July 1983 / Vol. 8, No. 7 / OPTICS LETTERS 389

Image reconstruction by the speckle-masking method

G. Weigelt and B. Wirnitzer

Physikalisches Institut der Universitdt, Erwin-Rommel-Strasse 1, 8520 Erlangen, Federal Republic of Germany

Received February 10, 1983

Speckle masking is a method for reconstructing high-resolution images of general astronomical objects from stellar

speckle interferograms. In speckle masking no unresolvable star is required within the isoplanatic patch of the

object. We present digital applications of speckle masking to close spectroscopic double stars. The speckle inter-

ferograms were recorded with the European Southern Observatory's 3.6-m telescope. Diffraction-limited resolu-

tion (0.03 arc see) was achieved, which is about 30 times higher than the resolution of conventional astrophotog-

raphy.

The turbulent atmosphere of the Earth limits the res-olution of conventional astrophotography to about 1 arcsec. This resolution is much lower than the theoreticaldiffraction limit of large telescopes. For example, thediffraction limit of a 3.6-m telescope is 0.03 arc see at X= 500 nm. This high diffraction-limited resolution canbe obtained by Labeyrie's famous speckle interferom-etry,' which yields the diffraction-limited object auto-correlation. During the last few years various speckletechniques have been developed that yield true imageswith diffraction-limited resolution 2 -18 in spite of imagedegradation by the atmosphere. One of these methodsis the speckle-masking method. 8 The speckle methodsdescribed in Refs. 1-18 are different. For example,shift-and-add1 0 is first-order interferometry, whereasthe Fienup method' is a phase-retrieval method thatis applied directly to the power spectrum of the object.Comparison of these methods is beyond the scope of thisLetter.

Speckle masking is a correlation of third order. Ityields the modulus and the phase of the object Fouriertransform for general objects. Correlation of thirdorder here means the following. We multiply eachspeckle interferogram by the same but shifted speckleinterferogram, and then this product is cross correlatedwith the same speckle interferogram. In specklemasking no point source is required within the isopla-natic patch of the object.

We describe below the principle of speckle maskingand then show applications of speckle masking to as-tronomical data. These experiments yielded diffrac-tion-limited images of close spectroscopic double starswithout any ambiguity. The magnitude difference wasalso obtained with good accuracy.

For speckle masking the same astronomical raw dataas for speckle interferometry, i.e., speckle interfero-grams, are evaluated. The intensity distribution In (x)of the nth recorded speckle interferogram (short-ex-posure photograph) can be described by the followingincoherent, space-invariant imaging equation:

Inx) = 0(x) *P (x), n = 1, 2,..., N, (1)

where x denotes a two-dimensional vector, 0(x) is the

object intensity distribution, the asterisk denotes con-volution, and P,(x) is the nth atmospheric point-spreadfunction (psf). Usually only a sequence of objectspeckle interferograms I, (x) can be recorded. Theindividual psf's are not known.

The first image-processing step of speckle maskingis the calculation of the following third-order correla-tion:

([Inx)* tW x - i) 0 In(x))= (SS[10(x') * I(x' - m)] I,,x' + x)dx'), (2)

where the circled cross denotes the cross-correlationoperator, the angular brackets denote an ensemble av-erage, and the vector m is the so-called masking vector.The masking vector m must be chosen suitably, as isdiscussed below. For example, in the case of a doublestar m is equal to the separation vector of the doublestar, which is known from the object autocorrelation.

The next step after Eq. (2) is to derive from thethird-order correlation of I, (x) the third-order corre-lation of the object O(x). This can be achieved bycompensating for the speckle-masking transfer func-tion. It is possible to perform this compensation in thecorrelation (image) plane, as described in Eq. (3). Be-fore this discussion, we give an intuitive graphical ex-planation of the third-order correlation [Eq. (2)] for thereconstruction of true images.

Figure I illustrates graphically the principle of thespeckle-masking method. As is shown in the figure, the(strongly simplified) atmospheric psf causes a replica-tion of the object. Therefore the masking product In(x)* n (x- m) is similar to the desired psf Pn (x) if a suit-able masking vector m is chosen. Suitable maskingvectors are vectors that produce diffraction-limitedsmall overlaps between the object replications. Theobject speckle interferograms I, (x) and the maskingproducts, which are similar to the P,,, represent enoughinformation to reconstruct the object O(x), as in speckleholography.2-4 Of course, a realistic atmospheric psfhas a higher speckle density. Therefore many unde-sired overlaps will be produced in addition to the de-sired overlaps, especially for complicated objects.However, the influence of these undesired overlaps is

0146-9592/83/070389-03$1.00/0 © 1983, Optical Society of America

390 OPTICS LETTERS / Vol. 8, No. 7 / July 1983

not disturbing. The effect is compensated for by aprocessing step following Eq. (2). This step is discussedbelow.

The speckle-masking transfer function can be com-pensated for by subtracting various bias terms from Eq.(2), i.e., we have to determine

( [Inx * I. (X - M] 0 In~x))PI(x) MOIx) In(X)

E b Bx l

I "(X)

Fig.1. Illustration of the masking step Ix) .In(x-m). Inthe above example we assumed a cornerlike object O(x) andan unrealistic simple atmospheric psf consisting of five smallspeckles. Of course, in actual atmospheric psf's the speckledensity is higher. The purpose of this example is to illustratethat the masking product In(x) - I (x - m) is approximatelyequal to the desired psf P,, (x) if the masking vector m waschosen suitably. The example also illustrates that additionalwrong points appear because of undesired overlaps. Fortu-nately, the effect of these undesired overlaps can be com-pensated for by the speckle-masking transfer function, asdescribed in the text. The information obtained about thepsf's permits reconstruction of the object.

- (.Im xP (x - m)]0 I.x))- ([I(x) .Im (x - m)] (I. x3 )- ([tx). t x - in)] 0 Im(X))+ 2 ([I,-, (x) *Ik(x - i)] 0 Im (x))

= [O(x) .0(x - i)] 00(x) - O(x), (3)

where different I indices indicate statistically inde-pendent speckle interferograms. The compensationterms in Eq. (3) are obtained from a rather long calcu-lation. First, Eq. (1) was inserted into Eq. (2). Theresulting moments are of third order in intensity andsixth order in amplitude. These moments were calcu-lated by assuming Gaussian statistics for the complexamplitude in the pupil plane of the telescope and byapplying the Reed theorem.19 The calculation is similarto the calculation of the speckle interferometry transferfunction by Dainty.20 We found that Gaussian statis-tics produce quite a good approximation, although it isknown that a log-normal model is a better descriptionof the atmosphere.20 We found experimentally thatthere must be a weighting factor C, which is almost 1,in front of the first term of Eq. (3) if we have nearly butnot exactly Gaussian statistics (see the discussionbelow). Elsewhere 2 ' we present the calculation leadingto Eq. (3) and describe a method for compensating forthe speckle-masking transfer function in the generalcase, i.e., without assuming Gaussian statistics. Inspeckle masking the subtraction of the four compen-sation terms described in Eq. (3) is important sincethese four terms are not simple soft-bias terms but havepeaklike structure.

If the four compensation terms in Eq. (3) are sub-tracted from the ensemble-average third-order corre-lation function of the In, then the third-order correla-

(a) (b) (c)

Fig. 2. Speckle-masking measurement of the close spectroscopic double star 4 Sagittarii. (a) One of the 150 speckle interfer-ograms evaluated (b), (c) show the reconstructed true image of 4 Sagittarii. We measured (epoch, 1982.378) separation, 0.184+ 0.004"; position angle, 105.6 + 1°; intensity ratio, 1.3 ± 0.2.

I"(2�-M) I'M-I"N-M)

July 1983 / Vol. 8, No.7 / OPTICS LETTERS 391

I

0.5"l U U(a) (b) (c)

Fig. 3. Speckle-masking measurement of the spectroscopic double stars (a)iq Virginis (1982.386; 0.137 ± 0.004"; 335.5 ± 1°; 1.2 ± 0.3), (b) 9 Puppis (1980.019;

0.159 + 0.004"; 38.9 + 1°; 1.3 + 0.3), (c) Q Leonis (1980.019; 0.463 ± 0.004"; 17.0+ 10; 1.6 ± 0.3). The numbers in parentheses are, from left to right, epoch,separation, position angle, and intensity ratio.

tion function [O(x)- 0(x - m)] 0 O(x) = 0O3) of theO(x) is obtained. From 0(3) the object O(x) can bederived if m is chosen such that 0(x) - 0(x - m) - (x)since I(x) ® O(x) = O(x) (@ is the cross-correlationoperator). For example, in the case of speckle-maskingmeasurements of a double star, the suitable maskingvector is equal to the separation vector (from star A tostar B or B to A). In the case of a resolvable star disk,the length of a possible m is equal to the star's diameterminus about one Airy-disk diameter. In the case of ageneral object, the especially useful masking vectorshave a length that is about one Airy-disk diametersmaller than the diameter of the object. The diameterinformation is obtained from the autocorrelation.Various m directions (and even m lengths) should beused in the case of general objects. Elsewhere2 l wepresent a method that uses the information obtainedfrom a whole set of masking vectors.

Figure 2 shows a speckle interferogram of the closespectroscopic double star 4' Sagittarii and the high-resolution image reconstructed by digital specklemasking. Figure 3 shows speckle-masking measure-ments of three additional double stars. The speckleinterferograms were recorded with the EuropeanSouthern Observatory's 3.6-m telescope. In all exper-iments we evaluated 150 speckle interferograms re-corded with an exposure time of about 1/60 sec. Thespeckle interferograms were digitized with 128 X 128 or256 X 256 pixels. In order to determine the constantC (rn), which was described above, we calculated Eq.(3) using a masking vector, which was perpendicular tothe separation vector of the double star. For this(perpendicular) masking vector [O(x) - O(x - m)] 0O(x) = 0, and therefore C can be determined from Eq.(3).

We thank A. W. Lohmann for many helpful discus-sions, the German Science Foundation for financing thisproject, and the European Southern Observatory for theobserving time.

This paper is based on data collected at the EuropeanSouthern Observatory, La Silla, Chile.

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