(1/\mathrm{MN}\right)\sum _{x}\sum _{y}f\left(x,y\right)$ is the mean of all pixel gray level, and *M* × *N* is the image size.

A high resolution image has clear edges and the gray level gradient expresses the edge information of an image, i.e. a bigger gray level gradient can offer us a better image. The second metric *F*
_{2} and the third *F*
_{3} are defined according to the gray level gradient information of each pixel.

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

The square operation in *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.

From the imaging theory of Fourier optics, we know that the sharpness of an image is closely related to the high frequency content of the image, which corresponds to the image edge details. A clearer image often includes more high frequency content and the total energy is also relatively higher. In simulation, the frequency information *F*(*u,v*) can be calculated through the Fast Fourier Transform of the image *f*(*x,y*), and then we can get the energy spectrum *P*(*u,v*) using the equation *P*(*u,v*) =|*F*(*u,v*)|^{2}. From *P*(*u,v*), we have the fourth metric *F*
_{4}:

To give emphasis to high frequency content, we weight the energy spectrum, *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

The high resolution imaging model with an AO system is shown in Fig. 2. The model includes an imaging system that records the image plane gray level *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:

where *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]:

where (*x _{j},y_{j}*) is the location of the

*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.

The SPGD algorithm [5] requires small random perturbations ∆*u* = {∆*u*
_{1},∆*u*
_{2},…∆*u*
_{61}} with fixed amplitude |∆*u _{j}*|=σ and random signs with equal probabilities for Pr(∆

*u*= ±σ) = 0.5 [10], to be applied to all 61 DM control channels simultaneously. Then for a given random ∆

_{j}*u*, the control signals are updated with the rule:

Where the scalar ∆*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

We perform the adaptation process over 50 phase realizations. The averaged evolution curves and the standard deviation evolution curves of the metric are the recorded simulation results. For comparing correction ability of the four image-quality metrics, we also give corresponding averaged SR evolution curves.

First, correction performance of Image A is investigated. Fig. 4 to Fig. 7 show simulation results when *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).

All image-quality metric curves have converged after 1500 iterations in Fig. 4(a) to Fig. 7(a). The iteration number needed for convergence increases and correction performance decreases with the growth of turbulence strength. From Fig. 4(b) to Fig. 7(b), we can see that *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 .

The same adaptation process is carried out for Image B. Results are very similar to those of Image A and aren’t given in the section.

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

Figure 8 gives Zernike coefficients 3-104 decomposed from the same 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.

From Fig. 8, we can see relatively low order content occupies the main part of the wave-front aberrations. Fig. 8 and Fig. 9 all show *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

Figure 10 presents averaged imaging results of Image A after different turbulence strength wave- front aberrations are corrected by using *F*
_{1} , *F*
_{2} , *F*
_{3} and *F*
_{4} respectively.

Figure 11 shows imaging results of Image B after different turbulence strength wave-front aberrations are corrected by using *F*
_{1} , *F*
_{2} , *F*
_{3} and *F*
_{4} respectively.

From Fig. 10 and Fig. 11, we can see four image-quality metrics can offer better imaging effect when *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

For comparing correction performance of four image-quality metrics, we examine the SR before and after correction by these metrics on Image A. Results are shown in Fig. 12. The result of the least squares fitting is also given in Fig. 12. From Fig. 12, we see that four image-quality metrics have strong correction ability when the turbulence strength is relatively weak. *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.

The correction ability of different metric can be explained by analyzing their definitions. From Eq. (3) and Eq. (4), we know *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.

From Fig. 13, we can see *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

We have simulated a high resolution imaging AO system with a 61-element deformable mirror and SPGD control algorithms. Numerical simulation results verify the performance metric offered is effective, which can provide us with guidance on how to choose appropriate performance metrics in extended object imaging: image metrics based on image global information are better than those based on image local information and the grey level variance function is the most promising image-quality metric. The compensation effect is very close to the optimal correction on combining the grey level variance function and SPGD control algorithm for a given DM. The AO system can restore those images degraded by different turbulence strength successfully. Note that the iteration number needed for convergence increases and correction performance decreases with the growth of turbulence strength. The system can be applied to static or slowly changing wave-front aberrations, while the convergence rate must be improved for rapidly changing aberrations. Further research should be focused on improving the convergence rate so that the system can be used for the dynamic environment.

## Acknowledgments

We express our thanks for the support of the National High Technology Project of China.

## References and links

**1. **R. K. Tyson, *Principle of Adaptive Optics* (Academic Press, 1991).

**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]

**3. **S. Zommer, E. N. Ribak, S. G. Lipson, and J. Adler. “Simulated annealing in ocular adaptive optics,” Opt. Lett . **31**, 1–3 (2000).

**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]

**7. **J. W. Goodman. *Introduction to Fourier Optics* (Publishing House of Electronics Industry, 2006).

**8. **N. Roddier. “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng . **29**, 1174–1180 (1990). [CrossRef]

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

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