No wavefront sensor adaptive optics system forcompensation of primary aberrations by software analysis
of a point source image. 1. Methods
Enrico Grisan,1,* Fabio Frassetto,1,2 Vania Da Deppo,1,2 Giampiero Naletto,1,2 and Alfredo Ruggeri1
1Department of Information Engineering, University of Padova, Via Gradenigo 6�b, I-35131 Padova, Italy2CNR-INFM-LUXOR, c�o Department of Information Engineering, Via Gradenigo 6�b, I-35131 Padova, Italy
In the early days, adaptive optics (AO) was usedmainly in astronomy to compensate for atmosphericturbulence. Recently, applications for ophthalmologyhave received growing attention [1–4]. Their mainobjectives are to obtain higher-resolution images fordiagnostic purposes, estimate high-order eye aberra-tions, or in experimental ophthalmology [5,6]. In par-ticular, it has been shown that the compensation ofthe optical aberrations introduced by the eye cangreatly improve the quality of retinal images. Thecore of every AO system is an optical device able tomodify the wavefront shape of the light entering thesystem. Ideally, by carefully controlling this device, itis possible to introduce a deformation to the wave-
front that is the exact complement of the one that thelight has undergone along its path, therefore impos-ing a perfectly spherical shape to the wavefront.
To perform such control, it is crucial to estimatethe shape of the incoming wavefront. For applica-tion in ophthalmology, this can be achieved in avariety of ways. The most widely used techniquesinclude the aberroscope, the Hartmann–Shackwavefront sensor [7], psychophysical methods [8],and double-pass methods to estimate the eye point-spread-function (PSF) [9].
Recently, no wavefront sensor systems that makeuse of model-based approaches [10,11] or stochasticoptimization techniques have been proposed. Inparticular, the latter search the state space of theAO to obtain the best configuration for reducingoptical aberrations [12–14] that are indirectly mea-sured by some image quality metrics [14–16]. Giventhe high dimensionality of the state space, and the
possible presence of local extrema for the objectivefunction, these methods provide variable resultsand need a large number of iterations to converge.In contrast, the model-based optimization scheme[10,11] makes use of only N � 1 measurements of aspot image for the estimation of N aberration coeffi-cients.
In this paper we demonstrate the feasibility, al-though in a simulated system, of estimating and cor-recting the wavefront shape by means of an iterativesoftware procedure, which exploits ad hoc cost func-tions and analyzes a small subset of states at a time.In this way, we reduce the dimensionality of thesearch space, achieving faster convergence and bettersensitivity. We also provide a comparison of the per-formance of the proposed algorithm with the directmaximization of [10].
2. Methods
Thanks to their characteristics, Zernike polynomialsare particularly attractive for wavefront representa-tion. They are a complete set of polynomials orthog-onal over the interior of the unit circle and they canbe written in polar coordinates as products of angularfunctions and radial polynomials [17]. To estimatethe aberrations of the wavefront, the Zernike coeffi-cients can be evaluated. Following the Air Force enu-meration scheme, used by the ray-tracing softwareZEMAX (ZEMAX Development Corporation, SanDiego, California), the coefficients we are able to es-timate using the developed algorithm are the z4 term,which represents defocus; z5 and z6, which are asso-ciated with astigmatism at 0° and 45°, respectively,plus defocus; z7, which represents the coma along thex plus x tilt; z8, which represents the coma along they axis plus y tilt; and finally z9, which representsthe spherical aberration. Hence, the objective is toestimate the coefficient vector Z � �z4, . . . , z9�.
Given an image of a point source, we demonstratethat it is possible to correct the aberrations intro-duced in the optical path from the source to theimage, making use of an AO �2, whose surface is de-fined by the Zernike coefficient vector Z�2
, describingits wavefront modification. This method does notmake use of wavefront sensors nor of the burdensomecomputation of the PSF of the optical system. For ageneric optical system, a detector positioned on itsimage plane will acquire an image I. When a pointsource S is present, the image of S on the image planeis P and its quantized and acquired version is Pq � I.The proposed method analyzes the image I, obtaininginformation on the aberrations that are present, andthen modifies �2 by varying its Zernike coefficients.Then a new image I is acquired to test the aberrationcorrection introduced, and then the algorithm isiterated until no residual aberration is detected. Todevelop such an algorithm, a hierarchy in the aber-rations is introduced, so that aberrations higher inthe hierarchy can be estimated without knowledgeof the aberrations lower in the hierarchy. By thismeans, the system achieves a dimensionality reduc-tion of the search space in addition to that obtained
by considering only six Zernike coefficients: in fact,the hierarchy will aim at the optimization of six one-dimensional problems instead of one six-dimensional.
A. Aberration Hierarchy
To obtain the estimation of the Zernike coefficientvector Z, a six-dimensional space must be searched,leading to a difficult optimization task. To increasethe convergence speed and decouple the aberrationinteractions, we would like to find a way to dividethe six-dimensional optimization problem intosome smaller-dimensional optimizations: Ideally,we would like to end up with six one-dimensionalproblems. In order to provide this dimensionalityreduction in the estimation of Z from I, it is impor-tant to find a way to obtain information on singleaberrations, without the procedure being influencedby the possible presence of other aberrations.
Even if this is not possible in general, a less strin-gent requirement can be met. A hierarchy among theaberrations can be devised: One aberration can beestimated regardless of the presence of aberrationslower in the hierarchy, whereas it cannot be esti-mated reliably as long as aberrations that are higherin the hierarchy are present. Among the Zernike co-efficients to be estimated here, the highest in thehierarchy is z9, related to the spherical aberration.Then there are z7 and z8 related to a coma aberration.In fact, when a coma is present but there is no spher-ical aberration, a characteristic tail is present (seeFig. 1) in Pq (the acquired image of S), which makesit clearly identifiable. At a lower step, z4 is found,related to focus shift, and finally the coefficients z5and z6 of astigmatism follow. Even if intuitively astig-matism should come before defocus in the hierarchy,it is easily shown that the estimation of astigmatismin the presence of an unknown defocus is not possible.The astigmatism axis changes with focus position(see Fig. 2), thus making it almost impossible to iden-tify the sign and amplitude of its two Zernike coeffi-cients without knowing the wavefront phase. Incontrast, after a focus shift correction, the astigma-tism would present itself in the circle of minimumconfusion. Even if no information on its axis can begathered at this point by simply analyzing I, we showthat the astigmatism can be corrected and the z5 andz6 coefficients can be estimated. With this hierarchy,and considering the unavoidable coupling between z7and z8, and between z5 and z6, we can reduce theoriginal problem of estimating Z � �6 into the se-
Fig. 1. Typical image of a point source image affected by coma.
quence of four problems of dimensionality at the mostin �2: First we will estimate z9 � �, then �z7, z8� � �2,subsequently z4 � �, and finally �z5, z6� � �2.
B. Aberration Correction
The described aberration ranking allows the devel-opment of a system that progressively corrects theaberrations, starting from those highest in the hier-archy and ending with the lowest. An outline of theproposed system is shown in Fig. 3: after the imageI is acquired, a spherical aberration is corrected byiteratively modifying the Zernike coefficient z9,�2
ofthe AO until no further improvement can be ob-tained. Then the algorithm detects and correctscoma, defocus, and finally astigmatism.
1. Spherical Correction StepSpherical aberrations cause the probe point image Pto spread, scattering the light power symmetricallyaround the center of P. To estimate the amount ofspread, we use two different metrics in a two-stageprocess: the first stage tries to maximize the amountof power present in the image by changing the coef-ficient z9 in the direction that locally increases the S1image metric described by Muller [15]:
S1 �� I�x, y�dxdy. (1)
In the second stage, the mean intensity value �I of thenonzero pixels of the image I is maximized: in thisway we obtain an image with the power as concen-trated as possible around Pq. To have a coherent ap-proach in the correction of the various aberrationsconsidered, we recast the maximization as a minimi-zation problem. Hence, the objective function to beminimized is
Js ��S1 for stage 1��I for stage 2. (2)
At every iteration i of the optimization procedure,we set
z9,�2�i� � z9,�2�i � 1� � �z�i�, (3)
where �z�i� is the optimization step varying the valueof the z9 coefficient. During the procedure, the �z�i�magnitude is decreased and its sign changed depend-ing on the values of S1�i � 1� and S1�i � 2� in the firststage, and on the values of �I�i � 1� and �I�i � 2� inthe second stage. The step update is then
2. Coma Correction StepA point source image of a point affected by a comais characterized by the presence of a tail along thecoma axis. The presence of a coma can be detectedby looking at the distance between the centroidc1 � �xc1
, yc1� of the nonzero pixels of I and the centroid
c2 � �xc2, yc2
� of the pixels weighted by their gray-levelvalue (related to the power density); the two centroidsare shown in Fig. 4 in a sample image. The more thetwo calculated centroids are apart, the more the comais assumed. However, since it may happen that partof the coma tail is outside the field of view, thuscreating local minima that impair the algorithm, wechose to weight this distance by the negative of thelight power in the image. The function to be mini-mized is
Fig. 2. Point source image affected by astigmatism at differentfocal plane positions. The position (b) corresponds to the so-called“circle of least confusion.”
Fig. 3. Flow chart of the proposed algorithm. Zernike coefficientsrelated to each aberration are hierarchically and separately esti-mated.
where � · �2 is the L2 norm. By looking at the covari-ance of the nonzero pixels (which represent Pq), theangle coma of the coma axis can be estimated from theeigenvector associated with its highest eigenvalue.The Zernike coefficients related with coma z7,�2
andz8,�2
are varied of a quantity �z so to compensate forthe aberration, they are applied to the adaptive opticssurface �2. At every iteration i:
z7,�2�i� � z7,�2�i � 1� � �z�i�cos�coma�i��, (7)
z8,�2�i� � z8,�2�i � 1� � �z�i�sin�coma�i��. (8)
In the case of the coma, since we want to minimize Jc,the decreasing strategy for the step �z is
3. Focus Shift Correction StepOnce the spherical and coma aberrations have beencorrected, the optical system might still be affectedby focus shift and astigmatism. The acquired imagePq of S, when affected by astigmatism alone, wouldappear as a circular spot with a dimension dependingon the astigmatism, whereas it would become elon-gated when some focus shift is present in the system.Therefore, in order to correct the focus shift, the al-gorithm aims at obtaining a Pq that is as circular aspossible, and as small as possible. This can beachieved by means of a min–max optimization of thecovariance matrix: The objective is to minimize themaximum value in the covariance matrix. A two-stage process similar to the one used for correctingthe spherical aberration is used. The first stage min-imizes the opposite of the image intensity in a smalldisk A centered on Pq, corresponding to the oppositeof the metric S5 in [13]:
S5 ��A
I�x, y�dxdy, (11)
whereas on the second stage the maximum of thecovariance matrix of the pixels in A, max(Cov(A)), isminimized. The objective function Jd to be minimizedfor defocus correction is then
Js �1 � S5 for stage 1max�Cov�A�� for stage 2, (12)
the coefficient update at the ith iteration is
z4,�2�i� � z4,�2�i � 1� � �z�i�, (13)
with the step �z update rules
Jd�i � 2� Jd�i � 1� ) �z�i� � �z�i�, (14)
Jd�i � 2� � Jd�i � 1� ) �z�i� � �0.5�z�i�. (15)
4. Astigmatism Correction StepFinally, the astigmatism has to be corrected. At thispoint, in which the focus shift has been corrected, noinformation on the astigmatism axis can be obtainedby looking at I. Therefore there is no possibility tohave a measure of the influence of the z5 and z6 coef-ficient separately. When the image is perfectly fo-cused, astigmatism causes Pq to appear as a circle, sothe objective of the correction is to reduce the dimen-sion of Pq. To evaluate the dimension of Pq, we use themaximum eigenvalue �max of the covariance matrix ofthe brightest pixels in I. Minimization of Ja � �max issupposed to yield the desired aberration correction.Since no information on the direction of the astigma-tism can be gathered explicitly, each iteration of theoptimization procedure has to evaluate the gradientsof Ja with respect to both z5 and z6 separately, mod-ifying the sign and magnitude of steps �z5
and �z6accordingly.
3. Simulation Setup
To test the proposed algorithm, a simple optical sys-tem has been simulated as shown in Fig. 5. Thissystem has been kept as simple as possible, in orderto evaluate the possibility of estimating the Zernikecoefficients by means of a software analysis of thepoint source image. In this way the algorithm perfor-mance is tested per se, without considering any otheraspect of wavefront correction and imaging.
S is an on-axis source at infinity, equivalent to asource with a finite dimension of 3.6 arc sec. �1 and �2
Fig. 5. Equivalent model of the simple optical system simulatedfor testing the aberration correction algorithm.
Fig. 4. (Color online) Coma detection: the white star is c2 and thewhite circle is c1, as defined in the text.
are ideal surfaces able to modify the shape of thewavefront that passes through them. This introducedmodification depends on the values of the Zernikecoefficient vectors Z�1
and Z�2that describe the sur-
faces. Finally, an ideal paraxial lens L with a 100 mmfocal length focuses the light beam on the detectorlocated on the focal plane. In the specific settingused, the detector samples the image with a reso-lution of 2 �m providing an image of 500 500pixels. Hence, due to the optical geometry of the sys-tem and the quantization introduced by the detector,the aberration-free image of S is represented on thedetector with an 8 8 pixels square Pq.
In this simple and ideal system, the surface �1introduces the fixed but “unknown” optical aberra-tions, whereas �2 acts as the aberration-correctionsystem. Each component zi, with i � 4, . . . , 9, of thecoefficient vector Z�1
describing the aberrations of the
system has been drawn from a uniform distribution���0.1, 0.1�. Fifty different Z�1
were generated totest the ability of the proposed algorithm to correctthe optical aberrations introduced by �1.
Along with the results obtained with the proposedmethods, the Zernike coefficients were also esti-mated through the direct maximization approachproposed in [10]. The coefficients vector Z�2
provid-ing the N � 1 measurements needed were located atthe vertices of a simplex distant 0.1 from the origin.In this way the N-dimensional simplex lay close tothe N-dimensional hypercube that the coefficientswere drawn from.
4. Results and Discussion
First, a qualitative evaluation of the performance ofthe algorithm in the 50 sample cases can be gatheredby comparing the plots of the coefficients describing
Fig. 6. Comparison plots of the Zernike coefficients of the fixed surface �1 versus the estimated coefficient describing the surface �2. InFig. 7(a) the plot for the z9 coefficient is represented, Fig. 7(b) for the z7 coefficient, Fig. 7(c) for the z8 coefficient, Fig. 7(d) for the z4
coefficient, Fig. 7(e) for the z6 coefficient, and finally Fig. 7(f) for the z5 coefficient.
�1 versus the opposite of those describing the AOsurface �2 at the end of the estimation procedure. InFig. 6, the comparison plots for the six coefficients areshown: the estimates lay around the identity lines,even if the estimation worsens for astigmatism anddefocus. Three indices are then used to evaluate theperformance of the proposed algorithm. The first isthe absolute sum between the six estimated Zernikecoefficients zi � Z�2
used to compensate the opticalaberrations and those zi � Z�1
of the aberrating sur-face �1. Since a perfect compensation of the aberra-tions is achieved when the two surfaces �1 and �2 arecomplementary, the estimated coefficient Z�2
shouldhave a value opposite that of the coefficient Z�1
. Theabsolute estimation error of the ith Zernike coeffi-cient is thus
�i � �zi � zi�. (16)
This shows the capability of the system to accuratelydescribe the unknown aberrations. The median val-
ues, along with their standard deviations, are re-ported in Table 1, whereas in Table 2 the same valuesobtained estimating the coefficients with the directmaximization are reported.
Additionally, in Fig. 8 the distributions of the esti-mation errors ei � zi � zi are shown for each Zernikecoefficient separately, and in Fig. 9, the distributionof the estimation errors obtained with the direct max-imization procedure are shown. At variance with theproposed algorithm, it is clear how the error distri-butions are similar for all coefficients, but the vari-ances of the error are greater than those provided bythe proposed method.
Both from Table 1 and from the figures it is appar-ent that the estimation error increases in the coeffi-cients that are estimated last. The reason for thisbehavior of the algorithm is twofold. First, when es-timating coefficients low in the hierarchy, residualerrors in the already estimated coefficients can mis-lead the estimation procedure. Second, as the size ofPq decreases, the sensitivity of the objective functions
Fig. 7. Progressive aberration estimation and correction. The xy plot scale is the same for all the images, however the color map isdifferent, so that each image is normalized to its maximum intensity to visualize the point source shape. (a) Original aberrated point image.Maximum intensity is 2 10�4, and the function f(c) described in [10] is 0.19. (b) Aberrated point image after spherical correction.Maximum intensity is 5 10�4, and the function f(c) described in [10] is 0.99. (c) Aberrated point image after coma correction. Maximumintensity is 6 10�4, and the function f(c) described in [10] is 0.98. (d) Aberrated point image after defocus correction. Maximum intensityis 16 10�4, and the function f(c) described in [10] is 0.90. (e) Aberrated point image after astigmatism correction. Maximum intensityis 16 10�4, and the function f(c) described in [10] is 0.90.
Table 1. Absolute Error on the Estimates of the Zernike Coefficients ofthe Aberrating Surface
�4 �5 �6 �7 �8 �9
Median 0.0127 0.0232 0.0209 0.0097 0.0091 0.0029Standard
used decreases, since the image resolution is not suf-ficient for reliable detection of the remaining aberra-tions.
The second index is the number of pixels n3 dB thatare above the �3 dB threshold (half-maximum) withrespect to the maximum power density value, at theend of the procedure. The optical system described inSection 3 provides a nominal nonaberrated and quan-tized image of S of 8 8 pixels. Hence, the closer thedimensions of the imaged point resulting at the end ofthe procedure are to the nominal ones, the better thecompensation of the aberration. The average n3 dBside dimension of the resulting Pq is nine pixels witha standard deviation of 1, showing an excellent abil-ity to reduce the spot size.
Finally, the third index is the number of iterationsnit required to achieve the convergence. This is a mea-sure of the speed of the algorithm, in order to assessthe possibility of using the proposed algorithm in areal system where convergence must be achieved in afew seconds: In our simulations, the number of iter-ations nit required by the algorithm to converge hasproved to be very limited, thus making viable anon-line implementation. The average nit is 78 witha standard deviation of 14, for estimating the sixZernike coefficients considered, and the maximumnumber observed in the 50 repetitions of the experi-ment was 117: in Fig. 10 the boxplots show the dis-tribution of the number of iterations separately foreach estimated coefficient, so that an idea of the con-vergence speed separately for each optimization canbe gathered. The deterministic nature of the algo-rithm ensures that it will converge to the same esti-mate at each run, given the same starting conditions,at variance with stochastic algorithms widely usedfor this kind of optimization, as simulated annealingor genetic algorithms. Moreover, its gradient-basednature will ensure that it will quickly reach the near-est (local) optimum of the objective function withouttrying random coefficient configurations that lay farfrom the direction of objective function decrease.
In Fig. 7 the progressive aberration correction op-erated by estimating the related Zernike coefficientsis shown in a sample setting. Looking at the shapes ofthe different point images, it is easily understandablethat a single cost function based on image qualitymetrics would never allow it to step from an image asthat in Fig. 6(a) to those in Fig. 6(c) or Fig. 6(b): froma purely imaging point of view, Fig. 6(c) or Fig. 6(b)appear as a worsening of the first image Fig. 6(a). Incontrast, adapting a specific objective function foreach aberration allows it to overcome this limitation,achieving a better aberration compensation and asmaller spot size.
Fig. 8. Error distribution in the estimation of the Zernike coeffi-cients. Each box has lines at the lower quartile, median, and upperquartile values. The whiskers are lines extending to the mostextreme value of the data inside 1.5 times the interquartile range.Outliers are data with values beyond the ends of the whiskers, andare plotted with a single cross.
Fig. 9. Error distribution in the estimation of the Zernike coeffi-cients obtained with the direct maximization proposed in [10].Each box has lines at the lower quartile, median, and upper quar-tile values. The whiskers are lines extending to the most extremevalue of the data inside 1.5 times the interquartile range. Outliersare data with values beyond the ends of the whiskers, and areplotted with a single cross.
Fig. 10. Number of iterations needed for the proposed algorithmto converge for each single type of aberration to be estimated. Eachbox has lines at the lower quartile, median, and upper quartilevalues. The whiskers are lines extending to the most extreme valueof the data inside 1.5 times the interquartile range. Outliers aredata with values beyond the ends of the whiskers, and are plottedwith a single cross.
To the best of our knowledge, a new method to correctaberrations in an adaptive optics system has beendescribed. It does not need any wavefront sensor northe knowledge or computation of the PSF of the sys-tem, but only the acquisition of images of a pointsource while the aberrations are corrected; it relies onimage analysis and the identification of a specificobjective function to be optimized for each Zernikecoefficient considered. A strategy for reducing the di-mensionality of the state space of the coefficients hasbeen devised, which together with the deterministic,gradientlike optimization procedure, has proved torequire very few iterations to converge, and to be ableto estimate the Zernike coefficients of the real aber-rations with remarkable accuracy. After the simula-tion test, the software has been tested on a test benchwith real optics (see the companion paper [18]), andfurther developments are planned to include the es-timations of tilt and higher-order aberrations.
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