estimation of binary star parameters by model fitting the bispectrum phase

Vol. 9, No. 4/April 1992/J. Opt. Soc. Am. A 543

Estimation of binary star parameters by model fitting thebispectrum phase

A. Glindemann, R.G. Lane, and J. C. Dainty

Optics Section, Blackett Laboratory, Imperial College, London SW7 2BZ, England

Received August 6, 1991; accepted October 2, 1991; final manuscript received November 7, 1991

The analysis of binary stars has to date been one of the major successes of speckle interferometry. A new tech-nique for estimating the parameters of a binary star is presented. Unlike earlier methods, the system does notrequire the measurement of a reference star to compensate for the speckle transfer function. The algorithmrelies on model fitting to the bispectrum phase and can obtain the separation, position angle, and relativebrightness of the two components.

1. INTRODUCTION

Since speckle interferometry was originally proposed byLabeyrie,' there have been numerous papers on its appli-cation to astronomical imaging. The important contribu-tion of Labeyrie was to realize that short-exposure imagestaken of stars through the turbulent atmosphere con-tained information up to the diffraction limit of the tele-scope. Unfortunately, the phase of the higher frequenciesis randomized by the atmosphere and consequently lost onaveraging. Labeyrie proposed averaging the power spec-trum of a large number of short-exposure images to ob-tain information concerning the power spectrum or,equivalently, the autocorrelation of the object. This tech-nique remains the most effective method for determiningthe object power spectrum.

The major difficulty with speckle interferometry is thatit does not give any phase information concerning the ob-ject spectrum. This problem has been overcome by theadvent of such techniques as Knox-Thompson and the bi-spectrum.2 4 The recent development of robust methodsfor least-squares phase reconstruction5' 7 has further en-hanced these methods.

To date the most successful application of speckle inter-ferometry has been in the observation of binary stars.8

The estimation of a binary star can be reduced to the esti-mation of three parameters: separation, position angle,and relative brightness. The a priori information thatthe star of interest is a binary means that model fittingshould yield significantly better results than a general-purpose imaging technique. Recently Christou9 investi-gated a number of techniques for binary estimationranging from the simple, such as shift-and-add,'0 to thequite involved, such as iterative deconvolution.

In this paper we aim to present an alternative proce-dure that is based on the bispectrum and that is both con-ceptually simple and practically robust. By simply fittingthe bispectrum phase of a binary star to the measuredbispectrum phase, we find that the use of a reference starbecomes unnecessary. It is also possible to determinethe relative brightness, a parameter that is difficult toestimate by earlier methods that rely on the use of thepower spectrum.

We consider two two-dimensional spaces called the im-age and the Fourier space in which arbitrary points areidentified by the position vectors x = (x, y) and u = (u, v).An image f(x) and its spectrum F(u) constitute a Fouriertransform pair:

F(u) = IF(u)1exp[ii(u)] = [f(x)]

= ff(x)exp(-i2vux)dx, (1)

where F(u)l is the modulus and +(u) is the phase of theobject spectrum F(u).

Thus the sequence of N short-exposure speckle imagescan be represented by

s.(x) = f(x) (Dh.(x) + c.(x), n = N, (2)

where f(x), h(x) and c(x) represent the true image, thedistortion introduced by the atmosphere and the tele-scope, and the additive noise, respectively. The symbol 0is used to indicate a two-dimensional convolution.Equation (2) can be Fourier transformed to yield the in-stantaneous image spectrum

Sn(u) = F(u)Hn(u) + Cn(u), n = 1,...,N. (3)

A. Power SpectrumAn estimate of the power spectrum can then be formed by

(IS(u)l') = IF(u)12 (IH(u)12) + (E(u)), (4)

where () is used to indicate the process of averaging overthe ensemble and E(u) is a real term containing all thecross products that include the additive noise. It is appar-ent that Eq. (4) does not yield the true power spectrum ofthe object directly, even when the additive noise term issmall, because the term (H(u) 2), known as the speckletransfer function (STF), must be corrected to obtain thetrue power spectrum. In practice the STF is estimatedby observing a point source for which IF(u)12 is con-stant and removing it from Eq. (4) by either division orWiener filtering.

The problem with this procedure for compensating forthe estimated power spectrum becomes apparent on ex-

0740-3232/92/040543-06$05.00 ©0 1992 Optical Society of America

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544 J. Opt. Soc. Am. A/Vol. 9, No. 4/April 1992

amination of a typical STE" The STF can be dividedinto two distinct areas, the seeing spike near the originand a high-frequency region where the transfer functionis proportional to the autocorrelation of the telescope aper-ture. The point of transition between these two regionsis determined by the Fried parameter." As the seeingimproves, the seeing spike widens and the high-frequencyregion of the STF increases.'2

The drawback to using a reference star measurement isthat it is never possible to match exactly the seeing condi-tions for the object of interest and the reference star.The effects of mismatching the object and the referenceseeing are most severe near the origin, where smallchanges in the width of the seeing spike can cause largechanges in the final estimate of the spectrum. This isparticularly unfortunate since this region contains datawith a high signal-to-noise ratio (SNR). Because of thevariability of seeing, complicated binning procedures arerequired. l2,3

B. BispectrumThe bispectrum of an object is defined as

F3(ul,u 2) = F(u1)F(u2)F(-u - U 2 )

= F 3(u1,u 2)1exp[iq/(u1,u 2)], (5)

where (ul,u 2 ) is the phase of the object bispectrum. Aswas shown by Lohmann et al.,' the bispectra of the indi-vidual speckle frames can be averaged in a manner simi-lar to that used with the power spectrum and in the limitof an infinite number of speckle frames,

(S3(Ul, U2) = F3 (ul, U 2) (H3(Ul, U2 )) (6)

Moreover, it can be shown that (H3 (u1,u2 )) is a functionwith zero phase, a property that is approximately indepen-dent of minor telescope aberrations.3 Thus

1(U1,U2) = M(u1 ) + (U2) - (u + U2), (7)where (u) is the true object phase as defined in Eq. (1).

The problem of reconstructing the unknown objectphase (u) from the bispectrum phase f(u1,u2) has beenthe subject of much research. The recursive methods3 4

have the advantage of simplicity but have poor noise per-formance, particularly when the higher frequencies arebeing reconstructed.

More recently a number of researchers have used least-squares techniques to recover the object phase.5 6"145 Ofthese techniques the procedure described by Haniff6 ap-pears to be the most logical and robust. We recently ex-tended Haniff's method to the two-dimensional case andapplied it to experimental data.7

Before introducing the procedure of fitting the bispec-trum phase with a binary star model, we briefly discussour least-squares method for object phase reconstruction,since this provides a simple introduction to the more spe-cialized application of estimating binaries. It should beemphasized that a full-phase reconstruction can be usedto verify any marginal results produced by model fitting.

2. OBJECT PHASE RECONSTRUCTION

A. General CaseIn practice the bispectrum phase is measured from afinite number of speckle frames. We denote this mea-

surement of the bispectrum phase by f. The problem ofreconstructing the object phase is simply a problem of de-termining which set of object phases is most consistentwith the measured bispectrum phase.

Given an initial estimate of the object phase k, it is pos-sible to estimate the difference between the phase at apoint in the measured bispectrum and that calculatedfrom our current estimate of the object phase . Thisdifference is given by

ii= ij - (Ai + 4j - i+), (8)

where i and j are the discrete coordinates in the two-dimensional phase array.

The simplest procedure is to minimize the weightedsum

M

E W(upj) 2j (9)

where the summation is over M selected points in the bi-spectrum and W(i, j) is a weighting that is assigned to eachequation. The variable weighting is necessary since notall the bispectral phases are measured to the same accu-racy. Furthermore, as was noted by Haniff,5 it is neces-sary to define the difference Aj modulo 27r

In our reconstructions we have employed the SNR ofthe bispectrum phase as the weighting function. Thecomputation of the SNR is not a simple procedure, and weemploy the method described by Ayers et al.4 in which theSNR of the phase is calculated by

SNR,

sinj + Up 2 2 - covImRe)sin(24i)[LIM2 os 2

e sr2 inS,) 2 esi(1rQ)

(10)

where Im2 and o'Re2 are the variance of the real and the

imaginary parts of the complex bispectrum phasor,cov(Im, Re) is the covariance between them, and is thephase of the mean bispectrum phasor.

We thus minimize

M

> {mod 27T[drqj - (4; + j - Xj+j)]12SNRij, (11)

where M is the total number of points computed in thebispectrum.

In practice it is not computationally feasible to use theentire bispectrum, and only those portions with a largeSNR are employed. We define a subplane to be the set ofall the points in the bispectrum obtained by fixing i at aconstant value and varying j. It has been noted by Ayerset al. 4 that those subplanes for which liJ2 is small usuallyhave a significantly higher SNR. Thus, when reconstruc-tions are quoted as being for a given number of subplanes,we use those subplanes for which lil2 is the smallest.

The final stage when performing the reconstruction isto combine the object magnitude obtained by speckle in-terferometry with the bispectral phase estimate. This iscomplicated by the fact that compensation of the STF re-moves the telescope modulation transfer function (MTF)and amplifies the noise in spatial frequencies near dif-fraction limit. It is thus necessary to choose some form

Glindemann et al.

Vol. 9, No. 4/April 1992/J. Opt. Soc. Am. A 545

of window function, for example, the telescope MTF, toprevent this noise from dominating the reconstruction. Adifficulty that arises, however, is that the estimated rela-tive brightness of the binary is dependent on this some-what arbitrary choice of window function.

It is desirable to have a method that does not require theuser to estimate the power spectrum, first because themeasurement of a reference star consumes valuable obser-vation time and second because the measurement of thepower spectrum is in many ways more difficult to do con-sistently than a bispectral phase estimate. Since the bi-spectrum phase contains sufficient information for thedetermination of the binary star parameters without theneed for the power spectrum, we thus propose to estimatethe binary parameters directly from the bispectrum phase.

B. Binary Star EstimationFor the special case of binary star estimations it is pos-sible to describe the object by

b(x) = (x) + A3(x - p), (12)

where the first star, of brightness 1, is located at x = 0and the second one, of brightness A, is located at x = p.Thus the spectrum of the binary is given by

B(u) = 1 + A exp(i2vup), (13)

and its bispectrum is given by

B3(ul,u 2 ) = B(u,)B(u 2 )B(-ul - u2)

= 1 - A - A2 + A3 + (A + A2)4 cos(7ruip)

x cos(7ru2 p)cos[7r(u1 + u2 )p] + i4(A2 - A)

x sin(rulp)sin(7ru 2 p)sin[ir(ui + u2)p]. (14)

The procedure advocated is to select p and A to mini-mize, in a weighted least-squares sense, the difference be-tween the bispectrum phase computed from theobservations and that obtained by using Eq. (14).

We thus minimizeM

> [mod 2 7r(qfi j - 83,,j)]2 SNRij, (15)

where ,,,j is the phase computed from the current esti-mates of the binary star parameters and the difference istaken modulo 27r.

It was found that the functional defined by expres-sion (15) has multiple minima, and the application of theminimization did not yield consistent results for the

parameters when it was started from an arbitrary initialestimate of the binary star parameters. Fortunately, it isrelatively easy to obtain a crude estimate of the relativeposition of the binary stars from the uncorrected powerspectrum of the speckle images. With this crude startingestimate, convergence to the expected solution occurredin nearly all the cases. Other simple processing tech-niques, such as shift-and-add, could also be employed toproduce a starting estimate.

3. RESULTS

Two binary data sets were used to test the algorithm de-scribed in this paper. The first data set consists of high-light-level images observed on the San Martir Observatory2.12-m telescope, Mexico, at A = 516 nm in October 1988.These data were provided by J. Ohtsubo, who also pre-sented a power spectrum analysis.'6"7 The second dataset, provided by E. K. Hege, is photon limited and wastaken on the Steward Observatory 2.3-m telescope, Uni-versity of Arizona, at A = 550 nm in October 1986, usingthe Stanford University MAMA detector'8 ; these datawere reconstructed previously by Meng et al.,'4 Prez-Ilzarbe and Nieto-Vesperinas, 9 and Glindemann et al.7

Initially, to ensure that a correct minimum of cost func-tion (15) was found, two different minimization routinesof the NAG library20 (EO4DGF and E04HFF) were used.Both routines always found the same minimum.

Using the power spectrum to form a crude initial esti-mate for the position vector p yields two possible positionson opposite sides of the star at the center. After bothwere used for the least-squares minimization, one of themwas always identifiable as the true position because it re-sulted in the smaller error sum. The initial estimate ofthe relative brightness A is not critical and was set toeither 0.5 or 0.9 without significant differences in thefinal solution.

An important variable is the number of bispectrum sub-planes used for the reconstruction. We observe that thequality and the stability of the reconstruction improveswith increasing the number of subplanes. This is similarto the behavior observed in the problem of reconstructingthe phase of a general object from the bispectrum.7 Adrawback of increasing the number of subplanes is the in-creased level of computation. We have found that, with aSUN-Sparc 1 + workstation, reconstructing the binaryparameters from the phase of the bispectrum for 6 sub-

Table 1. Results for 126-Tau with 4900 Framesa

600 Events/Frame 300 Events/Frame 150 Events/Frame

No. of Subpl. Rel. Br. Sep. (") Orient. (°) Rel. Br. Sep. (") Orient. (°) Rel. Br. Sep. ) Orient. (0)

6 0.28 0.349 240.3 0.29 0.349 240.6 0.36 0.345 241.610 0.28 0.349 240.2 0.29 0.351 240.5 0.34 0.346 240.918 0.27 0.351 240.4 0.27 0.352 241.4 0.31 0.343 241.728 0.26 0.351 240.5 0.27 0.354 241.3 0.27 0.342 240.844 0.26 0.353 240.7 0.27 0.352 241.1 0.28 0.342 240.060 0.26 0.353 240.3 0.27 0.353 240.7 0.28 0.343 239.980 0.26 0.353 240.3 0.28 0.355 240.7 0.28 0.349 240.1

'Additionally to the full data set with 600 events/frame, the results with the reduced data are presented by taking every second and every fourth photon.It is apparent that for a lower number of photons a higher number of subplanes (Subpl.) is required. Rel. Br., Relative brightness; Sep., separation; Orient.,orientation.

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546 J. Opt. Soc. Am. A/Vol. 9, No. 4/April 1992

Table 2. Results for Photon-Limited Binary DataaThis Study McAlister and Hartkopf8

Star Name No. of Events No. of Frames Rel. Br. Sep. (") Orient. () Sep. (") Orient. ()

,G-Del 685 4500 0.37 0.184 101.6 0.2 (0.172) 136.0 (85)A-Ori 433 7100 0.16 0.219 26.3 0.218 206.4ADS4299 244 12,850 0.31 0.121 138.1 0.135 125.2

'The data are the averages over reconstructions with 44, 60, and 80 subplanes of the bispectrum. Rel. Br., Relative brightness; Sep., separation; Orient.,orientation.

Table 3. Results for High-Light-Level Binary Data'

This Study Isobe et al.17Star Name Combined Magnitude No. of Frames Rel. Br. Sep. (") Orient. (0) Sep. (") Orient.

ADS2253 6.7 64 0.53 0.644 83.5 0.50 265ADS2980 7.4 74 0.06 0.770 148.5 0.60 325ADS3390 7.8 74 0.25 1.243 14.2 1.2 15ADS5871 6.4 74 0.73 1.607 319.7 1.20 320ADS15267 7.4 74 0.65 0.417 253.6 0.30 75ADS15281 4.1 74 0.76 0.263 101.7 0.22 96ADS15992 8.0 64 0.16 0.595 49.9 0.50 52ADS16836 5.0 74 0.55 0.675 87.1 0.50 89

aThe data are the averages over reconstructions with 44, 60, and 80 subplanes of the bispectrum. Rel. Br., Relative brightness; Sep., separation; Orient.,orientation.

planes typically takes 1 min, while reconstruction with80 subplanes takes approximately 30 min.

The photon-limited data were registered on a 256 x256 array, and the high-light-level data were registered ona 128 x 128 array. The bispectrum was taken from a64 x 64 (32 x 32 for the high-light-level data) object phasearray in Fourier space. Thus 80 subplanes means 5%(20% for the high-light-level data) of the whole bispectrum.

Table 1 shows the model fit of the parameters relativebrightness A, separation, and orientation for 126-Tau.Since the data are available in time-tagged photon event

form, the magnitude of the binaries was artificially re-duced by using only a subset of the measured photons.The full data set gives the same solution for any numberof subplanes, whereas the reduced data sets converge tothe same solution only for a high number of subplanes.

An accurate estimate for the accuracy of this methodcan be obtained only by analyzing a number of differentdata sets for the same star. It is, however, possible to geta crude estimate for the accuracy from the results inTable 1. Since only high numbers of subplanes give con-sistent results, measurements with 44, 60, and 80 sub-

ADS 152814- ADS 5871_ A

I ~ADS 15267

4-

I I I I

6 10 18 28 44

I ADS 16836j ADS 2253

I fP-Del

ADS 4299I II_ I 126-TauI _ ~ I ADS 3390

j j ADS 15992I i ,-Ori

I I

60 80

Number of subplanesFig. 1. Relative brightness (rel. brightness) as a function of the number of subplanes for all the reconstructed stars. The increase instability obtained by using more subplanes is readily apparent. The model fitted to the ADS15267 data required at least 28 subplanes toprovide a solution, and the fit to ADS4299 was not successful with 18 and 28 subplanes.

rel. brightness

0.8 .

0.7 -

0.6 -

0.5 -

0.4 -

0.3 -

0.2

0.1

i

. . . . . .

Glindemann et al.

Vol. 9, No. 4/April 1992/J. Opt. Soc. Am. A 547

Fig. 2. Reconstructions of a, ADS2980, which shows an appar-ent triple-star reconstruction; b, ADS4299, which indicates thedifficulties posed by poor-quality data; and c, 126-Tau, which canbe clearly identified as a double star. The noise level for a goodreconstruction (c) is at approximately two contour lines, i.e., 5%of the peak intensity.

planes, whereupon the standard deviations are betterthan ±0.02 for the relative brightness, better than±0.005" for the separation, and better than ±0.5° for theorientation, are considered. Similar consistencies are ob-tained for the reconstructions of the other stars presentedin this paper.

Table 2 shows the three parameters for -Del, ,u-Ori,and ADS4299. The two observations of B-Del quoted inthe McAlister-Hartkopf catalog were done before and af-ter the measurements that we use, and the observation of

ADS4299 quoted in the catalog was performed beforethese data were taken. The results for the high-light-level data are presented in Table 3. All the results wereobtained by using the average of model fits with 44, 60,and 80 subplanes of the bispectrum.

Figure 1 shows the relative brightness of the stars as afunction of the number of subplanes used. Differences inthe performance of the algorithm among different startsare apparent, with the model fit being particularly stablefor 126-Tau. The least-stable results were obtained forADS2980 and ADS4299. In the case of ADS2980, how-ever, the reconstruction of the object intensity by theleast-squares method described in Subsection 2.A did notreveal a simple double star but a possible triple-star struc-ture (see Fig. 2a). Thus it is not surprising that themodel fitted to the phase of the bispectrum did not pro-duce reliable results.

The reconstruction of the photon-limited binaryADS4299 (Fig. 2b) shows the difficulties posed by noisydata. In this case the phase was not reconstructed up tothe same spatial frequency as the power spectrum, re-sulting in a band where the modulus is still significantbut the phase is undetermined. This band contributesto artifacts in the reconstruction, in particular the ap-pearance of a weak ghost star. This echo weakens whena narrower window function is employed, i.e., the extentof the modulus is restricted to where the phase has beenreconstructed.

In Fig. 2c the reconstruction of 126-Tau is displayed asan example of a good reconstruction. For these data theparameter fit was stable with respect to both the varia-tion in the number of subplanes and the reduction in thenumber of photons.

4. CONCLUSION

We have presented a new approach to observing binarystars by model fitting to the bispectrum phase. The cal-culation of the bispectrum does require considerable com-putation, but with the inexorable increase in the power ofmodern computers this should not present an obstacle.The off-line computation of the average bispectrum is alsoan ideal candidate for parallel processing since the bispec-trum of each frame can be computed independently beforebeing averaged.

The technique presented provides a conceptually simplemeans of obtaining the parameters of a binary star systemto a high degree of accuracy without the need for a refer-ence star. The method provides reliable accurate mea-surements of the separation, the orientation, and therelative brightness, the last-named parameter being par-ticularly difficult to measure with power-spectrum-basedtechniques.

ACKNOWLEDGMENTS

We acknowledge the considerable assistance of E. K.Hege and J. Ohtsubo in providing the astronomical dataused in this paper and E. K. Hege for his constructivecomments on the original manuscript. The study waspartly funded under a Science and Engineering ResearchCouncil grant GR/F 75544. A. Glindemann thanks theDeutsche Forschungsgemeinschaft for financial support.

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