Hubble by Roddier

Combined approach to the Hubble SpaceTelescope wave-front distortion analysis

Claude Roddier and Frangois Roddier

Stellar images taken by the Hubble Space Telescope at various focus positions have been analyzed toestimate wave-front distortion. Rather than using a single algorithm, we found that better results wereobtained by combining the advantages of various algorithms. For the planetary camera, the mostaccurate algorithms consistently gave a spherical aberration of -0.290-11m rms with a maximumdeviation of 0.005 pum. Evidence was found that the spherical aberration is essentially produced by theprimary mirror. The illumination in the telescope pupil plane was reconstructed and evidence was foundfor a slight camera misalignment.

Key words: Space telescope, wave-front sensing, phase retrieval, aberrations.

1. Introduction

The observation of defocused stellar images has longbeen known as a sensitive test for mirror figureerrors. However, there have been few attempts toextract quantitative information from such images.Recently we proposed a method based on geometricaloptics, which is valid only for highly defocused images.'The method consists of taking the difference inillumination between two defocused images as a mapof the local wave-front Laplacian. The wave front isreconstructed by solving a Poisson equation. Themethod is comparable in sensitivity to a Hartmanntest2 and has been successfully used to control theoptical quality of telescopes on Mauna Kea.3 Becausethe method is based on the geometrical optics approx-imation, it works with broadband, extended lightsources such as stellar sources blurred by atmo-spheric turbulence.

In the framework of the Hubble Aberration Recov-ery Program (HARP) organized by the Jet PropulsionLaboratory (JPL), we requested that highly defocusedimages be taken in flight by the Hubble Space Tele-scope (HST) so that the method could be applied toestimate the exact amount of spherical aberration.Because defocusing the image also defocuses thetelescope tracking system, it was not possible toobtain images sufficiently defocused for the methodto apply. However, defocused images recorded by

The authors are with the Institute for Astronomy, University ofHawaii, 2680 Woodlawn Drive, Honolulu, Hawaii 96822.

Received 24 June 1992.

0003-6935/93/162992.17$06.00/0.o 1993 Optical Society of America.

the HST are not blurred by the atmosphere and canbe taken through narrow-band filters. In this casethe wave-front information is still preserved and canbe recovered by using phase-retrieval algorithms.As we shall see, the image blur that is due to telescopejitter then becomes a limitation.

The first practical phase-retrieval algorithm wasdescribed by Gerchberg and Saxton.4 This algo-rithm uses a single in-focus image rather than twodefocused images. In addition, the pupil transmis-sion function is assumed to be known. One startswith a first guess of the incoming wave-front phase(which can be random) and computes the diffractedamplitude and phase in the image plane by taking theFourier transform of the input complex wave front.The calculated amplitude is then replaced by theobserved amplitude (square root of the image intensi-ty). An inverse Fourier transform gives a new esti-mate of the incoming wave-front phase and amplitude.The calculated amplitude is again replaced by theknown incoming wave-front amplitude (given by thepupil transmission function) and the process is iter-ated. At each step the difference between the knownamplitude and the calculated amplitude gives a mea-sure of the current error. Iteration is stopped whenthe error is below an acceptable level.

At first the algorithm quickly converges but thentends to stagnate. Furthermore, the solution maynot always be unique. Fienup and Wackerman5

developed methods to avoid stagnation and appliedthe technique to reconstruct images from the modu-lus of their Fourier transform. Misell6-8 found that,by using both an in-focus image and an out-of-focusimage, ambiguities can be removed and better conver-

2992 APPLIED OPTICS / Vol. 32, No. 16 / 1 June 1993

gence can be achieved. The Misell algorithm is nowused to test millimetric radio antennas.9,10 As in theGerchberg-Saxton algorithm, the Misell algorithmstarts with a first guess for the complex wave front inthe telescope entrance pupil and computes the com-plex amplitude associated with the in-focus image.The calculated amplitude is replaced by the observedamplitude and the result is Fourier transformed backproducing a new estimate of the complex amplitude inthe telescope pupil plane. However, no constraint isapplied in the pupil plane other than by setting tozero any value outside the pupil. The complex ampli-tude in the pupil plane is multiplied by a quadraticphase factor, which introduces a defocus, and a newFourier transform is taken that gives an estimate ofthe complex amplitude in the defocused image.Again the computed amplitude is replaced by theobserved amplitude and the result is Fourier trans-formed back. The process is then iterated.

During the course of this study similar algorithmswere developed to adapt the tool to the specificproblem posed by HST data. These algorithms werethoroughly tested on simulated data. They are de-scribed below. Their advantages and drawbacks arediscussed. The results we obtained on the HST aresummarized in the following sections. One mustemphasize that similar algorithms were also devel-oped by other teams as part of the JPL HARP effort.The results have been published in the final HARPreport. Most of the results were presented at theOSA topical meeting on Space Optics for Astrophysicsand Earth and Planetary Remote Sensing, held inWilliamsburg. Some results have also been pre-sented at various SPIE Conferences (e.g., vols. 1542and 1567). Our results were generally found to be ingood agreement with those of others.

partially balance the spherical aberration and pro-duce a least-confused image beyond the paraxialfocus. When this happens, the phase can still beproperly sampled in the pupil plane and the complexamplitude in the defocused image can be computedwith a single Fourier transform. For defocused im-ages taken before the paraxial focus, adding a defocusterm increases the phase slopes, which in turn re-quires smaller sampling intervals. These consider-ations led us to modify the Misell algorithm."

The algorithm we used is summarized in Fig. 1. Itstarts with a first guess of the wave-front aberration(±0.5 plm) and of the pupil transmission function andcomputes the complex amplitude for an image (1)recorded beyond the paraxial focus. This can bedone by using a single Fourier transform with adefocus factor that is partially balanced by the tele-scope spherical aberration. The computed ampli-tude is replaced by the observed amplitude and theresult is Fourier transformed back with a differentdefocus factor that produces the complex amplitudein image (2) that was taken before paraxial focusrather than in the pupil plane. Again the computedamplitude is replaced by the observed amplitude andthe result is used to recompute the complex ampli-tude for image (1). The process continues to iterateback and forth between image (1) and image (2)without ever returning to the pupil plane. As for theGerchberg algorithm,' 2 faster convergence is ob-served when image (1) is multiplied by an apodizationwindow. The width of the window is increased ateach iteration and then the window is suppressed.

As for any Gerchberg-Saxton type algorithm, con-vergence slows down and stagnates after severaliterations. To avoid stagnation, we used the follow-

2. Algorithms

A. Modified Misell Algorithm

From our previous experience in reconstructing wavefronts it was clear that highly defocused images werestill necessary to retrieve the wave front with goodspatial resolution, because highly defocused imagesmake better use of the detector dynamic range, can beused with larger optical bandwidths, and are lesssensitive to telescope jitter. In the Misell algorithm,the complex amplitude in a defocused image is com-puted by multiplying the complex amplitude in thepupil plane by a quadratic phase factor and taking theFourier transform of the product. However, whenthe amount of required defocus is large, the associ-ated phase factor may become undersampled, produc-ing aliasing errors. Another possible approach is totake the Fourier transform of the complex wave frontfirst, multiply the result by a defocus phase factor,and Fourier transform back. This second approachgives better results for highly defocused images but itrequires the use of two fast Fourier transformsinstead of one. In the case of the HST, the phase ofthe wave front in the pupil plane is affected byspherical aberration. Adding a defocus term may

First guess of amplitude and phase in pupil plane

Compute complex Compute complex Compute complexamplitude in image amplitude in image amplitude in image(3) through Fresnel (1) through Fresnel (2) through Fresneltransform transform transform

Flag off Fl g on

Replace amplitude Set flag. Replace amplitudewith observed If convergence with observedamplitude N 13 stagnates switch flag amplitude N Estimate erroV Estimate error.

No

Compute complex Replace amplitude Compute complexamplitude in image with observed amplitude in Image(1) through Fresnel amplitude -N I, (1) through Fresneltransform Estimate erroc transform

Is error acceptable?

y Yes

Take the square of Compute complex Unwrap the phasethe amplitude amplitude in the pupil

< plane

Estimatedillumination in thepupil plane

Estimated wavefrontsurface

Fig. 1. Flow chart of the modified Misell algorithm.

1 June 1993 / Vol. 32, No. 16 / APPLIED OPTICS 2993

ing procedure. We omitted image (2) and used ourcurrent estimate of the complex amplitude for image(1) to compute the complex amplitude for image (3) byusing a single Fourier transform with a differentdefocus factor. The computed amplitude is replacedby the amplitude observed in image (3) and a newseries of iterations is done by iterating back and forthbetween image (1) and image (3). Convergence be-comes rapid again and then slows down. When itstagnates one switches back to iterations betweenimage (1) and image (2) and so forth. This procedurewas found to increase convergence but was also foundto be sensitive to both decentering and despace errorsin the three images. Indeed, as long as only twoimages are used a decenter error translates into awave-front tilt; a despace error translates into awave-front defocus term. Adding a third image re-quires perfect alignment and correct despace values;otherwise convergence is affected.

B. Original Misell Algorithm

One advantage of the original Misell algorithm, whichapparently has never before been used, is the possibil-ity of combining the information from defocusedimages taken at different wavelengths, because thecom- plex amplitude in the pupil plane is computed ateach iteration. One can simply rescale the wave-front phase error in the pupil plane for a differentwavelength and similarly compute the diffractionpattern in the image taken at this new wavelength.Also, we wanted to compare the result of our modifiedMisell algorithm with that of the original algorithm.This motivated us to implement the original Misellalgorithm.

It was possible to apply the original Misell algo-rithm without increasing the sampling frequency byusing data taken at longer wavelengths for whichsampling is not critical. On the same pairs of imagesboth algorithms gave consistent results. However,the original Misell algorithm was found to be sensitiveto despace and decenter errors. Because constraintsare also applied in the pupil plane, it is indeed equiv-alent to a three-image algorithm as described above,whereas our modified Misell algorithm is insensitiveto despace and decenter errors when only two imagesare used. For this reason our modified algorithmwas found to be superior to the original algorithm.

Because constraints are applied in three differentplanes, the pupil plane and two image planes, theoriginal Misell algorithm should be compared to ourmodified algorithm with constraints applied in threeimage planes. However, because the illumination inthe pupil plane is not known accurately enough, onlyloose constraints can be applied in the pupil plane(illumination outside the pupil is set at zero), whereasstrong constraints can be applied in three imageplanes. This again is an advantage of our modifiedalgorithm.

C. Single-Image Algorithms

We found that algorithms with constraints applied inthree different planes are sensitive to despace and

decenter errors. For this reason we preferred ourmodified algorithm with constraints applied in twoimage planes only. However, in the case of plane-tary camera (PC) images a problem still occurs be-cause the pupil gometry strongly depends on thelocation of the image in the camera and images arenever taken exactly at the same location. This led usto investigate the use of algorithms based on a singleimage, namely, Gerchberg-Saxton type algorithmswith full constraints on the amplitude in a singleobserved defocused image and looser constraints inthe pupil plane. We empirically developed the follow-ing procedure.

Initially an estimate is used for the large-scaleaberrations (here defocus and spherical) and thetelescope pupil is modeled as a uniformly illuminateddisk (no central obscuration or spider arms). AFourier transform is used to compute the complexamplitude in the image. The argument or modulusis replaced by the square root of the observed illumi-nation and an inverse Fourier transform is taken thatprovides a new estimate of the illumination in thepupil plane. Surprisingly this single loop is gener-ally sufficient for the central obscuration and thespider arms to appear as darker areas. Images takenat the secondary mirror position of -90 ,um gaveexcellent results. By varying the amount of defocusor decenter, one can improve the appearance of thespider arms until they become straight and fairlysharp. Decenter errors tend to distort the recon-structed spider arms, indicating the direction of thedecenter. Despace (i.e., focus) errors tend to blur thereconstructed image and also tend to produce anonuniform illumination in the reconstructed pupilintensity. Either the edge or the center of the pupilis brighter depending on the sign of the focus error.We empirically found this was an accurate way tocontrol the decenter and despace errors. The resultis rather insensitive to the estimated spherical aberra-tion. Best results were obtained by refining thecentering of the image first until straight spider armswere observed. Additional constraints were thenused by setting to zero the illumination in the ob-served obscured central area. The focus value isthen refined until both sharp spider arms and a fairlyuniform illumination are observed in the recon-structed pupil. Further constraints can then beoptionally applied by also setting to zero the illumina-tion inside the areas that are obscured by the spiderarms and the clamp holes in the primary mirror.

Once the pupil geometry is accurately modeled, thismodel is used to refine our estimate of the defocusand of the spherical aberration. This is done bycomparing the observed and the computed imagesand by minimizing their difference by using thewave-front fitting procedure described in Subsection2.D. This fitting procedure was found to be moreeffective than the use of iterative Fourier transformsto retrieve large-scale aberrations. For the sharpestimages it was also used to refine the wave-front tiltestimate. On the other hand, iterative Fourier trans-


forms are more effective in retrieving small-scalewave-front structures.

Once wave-front tilt, defocus, and spherical aberra-tion are carefully estimated, these values are used asa first wave-front guess in a regular Gerchberg-Saxton algorithm, together with the determined pupilgeometry. At each iteration the pupil geometry andthe observed image illumination are used as ampli-tude constraints, whereas the wave-front phase is leftfree to evolve. Experience shows that good conver-gence is observed after 50 iterations without anysignificant change in the amount of defocus or spheri-cal aberration. Only smaller scale details appear inthe reconstructed wave-front phase. After 50 itera-tions we found that convergence further improves ifthe constraint of a uniformly illuminated pupil isremoved. Approximately five additional iterationswith the constraint removed produced stable struc-tures on the pupil illumination. The structures aresimilar for all the images we processed and could bereal as discussed in Section 3.

D. Wave-Front Fitting Procedure

This procedure is applied after a careful determina-tion of the pupil geometry as described in Subsection2.C. By looking at the spider arms in the pupilimage reconstructed from a single-loop Gerchberg-Saxton iteration, one can determine the center of theimage within one pixel. This center coincides withthe center of the computed image when the compari-son is made. We assume that the main low-orderwave-front error terms are defocus (Z4) and sphericalaberration (Z11) and use a two-parameter fit. Whenthis comparison was made, we found it essential totake into account any blur in the observed image.Indeed, computed images look sharper than observedimages. This is clearly an effect of telescope jitter.Computer simulations showed that trying to match acomputed image with a blurred image gives biasedresults. An excess of defocus provides a better match.The excess of defocus is balanced by an underesti-mated spherical aberration. Computer simulationsalso showed that convergence is quite sensitive to thechoice of the norm that was used in the comparison.A detailed account of the image-fitting procedure isgiven in Appendix A.

E. Discussion of Algorithms

From the above descriptions it is clear that eachalgorithm has its own advantages and drawbacks.Algorithms that use several observed images as con-straints are expected to give the most reliable results.Unfortunately they are sensitive to any decenter ordespace error. Although less accurate, algorithmsbased on a single image were found to be extremelyuseful and helped us to retrieve missing informationsuch as the exact despace and decenter values and theexact pupil geometry for each of the images weprocessed. Ideally one should process single imagesfirst and determine for each of them the exact despaceand decenter value and pupil geometry. Imageswith the same pupil geometry can then be matched

and combined as constraints in a multiple-imagealgorithm by using accurate despace and decentervalues for optimum wave-front reconstruction. Thisleads us to a combined approach that profits fromeach algorithm advantage while the drawbacks areminimized.'3 This approach consists of the follow-ing steps:

(1) Determination of the pupil geometry by apply-ing the Gerchberg-Saxton algorithm to single im-ages.

(2) Determination of the main aberration termsby using an error minimization algorithm (wave-front fitting).

(3) Full wave-front reconstruction with a modi-fied Misell algorithm.

Lack of time prevented us from implementing thefull procedure on the same images.

We found that image blur that was due to telescopejitter was the main factor that limited the accuracy ofthe wave-front estimation. The methods we used toovercome this problem are described in detail inAppendix A. It has been claimed that the accuracyof direct pupil-plane to image-plane propagation algo-rithms is limited and that multiple-plane propagationalgorithms that incorporate all the telescope compo-nents must be used to get the required accuracy.We disagree with this statement. Direct propaga-tion algorithms simply reconstruct the wave frontthat, diffracted from infinity, would produce therecorded image regardless of how the image wasactually produced. Some inaccuracy may come froma priori assumptions made on the pupil transmissionfunction that may not apply to the actual pupil imageas seen from the focal plane through the telescope,because this image may be distorted. Single-imagereconstruction algorithms indeed require a prioriassumptions such as the existence of straight spiderarms and a fairly uniform pupil illumination. More-over these algorithms are necessarily blind to aberra-tions that are produced in a plane conjugate to theimage because these aberrations cannot affect theirillumination. However, as soon as more than oneimage is used, all the wave-front aberrations aretaken into account and no a priori assumption isneeded on the pupil configuration. Our approachwas to minimize such assumptions. When they weremade, the assumptions were used to boost algorithmconvergence and were later removed. We suspectthat multiple-plane propagation algorithms may seemto produce more consistent results simply becausethey tend to produce images with more blur, whichreduces the error that is due to telescope jitter.Neither direct nor multiple-plane propagation algo-rithms give information on the origin of the observedwave-front aberration unless several images are usedat different field positions (see Subsection 4.G).

3. Image Analysis

The data we used are displayed in Table 1. They aredivided into four main sets. The results obtained


Table 1. Hubble Space Telescope Data Used In This Study

JPLDate No. Camera Filter Focus Coordinates Defocus (um rms) Spherical (pum rms)

First set of data8/319/49/49/59/49/4

Second set of data10/2510/2510/2510/2510/2510/2510/2510/2510/2510/2511/3012/3

Third set of data12/312/312/312/310/2611/3011/3011/3011/3011/3011/3011/3011/30

Fourth set of data

w24318w24706w24730w24818w24710w24734

w29826w29830w29906w29910w29918w29826w29830w29906w29910w29930w33434w33714

w33702w33706w33710w33726w29938w33402w33406w33410w33414w33422w33426w33430w33434

FOC F486N -90FOC F486N +250

aThe most reliable values of spherical aberration.

from each of these four sets are described in separatesections. Each set is divided into subsets of imagesthat were processed together. The focus columngives the secondary mirror despace in micrometers.The star coordinates are given in pixel numbers onthe detector. The last two columns list the defocusterm that is estimated for each image and the spheri-cal aberration term estimated for each subset. Themost reliable values of the spherical aberration aremarked by an asterisk.

A. Analysis of the First Set of Data

We started our analysis with the images of the firstset of data taken at 0.487 m. The modified Misellalgorithm described in Fig. 1 was used. This algo-rithm uses thru images labeled (1), (2), and (3).Image (1) is used at each iteration. To limit theamount of computing time we used 256 x 256 arraysand found that the aberrated image at the paraxial

focus was already too large to be accurately repro-duced without aliasing errors by taking the Fouriertransform of the estimated complex amplitude in thepupil plane. We therefore decided to take image (1)near the circle of least confusion (focus position 250).Images (2) and (3) were taken as far as possible fromimage (1), at focus positions -90 and -260. Thesepositions are shown in Fig. 2. To minimize interpo-lation errors, we set the size of the telescope pupil at65.71 pixels, which gave us computed images with thefollowing sampling: 7.5 ,um pixels for image (1),which is half of the data pixel size; 15 ,um pixels forimage (2), which is the camera pixel size; 22.2 ,umpixels for image (3), the only image that requiredinterpolation.

Several runs were made by using slightly differentinitial guesses and by varying the order of iterations.In each case an estimate of the complex amplitude inthe telescope pupil plane was obtained. All the


PC6PC6PC6PC6PC6PC6

PC6PC6PC6PC6PC6PC6PC6PC6PC6PC6PC6PC6

PC6PC6PC6PC6PC6PC6PC6PC6PC6PC6PC6PC6PC6

F487NF487NF487NF487NF889NF889N

F889NF889NF889NF889NF889NF889NF889NF889NF889NF889NF631NF889N

F889NF889NF889NF889NF613NF613NF613NF631NF631NF631NF631NF631NF631N

-5+250-90

-260+250-90

-267-267-260-260+ 170-267-267-260-260+210-90

+333

+333+333+333+333+250-90-90-90-90-90-90-90-90

448, 381451, 382450, 378454, 378450, 382451, 379

532, 425532, 425530, 426531, 425537, 419532, 425532, 425530, 426531, 425538, 421713, 674671, 677

196, 228680, 158143, 674487, 495538, 422190, 221453, 336717, 337184, 595716, 412183, 670447, 672713, 674

-0.7+0.87-1.18-2.25+0.89-1.15

-2.32-2.32-2.27-2.27+0.2-2.32-2.32-2.27-2.27+0.42-1.12+ 1.25

+ 1.28+ 1.25+1.25+ 1.26+0.83-1.184-1.216-1.252-1.205-1.230-1.200-1.220-1.253

-1.10.8

-0.294a

-0.288a

-0.278

-0.270

-0.298

-0.305-0.294-0.300-0.302-0.292-0.292a-0.289a-0.295a-0.292a-0.288a-0.288a-0.286a-0.291a

-0.245-0.248

Caustic corresponding to HST primary mirror

Conic constant error = -0.012

Transverse scale expanded by factor 10

Direction of light travel

I ,, , ,' , I

-10 mmin 1/24 image space

250 500

Fig. 2. Ray tracing showing the effect of spherical aberration at the focal plane of the Hubble Space Telescope (courtesy A. Vaughan).The modified Misell algorithm was applied to images labeled (1), (2), (3) taken at 0.487 pm at secondary mirror positions of +250, -90, and-260.

results were quite similar. Here we give resultsobtained by averaging all the complex amplitudeestimates to produce our final wave-front estimate.From this estimate we computed the illumination ineach observed image and compared the result of thecomputation with the actually observed illumination.The agreement was found to be excellent (see alsoSubsection 4.F).

Note that, unlike the original Misell algorithm, ourmodified algorithm does not apply any constraint inthe pupil plane. It was still able to reproduce satis-factorily the illumination in the telescope pupil. Thereconstructed pupil image clearly showed the centralobscuration produced by the PC secondary mirrortogether with the three thick supporting arms.However, to our surprise they were not centered onthe pupil but shifted sideway. Since the images weretaken close to the optical axis, the shift showedevidence for a misalignment of the PC. Other evi-dence for this misalignment was independently foundby Burrows14 from images taken in the wide-fieldconfiguration of the camera. A detailed analysis ofthis misalignment is given in Subsection 4.E.

The illumination on the pupil was found to benonuniform, showing dark rings. It is not clearwhether these rings are real or artifacts. Computa-tion errors tend to produce dark circular zones in thepupil illumination. However, rings appear at nearlythe same location in the reconstructed wave-frontmap and correspond to grooves left by the polishingtool. Because the camera entrance pupil is notconjugate to the telescope pupil and because of theadditional telescope spherical aberration these grooves

may indeed diffract light outside the camera entrancepupil, thus producing the observed dark rings in thereconstructed pupil image.

The reconstructed wave front showed almost purespherical aberration. A least-squares fit of the recon-structed wave front was made with Zernike aberra-tion terms. For best results we used Zernike annu-lar polynomials'5 that are orthogonal over a uniformdisk with a 0.33 central obscuration diameter ratio.The fit was made over the illuminated pupil area withthe exception of a small region, a few pixels wide, nearthe pupil edges at which the uncertainty of thereconstructed wave front is quite large. The ampli-tude of the spherical aberration term was estimatedto be -0.294-jim rms with a dispersion of ±0.005.Other terms such as astigmatism, coma, and trefoilwere found to be below 4 x 10-3, that is, within theuncertainty of the estimate. However, smaller scalewave-front errors with an amplitude larger than theuncertainty appeared on the reconstructed wave front.These ringlike structures were found on all thereconstructed wave fronts independent of the algo-rithm used and are believed to be structures left bythe polishing tool on the primary and secondarymirrors. They are described in Subsection 4.D.

The same algorithm was then applied to a new setof two images taken at a longer wavelength (nextsubset in Table 1). These images were taken on 4September at 0.889 jIm at focus positions +250 and-90. Using images taken at a longer wavelengthhas the effect of relaxing constraints on the samplingof the quadratic phase factors, allowing us to keepworking with 256 x 256 arrays with a minimum of


-500 -260 -90 0Secondary mirror despace in u ir

lr

aliasing errors. On this set the spherical aberrationterm was estimated to be -0.228-jIm rms with adispersion of ±0.005. The other Zernike terms wereagain found to be all below 0.004, that is, within theuncertainty of the measurement. From the recon-structed pupil image we have estimated that thecamera spider arms should coincide with the tele-scope spider arms when the star image coordinatesare in pixels 210 ± 50 and 300 ± 50. A more detailedanalysis of the camera misalignment is given by thethird set of data used.

B. Analysis of the Second Set of Data

The original Misell algorithm was implemented andapplied to this set of data rather than our modifiedversion (see Subsection 2B). It was first applied tothe same two images as above (the last two images ofthe first set) with almost identical results. However,the convergence was found to be slower because of thesensitivity of the algorithm to decenter and despaceerrors. Since at each iteration this algorithm com-putes the complex wave front in the pupil plane, wetried to unwrap its phase and each time estimate thewave-front tilt and defocus. The convergence wasimproved but the algorithm was considerably sloweddown. Nevertheless it opened the possibility of us-ing a large set of images in order to maximize thenumber of constraints.

This was done on the first two subsets of imagesshown in Table 1, which are identical except for thelast image. Again the convergence was found to bepoor and the algorithm slow. The final estimatetended to depend on the initial guess used as an inputto the algorithm. A low value was found for thespherical aberration term (less than 0.28 jim). It isconsidered to be unreliable. There are several possi-ble explanations. One is that the estimations of thewave-front tilt and defocus made at each iteration arenot sufficiently accurate. The other is that the pupilgeometry may differ slightly from one image to theother although the images were taken with the starnearly at the same position in the field of view.

Our last attempt to apply the original Misell algo-rithm was on a pair of images taken at two differentwavelengths (0.631 and 0.899 jim) with the starimage almost at the same location (next subset inTable 1). Again the result tended to depend on theinitial guess. In this case a high value was found forthe spherical aberration (0.298 jim). It is not consid-ered to be reliable either. It is likely that the pupilgeometry for the two images was sufficiently differentto prevent good convergence.

C. Analysis of the Third Set of Data

The third set of data was analyzed by using thesingle-image algorithms described in Subsection 2.C.A single iteration was used to reconstruct an image ofthe telescope pupil. As discussed above, the recon-structed pupil image was found to be quite sensitiveto despace and decenter errors. It was also found tobe sensitive to the image blur that was due totelescope jitter, which varies from one image to

another. Figure 3 shows an example of a goodreconstructed pupil image.

The method was first applied to images taken at0.889 jim at focus position +333, which are the fivefirst images of the third set of data. In this case thecentral obscuration and the spider arms barely appear.Images taken before the focal plane are larger andless sensitive to image jitter and gave the mostreliable results. We concentrated our efforts on aseries of eight images taken on 30 November at 0.631Pim (the last eight images in Table 1). The firstiteration was used to determine the exact location andsize of the obscurations produced by both the tele-scope and the camera secondary mirrors. The pupilwas then modeled with uniform illumination and theestimated central obscuration, but still no spiderarms. This model was used as the input for a newiteration giving an improved pupil irfage with sharperspider arms that can be accurately located. Thisimage was used to recover entirely the pupil geometryto be used. This procedure was found to be bothsimple and effective and was used to map the relativeposition of the telescope and camera pupils as afunction of the star coordinates in the field of view.Table 2 gives the PC pupil offset in pupil-imagesample points (pupil radius is 103.05 sample point) asa function of the star coordinates in camera pixelunits. The uncertainty on the pupil offset is esti-mated to be + 1 pixel.

Once the pupil geometry was determined, we esti-mated defocus and spherical aberration as describedin Appendix A and we used 50 Gerchberg-Saxtoniterations to refine our estimate of the complexamplitude in the pupil plane. In addition, five itera-tions were made with the constraint of uniform pupilillumination removed as described in Subsection 2.C.Table 1 shows the defocus term and the sphericalaberration term that have been estimated for each ofthe eight reconstructed wave fronts.

The eight reconstructed wave fronts were thenaveraged together and a least-squares fit of theZernike terms was made on the average wave front asdescribed above. Apart from defocus and sphericalaberration, the amplitude of all the terms in betweenwere found to be below 10-3-jim rms. Figure 4 is amap of the reconstructed wave front after removal of

Fig. 3. Example of pupil image reconstructed from a singleGerchberg-Saxton iteration.


Table 2. Camera Pupil Offset as a Function of Star Coordinates

Image No. x Coordinates x Offset y Coordinates y Offset

w33042 190 -1.5 221 -2.5w33406 453 7 336 0.5w33410 717 17.5 337 0.5w33414 184 -2.5 595 10.5w33422 716 17.1 412 3w33426 183 -2.2 670 12.8w33430 447 7.5 672 12.8w33434 713 17.1 674 12.6

2

0

-1

-2

tilt, defocus, and spherical aberration. These resultsare consistent with those obtained with the first set ofdata and are considered to be our most reliablewave-front estimate.

D. Analysis of the Fourth Set of Data

The same single-image algorithm was used to analyzea few stellar images from the faint object camera.Table 1 shows the defocus and spherical aberrationterms we have estimated. The uncertainty on thewave front is estimated to be +0.01-im rms. Otherlow-order terms were found to be larger than that ofthe PC but still below the uncertainty. As expected,the reconstructed pupil illumination showed a differ-ent geometry from that of the PC (see Subsection4.E). Best results were obtained with the imagetaken at focus position - 90.

4. Results

A. Defocus

Table 1 lists the Zernike defocus term in micrometersrms as a function of the secondary mirror position.Theoretically a 1-mm motion of the secondary shouldproduce a 6.135-jim rms defocus error. A linearregression on the most reliable data (marked by anasterisk) gives a slope of 6.139 jim/mm. This agree-ment shows the remarkable accuracy of the modifiedMisell algorithm we used and gives us confidence thata similar level of accuracy can be achieved for thespherical aberration term. The linear regressioncurve shown in Fig. 5 gives the secondary mirrorposition for the estimated best focus as follows: bestfocus position: + 107 5 m.

-30 1 l , I .. l l l l I .. I . . I .. I, , ..l , I , I l-300 -200 -100 0 100 200 300

Fig. 5. Linear regression showing the observed defocus terms inmicrometers rms as a function of the secondary mirror position inmicrometers. Data taken at 0.487 pm (crosses), 0.889 im(squares), and 0.631 pum (triangles).

By using all the data in Table 1 taken on 4 and 5September, we estimated the defocus term to be- 1.18 ± 0.03-jim rms at a secondary mirror positionof +90. By using the data taken on 30 November(last 8 lines in Table 1), we estimated the defocusterm to be -1.22 ± 0.025-Pm rms at the sameposition. Evidence for telescope shrinkage is there-fore marginal. If there is any shrinkage it is of theorder of 2 jim/month. Given the uncertainty weinfer that telescope shrinkage is less than 5jim/month.

In Subsection 4.F it is shown that a single wave-front model accurately reproduces all the imagesobserved over a 3-month period, providing furtherevidence that telescope shrinkage is not significant.The FOC data are consistent with the PC results.

B. Spherical Aberration

Taking an average of the most reliable data markedby an asterisk in Table 1 gives for the PC images awave-front rms spherical aberration term of -0.290 ±0.0034-jim rms. The uncertainty given here is thestandard deviation. Note that the total deviation is+0.005 lm. The FOC data give a significantlysmaller value for the spherical aberration. It isestimated to be -0.245 001-jim rms.

Fig. 4. Reconstructed wave-front surface showing wave-fronterror residuals after removal of the spherical aberration.

C. Other Low-Order Aberration Terms

All the low-order aberration terms between defocusand spherical aberration (astigmatism, coma, trefoil)were found to have an amplitude below the uncer-tainty level and probably below 10-3-jim rms, show-ing the remarkable quality of the telescope optics.

D. Mirror Roughness

Figure 4 shows the residual wave-front errors afterremoval of spherical aberration. Small-scale wave-front residuals are clearly seen with a rms amplitudeof approximately 0.02 jim. These small-scale wave-front errors take the form of circular rings and appearto be structures left by the polishing tools both on theprimary and the secondary mirrors. A circular zone


I I I - - . I . . . . . ...,. . . . . .-1 . I . . . . I ' I ' I I I I I I I I I I I I I I I I I I I I-

can be seen with a mean radius that is equal to 0.65pupil radius and a width of approximately 0.1 pupilradius. It is 0.04 m above the surrounding surface.The same zone was independently found on wavefronts that were reconstructed from interferogramsof the HST primary mirror taken prior to the launch.'6It is therefore believed to be real. Smaller rings alsoappear and are believed to be produced by the tele-scope secondary.

E. Pupil Geometry

Figure 3 shows an example of reconstructed pupilimage. One can see three of the four arms thatsupport the telescope secondary mirror. They arethinner than the camera arms. The fourth arm onthe left is hidden behind the camera arm. Thedistance between the camera arms and the telescopearms is a measure of the camera misalignment. Thethree black dots at 1200 near the pupil outer edge areproduced by the three clamps that hold the telescopeprimary mirror in place. The bottom clamp is di-rectly behind one of the telescope spider arms.

The pupil geometry was reconstructed from suchdata and discrepancies were found between the recon-structed geometry and the data provided by JPL.The largest discrepancy was evidence for a decenter ofthe PC pupil with respect to the optical telescopeassembly pupil. The amount of decenter is esti-mated here.

A linear least-squares fit to the data displayed inTable 2 gives the following relationship between thestar coordinates (x, y) in camera pixels and the cam-era pupil offset (X, Y) in pupil image pixels (pupilradius is 103.05 pixels): x = 27.8X + 245, y =27.8Y + 310. According to JPL data the slopeshould be 28.9 instead of 27.8. Figure 6 shows onthe same plot the star coordinates effectively used andthose computed with the above relations. The uncer-tainty on the pupil offset is ± 1 pixel in the pupilimage that is approximately ± 28 camera pixels. Theerror boxes in Fig. 6 are 56 camera pixels wide. The

400 , , l C , l , l l l l l

- 1 F3 F E7 -600 - 7

400 - a _

_ E7200

n I I I I _0 200 400 600 800

Fig. 6. Star image positions actually measured (dots) and esti-mated from reconstructed pupil configurations (crosses with errorboxes).

offset goes to zero for a star at coordinates x = 245 +10, y = 310 + 10.

The error bar has been divided by the square root of8 since the results comes from eight independentestimates. The above relationship between the starcoordinates and the PC pupil offset was tested onpupil images reconstructed from data taken at second-ary mirror positions ranging from 333 to -267.The star coordinates predicted from the observedoffset were found to be consistent with the knowncoordinates every time the spider arms could beaccurately located.

Another discrepancy was found with regard to thelocation of the three clamps that hold the primarymirror. The coordinates of the center of the threeclamps were carefully measured on the pupil imagesthat were reconstructed from the same eight images.The rms dispersion for these coordinates is 0.5 pixelor 0.6 cm on the pupil. The rms error for an averageof eight measurements is therefore 2 mm. Theresults of these measurements are shown in Table 3.The coordinates are given in pupil radius units. Thecoordinates given by JPL were rotated to match thecoordinates measured on the lower clamp (see Fig. 3).Clearly the lower clamp is at the expected position,the upper right clamp is 16 mm higher, and the upperleft clamp is both 37 mm higher and 6 mm to the rightof the expected position. These offsets are wellabove the uncertainty of the measurement. In addi-tion, we note that the columns of the CCD camera arenot quite aligned with the telescope spider armcentered over the lower clamp. The angle is approx-imately 0.6°.

The pupil image reconstructed from the FOC datashows a different geometry. Compared to the PCthe telescope spider arms are clearly rotated 350clockwise. Since the camera has no pupil-obstruct-ing part its alignment cannot be checked. The loca-tion of the clamps approximately confirms the above-described offsets although data may be affected bydistortion that is due to the image intensifier.

F. Accuracy of the Reconstructed Wave Front

The above-listed results can be used as a model of theimage-forming wave front. This model is comprisedof our best reconstructed wave front and the above-determined relationship that gives the pupil offset asa function of the star position in the field of view.From this model it is possible to compute the illumina-tion at any wavelength and focus position for a staranywhere in the field of view. The results of the

Table 3. Location of the Three Clamps Holding the Primary Mirror

Clamp Coordinate Measured JPL Value A A (mm)

1 x -0.767 -0.772 0.005 6y 0.471 0.440 0.031 37

2 x 0.009 0.009 0.000 0y -0.889 -0.890 0.001 1

3 x 0.754 0.755 -0.001 1

y 0.481 0.468 0.013 16


computation can be used as a point-spread function todeconvolve images that are currently recorded by thetelescope. It is therefore important to estimate theaccuracy of the model. To check this accuracy wecomputed a large number of images and compared theresult with the actually observed image. The resultof this comparison is now shown.

First the comparison was made on the eight imagesthat were used to produce the model, that is, the lasteight images of the third set of data in Table 1. Asexpected, the agreement was found to be excellent.The observed images are clearly blurred by telescopejitter except for one image (w33410) that was noted tobe sharp; this image is shown in Fig. 7. The com-puted image is shown in Fig. 8. Image profiles areshown in Fig. 9 in which the computed image is

represented by a solid curve, the observed image by adotted curve.

Next we used the model to compute images over alarge range of secondary mirror positions spanningfrom -267 to +333. Figure 10 shows image w29822and Fig. 11 shows the same image as computed fromthe model. A cross section of the two images isshown in Fig. 12. Image w29822 was taken a monthearlier at position -267 and at a shorter wavelength(0.487 plm). In spite of these differences the agree-ment is still good. A similar agreement was alsofound for images w24818 and w24730. These twoimages were taken two months earlier at positions-260 and -90 and at the same shorter wavelength

Fig. 10. Image w29822 as observed.

Fig. 7. Image w33410 as observed.

Fig. 11. Image w29822 as computed from the model.

Fig. 8. Image w33410 as computed from the model.

0.003

0.002

0.001

-4 -2 0 2 4

Fig. 9. Comparison between observed (dotted curve) and com-puted (solid curve) image w33410. The horizontal scale is inarcseconds.

-4 -2 0 2 4

Fig. 12. Comparison between observed (dotted curve) and com-puted (solid curve) image w29822. The horizontal scale is inarcseconds.


a0010

0.001

40000

40007

0.0000

o.as

0.0004

00003

0.0002

0.0001

-4 -2 0 2 4 -4 -2 0 2 4

Fig. 13. Comparison between observed (dotted curve) and com-puted solid curve) image w22718. The computed image is eitherunblurred (left) or blurred (right). The horizontal scale is inarcseconds.

(0.487 m). They were used in our modified Misellalgorithm (first set in Table 1). The aberrationterms and pupil geometry that were obtained withthe two different methods were indeed in good agree-ment.

Figure 13 shows a profile of image w27718 (dottedcurve). This image was also taken two monthsearlier at the same shorter wavelength (0.487 pim)but at position + 10, that is, close to best focus. Thesolid curve shows a profile of the same image com-puted with our model. The agreement is good. Itbecomes excellent when the computed image is slightlyblurred, simulating the effect of telescope jitter (right).This shows that our model can probably be used toestimate the point-spread function of the PC at anywavelength and anywhere in the field of view.However, the effect of telescope jitter would probablyhave to be independently estimated by using differentstars in the field of view.

The effect of telescope jitter is clearly seen in acomparison with a series of images (w29914, w29922,and w29938) taken at 0.631 gum at positions + 170,+210, and +250. In these images the diffractionrings are heavily blurred by telescope jitter and mostof the attempts to use these images in a phase-retrieval algorithm were a failure (see Appendix A).

Table 4. Image Pairs Shifted in the x Direction

Image Pairs Coordinate Difference

JPL No. Ax Ay

w33406 w33410 264 1w33426 w33430 264 2w33430 w33434 266 2w33426 w33434 530 2

They are still fairly well reproduced by the model thatproduces sharper images. The effect of telescopejitter is particularly dramatic in Fig. 14, which showsimage w30014 taken at a shorter wavelength (0.487pum) at position + 333. The computed image is shownin Fig. 15. A blurred version of the computed imageis shown in Fig. 16 for better comparison with theobserved image.

The above-discussed model does not apply to FOCdata because of the different pupil geometry. Toreproduce FOC images we used the wave front recon-structed from the image taken at focus position -90(Table 1). Again good agreement was found betweenthe computed and the observed images at focuspositions ranging from - 260 to + 250.

G. Secondary Mirror Figure

An attempt was made to reconstruct the secondarymirror figure by subtracting wave fronts recon-structed from images taken with the star at differentpositions in the field of view. For this purpose weused the last eight images in the third set of data inTable 1. These images were taken with differentstar coordinates that are shown in Fig. 1. We usedthe wave fronts reconstructed after 50 iterations asdescribed in Subsection 3.C without any furthermodification. First we subtracted wave fronts recon-structed with star image pairs shifted in the x direc-tion as shown in Table 4. The wave-front differenceproduced by the last pair in Table 4 was divided bytwo to account for the two times larger Ax value andthe average of the four wave-front differences wastaken; this is displayed in Fig. 17. Next we sub-tracted wave fronts reconstructed with star imagepairs shifted in the y direction as shown in Table 5.

Fig. 14. Image w30014 as observed. Fig. 15. Image w30014 as directly com-puted from the model.

Fig. 16. Image w30014 as computed fromthe model with additional blur to simulatethe effect of telescope jitter.


...................1-111.111.1'..1,

. . . . . . . . .. . . . . . . . .

- , I . . . I . . . . . . I . . . I -

I . . . I . - . I . . . I . .

Fig.17. Secondary mirrorx slopes esti-mated from wave-front differences.

Fig. 18. Secondary mirrory slopes esti-mated from wave-front differences.

Fig. 19. Secondary mirror figure recon-structed from the slopes shown in Figs.17 and 18.

Again the average of the three wave-front differenceswas taken. Its amplitude was scaled down to matcha 265-pixel shift similar to the horizontal shift; this isdisplayed in Fig. 18.

Assuming that the primary mirror is in the pupilplane, its contribution to the wave-front errors shouldbe the same for all the images and should thereforecancel out when the differences are calculated. Onthe other hand, the contribution of the secondarymirror is slightly shifted in the direction of the imageand does not entirely cancel out. With a scale of0.043 arcsec/pixel for the PC camera, a 265-pixelshift corresponds to an angular deviation of 11.4arcsec. By assuming that the secondary is at 4.9 mfrom the primary, this translates into a displacementof a beam of 270 jim on the secondary. Since the103-pixel aperture diameter corresponds to an illumi-nated beam size of 267 mm on the secondary, onepixel corresponds to 2.59 mm. Hence the displace-ment is approximately one tenth of a pixel on thereconstructed wave front.

The question arises whether the structures on thereconstructed wave front reproduce themselves atthe same location with that accuracy. It may well bethe case. These structures diffract light that inter-feres with light diffracted by a sharp pupil. Any shiftof the structures with respect to the pupil appreciablychanges the interference pattern. Since a fixed pupilmask was used as a constraint over 50 Gerchberg-Saxton iterations, the location of the wave-frontstructures with respect to the pupil mask should befairly accurate. This is further supported by theaspect of the wave-front differences displayed in Figs.17 and 18, which closely resembles two knife-edgetest patterns taken with the knife edges at 900 fromeach other. We assume that these wave-front differ-

Table 5. Image Pairs Shifted in the y Direction

Image Pairs Coordinate Difference

JPL No. Ax Ay

w33402 w33414 6 374w33406 w33430 6 336w33410 w33434 4 337

ences map the wave-front slopes on the secondarymirror, and we reconstructed an estimate of thesecondary mirror figure by using a standard algo-rithm that reconstructs a wave front from its slopes.The result of the integration is shown in Fig. 19.The reconstructed mirror figure shows grooves simi-lar to those left by the polishing tool. Knowing thatthe 308-mm-diameter mirror is illuminated over onlya 267-mm-diameter area, we can see the same groovesnearly at the same position on secondary mirrorfigures that were obtained before the launch, givingus confidence in the accuracy of our reconstruction.The rms wave-front error in Fig. 19 was estimated tobe 0.028 Vim. This is to be compared with thesecondary mirror figure error estimated to be of theorder of 0.01-jim rms from data taken prior to thelaunch. One reflection on the secondary shouldtherefore produce wave-front errors of the order of0.02-jim rms in fair agreement with our result. A fitof Zernike terms to the reconstructed wave front gavevalues that were all below 0.01-jim rms; coma beingthe largest term. Although there is clearly a lot ofuncertainty in the accuracy of this reconstruction, webelieve it shows that the contribution of the second-ary mirror to the telescope spherical aberration isprobably less than 3% and perhaps of the order of theuncertainty on our estimate. In other words, spher-ical aberration is essentially produced by the tele-scope primary mirror.

5. Conclusion

The Hubble Space Telescope wave-front distortionhas been estimated from stellar images taken in flightat various focus positions. A combination of differ-ent phase-retrieval algorithms was used to minimizethe effect of despace and decenter errors and uncer-tainties on the star image position on the CCDcamera. Image blur produced by telescope jitterultimately limited the accuracy of our results.

A total of 30 different planetary camera imageswere analyzed by using groups of from one to fiveimages. For each group the wave-front surface wasreconstructed and fitted with Zernike aberrationterms. Different algorithms gave consistent results,and we believe we have determined spherical aberra-


tion for the planetary camera with the requiredaccuracy. Our best estimate is -0.290-jim rms witha maximum deviation of 0.005 jim. Other low-orderaberration terms were found to be smaller than thisdeviation and therefore not measurable. Howeversmall-scale wave-front errors were detected with arms amplitude of 0.02 jim. They appear as circularrings typically left by a polishing tool and are quiteconsistent with wave-front errors estimated by com-bining primary and secondary mirror figures recon-structed from interferograms taken before the launch.Analysis of a few FOC images gave similar results buta lower value for the spherical aberration, - 0.245-jimrms.

An attempt was made to estimate the contributionof the telescope secondary mirror to the aberrations.Evidence was found that the spherical aberration isessentially produced by the primary mirror. Wewere also able to reconstruct the illumination in thetelescope pupil plane. It shows the central obscura-tion and the spider arms of both the telescope and thecamera. These are not superimposed and their rela-tive position varies with the coordinates of the starimage on the camera. For PC-6 images the bestsuperposition was found to occur when the star imagecoordinates were approximately 245 and 310 insteadof coordinates 400 and 400 of the field center, show-ing evidence of camera misalignment.

The methods developed for this study appear to bepowerful and can probably be used to model thetelescope point-spread function as a function of starcoordinates to deconvolve images. The same meth-ods could also be used to check the telescope align-

.95

0.9

0.85

-0.31 -0.3 -0.29 -0.28

| I I I I | I I I I I I

0.95_

0.9_

0.85_

0.8

- I I I I I I I I I I I I I I- 0.31 - 0.3 - 0.29 - 0.28 -0.27

ment and control its optical performance on a regularbasis. For this application we recommend taking afew stellar exposures at 0.631 jim with the secondarymirror set at position -90 jim, conditions that gaveus the best results.

We have recently been able to apply the samemethods to a ground-based telescope by using shortexposures taken in the infrared (4 jlm). Seeingeffects were reduced to an acceptable level by averag-ing several reconstructed wave fronts.' 7

Appendix A

As discussed in Section 2, low-order aberration termswere estimated from a single image by comparingcomputed images with the observed image and search-ing for the minimum difference. A detailed descrip-tion of the procedure is given here together with adiscussion of the errors involved.

1. Effect of Telescope Jitter on the Wave-Front FittingProcedure

Three different norms were considered as a measureof the image difference. These are labeled N1, N2,and N3:

N, = Y.II - S,

N 2 = [Y(I - S)2]112

N3 = ( -) ,where I is the observed image and S is the computedimage. Figure 20 shows contour plots for N,, which

1, I -l I ; I # I {_

0.95

0.9

0.85

I {, fI, I, I R , ,-0.3 -0.29 -0.28 -0.27

71 I

0.95

0.9

0.8

-0.29 -0.28 - 0.27 -0.26Fig. ZQ. Contour plot§ Showingthe1 difference VI between the observed and the computed images as a function of Z,, and Z4, with (left) andwithout (right) taking telescope jitter into account. The top row shows simulations with a black dot at the correct values.


_l I I I I I I I I I I I I I I I I I I-

- -ccw

I I

-3-.29 _A-2

-- -28 -27

-.3 -Xs -. 2 -27

-2.3

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-2.3

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-La

-2.26

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-La

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-2.36

I xv

-.3 -29 -2 -27

I, ~ _

X _

-.3 -29 -. 2a -27

-. 3 -.29 -.2a -27-.3 -29 -.28 -. 27

Fig. 21. Zll and Z4 estimated at focus position -260 with norms N, (triangles), N2 (stars), and N3 (crosses) for various levels of blur in theobserved and computed images (see text). The black dot shows the exact Z 11 and Z4 values for simulations (left two columns).

is the norm mainly used throughout this work.Norm N, is plotted as a function of Z1, (horizontalaxis) and Z 4 (vertical axis) expressed in micrometersrms. The contours are for 0.1%, 0.5%, 1%, 2%, and3% excess over the minimum. Figure 20 demon-strates the effect of telescope jitter on the fittingprocedure. The upper two plots show the result of acomputer simulation. A defocused image was simu-lated with Z4 = 0.9 jim and Z1 = -0.29 jim. Theimage was then blurred by taking a two-dimensionalrunning mean over 2 x 2 pixel squares to simulatethe effect of telescope jitter. The upper right plotshows N, when we attempted to fit computed imagesto this blurred image. First, the minimum is shiftedby introducing systematic errors in the parameterestimates. Second, the minimum is shallow and thecontours are elongated in a direction different fromthe axes, producing random correlated errors. Theupper left plot shows N, when the computed imagesare similarly blurred before comparison. The mini-mum occurs close to the correct values of the parame-ters. It is much deeper and the errors are uncorre-lated. The lower two plots show N when weattempted to fit a real image (in this case w29938).In the lower right plot no attempt was made to taketelescope jitter into account. In the lower left plotcomputed images were blurred before comparisonwith the observed images. A similar effect is ob-served on the contour plots. Although in this casewe do not know the exact values of the parameters,

the coordinates of the minimum in the left plot areclearly more reliable.

We now discuss the effect of the norm and theestimation of telescope jitter on the error of thewave-front parameter estimates.

2. Estimation of the Fitting Error

Figures 21-23 are similar to Fig. 20, but we haveplotted only the value of the minimum for the differ-ent norms N, (triangles), N2 (stars), and N3 (crosses).The first two columns represent the results of simula-tions and the black dot indicates the correct value ofthe parameters. The rectangule is the estimatederror box for the fitting procedure. The last columncorresponds to real images and the correct value isunknown. Different figures are for different focuspositions: -260 (Fig. 21), -90 (Fig. 22), and +250(Fig. 23). The real images are, respectively, w29906,w33410, and w29938.

Two different types of image blur have been used,either running means or filters that match observedimage blurs. Filters were obtained as follows. Theobserved image is Fourier transformed and the modu-lus of its Fourier transform is divided by the modulusof the Fourier transform of an unblurred simulationof the same image. Since the true aberrations areunknown, it is difficult to simulate an image withexactly the same aberrations. What we did is simu-late several images with the same pupil geometry andwith aberrations Z 4 and Z in the estimated range


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X

-. 3 _,29 _,8 -_

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Fig. 22. Z and Z 4 estimated at focus positionobserved and computed images (see text). The

-1.15

-1.2

-aL

-1.3

-1.15

-1.2

-Las

-1.3-. 3 -. 29 -.2 -27 -. 3 -is -28 -27

-90 with norms N, (triangles), N2 (stars), and N3 (crosses) for various levels of blur in theblack dot shows the exact Z1 1 and Z 4 values for simulations (left two columns).

-.3 -s

-.3 -is -.28 -27

.. . . . . . . . ..1 . .1 . . . .

.9

.8

.9

.36.3

.06.9

.8

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.a

.96.a

.36.a

I. I . .. ---. 3 _,;" -. 8 -_

-.3 -29 -.28 -27

-. 3 -29 -.28

-. 3 -is -.28 -27

Fig. 23. Zll and Z4 estimated at focus position +250 with norms N, (triangles), N2 (stars), and N3 (crosses) for various levels of blur in theobserved and computed images (see text). The black dot shows the exact Z,, and Z4 values for simulations (left two columns).


-1.16

-1.2

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-. 3 - 2 -. -27

II - -- - --- -- - -- - - -z W B §

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7

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(±0.05 for Z4, 0.28-0.3 for Z11). The result of thedivision was then smoothed with a running meanover 10 x 10 pixels. All the results looked similar.We averaged them and took the inverse Fouriertransform to produce our estimated blur function.Phases were discarded because image blur essentiallyaffects the amplitudes, and the phase is not suffi-ciently accurately reproduced in the simulation.

In Figs. 21-23, the top row shows the results thatwere obtained when image blur is properly taken intoaccount. In the left plot unblurred simulated im-ages were matched with unblurred estimates. In thecenter plot simulated images blurred with a filterwere matched with estimated images blurred withthe same filter. In the right plot observed imageswere matched with computed images blurred withthe filter estimated from the same image. The nextthree rows are a double-input array. Different col-umns correspond to differently blurred input images.The left column is for simulated images with a 2 x 2pixel blur. The center column is for simulated im-ages blurred with the estimated filter. The rightcolumn is for the observed images. Different rowscorrespond to a different amount of blur applied tothe computer-estimated images. In the first row noblur was applied; a 2 x 2 pixel blur was applied in thesecond row; a 3 x 3 pixel blur was applied in the thirdrow.

3. Error Minimization

From observation of Figs. 21-23, the following conclu-sions can be drawn. The upper left plot is a measureof the intrinsic accuracy of the wave-front fittingprocedure. The error box is an estimate of thisaccuracy. The three other plots in the left columndescribe attempts to fit images blurred by a 2 x 2pixel running mean with images unblurred (firstrow), blurred by a 2 x 2 pixel (second row) or 3 x 3pixel (third row) running mean. This demonstratesthat

(1) Images blurred at the same level (second row)produce the smallest dispersion in the three figures.In addition, unblurred estimated images (first row)produce larger errors than overblurred estimates(third row). It is therefore essential to blur thecomputed images before comparison with observedimages. Overblurring is preferable to underblur-ring.

(2) Norm N2 (stars in the figures) is the poorestnorm when we deal with unmatched blurs. This isexpected because it emphasizes the importance oflarge differences such as those produced by differentblurs.

(3) The effect of telescope jitter is less severe onthe images taken inside focus (Figs. 21 and 22)compared with images taken outside focus (Fig. 23).This is also expected because images taken in focusare larger and display larger interference patterns(the distance between interfering rays is smaller onthe telescope pupil). For outside-focus position + 250(Fig. 23), note the excessively small spherical aberra-

tion and the excessively high defocus estimate givenby all the norms when telescope jitter is not takeninto account (upper of the left three plots). Clearly,image blur is matched by an excess of defocus, whichis in turn balanced by an underestimated sphericalaberration.

The four plots in the second column in Figs. 21-23confirm these results. The plot in the isolated upperrow shows results that were obtained when thesimulated image and the computer estimates areblurred by the same filter. The dispersion betweenthe values at the three focus positions is small. Inthe three lower plots the dispersion becomes larger.Poor aberration estimates are obtained at focus posi-tion +250, especially when telescope jitter is nottaken into account (upper of the central three plots inFig. 23). Again spherical aberration is underesti-mated with all norms. Norm N2 is the poorest;norms N, and N3 appear equivalent.

The last column in Figs. 21-23 shows results thatwere obtained on real images: w29906 taken atfocus position -260 (Fig. 21), w33410 taken at focusposition -90 (Fig. 22), and w29938 taken at focusposition +250 (Fig. 23). Although in this case thereal value of the aberrations is unknown, similareffects can be seen. The simulation results help usto understand these results and make better decisions.They tell us we should discard results that wereobtained with norm N2 (stars). We should alsodiscard results that were obtained without takingtelescope jitter into account (upper of the three rows),which, as expected, tend to underestimate sphericalaberration, especially out of focus (Fig. 23). For theinside-focus images, it is easy to give an estimate bytaking into account the three other plots and the tworemaining norms because the values are actuallyquite close. For the outside-focus image it is moredifficult, and our estimate (Z4 = 0.83, Z1 = -0.292)is considered unreliable (no asterisk in Table 1). Forimage w29906, we obtained the estimates given inTable 1 (Z4 = -2.27, Z1 = -0.278) with the originalMisell algorithm; these were also considered unreli-able. From the plots in Fig. 21 we see that Z 4 =

-2.27 is probably a fair estimate, but Z1 = -0.278 isclearly an underestimate in agreement with valuesthat were obtained when telescope jitter was nottaken into account (upper right plot of the 3 x 3 arrayin Fig. 21). From the right plot in the upper isolatedrow in Fig. 21, a more reliable estimate would be- 0.285, which is closer to our reliable values.

4. Refinements

Once defocus and spherical aberration are estimated,a similar procedure can be used to refine our wave-front tilt estimate and determine the image center asa fraction of pixel. These estimated low-orderZernike aberration terms determine the first inputwave front in our Gerchberg-Saxton iteration loop.After 50 iterations small-scale wave-front aberrationsappear, but, in general, the low-order Zernike terms


remain within the uncertainty in our first estimate.Only on a few occasions, when the sharpest imageswere processed, did we decide to change our firstestimate and start new iterations until the Zerniketerms in the output wave front agreed with those inthe input wave front.

This work was supported by the Jet PropulsionLaboratory under contract 958893.

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