October 1, 018 Page 750 Image Reconstruction Using Bispectrum Speckle Interferometry: Application and First Results Roberto Maria Caloi robime@iol.it Abstract: 1. Introduction For rapid prototyping and deployment of all required calculations, I have used Matlab, which has provided me a very good compromise between performance and ease of development.. Image Formation The image reconstruction process is essentially an inverse problem, since the main task is to recover the
October 1, 018 Page 751 original image based on the result of observations. Several noise sources make the recovering process subject to uncertainties and biases. In order to appreciate all the data and assumptions necessary to deal with these aspects, it helps to review the main physical processes involved in the image formation on the detector that are later considered to recover the original image. Other refinements could be evaluated, in particular on how to model the actual functioning of a real detector, but they are not considered here. x ' + y ' R '0 it ( x, y ) = o( x, y ) PSFt ( x, y ) [] Given the incoming intensity distribution itλ(x, y), the actual number of photons detected per pixel will depend also on the quantum nature of light and on the actual detector used, whose main characteristics are the following: level of dark current generation, quantum efficiency, electronic read-out noise, electronic biases, presence of defects and hot pixels. These effects are particularly important when the average total number of detected photons per speckle Nph is low, some hundreds or thousands to give an order of magnitude, and must be dealt with in order to avoid estimation biases (which do not disappear by simply increasing the number of recorded images used in the data reduction process). If itλ(x, y) is normalized so that its total value over the image plane is one, the expected number of photoelectrons itphe(x,y) is then given by itphe ( x, y ) = N ph o( x, y ) PSFt ( x, y ) + nth [3], i ( x, y ) = o ( x, y ) PSF ( x, y ) [1] summing all three previous components and dividing by the detector gain G, itadu ( ) ( ) P i phe ( x, y ) l ( x, y ) + N 0, + 1 e [4] ( x, y ) = t b ( x, y ) G DC
October 1, 018 Page 75 is assumed to be statistically independent. Under discrete Fourier transform conditions, which hold in our case, their proposed estimator becomes ( ) t ( ) = t ( ) t ( ) + e ( ) [5] x S u I u i x x 3. Power Spectrum Estimation A data cube {i t ADU (x)} M t=1 of M short exposure frames {d t ADU (x)} L t=1 obtained with the telescope aperture covered is needed. The first step in the image reconstruction process requires the computation of the power spectrum, both its average and variance. Each frame i t ADU (x) is first corrected for the gain, G, the bias b DC (x) and the flat field l(x), to yield the actual photo-electrons plus read-out noise per pixel i t ( x) ADU t It is subsequently cropped within a square N t ( ) DC ( ) l( x) Gi x b x = Here we assume that N t th and N t dth have the same expected values. The discrete Fourier transform I t (u) = DFT(i t (x)) where u indicates the biinherent in the detection process. Gordon and Buscher (01) derived a formula for a bias-free power spectrum estimator S t (u), when photon noise, thermal noise (dark current), and read-out noise are all taken into account. The main stated assumption in their study is that the various noise sources considered are independent and additive. Moreover the overall noise on different pixel positions where x indicates the pixel position, i t (x) the recorded intensity, and σ e (x) is the read-out noise variance. Under the additional assumption that the read-out noise does not depend on the detector position, i.e. σ e (x) = σ e, equation 5 becomes t ( ) t ( ) t pix e S u = I u N N [6] where N t = N ph th t + N t is the total number of photoelectrons and thermal electrons per frame and N pix is the total number of pixels. By averaging S t (u) we obtain an unbiased estimate of the power spectrum S = I( u) N N pix e [7] where the operator represents averaging over the entire sequence of specklegrams. The sample variance of such estimate var(s) is then used to calculate the frequency dependent power spectrum standard deviation σ S (u), ( ) S ( u ) = var S ( u ) / M [8] where M is the number of frames. The resulting power spectra has a central peak which is given by (Ñ ph + Ñ th ) where Ñ ph is the average number of photo-electrons per frame and Ñ th = Ñ dth is the average number of electrons generated by the thermal noise. By correcting the central peak component for the thermal noise, the final estimate of the power spectrum S(u) is obtained. No further correction is necessary outside the central peak if the thermal noise is spatially uncorrelated, so that its power spectrum is negligible for spatial frequencies different from zero. The same analysis just described is repeated for a nearby reference star, which must be observed with similar seeing conditions as the object under study. Once we have estimated the object power spectrum S obj (u) and its variance σ Sobj(u) as well as the reference star s power spectrum S ref (u) and its variance σ Sref(u), we can apply the standard procedure of speckle interferometry (Labeyrie, 1970) which yields an estimate of the true object power spectrum O(u) = S obj (u)/ S ref (u) up to the telescope cut-off frequency, and its variance σ O(u)
October 1, 018 Page 753 O (u ) S (u ) Sref (u ) = obj + O(u ) Sobj (u ) Sref (u ) [9] Bt (u, v) = I t (u ) I t (v) I t* (u + v) I t (u ) I t (v) I t (u + v) + Nt + 3N pix e + C (u, v) [10] C = 3Nt N pix e from which the image could be reconstructed by applying the inverse Fourier transform. 4. Bispectrum Estimation Let us first recall its definition. Given an intensity distribution i(x) and its Fourier transform I(u), the bispectrum is defined as I(3)(u,v) = complex conjugate of I(u). Similarly, for the true object intensity distribution o(x) we have O(3)(u, v) = O(u)O(v)O*(u + v) = O(3)(u,v) exp[βo(3)(u,v)]. It has been shown by Lohmann et al. (1983) that the phase of the time-averaged bispectrum of the atmospheric transfer function is close to zero while its modulus is not and, as a consequence, the phase of the observed average bispectrum βi(3)(u,v) is equal to the phase of the object bispectrum βo(3)(u,v). This relation holds for u, v, and u + v smaller than the telescope cut-off frequency. They also describe an iterative computation to recover φ(u) from βo(3)(u,v). This property highlights the importance of the bispectrum for image reconstruction. where, following the same convention used in the previous section, Nt = Σx it(x) is the total number of photoelectrons and thermal electrons recorded per frame, Npix is the total number of pixels, and σe is the read-out noise variance. By averaging (10) over t, we obtain an unbiased estimate of the bispectrum. Finally, taking into account (6), we get B(u, v) = I (u ) I (v) I * (u + v) S (u ) S (v) S (u + v) N + C * (u, v) lus [11] We can now combine the object s visibility modu- O (3) (u, v) = O(u ) O(v) O(u + v) ei (u,v ) 5. Image Reconstruction of distance dk. [1]
October 1, 018 Page 754 dk = Ok(3) (u, v) O (3) (u, v) O (3) (u,v ) for the bispectrum. du dv [13] O(3) (u,v ) = 0.5 S (u ) S (v) S (u + v) S (u ) S (v) S (u + v) S (u ) + S (v ) + [14] in this study, i.e. triple and binary stars. Because the image is built by changing one pixel at a time, some details that are not S (u + v) β(u, v) (u,v) O(u) O(v) O( u v) eiβ and its Smooth the resulting image with the theoretical PSF of the telescope to avoid super-resolution artifacts 6. Simulation O (3) (u,v ) = O(3) (u,v ) + O(3) (u, v) (u,v ) [15] which holds when both modulus and phase variances are relatively small. A better approximation, which takes into account the wrap for the preliminary results reported in the following sections. Pauls et al., (005) propose a different model, where σ O(3)(u,v) and σβ(u,v) are reported and used separately in order to arrive to a more general error model Ai ( x xi ) o( x) = i =1 n i =1 Ai n [16]
October 1, 018 Page 755 where xi represents the position and Ai the intensity of each star s component i. Imaging equation 1 is then used. The PSF of each simulated frame is generated using a complex transmission function with a modulus defined by the geometry of the telescope aperture (primary and secondary mirror diameters) tion in the entrance pupil, the phase screen, which represents the effect of the atmospheric turbulence. The algorithm used to generate the phase screen is based on the Fast Fourier Transform method (McGlamery, 1976). The FFT method computes the phase screen by means of the inverse Fourier transform of the product of a circular complex Gaussian random noise with zero mean and unit variance and the square root of the phase power spectrum density Wφ(f ). For this purpose we can use Kolmogorov s law for energy dissipation in a viscous medium W ( f ) = 0.03 ro5/3 f 11/3 mentation of the image reconstruction algorithm. (u,v) [17] where ro is the Fried s parameter. This model is assumed to hold for spatial frequencies 1/Lo < f < 1/lo, where Lo is the outer scale and lo is the inner scale of turbulence. 7. Comparison to Data The application of the previous simulation and data analysis steps to known objects through simulations, as well as the comparison with the results obtainable with other analysis methods in the special case of binary stars, has been very helpful to correct initial coding errors and to check the actual performance of my imple- As an additional test and in order to verify the performance of the reconstruction algorithm with a more complex object, I considered also the case of a triple star with components AB and 0.8 arcsec respectively, position angles θab=3o,
October 1, 018 Page 756 Table 1. Binary star parameters estimated from 00 frames generated under different simulation conditions: (a) N ph =30000, nth=15, σe=10, bdc =10, r 0 =.5 cm; (b) N ph =3000, nth=5, σe=7, bdc =15, r 0 =.5 cm; (c) N ph =1000, nth=5, σe=7, bdc =15, r 0 =5.0 cm; (d) N ph =500, nth=, σe=, bdc =5, r 0 =5.0 cm # ρ PS θ PS ρ BS 1. 3 0.55 1.16 1.7 0.71 1.17.1 0.63 1.17 1.9 0.59 1.3 10 0.5 1.5 10.7 0.7 1.30 06.0 0.41 1.3 08.1 0.0 1.3 10 0.5 1.1 11.1 0.1 1.6 11. 0.3 1.6 11.1 0.13 0.8 15 1.0 0.80 14. 0.98 0.81 11.3 0.87 0.80 11.5 0.76 θ AC =15 o, and with a magnitude difference Δm AB =0.55 and Δm AC =0.99. An example of a specklegram simulated by using these parameters is shown in Figure 1. The estimated power spectrum, after bias subtraction and calibration is shown in Figure. As can be seen, the actual usable region is smaller than the theoretical maximum one given by the telescope cut-off frequency - which in this case corresponds to a circle with radius of about 15 pixels. It is the measured variance that gives the weight to assign to different regions when the data reduction process is carried out. The result is in any case sufficiently defined to be somewhat different, as expected, from the typical fringe pattern of a simple binary star. The image reconstructed using the building block method is shown in Figure 3. The estimated values of the magnitude differences from the reconstructed image are m AB =0.54 and m AC =0.97, which are very close to the actual values of the simulated object. 8. Experimental Setup and Calibration. The image reconstruction requires some input parameters that must be calibrated before the algorithm is executed. These estimates are given in Table 3. D x D fc = = K f where D is the telescope primary diameter, Figure : Estimated power spectrum (log scale) of a triple star from a sequence of 00 simulated specklegrams.
October 1, 018 Page 757 Table. Observation Nominal Parameters Parameter Pixel size (square shape) 5.6 µm 5.6 µm Table 3. Calibrated parameters. Gain G is estimated for a nominal setting of 103 on the DMK1AU04.AS CCD camera. Figure 3. Reconstructed image of a simulated triple star. ( ) G e / ADU = c c where μc and σc are the mean and the variance respectively of the total counts, per given and requires measurement of μc and σc as before under different light conditions. The gain is estimated as the slope of the linear regression of μc vs σc. Parameter Symbol Value Image scale K 0.17 as/pixel Detector gain G.6 e-/adu Theoretical cutoff frequency fc 0.4594 9. Comparison to Measured Data I have applied the image reconstruction procedure to video sequences captured during three different nights with the equipment already described in section 8. Due to the small size of the telescope, all objects considered have been binary stars. More complex objects, like close triple stars could become interesting targets if at least a medium size telescope with an aperture greater than 40 cm were used.
October 1, 018 Figure 4: Average power spectrum of STF1909. engine available at http://stelledoppie. Figure 5: Reconstructed image of STF1909. Page 758
October 1, 018 Page 759 Table 4: Estimated binary star parameters vs. known values from the WDS catalog using image reconstruction (BBM) and model fitting both the power spectrum (PS) and bispectrum (BS). The number of observations per night N obs is given in the last column. Disc Date WDS BBM PS BS ρ(") θ(º) Δm ρ(") θ(º) Δm ρ(") θ(º) Δm ρ(") θ(º) Δm STT 413AB 016.479 0.9 359.5 1.53 0.9 359. 1.63 0.94 1.44 1.36 0.9 1.6 1.13 6 STF1998AB 016.479 1.09 6.3 0.9 1.09 4.9 0. 1.11 6.6 0.31 1.09 6. 0.01 4 STF3050AB 016.83.41 340.6 0.6.44 340.8 0.85.44 340.8 0.99.44 340.9 0.84 STF 8 016.83 0.68 30.1 0.65 0.66 305.8 1.53 0.63 303.6 1.11 0.63 303.4 1.1 STF 8 018.078 0.64 304.6 0.65 0.6 304. 1.6 0.7 305.1 1. 0.71 303.5 0.74 STF 333AB 018.078 1.33 09.9.65 1.3 09.7.4 0.88 05.8 0 0.89 04.48 0 STT 15 018.078 1.57 177.89 0.1 1.5 175.1 1.5 1.54 173.7 1.8 1.48 174.8 1.1 STF1687AB 018.078 1.18 00.16 1.93 1.16 197.7.36 1.08 194.1 0.7 1.08 199.8 0.58 STF1909 018.078 0.5 80.84 0.9 0.48 80 0.8 0.41 77.7 0.9 0.44 77.5 0.9 Nobs tion can be stopped once the rate of decrease of dk reduces significantly. 10. Conclusions Figure 6. Residual value of the cost function dk 1. References 11. Acknowledgements The author wishes to thank Carlo Perotti for his support during observations and Karl-Ludwig Bath for useful discussions in the early stages of this work. This research has made use of Cartes du Ciel, the Washington Double Star Catalog, and the double star database Stelle Doppie. Caloi, R. M., 008,, 4-3, 111-118. Genet, R. M., 015,, 11-1S, 66-76. Astrophys., 541, A46. Astron.
October 1, 018 Page 760 Astron. Astrophys., 78, 38. Labeyrie, A., 1970, Astron. Astrophys., 6, 85. McGlamery, B. L., 1976, Proc. SPIE 0074 Image Processing. Stelle Doppie,. 017, Soc. Am. A, 34, 6. Weigelt, G., 1977, Opt. Commun., 1, 55. Wirnitzer, B., 1985, J. Opt. Soc. Am. A,,14.