Prescription retrieval is a technique for directly estimating optical prescription parameters fromimages. We apply it to estimate the value of the Hubble Space Telescope primary mirror conicconstant. Our results agree with other studies that examined primary-mirror test fixtures andresults. In addition they show that small aberrations exist on the planetary-camera repeater optics.
Introduction
Initial images from the wide field planetary camera(WFPC) and the faint-object camera indicated thatthe primary mirror of the Hubble Space Telescope(HST) was figured incorrectly. Two sets of studiesinvolving numerous investigators were performed todetermine the primary-mirror conic-constant value.They derived apparently inconsistent results. Thearchival data studies examined the test hardware andresults to determine problems caused by fabricationerrors, deriving values for a primary-mirror conicconstant of -1.01378 ± 0.00031. The phase-re-trieval studies2 3 processed images to determine exit-pupil phases by using a variety of techniques. Weanalyzed the converged phase solution to determinethe degree of spherical aberration. The conic con-stant is then estimated on the critical assumptionthat the spherical aberration is due to only theprimary mirror. The resulting estimate of the conicconstant (averaged among seven different study re-sults) is - 1.0148 ± 0.00038.
We employed a third approach, which Si Brewer(PAGOS Corporation, San Simeon, Calif.) called pre-scription retrieval. Like phase retrieval, prescrip-tion retrieval is an image-inversion technique, whereparameters are varied in a mathematical model of theoptical system until that model produces images thatmatch the actual images. It differs from phaseretrieval in the nature of the model: prescriptionretrieval uses a full ray trace and diffraction modelrather than a single-plane exit-pupil phase model.The model is parameterized with prescription vari-
All authors are with the Jet Propulsion Laboratory, CaliforniaInstitute of Technology, Pasadena, California 91109. D. Reddingis with the Charles Stark Draper Laboratory.
ables such as the conic constants and alignments ofthe 13 optical elements, which enable the directestimation of these parameters from the images.
We estimate the primary mirror conic constant tobe -1.01386 ± 0.0002 based on a limited set ofimages taken with planetary camera 6 (PC6). Thisresult is in agreement with the archival-data studiesbut not with the phase-retrieval studies. Our re-sults also show that significant aberrations exist inthe relay-camera primary and secondary mirrors.These changes impact the pupil phase map. Esti-mates of other key design parameters are listedbelow.
To compare our results with the phase-retrievalstudies, we computed the total spherical aberrationz11 in the exit pupil based on the converged prescrip-tion results for a particular image set. The resultingz11 is -0.2943 Am rms. This agrees with an averagevalue of spherical aberration of -0.2906 ± 0.0061computed by the Shao group.3 The phase derivedfrom our converged prescription agrees with thephase-retrieval code to within a 0.0002 level of theequivalent primary-mirror conic constant.
Our conclusion is that the archival data studiesderived the correct conic constant. We believe thatthe phase-retrieval studies correctly determined theexit-pupil phase map but that the key assumptionthat the spherical aberration component was due toonly the primary mirror is in error. Estimates of theconic constant derived on that basis are in error.Our study was not completed. Further work wouldimprove the estimates of the primary-mirror conicconstant and other parameters.
Prescription Retrieval
Problem Statement
Given a set of images and a good initial estimate of theoptical prescription of the HST/VVFPC system, onecan determine the HST primary-mirror conic con-
stant to an accuracy of 0.0005. This level ofaccuracy is required for the design of the replacementWFPC2, which is to be flown in 1993.
Solution Approach
Our prescription-retrieval approach is a model-matching technique based on a highly detailed com-puter model of the HST/WFPC system. This modelgenerates images based directly on the optical designvariables of the HST/WFPC6 system. The solutioncode iterates these parameters, i.e., the optical pre-scription, until the model accurately predicts theobserved images (Fig. 1). The computer model usesa hybrid ray trace and diffraction approach. The raytrace is the only way to determine accurately thesystematic aberrations induced by field position orelement misalignments. The diffraction producesthe actual images. The model includes the effects ofdetector sampling, detector noise, and spacecraftjitter.
Perturbations of the elements that induce aberra-tions (e.g., secondary-mirror misalignment producingcoma and astigmatism) are determined explicitly soas to separate them from the effects of the primary-mirror figure error. The starting point for our solu-tions was provided by the Code V as-built prescriptiondetermined for the on-orbit system by the HubbleAberration Recovery Program (HARP). This pre-scription uses element parameters taken from piece-part drawings. We derived element positions froman alignment procedure, as simulated using Code V.The primary conic was set by early results of thearchival and phase-retrieval studies.
We coded the model by using the SubroutineControlled Optics Modeling Package (SCOMP) de-scribed below. A total of 65,000 rays are tracedthrough the HST/PC6 system for each 512 x 512image. The diffracted point-source images are com-puted by SCOMP based on phase and obscurationinformation derived from the ray trace.
The solution code that manipulates the model tomatch the images is based on the Levenberg-Marquardt implementation of a nonlinear least-squares (NLLS) algorithm. NLLS finds the best fitbetween simulated images produced by a parameter-ized model of the optical system and on-orbit images.The best fit is defined in the least-squares sense byminimizing the sum of the square of pixel differencesbetween simulated and on-orbit images.
Optical Model of the HST/WFPC
HST/WFPC6 Optical System
The images that we used were taken specifically fordiagnostic purposes as part of HARP. They weretaken on both sides of focus (the focus setting isaltered by the piston motion of the secondary mirror)and in several different field positions to expose imagefeatures such as the secondary-mirror spiders, theprimary-mirror pads, the repeater-camera spiders,and the spherical aberration rings. Images weretaken in several of the WFPC's, but those taken withPC6 were the clearest. We used several PC6 images.
The basic optical prescription we used as a startingpoint for our solutions was derived from the as-flownprescription developed for the HARP.4 TheHST/WFPC6 beam train is sketched in Fig. 2. Itconsists of 13 optical elements modeled by using 16surfaces as listed in Table 1.
The main system stop is at the primary mirror.The light beams from objects at different field pointsare represented as collimated bundles of rays (planarwave fronts) centered on the primary mirror. Thebeam is obscured at the secondary-mirror spiders, theprimary-mirror pads, and at the relay secondary-mirror spiders, which are three-sided. These obscu-rations are illustrated in the pupil intensity mapsgiven below (Figs. 3-6). The aberrated HST/WFPC6system exit pupil is located 66 mm in front of theCCD detector.
The HARP images that we used for our solutionswere taken at three wavelengths: 487, 631, and 889nm. The images were provided as cropped 256 x256 sections of the full 800 x 800 CCD field.
The size of the image pixels (15 plm) plus the needto sample the aberrated wave front adequately at theexit pupil set the optical model code size parameters.The final far-field diffraction propagation is requiredto scale to a wave-front grid size that is smaller than
the actual pixel size. This plus the need to fill the256 x 256 pixel array led to a requirement that theoptical model use a 512 x 512 diffraction grid.Experiments with the guard bands used to avoidaliasing in the diffraction calculations showed thatthe best compromise between aliasing and samplingrequirements left a central 283 x 283 for the beam.The optical modeling code was therefore set up totrace 66,000 rays to determine beam phases for the512 x 512 diffraction calculations.
Controlled Optics Modeling Package
For prescription retrieval the optical modeling code iscalled many times to generate images and pixel-by-pixel numerical derivatives of images. This, plus ourdesire that the code run on desktop work stations,imposes a requirement that the modeling code be ascomputationally efficient as possible.
Fig. 4. Intensity at PC6 repeater spiders.
We used the controlled optics modeling package(COMP)5,6 to provide the optical model. COMP is astand-alone optical modeling computer code whosecapabilities include geometric optics, physical opticsincluding multiplane diffraction, image simulation,differential ray tracing, sensitivity analysis, and lin-ear optical model building. We used an in-line sub-routine version (SCOMP) to generate images for ourparameter solver program.
COMP provides a computationally efficient generalray-trace capability through the use of a set ofcoordinate-free ray-trace equations.7 The computa-
Fig. 3. Intensity at the HST secondary mirror. Fig. 5. Intensity at the PC6 exit pupil.
Image intensity at the PC6 detector (five-plane model, no
tional efficiency results from working in a singlecoordinate system, elimination of transcendental func-tions, elimination of extraneous surfaces, and anoptimized code.
To generate images COMP defines a mapping froma bundle of rays to pixels in a complex amplitudematrix that represents the light beam. The propaga-tion of the complex amplitude is driven by the exactphases determined from the ray trace. This ap-proach correctly computes induced aberrations andpupil obscurations. COMP allows the user to specifythe type of diffraction propagation that is to be usedat each point in the beam train. It provides Fresnelnear-field and far-field propagators as well as a geomet-ric propagator, which updates phase only based onthe optical path length and is used in areas where theFresnel assumptions are violated. The user sets upreference surfaces to define different propagationregions. The user can define a full surface-to-surface multiplane diffraction model or a single-planeexit-pupil diffraction model to generate images.
Comparison of Single- and Multiple-Plane Models
We computed most of the results presented here byusing a single-plane model. The single-plane modelswere used mainly because we ran out of time beforecompletely implementing the multiplane version ofSCOMP. In this section we compare an image gener-ated by the single-plane model with one generated bya more rigorous multiplane model. The two imagesmatch quite closely. Nonetheless, as Fienup 2 showed,small differences resulting from diffraction effectscan have an appreciable effect on estimates of theconic constant.
The single-plane model uses a single far-field diffrac-tion calculation from the exit pupil to the detector.The multiple-plane model uses five planes of diffrac-tion: four near-field propagations in the beam trainand a final far-field propagation. The SCOMP com-putation sequence for each model is as follows (thisprocess corresponds to the Generate Images functionof Fig. 1):
(1) Apply element perturbations to the base pre-scription. Exact kinematical relations are used torotate and translate the optical elements and inputbeam, and deformations and other effects are appliedin response to prescription parameter changes deter-mined by the prescription-retrieval code.
(2) Compute a new exit pupil location.(3) (Multiplane model) compute new diffraction
reference surfaces.(4) Trace rays and propagate the complex ampli-
tude matrix.(5) Resample the image to the detector pixels and
add jitter. Noise is added for display purposes only.For images generated during the solution process, thenoise is used only in the calculation of the x2 fittingerror metric.
Figures 3-6 illustrate the result of this process fora particular image. They show the light intensitiesat three locations in the multiplane model, after eachobscuring surface and the image at the detector.The multiplane image is to be compared with thesingle-plane image of Fig. 7. These images werecreated by using a 1024 x 1024 version of COMP.(Only the central 512 x 512 is displayed.) No noiseor jitter is added.
Fig. 7. Image intensity at the PC6 detector (one-plane model, nonoise).
Noise ModelsThe HARP images are corrupted by noise and jitter.We explicitly account for these effects so as to mini-mize the distortion of image features. The threephenomena included in our noise model are thedetector sampling, telescope pointingjitter, and back-ground noise.
The on-orbit images are recorded with a CCDdetector. The actual CCD detectors have a fixedpixel size (15.24 mm and square), which in general isnot equal to the SCOMP complex amplitude meshsize. There is also an offset between the centers ofthe wave-front grid and the detector pixel grid.These effects are accounted for by resampling thecomputed intensity pattern onto a physically correctpixel grid. The centroid of the image relative to thepixel grid is optimized by the parameter solver.
Jitter, or the random or quasirandom vibration ofthe Hubble telescope during an exposure, results insmearing the detected image with a consequent lossof high-spatial-frequency information. If the tempo-ral frequency of the jitter is high compared with theimage sampling/integration rate, we can simulate thesmearing using convolution methods. This ap-proach assumes that the jitter pointing error is azero-mean Gaussian random process.
We used this approach by computing the standarddeviation for the probability density function (PDF)of the jitter distribution from fine guidance sensordata. The two-dimensional function used to repre-sent the jitter PDF is
P(x,y) = exp[y2 xy
The jittered image is the convolution of the jitter PDFwith the instantaneous point spread function pro-duced by the optical system.
The background noise includes effects associatedwith image preprocessing (variances in the flat-fieldcorrection and the UV flood illumination of the CCD),the CCD read noise, and the Poisson noise of thedetected signal in each pixel from the star. Thenoise in each pixel is estimated from
s(DN) = [(DN + 2P + 2R)
+ var(F)(DN2 + DN + 2P + 2R)]0.5, (2)
where DN is the signal count in each CCD pixel, P isthe flood illumination value, R is the CCD read noisevalue, and F is the flat-field factor. This noise termis not added directly to the images. Rather it is usedto set the uncertainty values used to compute the x2
fitting error metric.
Nonlinear Least-Squares Parameter Solver
Algorithm
Images are nonlinear functions of perturbations tothe optical prescription, even though the complexamplitudes from which the images are computed are
nearly linear functions of those perturbations. Thisis because the intensity is the modulus squared of theamplitude, and all phase information is lost in thesquaring. As a result an iterative algorithm is re-quired to determine prescription parameters fromintensity data.
For prescription retrieval we use a NLLS approachbased on the Levenberg-Marquardt algorithm.3 8
NLLS, like linear least squares, is based on themaximum-likelihood hypothesis, which states thatthe most likely values of a set of parameters describ-ing a system (in our case the HST/WFPC opticalsystem) are those that minimize the weighted squareddifferences in the computed and actual images. Theweighting is supplied by the variance of the signal ineach datum. The metric to be minimized is the x2:
X 2 = 1 yi - y modi] (3)
Here yi is the intensity value of the ith real imagepixel;y mod, is the intensity value of the ith simulatedimage pixel (computed by SCOMP), a function of thecurrent parameter vector a; mr is the estimate of thevariance in the ith image pixel based on the noisecalculations described above; and N is the totalnumber of image pixels (N = number of images xnumber of pixels/image).
The NLLS algorithm assumes that the problem inthe neighborhood of the current parameter valuestakes a quadratic form.8 The partials ofx2 are takenwith respect to each parameter to set up a system oflinear equations, which is solved for corrections to thecurrent parameter vector estimate. In practice thisworks well if one is near the correct solution or if theproblem is fairly quadratic. The computation of thepartials of the x2 is extremely computationally inten-sive; it requires the regeneration of each image in thesolution set for each parameter being solved for.This occurs during the Generate Images and Correc-tions block of Fig. 2.
The Levenberg-Marquardt implementation ofNLLS introduces a scaling factor that changes thebehavior of the algorithm according to the localbehavior of the problem. If the quadratic solutionfails to improve the x2, the scaling factor is increased,which emphasizes the linear term in the estimationstep. The algorithm then behaves more as a steep-est descent algorithm does, which is a slower butmore robust approach. As the solution nears thecorrect answer, as indicated by decreasing x2, thisscaling term is decreased, so that the algebra yieldsquadratic minimum-seeking steps for the parametercorrection set. This process is sketched in Fig. 2 andreviewed in detail in Ref. 8.
Potential Problems
Several potential pitfalls must be avoided when aNLLS algorithm is used:
(1) Local minima. The NLLS algorithm cannotescape from a local minimum. The likelihood of
being trapped in a local minimum is reduced with asufficiently diverse data set and a good initial guess.
(2) Inseparable parameters. If two parametershave an identical (or proportionally similar) effect onthe images, the contribution of one cannot be distin-guished from that of the other. These effects arereduced when we look at different field angles andfocus positions.
(3) Inadequate model. Our problem is that notall prescription parameters can be varied at oncebecause of computer-size limits. Such variables asthe widths of obscurations were not solved for but canhave a significant effect on the image and especiallyon the value of the x2 metric. Similarly, unmodeledchanges in the actual HST/WFPC system that arenot common to all images in a set add confusion andreduce the quality of the solutions. This type ofproblem can occur if images are taken at differenttimes, during which the structure of the telescope haschanged because of thermal or other effects.
(4) Meaningless parameters. Care must be takennot to confuse important image structure with noiseor jitter effects. Solving for these parameters canproduce good x2 values, but since they carry noinformation about the optical system, they do notimprove the estimates of the prescription parameters.
Results
We obtained results for three sets of images. Thefirst set consists of two 487-nm wavelength HARPimages with serial numbers PC6f487n ol andPC6f487nmsl. These images are compared with thecomputer-generated images in Fig. 8-11. The esti-mated parameter values are given in Table 2. Theestimate of the primary-mirror conic K is - 1.0139888;the x2 associated with it is 876,534.
The second set consists of two 889-nm wavelengthHARP images, PC6f889n o1 and PC6f889nmql.
Fig. 8. Simulated image: PC6f487-ol.
Fig. 9. Actual image: PC6f487ol.
These images and the computer-generated images arecompared in Figs. 12-15. The estimated parametervalues are given in Table 3. The estimate of Kc is- 1.0137195; the x2 associated with it is 1,316,730.
The third set consists of one 487-nm wavelengthimage (PC6f487n-ol) and one 889-nm image from theHARP set, w66407. The estimated parameter val-ues are given in Table 4. The estimate of Kc is-1.0137803; the x2 associated with it is 2,079,117.To produce a single number estimate of the primaryconic constant, we use an average weighted by the x2
Fig. 12. Simulated image: PC6f889-ol.Fig. 11. Actual image: PC6f487.s1.
values:
K= (Kj/X12 ) + (K2 /X22 ) + (K3 /Y32 )
1/X12 + l/X22 + /X32
The resulting estimate of K is - 1.0138608.
Discussion
How Good are These Results?
The values of x2 for these images are high comparedwith their ideal values. This indicates that thesolutions have not fully converged. A detailed exam-
ination of the images shows good overall agreementwith the actual images, but some details are notperfectly matched. The formal errors for primarymirror conic constant K are quite low (< 10-5 in eachcase). This is much smaller than the differencesseen when we compare the values of parameters fromone image set with those derived from another imageset. This also indicates incomplete convergence.
The low formal error is an indication that K can bemeaningfully separated from the other parameters bythe solution code. This might seem unexpected,since the conic constants of all the mirrors HST/
Table 2. Images Set 1 Estimated Parameters
Estimated Nominal EstimatedParameter Change Value Value
OTA primary- -0.4888453858-3 -1.0135 -1.0139888mirror conic
WFPC should affect the images in comparable waysin an aligned, on-axis system. We used images thatwere not taken at the same field point, and theprimary mirror aberrations indicate that the opticsdo not reimage each other. As a result conic changeson other optics do not have a symmetric effect and canbe distinguished from primary-mirror conic K.
How good our estimate is cannot be determinedfrom the formal error calculation until all parametersand images can be included in the same solution run.An approximate bound can be guessed based on thestandard deviation of the differences in estimatesbetween runs. This value is 0.0002.
Convergence can be improved in a number of ways:
(1) More of the prescription parameters can beadded to the solution list, which enables a better fitwhen the existing model is used.
(2) More images can be processed at the sametime.
(3) We can improve the model by incorporatingmore details, such as primary-mirror zonal aberra-tions that have been determined from interferometrictest data. Inclusion of the zones was found to reduceX2 by one-third in some of the phase-retrieval cases.3
(4) The multiplane diffraction model could beused.
(5) Improved initial guesses could be derived frommore recent archival data analyses.
(6) To avoid local minima, one can employ an-other solution algorithm such as simulated annealing.
Comparison with the Archival Data Studies
Our estimate of the HST primary-mirror conic con-stant is -1.01386 ± 0.0002, which agrees with thevalue determined by the archival data studies. Eachof the solved values for K falls within one standarddeviation of the archival estimates.
Comparison with the Phase-Retrieval Studies
Our estimate of the conic constant does not agreewith estimates derived from the results of the phase-retrieval studies, even though the images we generatecompare well (qualitatively) with images generated bythe phase-retrieval studies. To investigate this fur-ther we picked as a sample case the first 487-nmimage set as described above (Figs. 8 and 9). Wesolved for the spherical aberration component of thepupil phase by using the converged prescription.That value was -0.2943,um rms. This result agreeswith the value for spherical aberration of -0.2906 ±0.0061 that we derived using the Shao group's phase-retrieval code.3
If we compute the conic constant by using theassumption that the spherical aberration is due solelyto the primary-mirror conic-constant error, we obtainan estimate for K of -1.01520. Again this agreeswith the estimate for K of - 1.01504 ± 0.00027 thatwe derived from the phase-retrieval results using thesame assumption. It disagrees with the value of- 1.0139888 estimated directly (Table 2).
The difference lies in the other elements of thebeam train. The prescription retrieval results indi-cate that aberrations that contribute to the totalexit-pupil spherical aberration exist not only on theprimary mirror but on the relay-camera primary andsecondary mirrors as well. This contradicts theassumption that we used in converting the phase tothe estimated conic constant.
To sum up, the phase-retrieval studies seem tohave correctly determined the exit pupil phase. The
Fig. 15. Actual image: PC6f889.ql.
Table 4. Image Set 3 Estimated Parameters
Estimated Nominal EstimatedParameter Change Value Value
formula used to convert the phase to the primary-mirror conic constant is in error, however, causingthe phase-retrieval-based estimates of the conic con-stant to be in error.
Using Prescription Retrieval for Optical SystemAlignment
We have investigated prescription retrieval for use inaligning general optical systems. We ran a series ofsimulated tests on a sample system that consisted ofthe relay barrel for one of the WFPCI's. Using a setof seven images taken at four field points and twofocus positions, we can recover all 10 degrees offreedom of the two-mirror system. We terminatedthis activity before addressing some key issues, suchas detector-noise effects. Nonetheless indicationsare that it can provide a powerful alternative toconventional interferometric alignment techniques.
The advantages of prescription-retrieval image-based alignment include the following:
(1) It does not require null lenses, interferome-ters, or other test optics to align aberrated or asphericsystems.
(2) As a result it is not susceptible to errorsintroduced by test optics.
(3) It is an end-to-end technique that is capable ofworking in the nominal operational configuration ofan instrument including using the science detector.
(4) It provides alignment over the full operationalrange of an instrument, i.e., over the full field of view,focus settings, and zoom positions.
(5) It can be performed remotely, such as duringenvironmental testing of an instrument or duringoperation of a space instrument.
Conclusion
By using a direct prescription-retrieval image-inver-sion technique, we have estimated the value of theHST conic constant to be -1.01386 ± 0.0002. Thisresult agrees with estimates derived from archivaldata studies. In addition we determined that therelay cameras are slightly aberrated. These aberra-tions account for discrepancies between the archival
data studies and other studies that used phase-retrieval techniques.
We thank the other members of the Jet PropulsionLaboratory image-inversion team, Mike Shao, MartyLevine, Mark Colavita, Brad Hines, Rich Dekany, andTony Decou, for their contributions of code and ideas.Others at the Jet Propulsion Laboratory, notably BobKorechoff and Jim McGuire, were extremely helpfulin this work. Si Brewer was extremely generouswith his knowledge and insights. This work wasperformed under contract with NASA.
References
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