bispectrum phase Estimation of binary star parameters by model fitting the = ff(x)exp(-i2vux)dx, (1) s.(x) = f(x) (Dh.(x) + c.

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1 Vol. 9, No. 4/April 1992/J. Opt. Soc. Am. A 543 Estimation of binary star parameters by model fitting the bispectrum phase A. Glindemann, R.G. Lane, and J. C. Dainty Optics Section, Blackett Laboratory, mperial 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 technique for estimating the parameters of a binary star is presented. Unlike earlier methods, the system does not require the measurement of a reference star to compensate for the speckle transfer function. The algorithm relies on model fitting to the bispectrum phase and can obtain the separation, position angle, and relative brightness of the two components. 1. NTRODUCTON Since speckle interferometry was originally proposed by Labeyrie,' there have been numerous papers on its application to astronomical imaging. The important contribution of Labeyrie was to realize that short-exposure images taken of stars through the turbulent atmosphere contained information up to the diffraction limit of the telescope. Unfortunately, the phase of the higher frequencies is randomized by the atmosphere and consequently lost on averaging. Labeyrie proposed averaging the power spectrum of a large number of short-exposure images to obtain information concerning the power spectrum or, equivalently, the autocorrelation of the object. This technique remains the most effective method for determining the object power spectrum. The major difficulty with speckle interferometry is that it does not give any phase information concerning the object spectrum. This problem has been overcome by the advent of such techniques as Knox-Thompson and the bispectrum. 2 4 The recent development of robust methods for least-squares phase reconstruction 5 ' 7 has further enhanced these methods. To date the most successful application of speckle interferometry has been in the observation of binary stars. 8 The estimation of a binary star can be reduced to the estimation of three parameters: separation, position angle, and relative brightness. The a priori information that the star of interest is a binary means that model fitting should yield significantly better results than a generalpurpose imaging technique. Recently Christou 9 investigated a number of techniques for binary estimation ranging from the simple, such as shift-and-add,' to the quite involved, such as iterative deconvolution. n this paper we aim to present an alternative procedure that is based on the bispectrum and that is both conceptually simple and practically robust. By simply fitting the bispectrum phase of a binary star to the measured bispectrum phase, we find that the use of a reference star becomes unnecessary. t is also possible to determine the relative brightness, a parameter that is difficult to estimate by earlier methods that rely on the use of the power spectrum. We consider two two-dimensional spaces called the image and the Fourier space in which arbitrary points are identified by the position vectors x = (x, y) and u = (u, v). An image f(x) and its spectrum F(u) constitute a Fourier transform pair: F(u) = F(u)1exp[ii(u)] = /92/4543-6$ Optical Society of America [f(x)] = ff(x)exp(-i2vux)dx, (1) where F(u)l is the modulus and +(u) is the phase of the object spectrum F(u). Thus the sequence of N short-exposure speckle images can 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, the distortion introduced by the atmosphere and the telescope, and the additive noise, respectively. The symbol is used to indicate a two-dimensional convolution. Equation (2) can be Fourier transformed to yield the instantaneous image spectrum Sn(u) = F(u)Hn(u) + Cn(u), n = 1,...,N. (3) A. Power Spectrum An estimate of the power spectrum can then be formed by (S(u)l') = F(u)1 2 (H(u)12) + (E(u)), (4) where () is used to indicate the process of averaging over the ensemble and E(u) is a real term containing all the cross products that include the additive noise. t is apparent that Eq. (4) does not yield the true power spectrum of the object directly, even when the additive noise term is small, because the term (H(u) 2), known as the speckle transfer function (STF), must be corrected to obtain the true power spectrum. n practice the STF is estimated by observing a point source for which F(u)1 2 is constant and removing it from Eq. (4) by either division or Wiener filtering. The problem with this procedure for compensating for the estimated power spectrum becomes apparent on ex-

2 544 J. Opt. Soc. Am. A/Vol. 9, No. 4/April 1992 amination of a typical STE" The STF can be divided into two distinct areas, the seeing spike near the origin and a high-frequency region where the transfer function is proportional to the autocorrelation of the telescope aperture. The point of transition between these two regions is determined by the Fried parameter." As the seeing improves, the seeing spike widens and the high-frequency region of the STF increases.' 2 The drawback to using a reference star measurement is that it is never possible to match exactly the seeing conditions for the object of interest and the reference star. The effects of mismatching the object and the reference seeing are most severe near the origin, where small changes in the width of the seeing spike can cause large changes in the final estimate of the spectrum. This is particularly unfortunate since this region contains data with a high signal-to-noise ratio (SNR). Because of the variability of seeing, complicated binning procedures are required. l2,3 B. Bispectrum The bispectrum of an object is defined as F 3 (ul,u 2 ) = F(u1)F(u 2 )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. As was shown by Lohmann et al.,' the bispectra of the individual speckle frames can be averaged in a manner similar to that used with the power spectrum and in the limit of an infinite number of speckle frames, (S3(Ul, U2) = F 3 (ul, U 2 ) (H 3 (Ul, U 2 )) (6) Moreover, it can be shown that (H 3 (u1,u 2 )) is a function with zero phase, a property that is approximately independent of minor telescope aberrations. 3 Thus 1(U1,U2) = M(u 1 ) + (U2) - (u + U2), (7) where (u) is the true object phase as defined in Eq. (1). The problem of reconstructing the unknown object phase (u) from the bispectrum phase f(u1,u 2 ) has been the subject of much research. The recursive methods 3 4 have the advantage of simplicity but have poor noise performance, particularly when the higher frequencies are being reconstructed. More recently a number of researchers have used leastsquares techniques to recover the object phase. 5 6 "14 5 Of these techniques the procedure described by Haniff 6 appears to be the most logical and robust. We recently extended Haniff's method to the two-dimensional case and applied it to experimental data. 7 Before introducing the procedure of fitting the bispectrum phase with a binary star model, we briefly discuss our least-squares method for object phase reconstruction, since this provides a simple introduction to the more specialized application of estimating binaries. t should be emphasized that a full-phase reconstruction can be used to verify any marginal results produced by model fitting. 2. OBJECT PHASE RECONSTRUCTON A. General Case n practice the bispectrum phase is measured from a finite number of speckle frames. We denote this measurement of the bispectrum phase by f. The problem of reconstructing the object phase is simply a problem of determining which set of object phases is most consistent with the measured bispectrum phase. Given an initial estimate of the object phase k, it is possible to estimate the difference between the phase at a point in the measured bispectrum and that calculated from our current estimate of the object phase. This difference is given by ii= ij - (Ai + 4j - i+), (8) where i and j are the discrete coordinates in the twodimensional phase array. The simplest procedure is to minimize the weighted sum M E W(upj) 2j (9) where the summation is over M selected points in the bispectrum and W(i, j) is a weighting that is assigned to each equation. The variable weighting is necessary since not all the bispectral phases are measured to the same accuracy. Furthermore, as was noted by Haniff, 5 it is necessary to define the difference Aj modulo 27r n our reconstructions we have employed the SNR of the bispectrum phase as the weighting function. The computation of the SNR is not a simple procedure, and we employ the method described by Ayers et al. 4 in which the SNR of the phase is calculated by SNR, 2 [LM2 os 2 e sr + Up 2 2 sinj - covmre)sin(24i) ins,) 2 esi(1rq) (1) where m 2 and o're 2 are the variance of the real and the imaginary parts of the complex bispectrum phasor, cov(m, Re) is the covariance between them, and is the phase of the mean bispectrum phasor. We thus minimize M > {mod 2 7T[drqj - (4; + j - Xj+j)]12SNRij, (11) where M is the total number of points computed in the bispectrum. n practice it is not computationally feasible to use the entire bispectrum, and only those portions with a large SNR are employed. We define a subplane to be the set of all the points in the bispectrum obtained by fixing i at a constant value and varying j. t has been noted by Ayers et al. 4 that those subplanes for which lij 2 is small usually have a significantly higher SNR. Thus, when reconstructions are quoted as being for a given number of subplanes, we use those subplanes for which lil 2 is the smallest. The final stage when performing the reconstruction is to combine the object magnitude obtained by speckle interferometry with the bispectral phase estimate. This is complicated by the fact that compensation of the STF removes the telescope modulation transfer function (MTF) and amplifies the noise in spatial frequencies near diffraction limit. t is thus necessary to choose some form

3 Glindemann et al. Vol. 9, No. 4/April 1992/J. Opt. Soc. Am. A 545 of window function, for example, the telescope MTF, to prevent this noise from dominating the reconstruction. A difficulty that arises, however, is that the estimated relative brightness of the binary is dependent on this somewhat arbitrary choice of window function. t is desirable to have a method that does not require the user to estimate the power spectrum, first because the measurement of a reference star consumes valuable observation time and second because the measurement of the power spectrum is in many ways more difficult to do consistently than a bispectral phase estimate. Since the bispectrum phase contains sufficient information for the determination of the binary star parameters without the need for the power spectrum, we thus propose to estimate the binary parameters directly from the bispectrum phase. B. Binary Star Estimation For the special case of binary star estimations it is possible to describe the object by b(x) = (x) + A3(x - p), (12) where the first star, of brightness 1, is located at x = and 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 B 3 (ul,u 2 ) = B(u,)B(u 2 )B(-ul - u 2 ) = 1 - A - A 2 + A 3 + (A + A 2 )4 cos(7ruip) x cos(7ru 2 p)cos[7r(u1 + u 2 )p] + i4(a 2 - A) x sin(rulp)sin(7ru 2 p)sin[ir(ui + u 2 )p]. (14) The procedure advocated is to select p and A to minimize, in a weighted least-squares sense, the difference between the bispectrum phase computed from the observations and that obtained by using Eq. (14). We thus minimize M > [mod 2 7r(qfi j - 83,,j)] 2 SNRij, (15) where,,,j is the phase computed from the current estimates of the binary star parameters and the difference is taken modulo 27r. t was found that the functional defined by expression (15) has multiple minima, and the application of the minimization did not yield consistent results for the parameters when it was started from an arbitrary initial estimate of the binary star parameters. Fortunately, it is relatively easy to obtain a crude estimate of the relative position of the binary stars from the uncorrected power spectrum of the speckle images. With this crude starting estimate, convergence to the expected solution occurred in nearly all the cases. Other simple processing techniques, such as shift-and-add, could also be employed to produce a starting estimate. 3. RESULTS Two binary data sets were used to test the algorithm described in this paper. The first data set consists of highlight-level images observed on the San Martir Observatory 2.12-m telescope, Mexico, at A = 516 nm in October These data were provided by J. Ohtsubo, who also presented a power spectrum analysis.' 6 " 7 The second data set, provided by E. K. Hege, is photon limited and was taken on the Steward Observatory 2.3-m telescope, University of Arizona, at A = 55 nm in October 1986, using the Stanford University MAMA detector' 8 ; these data were reconstructed previously by Meng et al.,' 4 Prez- lzarbe and Nieto-Vesperinas, 9 and 7 nitially, to ensure that a correct minimum of cost function (15) was found, two different minimization routines of the NAG library 2 (EO4DGF and E4HFF) were used. Both routines always found the same minimum. Using the power spectrum to form a crude initial estimate for the position vector p yields two possible positions on opposite sides of the star at the center. After both were used for the least-squares minimization, one of them was always identifiable as the true position because it resulted in the smaller error sum. The initial estimate of the relative brightness A is not critical and was set to either.5 or.9 without significant differences in the final solution. An important variable is the number of bispectrum subplanes used for the reconstruction. We observe that the quality and the stability of the reconstruction improves with increasing the number of subplanes. This is similar to the behavior observed in the problem of reconstructing the phase of a general object from the bispectrum. 7 A drawback of increasing the number of subplanes is the increased level of computation. We have found that, with a SUN-Sparc 1 + workstation, reconstructing the binary parameters from the phase of the bispectrum for 6 sub- Table 1. Results for 126-Tau with 49 Framesa 6 Events/Frame 3 Events/Frame 15 Events/Frame No. of Subpl. Rel. Br. Sep. (") Orient. ( ) Rel. Br. Sep. (") Orient. ( ) Rel. Br. Sep. ) Orient. () 'Additionally to the full data set with 6 events/frame, the results with the reduced data are presented by taking every second and every fourth photon. t 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.

4 J. Opt. Soc. Am. A/Vol. 9, No. 4/April 1992 Table 2. Results for Photon-Limited Binary Dataa This Study McAlister and Hartkopf8 Star Name No. of Events No. of Frames Rel. Br. Sep. (") Orient. () Sep. (") Orient. (),G-Del (.172) 136. (85) A-Ori ADS , 'The data are the averages over reconstructions with 44, 6, and 8 subplanes of the bispectrum. Rel. Br., Relative brightness; Sep., separation; Orient., orientation. Table 3. Results for High-Light-Level Binary Data' This Study sobe et al.17 Star Name Combined Magnitude No. of Frames Rel. Br. Sep. (") Orient. () Sep. (") Orient. ADS ADS ADS ADS ADS ADS ADS ADS athe data are the averages over reconstructions with 44, 6, and 8 subplanes of the bispectrum. Rel. Br., Relative brightness; Sep., separation; Orient., orientation. planes typically takes 1 min, while reconstruction with 8 subplanes takes approximately 3 min. The photon-limited data were registered on a 256 x 256 array, and the high-light-level data were registered on a 128 x 128 array. The bispectrum was taken from a 64 x 64 (32 x 32 for the high-light-level data) object phase array in Fourier space. Thus 8 subplanes means 5% (2% for the high-light-level data) of the whole bispectrum. Table 1 shows the model fit of the parameters relative brightness 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 reduced by using only a subset of the measured photons. The full data set gives the same solution for any number of subplanes, whereas the reduced data sets converge to the same solution only for a high number of subplanes. An accurate estimate for the accuracy of this method can be obtained only by analyzing a number of different data sets for the same star. t is, however, possible to get a crude estimate for the accuracy from the results in Table 1. Since only high numbers of subplanes give consistent results, measurements with 44, 6, and 8 sub- rel. brightness ADS ADS 5871 _ A ~ADS i 4- ADS j ADS fp-del ADS 4299 _ 126-Tau _ ~ ADS 339 j j ADS i,-ori Number of subplanes Fig. 1. Relative brightness (rel. brightness) as a function of the number of subplanes for all the reconstructed stars. The increase in stability obtained by using more subplanes is readily apparent. The model fitted to the ADS15267 data required at least 28 subplanes to provide a solution, and the fit to ADS4299 was not successful with 18 and 28 subplanes.

5 . a c n C (((V ' Lw// ( JJ ( `. C) <D C \ ; Fig. 2. Reconstructions of a, ADS298, which shows an apparent triple-star reconstruction; b, ADS4299, which indicates the difficulties posed by poor-quality data; and c, 126-Tau, which can be clearly identified as a double star. The noise level for a good reconstruction (c) is at approximately two contour lines, i.e., 5% of the peak intensity. planes, whereupon the standard deviations are better than ±.2 for the relative brightness, better than ±.5" for the separation, and better than ±.5 for the orientation, are considered. Similar consistencies are obtained for the reconstructions of the other stars presented in this paper. Table 2 shows the three parameters for -Del,,u-Ori, and ADS4299. The two observations of B-Del quoted in the McAlister-Hartkopf catalog were done before and after the measurements that we use, and the observation of C Vol. 9, No. 4/April 1992/J. Opt. Soc. Am. A 547 ADS4299 quoted in the catalog was performed before these data were taken. The results for the high-lightlevel data are presented in Table 3. All the results were obtained by using the average of model fits with 44, 6, and 8 subplanes of the bispectrum. Figure 1 shows the relative brightness of the stars as a function of the number of subplanes used. Differences in the performance of the algorithm among different starts are apparent, with the model fit being particularly stable for 126-Tau. The least-stable results were obtained for ADS298 and ADS4299. n the case of ADS298, however, the reconstruction of the object intensity by the least-squares method described in Subsection 2.A did not reveal a simple double star but a possible triple-star structure (see Fig. 2a). Thus it is not surprising that the model fitted to the phase of the bispectrum did not produce reliable results. The reconstruction of the photon-limited binary ADS4299 (Fig. 2b) shows the difficulties posed by noisy data. n this case the phase was not reconstructed up to the same spatial frequency as the power spectrum, resulting in a band where the modulus is still significant but the phase is undetermined. This band contributes to artifacts in the reconstruction, in particular the appearance of a weak ghost star. This echo weakens when a narrower window function is employed, i.e., the extent of the modulus is restricted to where the phase has been reconstructed. n Fig. 2c the reconstruction of 126-Tau is displayed as an example of a good reconstruction. For these data the parameter fit was stable with respect to both the variation in the number of subplanes and the reduction in the number of photons. 4. CONCLUSON We have presented a new approach to observing binary stars by model fitting to the bispectrum phase. The calculation of the bispectrum does require considerable computation, but with the inexorable increase in the power of modern computers this should not present an obstacle. The off-line computation of the average bispectrum is also an ideal candidate for parallel processing since the bispectrum of each frame can be computed independently before being averaged. The technique presented provides a conceptually simple means of obtaining the parameters of a binary star system to a high degree of accuracy without the need for a reference star. The method provides reliable accurate measurements of the separation, the orientation, and the relative brightness, the last-named parameter being particularly difficult to measure with power-spectrum-based techniques. ACKNOWLEDGMENTS We acknowledge the considerable assistance of E. K. Hege and J. Ohtsubo in providing the astronomical data used in this paper and E. K. Hege for his constructive comments on the original manuscript. The study was partly funded under a Science and Engineering Research Council grant GR/F A. Glindemann thanks the Deutsche Forschungsgemeinschaft for financial support.

6 548 J. Opt. Soc. Am. A/Vol. 9, No. 4/April 1992 REFERENCES AND NOTES 1. A. Labeyrie, 'Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images," Astron. Astrophys. 6, (197). 2. K. T. Knox and B. J. Thompson, "Recovery of atmospherically degraded short-exposure images," Astrophys. J. 193, (1974). 3. A. W Lohmann, G. Weigelt, and B. Wirnitzer, "Speckle masking in astronomy: triple correlation theory and applications," Appl. Opt. 22, (1983). 4. G. R. Ayers, M. J. Northcott, and J. C. Dainty, "Knox- Thompson and triple-correlation imaging through atmospheric turbulence," J. Opt. Soc. Am. A 5, (1988). 5. K.-H. Hofmann and G. Weigelt, "mage reconstruction from the bispectrum using an iterative algorithm and applications of the method to astronomical objects," in Digital mage Synthesis and nverse Optics, A. F. Gmitro, P. S. dell, and. J. LaHaie, eds., Proc. Soc. Photo-Opt. nstrum. Eng. 1351, (199). 6. C. A. Haniff, "Least-squares Fourier phase estimation from the modulo 2r bispectrum phase," J. Opt. Soc. Am. A 8, (1991). 7. A. Glindemann, R. G. Lane, and J. C. Dainty, "Least squares reconstruction of the object phase from the bispectrum," in Digital Signal Processing-91, V. Cappellini and A. G. Constantinides, eds. (Elsevier, Amsterdam, 1991), pp H. A. McAlister and W. Hartkopf, Second Catalog of nterferometric Measurements of Binary Stars (Center for High Angular Resolution Astronomy, Georgia State University, Atlanta, Ga., 1988). 9. J. C. Christou, "nfrared speckle imaging: data reduction with application to binary stars," Exp. Astron. 2, (1991). 1. R. H. T. Bates and F. M. Cady, "Towards true imaging by wideband speckle interferometry," Opt. Commun. 33, (198). 11. F. Roddier, "The effects of atmospheric turbulence in optical astronomy," in Progress in Optics XX, E. Wolf, ed. (North- Holland, Amsterdam, 1981), pp J. C. Christou, A. Y. S. Cheng, E. K. Hege, and C. Roddier, "Seeing calibration of optical astronomical speckle interferometric data," Astron. J. 9, (1985). 13. M. Haas, "Seeing diagnosis for calibration in speckle interferometry," Astron. Astrophys. 236, (199). 14. J. Meng, G. J. M. Aitken, E. K. Hege, and J. S. Morgan, "Triple-correlation subplane reconstruction of photon address stellar images," J. Opt. Soc. Am. A 7, (199). 15. J. C. Marron, P. P. Sanchez, and J. C. Sullivan, "Unwrapping algorithm for least-squares phase recovery from modulo 2r bispectrum phase," J. Opt. Soc. Am. A 7, 14-2 (199). 16. S. sobe, Y Norimoto, M. Noguchi, J. Ohtsubo, N. Baba, N. Miura, H. Yanaka, and T. Tanaka, "Speckle observations of visual and spectroscopic binaries," Publ. Natl. Astron. Obs. Jpn. 1, (199). 17. S. sobe, Y. Norimoto, M. Noguchi, J. Ohtsubo, N. Baba, N. Miura, H. Yanaka, and T. Tanaka, "Speckle observations of visual and spectroscopic binaries," Publ. Natl. Astron. Obs. Jpn. 1, (199). 18. J. G. Timothy and J. S. Morgan, "maging by time-tagging photons with the multianode microchannel array detector system," in nstrumentation in Astronomy V, D. L. Crawford, ed., Proc. Soc. Photo-Opt. nstrum. Eng. 627, (1986). 19. M. J. Perez-lzarbe and M. Nieto-Vesperinas, "Phase retrieval of photon-limited stellar images from information of the power spectrum only," J. Opt. Soc. Am. A 8, (1991). 2. The NAG library is available from NAG nc., 14 Opus Place, Suite 2, Downers Grove, ll