Vol. 7, No. 7/July 1990/J. Opt. Soc. Am. A 1237

High-resolution imaging using pupil segmentation

J. Sebag and J. Arnaud

CFHT Corporation, P.O. Box 1597, Kamuela, Hawaii 96743

G. Lelievre

Observatoire de Paris, Unit6 Associ6e Centre National de la Recherche Scientifique 335, 61 avenue del'Observatoire, 75014 Paris, France

J. L. Nieto

Observatoire Midi-Pyr6n6es, Unitb Associ6e Centre National de la Recherche Scientifique 205, 14 avenueEdouard Belin, 31400 Toulouse, France

E. Le Coarer

Observatoire de Marseille, 2 Place Le Verrier, 13012 Marseille, France

Received August 24, 1989; accepted February 2, 1990

Recentering and selecting short-exposure images result in significant improvements in spatial resolution comparedwith that for classical long-exposure images. A maximum gain in resolution of the order of 3 is possible for a Dirobetween 3 and 6, depending on the selection rates. A pupil-segmentation experiment has been performed at theCassegrain focus of the Canada-France-Hawaii telescope to match the optimum values of Diro. A photon-countingcamera records short-exposure images. The software processing is made after acquisition of the data. Recenteringis made by cross correlation of the short-exposure images with a long-exposure image of a star or a contrasted object.We present results obtained on the gravitational lens Q2237+030. A full width at half-maximum of 0.29 arcsec hasbeen achieved with this method. The gain in resolution is approximately 2.3 compared with that for classicalimaging. However, we need to count enough photons for reliable recentering; this prevents us from achieving thetheoretical gain in resolution. We expect to overcome this limitation in the case of crowded fields or with the use ofnew object-reconstruction methods.

1. INTRODUCTION

The mean value of the full width at half-maximum (FWHM)observed at the Canada-France-Hawaii telescope (CFHT)is 0.85 arcsec. This value is due to atmospheric turbulenceand telescope limitations (guiding errors, optical aberrationsetc.). Technical improvements are conducted to reduce theinfluence of local effects. Moreover, scientific experimentsare developed to obtain high-resolution images. Real-timedevices such as the high-resolution camera ' 2 consist of animage motion compensator and a fast shutter to record themoments of best image quality on a charge-coupled device.

An experiment called segmented pupil imagery (SPI) hasbeen designed as a part of CFHT efforts toward high resolu-tion for ground-based imaging for a large field (-40 arcsec).It relies on a spatial segmentation of the pupil and a tempo-ral recentering and selection of short-exposure images byusing off-line software processing. 3 From theoretical con-siderations, 4' 5 we expect a maximum gain in resolution for atelescope diameter D of 3-6 ro, where ro is the Fried parame-ter.6

The principle of the experiment is presented in Section 2.We describe in Section 3 the instrument built for the f/8Cassegrain focus at the CFHT. In Section 4 we discuss thereduction procedures, which are illustrated in Section 5 withan application to the gravitational lens Q2237+030.7-9 Fi-

nally, we discuss the limitations and the improvements thatcan be achieved.

2. PUPIL-SEGMENTATION PRINCIPLE

The degradation of the resolution for an astronomical long-exposure image comes from image motions and blurringintroduced by the atmospheric turbulence and telescopedefects.

The image motions correspond to the random variationsof the tilt of the wave front. During the recording of a long-exposure image, the image is spread by random motion. Onthe contrary, a short-exposure image will be insensitive tothe tilt if the images are recorded at a high enough frequen-cy. Fried6 first computed the improved resolution after theremoval of image motions (Table 1). The maximum angu-lar-resolution improvement is of the order of a factor of 2and occurs for Diro - 3.5.

The blurring corresponds to a spread of the image and isrelated to the distortion of the wave front. The probabilityof getting a good short-exposure image has been theoretical-ly investigated by Fried.4 He obtained the expression Prob-5.6 exp[-0.1557(D/ro)2 ] forD/ro 2 3.5. For example, if D/ro = 4, there are 46 good images in 100. Fried defined a goodimage as one that is formed when the wave-front distortionover the aperture is less than 1 rad rms. Hecquet and

0740-3232/90/071237-06$02.00 © 1990 Optical Society of America

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1238 J. Opt. Soc. Am. A/Vol. 7, No. 7/July 1990

Table 1. Dependence of the Theoretical Gain inAngular Resolution as Defined by Fried6 with the

Ratio D/roa

Gain inD/ro Resolution

2 1.843 2.043.5 2.053.8 2.044 2.025 1.917 1.66

10 1.48

a After Table 1 of Ref. 6.

Table 2. Dependence of the Theoretical Gain inAngular Resolution with the Ratio Diro for

Recentered and Selected Short-Exposure ImageaD/ro No Selection 10% Selection Rate

3 2 2.74 2 3.15 1.9 3.17 1.7 2.8

10 1.5 2.4

a After Table 5 of Ref. 5; the maximum gain in resolution occurs for valuesof D/r,, increasing with the selection rate.

Coupinot5 pointed out that the equivalent resolution is ap-proximately 1.7 times the diffraction limit resolution for aFried's good image. They computed the improved angularresolution for different selection rates of short-exposure im-ages and different values of Diro (Table 2).

Following this approach, we obtain high-resolution imagesfor solar observations by selecting the best short-exposureimages.'0 For faint objects, composite images are generatedby adding many recentered and selected frames. The reso-lution of a composite long-exposure image without selectionis the mean resolution reached by 76% of the short-exposureimages. The maximum gain is then 2.05 for Diro - 3.5.Combined with a 10% selection rate, this method shouldreach a gain of as much as 3.1 for Diro - 5.

At the CFHT, commonly observed values of Diro are gen-erally greater than 10, quite far from the optimum values.Thus the segmentation of the pupil into elements equivalentto smaller diameter telescopes, with more optimal Diro, willlead to the highest possible resolution gain by using thismethod.

3. SEGMENTED-PUPIL-IMAGERY EXPERIMENT

A SPI experiment has been designed at the CFHT. After afeasibility experiment successfully tested at the f/20 coud6focus,3 a dedicated instrument was built for the ff8 Casse-grain focus.

A segmented mirror divided into eight subarrays makesthe segmentation (Fig. 1). The telescope pupil is reimagedon thii mirror, which provides one image from each of theeight segments (Fig. 2). Each segment of the mirror can be

tilted independently in order to permit fitting the eightimages on different detectors. This configuration is equiva-lent to eight telescopes of 1.14-m diameter. A plain mirrorcan also be used to obtain full-pupil images for estimatingthe resolution gain owing to segmentation.

Because of the optical aberrations of the telescope, thebest focus for each subpupil is obtained for slightly differentpositions of the camera lens. To overcome this limitation,which is important at high resolution, a segmented lens hasbeen built to permit independent focusing on the singledetector.

The detector used is a photon-counting camera calledCP40," having a 3024 X 2048 format, a logical pixel size of 13gm, and a time resolution of 20 msec. The necessity offitting eight images on the same detector limits the angularfield of approximately 45 X 25 arcsec for each image with fi18.5 output aperture. The pixel size is then 0.045 arcsec.This is a good sampling for the expected resolution between0.2 and 0.4 arcsec. A focal doubler can be used to extend thedynamics. A point source (not represented in Fig. 2) isavailable for assistance in rough focusing and aligning theexperiment.

4. DATA REDUCTION

Recentering and selection algorithms have been developedat the CFHT on a SUN 4/280 computer. Recentering ismade by cross correlation of short-exposure images with along-exposure image of a star or a contrasted object to re-duce the influence of the sky photons. This procedure isused for each of the eight subpupil images.

To obtain the first long-exposure image, we select a sub-raster centered on the object used for recentering. We com-pute the centroid of the photons present in this window anduse it to recenter the short-exposure images. Photons fromthe sky in this window limit the precision of this recentering.However, a higher resolution is obtained for this image thanfor the integrated long-exposure image.

Then, the cross-correlation method is used to reduce the

Fig. 1. Eight subaperture-segmented mirror. Each segment canbe tilted to fit the eight images on the detector.

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Vol. 7, No. 7/July 1990/J. Opt. Soc. Am. A 1239

density and

segmented

Collimator

Cassegrain focusf/a

photon-countingcamera

Fig. 2. Optical configuration of the SPI experiment.

influence of sky photons, as it will generally give a lowerweight to them than to a photon from the object. Themaximum of the cross correlation determines the shift vec-tor used for recentering. This process is iterated by usingthe resulting long-exposure image to correlate, and a fastconvergence is observed. Moreover, a selection procedurecan be applied depending on the value of the maximum ofthe cross correlation, which is higher when the exposure is

sharper.The smaller the short-exposure times are, the more effi-

ciently we can expect to correct the image motions. Howev-er, we need to have enough photons on each short exposurefor a reliable recentering. Trying different short-exposuretimes during the data-reduction process, we can get the bestcompromise between an efficient correction of image mo-tions and a precise recentering. In practice, this compro-

mise depends on the morphology of the object used, itscontrast against the sky, and its location in the field. More-over, the recentering procedure described above can beadapted, depending on the observed field.

Eventually, we sum all eight of the subpupil images.Thus we keep the full light-collecting power of the telescope.

5. ILLUSTRATION OF THE METHOD ONTHE GRAVITATIONAL LENS Q2237+030

To illustrate this technique, we used the field of the gravita-tional lens Q2237+030. Huchra et al.7 discovered severalquasi-stellar objects images in the center of a Zwicky galaxyand suggested a gravitational lens effect. This result wasconfirmed later by high-resolution images8 and spectroscop-ic studies. 9

0.33

0.22

0.11

0

0

-2.. .

0 200 400 600 0 200 400 600

(a) time (sec) (b) time (sec)

Fig. 3. Motions of the image with 1-sec short-exposure images: (a) high-frequency movements from the atmospheric turbulence, (b) low-

frequency image motions due to telescope guiding and instrument flexures. (b) Was obtained by averaging the eight subpupil image motions;

(a) was obtained by subtracting (b) from the image motion of the subpupil number 3.

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1240 J. Opt. Soc. Am. A/Vol. 7, No. 7/July 1990

(a)

(b) " _ \ / - -to a X

Fig. 4. Isocontours of two long-exposure images of the gravitational lens Q2237+030: (a) no-recentered-integrated image, (b) the recenteredimage. The field for both images is 3.2 arcsec.

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Vol. 7, No. 7/July 1990/J. Opt. Soc. Am. A 1241

The data were obtained at the Cassegrain focus of theCFHT on September 16, 1988, with the SPI experiment.We plot in Fig. 3 the evolution with time of the image motionfor short-exposure images of 1-sec integration time. Figure3(a) shows high-frequency image motions due to the atmo-spheric turbulence. Figure 3(b) displays the low-frequencydrift from the guiding of the telescope and from instrumen-tal flexures.

In Fig. 4 we present two contour maps of long-exposureimages (700 sec) of this gravitational lens in the R band (X0 =0.65 um) obtained with pupil segmentation. Figure 4(a)represents the no-recentered-integrated image and Fig. 4(b)the recentered one. The respective FWHM's are 0.51 and0.29 arcsec.

The data were processed by using 1-sec short-exposureimages containing approximately 20 photons from the im-ages of the four quasi-stellar objects surrounding the core ofthe Zwicky galaxy. The distance is 1.68 arcsec between thetwo brightest quasars and 0.55 arcsec between the galaxynucleus and the closest quasar image. Such high-resolutionimages permit a better knowledge of the geometry of thegravitational system, which gives new constraints for thetheoretical models.

In this image, the object itself has been used for recenter-ing (there is no other suitable object in the field to be usedfor reference). The magnitudes of the quasar images are17.6 for the brightest images to 18.4 for the fainter imagesthat correspond, for the recentering, to a total magnitude of16.5.

6. DISCUSSION

This technique achieves gains in resolution of the order of 2.Besides reaching optimal values of the Diro ratio for recen-tering of short exposures, other gains are achieved by themethod:

(1) A better optical quality of each subpupil comparedwith the whole mirror

(2) A correction of telescope guiding imperfections andinstrument flexures by recentering.

The optical aberrations of the telescope have been foundto be of the order of 0.4 arcsec.12 That means that the imagequality of a full-pupil long exposure should have been ap-proximately 0.66 arcsec (in the same conditions as the obser-vation of Fig. 4), assuming that optical aberrations becomeof the order of 0.1 arcsec for subpupils images. So, in thiscase, the actual gain in resolution obtained by SPI, com-pared to a classical long exposure, is 2.3.

Image motion curves (Fig. 3) show a drift of the imageduring the 10-min exposure (0.14 arcsec in declination and0.16 arcsec total). It is due to instrumental flexures. Theimage elongation due to this drift increases the meanFWHM of the nonrecentered long exposure by 0.08 arcsec.Taking this into account and deconvoluting from spreadsdue to guiding errors and optical aberrations (Table 3), wefind an atmospherical seeing contribution of 0.43 arcsec onthe long-exposure image. We estimate that the remainingspreads due to optics and guiding are approximately 0.1 arc-

Table 3. Budget of the Image Quality for Q2237+030a

Flexures Guiding Optics Atmosphere Total

Full-pupil 0.08 0.05 0.4 0.43 0.66long exposure

Segmented-pupil 0.1 0.27 0.29recentered longexposure

I The image degradation is divided among optics deformations, instrumen-tal flexures, guiding errors, and atmospheric turbulence. The full-pupil longexposure corresponds to the image obtained with the plain mirror. All thevalues are in seconds of arc.

sec on the recentered image, yielding an atmospheric contri-bution of 0.27 arcsec. This corresponds to an observationalgain on the atmosphere of 1.6.

The long-exposure FWHM (in radians) equals 0.987X/ro(this can be computed from Ref. 13). So the ro value is 31 cmfor the discussed observation. For the corresponding ratioDiro = 3.7 the theoretical gain in resolution is 2.04 (Table 1).As we said above, the necessity to use short-exposure imagesof the order of 1 sec limits the gain on the atmosphere, thistime being too long to permit efficient correction of theimage motions due to the turbulence.

The efficiency of the selection is also reduced because wehave fewer image-quality variations among these 1-sec shortexposures than for shorter exposures due to integration overimportant variations. Actually, for Q2237+030 we did notthrow out any image because the small gain in resolutionobtained with selection did not compensate for the loss insignal-to-noise ratio, owing to the relatively poor signal-to-noise ratio of the individual images.

In the case of crowded astronomical fields (for exampleglobular clusters), a straightforward improvement could beexpected with the use of many objects for recentering. Oth-erwise, object-reconstruction techniques 14 using short-expo-sure images, with many fewer photons than those used atpresent, could permit a further significant improvement inspatial resolution with segmented pupil imagery.

ACKNOWLEDGMENTS

We are indebted to A. Blazit and R. Foy for providing andoperating the CP40 during those observations. It is a plea-sure to thank the Director of CFHT, R. McLaren, for allocat-ing the discretionary time during which these observationswere made. Sincere thanks also go to J. Brewster, whoprovided assistance in transferring the data from CP40 car-tridges. This paper is based on observations obtained withthe CFHT operated by the National Research Council ofCanada, the Centre National de la Recherche Scientifiquede France, and the University of Hawaii.

J. Sebag is also with the Observatoire de Paris, Unit6Associ6e Centre National de la Recherche Scientifique 335,61 avenue de l'Observatoire, 75014 Paris, France.

J. Arnaud is also with the Observatoire Midi-Pyr6n6es,Unit6 Associee Centre National de la Recherche Scientifi-que 205, 14 avenue Edouard Belin, 31400 Toulouse, France.

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