Research Article Research on Adaptive Optics...

Research ArticleResearch on Adaptive Optics Image Restoration Algorithm byImproved Expectation Maximization Method

Lijuan Zhang12 Dongming Li3 Wei Su4 Jinhua Yang1 and Yutong Jiang15

1 School of Opto-Electronic Engineering Changchun University of Science and Technology Changchun 130022 China2 College of Computer Science and Engineering Changchun University of Technology Changchun 130012 China3 School of Information Technology Jilin Agriculture University Changchun 130118 China4 Informatization Center Changchun University of Science and Technology Changchun 130022 China5 College of Electrical and Mechanical Engineering Beijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Wei Su swcusteducn

Received 18 April 2014 Revised 5 June 2014 Accepted 6 June 2014 Published 6 July 2014

Academic Editor Fuding Xie

Copyright copy 2014 Lijuan Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To improve the effect of adaptive optics imagesrsquo restoration we put forward a deconvolution algorithm improved by the EMalgorithm which joints multiframe adaptive optics images based on expectation-maximization theory Firstly we need to makea mathematical model for the degenerate multiframe adaptive optics images The function model is deduced for the points thatspread with time based on phase errorThe AO images are denoised using the image power spectral density and support constraintSecondly the EM algorithm is improved by combining the AO imaging system parameters and regularization technique A costfunction for the joint-deconvolution multiframe AO images is given and the optimization model for their parameter estimationsis built Lastly the image-restoration experiments on both analog images and the real AO are performed to verify the recoveryeffect of our algorithm The experimental results show that comparing with the Wiener-IBD or RL-IBD algorithm our iterationsdecrease 143 and well improve the estimation accuracy The model distinguishes the PSF of the AO images and recovers theobserved target images clearly

1 Introduction

Adaptive optics (abbreviated as AO) is developed in orderto overcome the interference of atmospheric turbulence onthe telescope imaging and it can measure and correct theoptical wavefront phase of atmospheric turbulence in real-time The technology matures its application is extendingto civilian fields besides the fields of large telescopes andlaser engineering [1] In recent years with the development ofadaptive optics technology many large ground-based opticaltelescopes are equipped with adaptive optics system [2] in theworld

When surface-to-air object imaging (eg astral target) isobserved by ground-based optical telescopes adaptive opticssystem can only realize part of the wavefront error correctionand the information of high frequency on target suppressionand attenuation seriously [3 4] There are many factors such

as atmospheric turbulence the AO system error and photonnoise on imaging path Therefore we acquire more clearimages carrying on image restoration before adaptive opticscorrection At present there aremany image restoration algo-rithms to overcome the influence of atmospheric turbulencesuch as iterative blind deconvolution algorithm (abbreviatedas IBD) [4 5] IBD based on Richardson-Lucy iteration(abbreviated as RL-IBD) IBD based on theWiener algorithm(abbreviated as Wiener-IBD) and the algorithm based onthe expectationmaximization algorithm (abbreviated as EM)[6 7]

This paper proposes a blind image restoration method bymultiframe iteration with improved adaptive optical imagesbased on expectation maximization algorithm Section 2introduces the method of adaptive optics image restorationin detail including constructing adaptive optics model ofimaging process deducingmodel of point spread function on

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 781607 10 pageshttpdxdoiorg1011552014781607

2 Abstract and Applied Analysis

Atmospheric turbulence

DM

WFS

Controller

Separating slide

Corrected wavefront

Scientific camera

Flat wavefrontDeformed wavefront

Telescopersquos pupil

Figure 1 Works of adaptive optical system for imaging compensa-tion

AO image and researching the denoise processingmethod ofAO image Finally it is given a deconvolution method usingmultiframe AO images by combining AO imaging systemparameters with the regularization technique on the basis ofexpectation maximization algorithm Section 3 verifies thevalidity of the algorithm in this paper by using simulate AOimages actual observation binary AO images and stellar AOimages Section 4 concludes the paper

2 Restoration Algorithm ofAdaptive Optics Images

Adaptive optics system can measure control and correct theerror of optical wavefront in real-time Figure 1 shows theprinciple diagram of an adaptive optical system for imagingcompensationThe system includes wavefront sensor (abbre-viated as WFS) the deformable mirror (abbreviated as DM)high speed processor for wavefront and other parts Thesystem can measure the distortion of optical wavefront inreal-time which is caused by the influence of atmosphericturbulence and convert it to corresponding controlling signaladding to the wavefront correction element which cancompensate distortion of wavefront in real-time and theoptical telescope would be able to get close to target imagingfor diffraction AO images are corrected by AO system butits compensation or correction is not sufficient in order toget clearer target images AO images restoration algorithm isresearched

21 The Imaging Process of Adaptive Optics System Con-sidering the normal optical imaging model first of all theproper adaptive optics imagingmodel is set up which lays the

foundation of restoring high-quality adaptive optical imagesThe acquisition process of adaptive optics image model canconsist of a degradation function and an additive noise[5] the imaging process of AO in spatial frequency can beexpressed as follows

119892 (119909) = (119891 lowast ℎ) (119909) + 119899 (119909)

= intΩ

119891 (119910) ℎ (119909 minus 119910) 119889119910 + 119899 (119909) 119909 119910 isin Ω

(1)

where 119891(119909) is the original image for energy concentrationℎ(119909) is the point spread function (abbreviated as PSF) orimpulse response for image system 119892(119909) is output observa-tion images for the AO system 119899(119909) is the additive noise andΩ is the image area

In application we use multiple frames of short exposureimages 119892

119898119872

119898=1for one target to restore the target image 119891

119898 is the number of frames The model of multiframe shortexposure image with degradation can be written as

119892119898 (119909) = 119891 (119909) lowast ℎ119898 (119909) + 119899 (119909) 1 lt 119898 le 119872 (2)

where 119892119898(119909) and ℎ

119898(119909) represent the 119898th frame of short

exposure AO image with degradation for the target andits corresponding PSF respectively This paper adopts theselection techniques of frames referred to in [6] as the imagepreprocessing which picks up the images with better qualityfrom image sequence of short exposure images in order toimprove the solution of blind convolution and convergenceof iteration

22 Model of Point Spread Function on AO Images In adap-tive optics system the residual error of wavefront correctionmainly consists of incomplete correction error by turbulenceand noise of closed-loop system

When AO system works in closed-loop the transfererror function is the transfer function of wavefront phaseperturbation caused by atmospheric turbulence and thenoise of system introduced bywavefront sensor is closed-looptransfer function [4] Theoretically the mathematical modelof the PSF after closed-loop correction of adaptive opticssystem is shown as follows

ℎ (119910) =

1003816100381610038161003816100381610038161003816int 119875(119906)119890

minus119895sdot(2120587120582119897)119910sdot119906119889119906

1003816100381610038161003816100381610038161003816

2

(3)

In the formula

119875 (119906) = 1 within the lens aperture0 others

(4)

is the pupil function of telescope 119906 is the coordinates for thepupil plane 120582 is the central wavelength of imaging beam 119897 isthe focal length of the optical system 119910 is the coordinates oftwo-dimensional focal plane

Due to the error of focal length and optical deviationin optical system the pupil function should be rectified asfollows

119901 (119906) = 119875 (119906) 119890119895120579(119906) (5)

Abstract and Applied Analysis 3

where 120579(sdot) is the wavefront phase error caused by the error offocal length or optical deviation

So the mathematical model of PSF can be written as

ℎ (119910 120579) =

1003816100381610038161003816100381610038161003816int 119901(119906)119890

minus119895sdot(2120587120582119897)sdot119910sdot119906119889119906

1003816100381610038161003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816int 119875(119906)119890

119895120579(119906)119890minus119895sdot(2120587120582119897)sdot119910sdot119906

119889119906

1003816100381610038161003816100381610038161003816

2

(6)

The wavefront phase error is caused by the focal lengtherror or optical deviation due to the influence of atmosphericturbulence the PSFmodel changes with the focal length erroror optical deviation over time as follows

ℎ (119910 120579119905) =

1003816100381610038161003816100381610038161003816int 119875(119906)119890


119889119906

1003816100381610038161003816100381610038161003816

2

(7)

In the formula 120579119905(sdot) is the phase error caused by atmospheric

turbulence changing over timeIn terms of AO system the exposure time of observing

images is shorter than turbulent fluctuation If the intensity ofthe target is kept unchanged during exposure time themodelfor series images is

119892 (119910 120579119905119898 119891) = int ℎ (119910 minus 119909 120579

119905119898) 119891 (119909) 119889119909 (8)

In the formula 120579119905119898

is the phase error of turbulence for the119898th frame short exposure AO image

23 The Denoising Processing of AO Images In order toreduce the noise of AO images Chinese researchers putforward themethod to suppress noise based on total variation[8 9] This paper proposes a method to suppress noise that isbased on the power spectral density (abbreviated as PSD) ofimage and the supporting domain field of constraints imagesReference [10] analyzed the method to minimize the noisein symmetrical support domain field of AO images Theanalysis for this method is based on the variance ratio of realimages In this paper the assessment about denoise resultusing noise changed function The denoising process of AOimage is actual a kind of image noise corrected process inFourier spectral domain If theAO image has the symmetricalsupport field its noise mainly consist by CCD read-in noiseand AO noise and then noise power spectral density can beresearched Therefore suppress noise for observation images119892(119909) and 119892

119888(119909) is constrained by field of support domain

119892119888 (119909) = 119892 (119909)119908 (119909) (9)

In the formula 119908(119909) is weighting function in support fieldthe image spectrum 119866

119888(119906) is

119866119888 (119906) = int119866 (V)119882 (119906 minus V) 119889119906 (10)

where 119906 is the spatial frequency of image119866(119906) = 119878(119906)lowast119873(119906)119878(119906) is the signal frequency spectrum function 119873(119906) is thenoise spectrum function and bandwidth function 119882(119906) =119866(119906) lowast 120575 In the limited support domain field119882(119906) is verynarrow which nearly equals the 120575 times of the width of 119866(119906)

According to the definition of J M Conan the PSD ofan image is the Fourier transformation of its covariancedefining the power spectrum of noise and power spectrumof distortion image [11]

PSD(119906)119899119899 = ⟨10038161003816100381610038161003816Re 119866 (119906) minus Re 119866 (119906)10038161003816100381610038161003816

2

⟩

PSD(119906)119891119891 = ⟨10038161003816100381610038161003816Im 119866 (119906) minus Im 119866 (119906)10038161003816100381610038161003816

2

⟩

(11)

In the formula 119866(119906) is the Fourier transformation of image119892(119909) 119866(119906) is the mean value of sample 119866(119906)

Regard the circle as target supporting domain field itsdiameter is from 8 pixels to 185 pixels The definition of thechanging image noise function 119865

119899(119866) is

119865119899 (119866) =

int (PSD(119906)119888119899119899minus PSD(119906)119888

119899119899) 119889119906

intPSD(119906)119888119899119899119889119906

(12)

In the formula PSD(119906)119888119899119899is the power spectral density of the

image with constraints of support domain field and PSD(119906)119888119899119899

is the power spectral density of the image without constraintsof support domain field

24 Improved Expectation Maximization Algorithm Refer-ence [12] deduced the point spread function of multiframedegraded images with turbulence and the iteration formulaof target image based on maximum likelihood criterion butusing thismethod leads to the larger solution space Howeverwe put forward the technique of regularization to modify themaximum likelihood estimation algorithm introduce con-straints in accordance with AO image in physical reality andadopt the improved EM algorithm onmultiframe iteration ofAO images to acquire ideal 119891 and ℎ

119896

241 EM Algorithm by Combining Regularization TechniqueEM algorithm is put forward by Dempster et al [13] forparameters of maximum likelihood estimate of the iterativealgorithm which is iteratively going on two basic stepscalculation steps E (Expectation) and M (Maximization) Inthis paper a new regularizationmethod combined with priorknowledge is applied to the EMalgorithm realizing amethodof multiframe AO image restoration

Based on prior knowledge of target image and PSF jointprobability distribution function defining the maximizationfunction will get target image estimation 119891 and PSF estima-tion ℎ that is

[119891 ℎ] = max119891ℎ

(119901 (119892 | 119891 ℎ 119886)) = max119891ℎ

(exp(minus119869 (119891 ℎ))

119911)

(13)

In the formula 119911 is the normalization factor The costfunction is

[119891 ℎ] = max119891ℎ

(119901 (119892 | 119891 ℎ 119886)) = max119891ℎ

(exp(minus119869 (119891 ℎ))

119911)

(14)


So the problem is converted to minimize cost function119869(119891 ℎ) that is [119891 ℎ] = min

119891ℎ(119869(119891 ℎ)) The cost function

of AO images and the optimization model of its parameterestimation are shown as follows

119869 (119891 ℎ) = 119869119899(119892 | 119891 ℎ 119886) + 119869

ℎ (ℎ | 119886) + 119869119891 (119891 | 119886)

st 119869119899(119892 | 119891 ℎ 119886) = sum

119894119895

1

1205902119899

[119892 (119894 119895) minus (ℎ lowast 119891) (119894 119895)]2

119869ℎ (ℎ | 119886) =

120579119905

2sum

119906

10038161003816100381610038161003816(119906) minus 119867(119906)

10038161003816100381610038161003816

2

PSDℎ (119906)

119869119891(119891 | 119886) = 120572sum

119894119895

Φ(119903 (119891 120574 (119894 119895)))

(15)

Symbol explanation the first item represents the noise costfunction the second item represents the constraints costfunction of PSF and the third item is the cost functionswith target edge constraint 119886 is prior information 1205902

119899is the

variance of mixed noise 120579119905is the wavefront phase error

the power spectral density of the point spread function isPSD(119906)

ℎ= ⟨|(119906) minus 119867(119906)|

2

⟩ (119906) is the Fourier transfor-mation of ℎ(119909) 119867(119906) is the Fourier transform of averagePSF the estimation of PSD

ℎ(119906) can be estimated through the

AO system closed-loop control data and 120572 and 120574(nabla119909) areregularization adjustment parameters using the target edgeconstraint theory of Mugnier isotropic [14] as

Φ(119901) = 119901 minus ln (1 + 119901)

119901 (119891 120574 (nabla119909)) =

10038171003817100381710038171003817nabla119891 (119894 119895)

10038171003817100381710038171003817

120574 (nabla119909)

10038171003817100381710038171003817nabla119891 (119894 119895)

10038171003817100381710038171003817= ((nabla119891

119909(119894 119895))

2+ (nabla119891

119910(119894 119895))

2

)

12

(16)

where 120572 is a constant coefficient 120574(sdot) is a regularized functionthat has variable regularization coefficient associated with thegradient of each point 120574(nabla119909) = expminus|nabla119909|

221205852

is based on [15]0 lt 120574(nabla119909) le 1 parameter 120585 is the controlled smoothingcoefficient for selection and nabla119909 is the gradient amplitudein four directions for the point 119909(119894 119895) (where 119894 119895 are pixelcoordinates and 119909 is gray value of pixel)

This paper will apply all the prior information on theestimation of target and PSF in order to get the optimalestimation of target and the point spread function (PSF)Combine formulas (2) (8) and (15) to establish the costfunction for joint deconvolutionwithmultiframeAO images

119869multi (119891 ℎ) = 119869119899 (119892119898 | 119891 ℎ119898 119886) + 119869ℎ (ℎ119898 | 119886) + 119869119891 (119891 | 119886)

(17)

where119898 is the number of frames

242 Implement of Image Restoration Algorithm This paperis combining with the basic idea of the EM algorithm [13]through limited times of cycle calculations find the mostsuitable PSF estimation on the purpose of restoring target

image drastically modify the cost function (17) and make itcorrespond to full data [15] if 119892

119896(119910 | 119909) is full data set of 119896

times observation results 119892119896(119910) of observed image 119892

119896(119910) and

the expectation of cost function are as follows

119864 (119892119896(119910)) = 119864(sum

119909

119892119896(119910 | 119909))

= sum

119909

ℎ119896(119910 minus 119909) 119891 (119909)

119864 (119869multi (119891 ℎ)) = 119864(119869119899 (sum119909

119892119896| 119891 ℎ119898 119886))

+ 119864 (119869ℎ(ℎ119898| 119886)) + 119864 (119869

119891(119891 | 119886))

(18)

In the iteration process it is supposed that the 119899 minus1 iteration result is known 119864119899(119869multi(119891 ℎ)) represents thecurrent expectation of the cost function In order tominimizethe expectation of the cost function formula (18) can be dif-ferentiated by ℎ119899

119896(119909) and 119891119899(119909) then make the differentiation

equal to zero that is

120597119864119899(119869multi (119891 ℎ))

120597ℎ119899

119896(119909)

= 0120597119864119899(119869multi (119891 ℎ))

120597119891119899 (119909)= 0 (19)

The derivation process in detail is omitted so the finalsolutions are

119891119899+1

119894(119909) = 119891

119899

119894(119909)

+120579119905

120572 lowast Φ (119903 (119891119899

119894(119909) 120574 (119909)))

times sum

119910isin119884

sum

119898

ℎ119899

119894(119910 minus 119909)

times (119892119898minus sum119911isin119883ℎ119899

119894(119910 minus 119911) 119891

119899

119894(119911)

1205902119894

)

(20)

ℎ119899+1

119894(119909) = ℎ

119899

119894(119909)

+120579119905

2sum

119906

10038161003816100381610038161003816119894(119906) minus 119867

119894(119906)10038161003816100381610038161003816

2

119875119878119863ℎ 119894 (119906)

times sum

119910isin119884

119891119899+1

119894(119910 minus 119909)


119894(119910 minus 119911) 119891

119899

119894(119911)

1205902119894

)

(21)

In this paper the specific steps of image restorationalgorithm are as follows

Step 1 (initialization operation)

(1) According to the method of Section 23 constrainthe support domain field of observed image and dealwith the noise and so forth regard the result of onetime Wiener filtering as the initial value of targetestimation119891

0


(a) Original image (b) Degraded image frame 1 (c) Degraded image frame 2 (d) Degraded image frame 3

(e) Restored image by Wiener-IBD

(f) Restored image by RL-IBD (g) Restored image by our algo-rithm

Figure 2 Degraded images in experiment and restoration comparison of three algorithms

(a) Degraded image frame 1 (b) Degraded image frame 2 (c) Degraded image frame 3 (d) Degraded image frame 4

(e) Estimated PSF by ouralgorithm

(f) Restored image by our algo-rithm

(g) Restored image by RL-IBDalgorithm

(h) Restored image by Weiner-IBD algorithm

Figure 3 Multiframe original images and restoration comparison of three algorithms

(2) According to the method of Section 22 for the initialPSF estimation ℎ

0 normalized processing should be

taken at same time

(3) Set theMax iteration of extrinsic cycles

(4) SetMax count of target and PSF estimation

(5) Set the inner loop counter variable h count by PSFestimation variable f count of target inner cyclecount

Step 2 (the calculation of parameter values)

(1) According to the formulas of Section 23 the variance1205902

119899of mixed noise of each degradation image is

estimated

(2) According to the method and calculation formulasfrom Section 22 calculate the power spectral densityPSDℎ(119906) of PSF and estimate the wavefront phase

error 120579119905


0

50

100

150

200

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y

(a) Energy spectrum of observed image in Figure 3(a)

0

50

100

150

0

50

100

150

200

250

0

50

100

150

Y (pixel) X(pixel)

Gra

y

(b) Energy spectrum of our algorithm restored image in Figure 3(f)

0

50

100

150

200

250

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y

(c) Energy spectrum of RL-IBD algorithm restored image in Figure 3(g)

0

100

100

150

00

50

50

100

150

200

250

2040

6080

Y (pixel)

X(pixel)

Gra

y

120

(d) Energy spectrum of Wiener-IBD algorithm restored image inFigure 3(h)

Figure 4 Energy spectrum of observed image and restoration comparison of three algorithms

(3) According to the scheme and formulas ofSection 241 estimate the regularization parameters120572 and 120574(nabla119909) of target edge

Step 3 (iterate calculation 119894 = 0 1 119872119886119909 119894119905119890119903119886119905119894119900119899)

(1) The inner loop count variable of PSF h count = 0(2) The iteration process of PSF estimation 119895 = 0 1

119872119886119909 119888119900119906119899119905

(a) Complete the PSF estimation ℎ(119894) with the con-jugate gradient method using optimized for-mula (20)

(b) Consider h count++ j++(c) Judge the value of the loop variable 119895 if 119895 lt119872119886119909 119888119900119906119899119905 continue otherwise turn to (3)

(3) The inner loop counter variable of target estimation119891 119888119900119906119899119905 = 0

(4) The iteration process of target estimation 119896 =

0 1 119872119886119909 119888119900119906119899119905

(a) The conjugate gradient method was used tooptimize formula (21) to complete target imageestimation 119891(119894)

(b) consider f count++ 119896 + +(c) judge loop variable 119896 if 119896 lt 119872119886119909 119888119900119906119899119905 con-

tinue otherwise turn to (5)

(5) Judge whether the extrinsic loop is over if it is 119894 gt119898119886119909 119894119905119890119903119886119905119894119900119899 then turn to Step 4

(6) 119894 = 119894 + 1 return to (1) and continue to loop

Step 4 (the algorithm is in the end) The cycle does not stopuntil the algorithm converges and the output target imageestimation is 119891 that is over



(e) Restored image by Weiner-IBD algorithm

(f) Restored image by RL-IBDalgorithm

(g) Estimated PSF by ouralgorithm

(h) Restored image by our algo-rithm

0

0

2

4

6

8

100

100

200200

300

400

150

250

times104

50

0

Y (pixel)X

(pixel)

Gra

y

(i) Energy spectrum of observed image in Figure 5(a)

0

0

0

20

40

60

80

100

50

100

150

2

4

6

8

times104

Y (pixel)

X (pixel)

Gra

y

(j) Energy spectrum of our algorithm restored image in Figure 5(h)

01

02

03

04

05

06

07

08

Inte

nsity

Our algorithm

Wiener-IBDRL-IBD

(k) A line plot taken access the horizontal axis of image restored byWeiner-IBD RL-IBD and our algorithm

Figure 5 The multiframe observed stellar images and restoration comparison of three algorithms


3 Results and Analysis of AO ImageRestoration Experiments

31 The Restoration Experiment on Simulate AO Images Thealgorithm proposed in this paper is used to do the simulationexperiment on degradation spot images the evaluation ofimage restoration quality is judged by using rootmean squareerror (RMSE) and improved ΔSNR Consider

RMSE = radic1

11987311198732sum

119894119895

(119891 (119894 119895) minus 119891 (119894 119895))2

ΔSNR = 100 log10

1003817100381710038171003817119892 minus 11989110038171003817100381710038172

10038171003817100381710038171003817119891 minus 119891

10038171003817100381710038171003817

2

(22)

The evaluation of estimation accuracy is defined as fol-lows

120576estimate =

radicsum119894119895isin119878ℎ

10038161003816100381610038161003816ℎ(119894 119895) minus ℎ(119894 119895)

10038161003816100381610038161003816

2


1003816100381610038161003816ℎ(119894 119895)10038161003816100381610038162

(23)

In the formula 1198731 and 1198732 represent the total number ofpixels on 119909-axis and 119910-axis on the image respectively (119894 119895)represents a pixel point on image and 119891(119894 119895) represents thegray value of the point (119894 119895) on ideal image

In the simulation experiments of this paper the originalimage is infrared simulation imaging with spots for multipletargets with long distance 10 frames of simulation degradedimage series are generated by image degradation softwarefrom the key optical laboratory of Chinese Academy ofSciences and then add Gaussian white noise and make thesignal-to-noise ratio SNR of image for 20 dB Set parametersthe same as adaptive optical telescope of 12m on observatoryin YunnanThemain parameters of telescope imaging systemare the atmospheric coherence length 119903

0= 13 cm diameter

of telescope 119863 = 108m comprehensive focal length ofoptical system 119897 = 2242m the wavelength in center 120582 =700 nm and the pixel size of CCD is 74 120583m Experiments onIBD algorithm based on RL iterative [16] (RL-IBD) iterativeblind restoration algorithm based on theWiener filtering [17](Wiener-IBD) and the proposed algorithm are compared

Figure 2 is the compared results of the original imagesimulated multiframe degradation images and restorationimages based on three kinds of algorithm the image size is217 times 218 Figure 2(a) is the original target image and tenframes turbulence degraded image sequences are generatedby using image degradation simulation software as shownin Figures 2(b) 2(c) and 2(d) (to save space we show onlythree frames the rest is omitted) Figure 2(e) is the Wiener-IBD restored image RMSE = 1635 Figure 2(f) is the RL-IBD restored image the number of iterations is 800 timesRMSE = 1031 Figure 2(g) is restored image used methodin this paper the variance of wavefront phase error 120579

119905(sdot) is

614 times 10minus3 ℎ0= 826 times 10

minus3 and 1205902119899= 11 times 10

minus5 that theregularization parameters in the experiments are 120572 = 125the parameter of regularization function is 120585 = 134 extrinsiciterations for 100 times inner loop of the PSF for 4 image

Table 1 Comparison of ΔSNR and 120576estimate values among threealgorithms

Algorithmname ΔSNR (dB) 120576estimate Iteration Time-consuming

Wiener-IBD 1982 362 800 20645 sRL-IBD 3046 449 800 17236 sOur algorithm 4164 510 600 10923 s

cycles for 2 namely the total times of cycling is 600 RMSE =1005 Comparing with ΔSNR and precision from Figures2(e) 2(f) and 2(g) are shown in Table 1 Comparison resultsshow that the proposed algorithm in this paper is better thanWiener-IBD algorithm and RL-IBD algorithm and needs lesstime of iterations

32 Restoration Experiments about Binary AO Images With-out the reference of ideal image using the objective qualityevaluation with no reference image the Full Width HalfMaximum (abbreviated as FWHM) [18] is used as objectiveevaluation for image restoration in this paper FWHM refersto imaging peak pixels is two times of half peak includingthe FWHM on 119909 direction and 119910 direction for the betterevaluation in astronomical observation areas computationformula is shown as follows

FWHM = radicFWHM 1199092 + FWHM 1199102 (24)

The experiment for the binary image restoration withoutreference images is carried out by the algorithm proposed inthis paper The binary images for experiments were taken bythe 12m AO telescope from Chinese Academy of Sciencesin Yunnan observatory on December 3 2006 The size forimaging CCD is 320 times 240 pixels array the main parametersof imaging system are as follows the atmospheric coherentlength 119903

0= 13 cm telescope aperture 119863 = 103 all

imaging observation band is 07sim09 center wavelength 120582 =072 the CCD pixel size is 67 120583m and optical imagingfocal length 119897 = 20 Recovery experiments based on RLIBD algorithm Wiener-IBD algorithm and the proposedalgorithm in this paper are compared the frame selectiontechnology is taken in this paper 40 frames are picked from100 frames of degradation images as blind convolution imagesto be processed

Figure 3 is the comparison results ofmultiframedegradedimages and restoration images based on three kinds ofalgorithms Figures 3(a) 3(b) 3(c) and 3(d) are for theobservation of the binary image (to save space we show onlyfour frames the rest is omitted) Figure 3(e) is the estimationof the point spread function (PSF) based on the algorithmin this paper Figure 3(f) is the restored image based on themethod proposed in this paper the variance of wavefrontphase error 120579

119905(sdot) is 604 times 10minus3 ℎ

0= 903 times 10

minus3 and1205902

119899= 132 times 10

minus5 so that the regularization parameter inthis experiment is 120572 = 115 the parameter of regularizationfunction is 120585 = 134 extrinsic iterations for 50 times the inner


loop of PSF for 4 times and image loop for 2 times namelyThe total times of iteration are 300 FWHM = 617 pixels andthe restoration results are very close to the diffraction limit of12m AO telescope Figure 3(g) is the restoration result basedonRL-IBD algorithm FWHM=562 pixels Figure 3(h) is therestoration result based onWiener-IBD algorithm FWHM=476 pixels Figure 4 is the sketch of energy for the observationimage and three kinds of restored images based on threerestoration algorithms With the comparison it shows thatthe proposed algorithm in this paper is better than Wiener-IBD algorithm and RL-IBD algorithm and needs less time ofiterations

33 Restoration Experiments about Stellar AO Images Alsothe restoration experiment of stellar images was carriedout using the algorithm in this paper the stellar imagesfor experiment were taken by 12m AO telescope from theChinese Academy of Sciences in Yunnan observatory onDecember 1 2006 collection The parameters of the opticalimaging system are the same as that of binary images andthe method of parameters selection for experiments in thispaper is nearly the same Figure 5 is the comparison ofmultiframe stellar observation images and restoration resultsbased on the three algorithms Figures 5(a) 5(b) 5(c) and5(d) are the observation of stellar images (to save space weshow only 4) Figure 5(e) is for the Wiener-IBD restoredimage Figure 5(f) is the RL-IBD restored image Figure 5(g)is the estimation of the point spread function (PSF) basedon algorithm in this paper the iterative initial value ℎ

0=

956 times 10minus3 PSF iterative for six times Figure 5(h) is the

restored image based on algorithm in this paper Figure 5(i)is the energy of the observed image for stellar Figure 5(j) isthe energy of restoration image based on the algorithm in thispaper Figure 5(k) is the graph of comparison results for threerestoration algorithms

4 Conclusions

In this paper according to the characteristic of adaptiveoptics image we establish a degradation image model formultiframe exposure image series and the mathematicalmodel of the point spread function (PSF) for adaptive opticalsystem after closed loop correction is derived the PSF modelbased on phase error and changing over time is solved andwe use the power spectral density of image and constrainsupport domain method for AO image denoising processingFinally this paper applied the actual AO imaging systemparameters as prior knowledge combined with regularizationtechnique to improve the EM blind deconvolution algo-rithm Experimental results show that the established blinddeconvolution algorithm based on improved EM methodcan effectively eliminate the atmospheric turbulence andeliminate the noise pollution brought by the AO closed-loopcorrection which improves the resolution of the adaptiveoptical image The image process of AO observed imagesshows that the restoration quality is better than the samekind of IBD algorithm It also shows that the restoration ofbinary images using the algorithmof this paper has carried on

the identification of PSF and completed the AO image afterprocessing the results of actual AO image restoration haveimportant application value

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the Scientific and TechnologicalResearch Project of Jinlin province Department of Edu-cation China (no 2013145 and no 201363) and the basisstudy project is supported by Jinlin Provincial Science ampTechnology Department China (no 201105055)

References

[1] J Wenhan ldquoAdaptive optical technologyrdquo Chinese Journal ofNature vol 28 pp 7ndash13 2005

[2] G R Ayers and J C Dainty ldquoIterative blind deconvolutionmethod and its applicationsrdquo Optics Letters vol 13 no 7 pp547ndash549 1988

[3] L Dayu H Lifa and M Quanquan ldquoA high resolution liquidcrystal adaptive optics systemrdquo Acta Photonica Sinica vol 37pp 506ndash508 2008

[4] C Rao F Shen andW Jiang ldquoAnalysis of closed-loopwavefrontresidual error of adaptive optical system using the method ofpower spectrumrdquo Acta Optica Sinica vol 20 no 1 pp 68ndash732000

[5] B Chen The theory and algorithms of adaptive optics imagerestoration [PhD thesis] Information Engineering UniversityZhengzhou China 2008

[6] T Yu R Changhui and W Kai ldquoPost-processing of adaptiveoptics images based on frame selection and multi-frame blinddeconvolutionrdquo inAdaptive Optics Systems vol 7015 of Proceed-ings of SPIE Marseille France June 2008

[7] T J Schulz ldquoNonlinear models and methods for space-objectimaging through atmospheric turbulencerdquo in Proceedings ofthe IEEE International Conference on Image Processing vol 9Lausanne Switzerland 1996

[8] L Yaning G Lei and L Huihui ldquoTotal variation basedband-limited sheralets transform for image denoisingrdquo ActaPhotonica Sinica vol 242 pp 1430ndash1435 2013

[9] J Zhao and J Yang ldquoRemote sensing image denoising algorithmbased on fusion theory using cycle spinning contourlet trans-form and total variation minimizationrdquo Acta Photonica Sinicavol 39 no 9 pp 1658ndash1665 2010

[10] C L Matson and M C Roggemann ldquoNoise reduction inadaptive-optics imagery with the use of support constraintsrdquoApplied Optics vol 34 no 5 pp 767ndash780 1995

[11] DWTyler andC LMatson ldquoReduction of nonstationary noisein telescope imagery using a support constraintrdquoOptics Expressvol 1 no 11 pp 347ndash354 1997

[12] L Zhang J YangW Su et al ldquoResearch on blind deconvolutionalgorithm of multiframe turbulence-degraded imagesrdquo Journalof Information and Computational Science vol 10 no 12 pp3625ndash3634 2013


[13] A P Dempster N M Laird and D B Rubin ldquoMaximumlikelihood from incomplete data via the EM algorithmrdquo Journalof the Royal Statistical Society B vol 39 no 1 pp 1ndash38 1977

[14] L M Mugnier J Conan T Fusco and V Michau ldquoJoint maxi-mum A posteriori estimation of object and PSF for turbulence-degraded imagesrdquo in Bayesian Inference for Inverse Problemsvol 3459 of Proceedings of SPIE pp 50ndash61 San Diego CalifUSA July 1998

[15] H Hanyu Research on Image Restoration Algorithms in ImagingDetection System HuazhongUniversity of Science and Technol-ogy Wuhan China 2004

[16] F Tsumuraya ldquoIterative blind deconvolution method usingLucyrsquos algorithmrdquo Astronomy amp Astrophysics vol 282 no 2 pp699ndash708 1994

[17] S Zhang J Li Y Yang and Z Zhang ldquoBlur identification ofturbulence-degraded IR imagesrdquoOptics and Precision Engineer-ing vol 21 no 2 pp 514ndash521 2013

[18] B Chen C Cheng S GuoG Pu andZGeng ldquoUnsymmetricalmulti-limit iterative blind deconvolution algorithm for adaptiveoptics image restorationrdquoHigh Power Laser and Particle Beamsvol 23 no 2 pp 313ndash318 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Atmospheric turbulence

DM

WFS

Controller

Separating slide

Corrected wavefront

Scientific camera

Flat wavefrontDeformed wavefront

Telescopersquos pupil

Figure 1 Works of adaptive optical system for imaging compensa-tion

AO image and researching the denoise processingmethod ofAO image Finally it is given a deconvolution method usingmultiframe AO images by combining AO imaging systemparameters with the regularization technique on the basis ofexpectation maximization algorithm Section 3 verifies thevalidity of the algorithm in this paper by using simulate AOimages actual observation binary AO images and stellar AOimages Section 4 concludes the paper

2 Restoration Algorithm ofAdaptive Optics Images

Adaptive optics system can measure control and correct theerror of optical wavefront in real-time Figure 1 shows theprinciple diagram of an adaptive optical system for imagingcompensationThe system includes wavefront sensor (abbre-viated as WFS) the deformable mirror (abbreviated as DM)high speed processor for wavefront and other parts Thesystem can measure the distortion of optical wavefront inreal-time which is caused by the influence of atmosphericturbulence and convert it to corresponding controlling signaladding to the wavefront correction element which cancompensate distortion of wavefront in real-time and theoptical telescope would be able to get close to target imagingfor diffraction AO images are corrected by AO system butits compensation or correction is not sufficient in order toget clearer target images AO images restoration algorithm isresearched

21 The Imaging Process of Adaptive Optics System Con-sidering the normal optical imaging model first of all theproper adaptive optics imagingmodel is set up which lays the

foundation of restoring high-quality adaptive optical imagesThe acquisition process of adaptive optics image model canconsist of a degradation function and an additive noise[5] the imaging process of AO in spatial frequency can beexpressed as follows

119892 (119909) = (119891 lowast ℎ) (119909) + 119899 (119909)

= intΩ

119891 (119910) ℎ (119909 minus 119910) 119889119910 + 119899 (119909) 119909 119910 isin Ω

(1)

where 119891(119909) is the original image for energy concentrationℎ(119909) is the point spread function (abbreviated as PSF) orimpulse response for image system 119892(119909) is output observa-tion images for the AO system 119899(119909) is the additive noise andΩ is the image area

In application we use multiple frames of short exposureimages 119892

119898119872

119898=1for one target to restore the target image 119891

119898 is the number of frames The model of multiframe shortexposure image with degradation can be written as

119892119898 (119909) = 119891 (119909) lowast ℎ119898 (119909) + 119899 (119909) 1 lt 119898 le 119872 (2)

where 119892119898(119909) and ℎ

119898(119909) represent the 119898th frame of short

exposure AO image with degradation for the target andits corresponding PSF respectively This paper adopts theselection techniques of frames referred to in [6] as the imagepreprocessing which picks up the images with better qualityfrom image sequence of short exposure images in order toimprove the solution of blind convolution and convergenceof iteration

22 Model of Point Spread Function on AO Images In adap-tive optics system the residual error of wavefront correctionmainly consists of incomplete correction error by turbulenceand noise of closed-loop system

When AO system works in closed-loop the transfererror function is the transfer function of wavefront phaseperturbation caused by atmospheric turbulence and thenoise of system introduced bywavefront sensor is closed-looptransfer function [4] Theoretically the mathematical modelof the PSF after closed-loop correction of adaptive opticssystem is shown as follows

ℎ (119910) =

1003816100381610038161003816100381610038161003816int 119875(119906)119890

minus119895sdot(2120587120582119897)119910sdot119906119889119906

1003816100381610038161003816100381610038161003816

2

(3)

In the formula

119875 (119906) = 1 within the lens aperture0 others

(4)

is the pupil function of telescope 119906 is the coordinates for thepupil plane 120582 is the central wavelength of imaging beam 119897 isthe focal length of the optical system 119910 is the coordinates oftwo-dimensional focal plane

Due to the error of focal length and optical deviationin optical system the pupil function should be rectified asfollows

119901 (119906) = 119875 (119906) 119890119895120579(119906) (5)




ℎ (119910 120579) =

1003816100381610038161003816100381610038161003816int 119901(119906)119890


1003816100381610038161003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816int 119875(119906)119890


119889119906

1003816100381610038161003816100381610038161003816

2

(6)


ℎ (119910 120579119905) =

1003816100381610038161003816100381610038161003816int 119875(119906)119890


119889119906

1003816100381610038161003816100381610038161003816

2

(7)




119892 (119910 120579119905119898 119891) = int ℎ (119910 minus 119909 120579

119905119898) 119891 (119909) 119889119909 (8)

In the formula 120579119905119898




119892119888 (119909) = 119892 (119909)119908 (119909) (9)


119888(119906) is

119866119888 (119906) = int119866 (V)119882 (119906 minus V) 119889119906 (10)



PSD(119906)119899119899 = ⟨10038161003816100381610038161003816Re 119866 (119906) minus Re 119866 (119906)10038161003816100381610038161003816

2

⟩

PSD(119906)119891119891 = ⟨10038161003816100381610038161003816Im 119866 (119906) minus Im 119866 (119906)10038161003816100381610038161003816

2

⟩

(11)



119899(119866) is

119865119899 (119866) =

int (PSD(119906)119888119899119899minus PSD(119906)119888

119899119899) 119889119906

intPSD(119906)119888119899119899119889119906

(12)





119896



[119891 ℎ] = max119891ℎ

(119901 (119892 | 119891 ℎ 119886)) = max119891ℎ

(exp(minus119869 (119891 ℎ))

119911)

(13)


[119891 ℎ] = max119891ℎ

(119901 (119892 | 119891 ℎ 119886)) = max119891ℎ

(exp(minus119869 (119891 ℎ))

119911)

(14)





119869 (119891 ℎ) = 119869119899(119892 | 119891 ℎ 119886) + 119869

ℎ (ℎ | 119886) + 119869119891 (119891 | 119886)

st 119869119899(119892 | 119891 ℎ 119886) = sum

119894119895

1

1205902119899

[119892 (119894 119895) minus (ℎ lowast 119891) (119894 119895)]2

119869ℎ (ℎ | 119886) =

120579119905

2sum

119906

10038161003816100381610038161003816(119906) minus 119867(119906)

10038161003816100381610038161003816

2

PSDℎ (119906)

119869119891(119891 | 119886) = 120572sum

119894119895

Φ(119903 (119891 120574 (119894 119895)))

(15)


119899is the



ℎ= ⟨|(119906) minus 119867(119906)|

2




Φ(119901) = 119901 minus ln (1 + 119901)

119901 (119891 120574 (nabla119909)) =

10038171003817100381710038171003817nabla119891 (119894 119895)

10038171003817100381710038171003817

120574 (nabla119909)

10038171003817100381710038171003817nabla119891 (119894 119895)

10038171003817100381710038171003817= ((nabla119891

119909(119894 119895))

2+ (nabla119891

119910(119894 119895))

2

)

12

(16)


221205852



119869multi (119891 ℎ) = 119869119899 (119892119898 | 119891 ℎ119898 119886) + 119869ℎ (ℎ119898 | 119886) + 119869119891 (119891 | 119886)

(17)






119896(119910) and


119864 (119892119896(119910)) = 119864(sum

119909

119892119896(119910 | 119909))

= sum

119909

ℎ119896(119910 minus 119909) 119891 (119909)

119864 (119869multi (119891 ℎ)) = 119864(119869119899 (sum119909

119892119896| 119891 ℎ119898 119886))

+ 119864 (119869ℎ(ℎ119898| 119886)) + 119864 (119869

119891(119891 | 119886))

(18)




120597119864119899(119869multi (119891 ℎ))

120597ℎ119899

119896(119909)

= 0120597119864119899(119869multi (119891 ℎ))

120597119891119899 (119909)= 0 (19)


119891119899+1

119894(119909) = 119891

119899

119894(119909)

+120579119905

120572 lowast Φ (119903 (119891119899

119894(119909) 120574 (119909)))

times sum

119910isin119884

sum

119898

ℎ119899

119894(119910 minus 119909)


119894(119910 minus 119911) 119891

119899

119894(119911)

1205902119894

)

(20)

ℎ119899+1

119894(119909) = ℎ

119899

119894(119909)

+120579119905

2sum

119906

10038161003816100381610038161003816119894(119906) minus 119867

119894(119906)10038161003816100381610038161003816

2

119875119878119863ℎ 119894 (119906)

times sum

119910isin119884

119891119899+1

119894(119910 minus 119909)


119894(119910 minus 119911) 119891

119899

119894(119911)

1205902119894

)

(21)




0














taken at same time







estimated


error 120579119905


0

50

100

150

200

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

50

100

150

0

50

100

150

200

250

0

50

100

150

Y (pixel) X(pixel)

Gra

y


0

50

100

150

200

250

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

100

100

150

00

50

50

100

150

200

250

2040

6080

Y (pixel)

X(pixel)

Gra

y

120






119872119886119909 119888119900119906119899119905





0 1 119872119886119909 119888119900119906119899119905













0

0

2

4

6

8

100

100

200200

300

400

150

250

times104

50

0

Y (pixel)X

(pixel)

Gra

y


0

0

0

20

40

60

80

100

50

100

150

2

4

6

8

times104

Y (pixel)

X (pixel)

Gra

y


01

02

03

04

05

06

07

08

Inte

nsity

Our algorithm

Wiener-IBDRL-IBD






RMSE = radic1

11987311198732sum

119894119895

(119891 (119894 119895) minus 119891 (119894 119895))2

ΔSNR = 100 log10

1003817100381710038171003817119892 minus 11989110038171003817100381710038172

10038171003817100381710038171003817119891 minus 119891

10038171003817100381710038171003817

2

(22)


120576estimate =


10038161003816100381610038161003816ℎ(119894 119895) minus ℎ(119894 119895)

10038161003816100381610038161003816

2


1003816100381610038161003816ℎ(119894 119895)10038161003816100381610038162

(23)



0= 13 cm diameter



119905(sdot) is


minus3 and 1205902119899= 11 times 10













0= 903 times 10

minus3 and1205902

119899= 132 times 10





0=



4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












ℎ (119910 120579) =

1003816100381610038161003816100381610038161003816int 119901(119906)119890


1003816100381610038161003816100381610038161003816

2

=

1003816100381610038161003816100381610038161003816int 119875(119906)119890


119889119906

1003816100381610038161003816100381610038161003816

2

(6)


ℎ (119910 120579119905) =

1003816100381610038161003816100381610038161003816int 119875(119906)119890


119889119906

1003816100381610038161003816100381610038161003816

2

(7)




119892 (119910 120579119905119898 119891) = int ℎ (119910 minus 119909 120579

119905119898) 119891 (119909) 119889119909 (8)

In the formula 120579119905119898




119892119888 (119909) = 119892 (119909)119908 (119909) (9)


119888(119906) is

119866119888 (119906) = int119866 (V)119882 (119906 minus V) 119889119906 (10)



PSD(119906)119899119899 = ⟨10038161003816100381610038161003816Re 119866 (119906) minus Re 119866 (119906)10038161003816100381610038161003816

2

⟩

PSD(119906)119891119891 = ⟨10038161003816100381610038161003816Im 119866 (119906) minus Im 119866 (119906)10038161003816100381610038161003816

2

⟩

(11)



119899(119866) is

119865119899 (119866) =

int (PSD(119906)119888119899119899minus PSD(119906)119888

119899119899) 119889119906

intPSD(119906)119888119899119899119889119906

(12)





119896



[119891 ℎ] = max119891ℎ

(119901 (119892 | 119891 ℎ 119886)) = max119891ℎ

(exp(minus119869 (119891 ℎ))

119911)

(13)


[119891 ℎ] = max119891ℎ

(119901 (119892 | 119891 ℎ 119886)) = max119891ℎ

(exp(minus119869 (119891 ℎ))

119911)

(14)





119869 (119891 ℎ) = 119869119899(119892 | 119891 ℎ 119886) + 119869

ℎ (ℎ | 119886) + 119869119891 (119891 | 119886)

st 119869119899(119892 | 119891 ℎ 119886) = sum

119894119895

1

1205902119899

[119892 (119894 119895) minus (ℎ lowast 119891) (119894 119895)]2

119869ℎ (ℎ | 119886) =

120579119905

2sum

119906

10038161003816100381610038161003816(119906) minus 119867(119906)

10038161003816100381610038161003816

2

PSDℎ (119906)

119869119891(119891 | 119886) = 120572sum

119894119895

Φ(119903 (119891 120574 (119894 119895)))

(15)


119899is the



ℎ= ⟨|(119906) minus 119867(119906)|

2




Φ(119901) = 119901 minus ln (1 + 119901)

119901 (119891 120574 (nabla119909)) =

10038171003817100381710038171003817nabla119891 (119894 119895)

10038171003817100381710038171003817

120574 (nabla119909)

10038171003817100381710038171003817nabla119891 (119894 119895)

10038171003817100381710038171003817= ((nabla119891

119909(119894 119895))

2+ (nabla119891

119910(119894 119895))

2

)

12

(16)


221205852



119869multi (119891 ℎ) = 119869119899 (119892119898 | 119891 ℎ119898 119886) + 119869ℎ (ℎ119898 | 119886) + 119869119891 (119891 | 119886)

(17)






119896(119910) and


119864 (119892119896(119910)) = 119864(sum

119909

119892119896(119910 | 119909))

= sum

119909

ℎ119896(119910 minus 119909) 119891 (119909)

119864 (119869multi (119891 ℎ)) = 119864(119869119899 (sum119909

119892119896| 119891 ℎ119898 119886))

+ 119864 (119869ℎ(ℎ119898| 119886)) + 119864 (119869

119891(119891 | 119886))

(18)




120597119864119899(119869multi (119891 ℎ))

120597ℎ119899

119896(119909)

= 0120597119864119899(119869multi (119891 ℎ))

120597119891119899 (119909)= 0 (19)


119891119899+1

119894(119909) = 119891

119899

119894(119909)

+120579119905

120572 lowast Φ (119903 (119891119899

119894(119909) 120574 (119909)))

times sum

119910isin119884

sum

119898

ℎ119899

119894(119910 minus 119909)


119894(119910 minus 119911) 119891

119899

119894(119911)

1205902119894

)

(20)

ℎ119899+1

119894(119909) = ℎ

119899

119894(119909)

+120579119905

2sum

119906

10038161003816100381610038161003816119894(119906) minus 119867

119894(119906)10038161003816100381610038161003816

2

119875119878119863ℎ 119894 (119906)

times sum

119910isin119884

119891119899+1

119894(119910 minus 119909)


119894(119910 minus 119911) 119891

119899

119894(119911)

1205902119894

)

(21)




0














taken at same time







estimated


error 120579119905


0

50

100

150

200

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

50

100

150

0

50

100

150

200

250

0

50

100

150

Y (pixel) X(pixel)

Gra

y


0

50

100

150

200

250

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

100

100

150

00

50

50

100

150

200

250

2040

6080

Y (pixel)

X(pixel)

Gra

y

120






119872119886119909 119888119900119906119899119905





0 1 119872119886119909 119888119900119906119899119905













0

0

2

4

6

8

100

100

200200

300

400

150

250

times104

50

0

Y (pixel)X

(pixel)

Gra

y


0

0

0

20

40

60

80

100

50

100

150

2

4

6

8

times104

Y (pixel)

X (pixel)

Gra

y


01

02

03

04

05

06

07

08

Inte

nsity

Our algorithm

Wiener-IBDRL-IBD






RMSE = radic1

11987311198732sum

119894119895

(119891 (119894 119895) minus 119891 (119894 119895))2

ΔSNR = 100 log10

1003817100381710038171003817119892 minus 11989110038171003817100381710038172

10038171003817100381710038171003817119891 minus 119891

10038171003817100381710038171003817

2

(22)


120576estimate =


10038161003816100381610038161003816ℎ(119894 119895) minus ℎ(119894 119895)

10038161003816100381610038161003816

2


1003816100381610038161003816ℎ(119894 119895)10038161003816100381610038162

(23)



0= 13 cm diameter



119905(sdot) is


minus3 and 1205902119899= 11 times 10













0= 903 times 10

minus3 and1205902

119899= 132 times 10





0=



4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119869 (119891 ℎ) = 119869119899(119892 | 119891 ℎ 119886) + 119869

ℎ (ℎ | 119886) + 119869119891 (119891 | 119886)

st 119869119899(119892 | 119891 ℎ 119886) = sum

119894119895

1

1205902119899

[119892 (119894 119895) minus (ℎ lowast 119891) (119894 119895)]2

119869ℎ (ℎ | 119886) =

120579119905

2sum

119906

10038161003816100381610038161003816(119906) minus 119867(119906)

10038161003816100381610038161003816

2

PSDℎ (119906)

119869119891(119891 | 119886) = 120572sum

119894119895

Φ(119903 (119891 120574 (119894 119895)))

(15)


119899is the



ℎ= ⟨|(119906) minus 119867(119906)|

2




Φ(119901) = 119901 minus ln (1 + 119901)

119901 (119891 120574 (nabla119909)) =

10038171003817100381710038171003817nabla119891 (119894 119895)

10038171003817100381710038171003817

120574 (nabla119909)

10038171003817100381710038171003817nabla119891 (119894 119895)

10038171003817100381710038171003817= ((nabla119891

119909(119894 119895))

2+ (nabla119891

119910(119894 119895))

2

)

12

(16)


221205852



119869multi (119891 ℎ) = 119869119899 (119892119898 | 119891 ℎ119898 119886) + 119869ℎ (ℎ119898 | 119886) + 119869119891 (119891 | 119886)

(17)






119896(119910) and


119864 (119892119896(119910)) = 119864(sum

119909

119892119896(119910 | 119909))

= sum

119909

ℎ119896(119910 minus 119909) 119891 (119909)

119864 (119869multi (119891 ℎ)) = 119864(119869119899 (sum119909

119892119896| 119891 ℎ119898 119886))

+ 119864 (119869ℎ(ℎ119898| 119886)) + 119864 (119869

119891(119891 | 119886))

(18)




120597119864119899(119869multi (119891 ℎ))

120597ℎ119899

119896(119909)

= 0120597119864119899(119869multi (119891 ℎ))

120597119891119899 (119909)= 0 (19)


119891119899+1

119894(119909) = 119891

119899

119894(119909)

+120579119905

120572 lowast Φ (119903 (119891119899

119894(119909) 120574 (119909)))

times sum

119910isin119884

sum

119898

ℎ119899

119894(119910 minus 119909)


119894(119910 minus 119911) 119891

119899

119894(119911)

1205902119894

)

(20)

ℎ119899+1

119894(119909) = ℎ

119899

119894(119909)

+120579119905

2sum

119906

10038161003816100381610038161003816119894(119906) minus 119867

119894(119906)10038161003816100381610038161003816

2

119875119878119863ℎ 119894 (119906)

times sum

119910isin119884

119891119899+1

119894(119910 minus 119909)


119894(119910 minus 119911) 119891

119899

119894(119911)

1205902119894

)

(21)




0














taken at same time







estimated


error 120579119905


0

50

100

150

200

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

50

100

150

0

50

100

150

200

250

0

50

100

150

Y (pixel) X(pixel)

Gra

y


0

50

100

150

200

250

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

100

100

150

00

50

50

100

150

200

250

2040

6080

Y (pixel)

X(pixel)

Gra

y

120






119872119886119909 119888119900119906119899119905





0 1 119872119886119909 119888119900119906119899119905













0

0

2

4

6

8

100

100

200200

300

400

150

250

times104

50

0

Y (pixel)X

(pixel)

Gra

y


0

0

0

20

40

60

80

100

50

100

150

2

4

6

8

times104

Y (pixel)

X (pixel)

Gra

y


01

02

03

04

05

06

07

08

Inte

nsity

Our algorithm

Wiener-IBDRL-IBD






RMSE = radic1

11987311198732sum

119894119895

(119891 (119894 119895) minus 119891 (119894 119895))2

ΔSNR = 100 log10

1003817100381710038171003817119892 minus 11989110038171003817100381710038172

10038171003817100381710038171003817119891 minus 119891

10038171003817100381710038171003817

2

(22)


120576estimate =


10038161003816100381610038161003816ℎ(119894 119895) minus ℎ(119894 119895)

10038161003816100381610038161003816

2


1003816100381610038161003816ℎ(119894 119895)10038161003816100381610038162

(23)



0= 13 cm diameter



119905(sdot) is


minus3 and 1205902119899= 11 times 10













0= 903 times 10

minus3 and1205902

119899= 132 times 10





0=



4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















taken at same time







estimated


error 120579119905


0

50

100

150

200

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

50

100

150

0

50

100

150

200

250

0

50

100

150

Y (pixel) X(pixel)

Gra

y


0

50

100

150

200

250

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

100

100

150

00

50

50

100

150

200

250

2040

6080

Y (pixel)

X(pixel)

Gra

y

120






119872119886119909 119888119900119906119899119905





0 1 119872119886119909 119888119900119906119899119905













0

0

2

4

6

8

100

100

200200

300

400

150

250

times104

50

0

Y (pixel)X

(pixel)

Gra

y


0

0

0

20

40

60

80

100

50

100

150

2

4

6

8

times104

Y (pixel)

X (pixel)

Gra

y


01

02

03

04

05

06

07

08

Inte

nsity

Our algorithm

Wiener-IBDRL-IBD






RMSE = radic1

11987311198732sum

119894119895

(119891 (119894 119895) minus 119891 (119894 119895))2

ΔSNR = 100 log10

1003817100381710038171003817119892 minus 11989110038171003817100381710038172

10038171003817100381710038171003817119891 minus 119891

10038171003817100381710038171003817

2

(22)


120576estimate =


10038161003816100381610038161003816ℎ(119894 119895) minus ℎ(119894 119895)

10038161003816100381610038161003816

2


1003816100381610038161003816ℎ(119894 119895)10038161003816100381610038162

(23)



0= 13 cm diameter



119905(sdot) is


minus3 and 1205902119899= 11 times 10













0= 903 times 10

minus3 and1205902

119899= 132 times 10





0=



4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

50

100

150

200

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

50

100

150

0

50

100

150

200

250

0

50

100

150

Y (pixel) X(pixel)

Gra

y


0

50

100

150

200

250

0

50

100

150

0

50

100

150

Y (pixel)X (pixel)

Gra

y


0

100

100

150

00

50

50

100

150

200

250

2040

6080

Y (pixel)

X(pixel)

Gra

y

120






119872119886119909 119888119900119906119899119905





0 1 119872119886119909 119888119900119906119899119905













0

0

2

4

6

8

100

100

200200

300

400

150

250

times104

50

0

Y (pixel)X

(pixel)

Gra

y


0

0

0

20

40

60

80

100

50

100

150

2

4

6

8

times104

Y (pixel)

X (pixel)

Gra

y


01

02

03

04

05

06

07

08

Inte

nsity

Our algorithm

Wiener-IBDRL-IBD






RMSE = radic1

11987311198732sum

119894119895

(119891 (119894 119895) minus 119891 (119894 119895))2

ΔSNR = 100 log10

1003817100381710038171003817119892 minus 11989110038171003817100381710038172

10038171003817100381710038171003817119891 minus 119891

10038171003817100381710038171003817

2

(22)


120576estimate =


10038161003816100381610038161003816ℎ(119894 119895) minus ℎ(119894 119895)

10038161003816100381610038161003816

2


1003816100381610038161003816ℎ(119894 119895)10038161003816100381610038162

(23)



0= 13 cm diameter



119905(sdot) is


minus3 and 1205902119899= 11 times 10













0= 903 times 10

minus3 and1205902

119899= 132 times 10





0=



4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















0

0

2

4

6

8

100

100

200200

300

400

150

250

times104

50

0

Y (pixel)X

(pixel)

Gra

y


0

0

0

20

40

60

80

100

50

100

150

2

4

6

8

times104

Y (pixel)

X (pixel)

Gra

y


01

02

03

04

05

06

07

08

Inte

nsity

Our algorithm

Wiener-IBDRL-IBD






RMSE = radic1

11987311198732sum

119894119895

(119891 (119894 119895) minus 119891 (119894 119895))2

ΔSNR = 100 log10

1003817100381710038171003817119892 minus 11989110038171003817100381710038172

10038171003817100381710038171003817119891 minus 119891

10038171003817100381710038171003817

2

(22)


120576estimate =


10038161003816100381610038161003816ℎ(119894 119895) minus ℎ(119894 119895)

10038161003816100381610038161003816

2


1003816100381610038161003816ℎ(119894 119895)10038161003816100381610038162

(23)



0= 13 cm diameter



119905(sdot) is


minus3 and 1205902119899= 11 times 10













0= 903 times 10

minus3 and1205902

119899= 132 times 10





0=



4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












RMSE = radic1

11987311198732sum

119894119895

(119891 (119894 119895) minus 119891 (119894 119895))2

ΔSNR = 100 log10

1003817100381710038171003817119892 minus 11989110038171003817100381710038172

10038171003817100381710038171003817119891 minus 119891

10038171003817100381710038171003817

2

(22)


120576estimate =


10038161003816100381610038161003816ℎ(119894 119895) minus ℎ(119894 119895)

10038161003816100381610038161003816

2


1003816100381610038161003816ℎ(119894 119895)10038161003816100381610038162

(23)



0= 13 cm diameter



119905(sdot) is


minus3 and 1205902119899= 11 times 10













0= 903 times 10

minus3 and1205902

119899= 132 times 10





0=



4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0=



4 Conclusions





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Research on Adaptive Optics...

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Transcript of Research Article Research on Adaptive Optics...