high-resolution reconstruction of photon-limited stellar images using phase gradients

7
High-resolution reconstruction of photon-limited stellar images using phase gradients G. J. M. AITKEN AND R. JOHNSON Departnletlt of Electrical Engineering, Queetl'.~ University, Kingston, Ont., Canada K7L 3N6 Received January 29, 1988 Phase-gradient and Knox-Thompson processes for determining the gradients of a stellar object's phase from speckle images are compared with respect to their behaviour at low photon levels. The phase-gradient process is immune to the degrading effects of image wandering and image recentering, and is computationally simpler and faster than the Knox-Thompson process. Examples of reconstructions from simulated and real speckle images are presented. Les mCthodes phase-gradient et Knox-Thompson pour dCterminer les gradient: de phase d'un objet stellaire a partir d'images speckle sont comparCes par rapport a leur fonctionnement pour de faibles nombres de photons. On trouve que la premikre mCthode est exempte des effets de degradation dus aux mouvements erratiques et au recentrage des images, et qu'elle est plus simple et plus rapide, du point de vue calcul, que la methode Knox-Thompson. Des exemples de reconstruction a partir d'images speckle rCelles et simulCes sont prCsentCs. [Traduit par la revue] Can. 1. Phys. 66, 847 (1988) 1. Introduction Atmospheric turbulence is the primary obstacle to high- resolution optical imaging of astronomical objects from the ground (1). Wavefronts arriving at a ground-based telescope &e severely distorted by the spatially and temporally varying refractive index of the atmosphere. Consequently, the instanta- neous point-spread function (PSF) of the telescope-atmosphere combination has an envelope of angular size A/ro, where A is the wavelength and ro is the coherence distance on the wave- front, typically about 10cm at a good site. When the telescope diameter D is much larger than ro, the PSF contains a fine structure of bright that are the size of the diffraction- limited PSF, -X/D, within the broader envelope. This pattern of bright patches, called speckles, changes within the envelope about every 10 ms. Conventional long-exposure images have a smooth PSF of width -A/ro and contain no finer angular infor- mation about the object's structure. Thus, a 4 m telescope like the Canada-France-Hawaii telescope has a very large light- collecting area but a resolving capability 40 times less than its theoretical diffraction limit. In 1970, Labeyrie (2) demonstrated that diffraction-limited information about the complex spatial spectrum of an object could be recovered from a series of short-exposure speckle images, whose exposure times were short enough, - 10 ms, to freeze the motion of the atmosphere. His speckle interferome- try (SI) measured the diffraction-limited modulus of the spatial spectrum and has been successfully applied to astronomical measurements where modulus alone is sufficient; for example, the case of binary or simple symmetric objects. The reconstruction of the image of an arbitrary object from its measured spectrum requires an estimate of phase as well as modulus. Phase-retrieval algorithms, such as those developed by Fienup (3), make use of other a priori information about the source, such as the positivity of an intensity distribution and the support of the image autocorrelation function, the latter being the Fourier transform of the modulus squared. While goodresults have been obtained with these techniques, particu- larly in two-dimensional (2-D) imaging, as yet there is no guarantee that the solutions are unique. The direct measure- ment of phase by phase tracking is quite impractical because of the rate and magnitude of the phase fluctuations, which have a standard deviation of several 2~ revolutions. A much more phase from a set of speckle images and then integrate to obtain the phase. The Knox-Thompson (KT) algorithm (4), intro- duced in 1974, finds a 2-D array of the phase differences between adjacent spectral components. This is equivalent to approximating the gradient of the phase function. More recently we have developed the phase-gradient (PG) algorithm (5, 6\, which estimates the true spatial derivatives of the phase from the speckle images. One advantage of the PG process is that it does not experience the degradation from image wandering that affects the KT process. Because speckle images must be recorded in very narrow band of light, AA/X < 0.05, with exposure times of 10 ms, most stellar objects of interest produce a relatively small num- ber of photon events per image. At these low photon rates, it is advantageous to use detectors that deliver the addresses of the individual photon events instead of intensity, and to execute the necessary computations in the image domain. In this paper the properties of the photon-address implementations of the PG and KT algorithms are compared in low-photon applications. The PG process is more computationally efficient than the KT process and gives less noisy images in the photon-address mode. 2. Phase-gradient processes The basic principles of SI, KT, and PG have been described elsewhere (2, 4, 5); therefore, only a summary is given here as a background to the comparison of properties presented later. As is customary, it is assumed that the object lies within the 2"-4" isoplanatic patch of the atmosphere, so the convolution relationship in = o * s, applies, where in is the nth image in a set of N speckle images, o is the object-intensity distribution, and s, is the PSF of the telescope and time-varying atmosphere combined. In the Fourier domain, I, = 0 .S, , where I,, 0, and S, are the Fourier transforms of i,, , o, and s,, respectively. The object spectrum has magnitude 101 and phase a. The SI, KT, and PG can be implemented as products of spectra or image cross-correlations on the individual speckle images, the former being most appropriate for strong sources while the latter has advantages at low photon fluxes. The SI estimates the average power spectrum of the speckle images, (11,12), by computing either the average of the modu- lus squared, successful approach has been to estimate the gradient of the [I] M = (11~1~)~ Can. J. Phys. 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Page 1: High-resolution reconstruction of photon-limited stellar images using phase gradients

High-resolution reconstruction of photon-limited stellar images using phase gradients

G. J. M. AITKEN AND R. JOHNSON Departnletlt of Electrical Engineering, Queetl'.~ University, Kingston, Ont., Canada K7L 3N6

Received January 29, 1988

Phase-gradient and Knox-Thompson processes for determining the gradients of a stellar object's phase from speckle images are compared with respect to their behaviour at low photon levels. The phase-gradient process is immune to the degrading effects of image wandering and image recentering, and is computationally simpler and faster than the Knox-Thompson process. Examples of reconstructions from simulated and real speckle images are presented.

Les mCthodes phase-gradient et Knox-Thompson pour dCterminer les gradient: de phase d'un objet stellaire a partir d'images speckle sont comparCes par rapport a leur fonctionnement pour de faibles nombres de photons. On trouve que la premikre mCthode est exempte des effets de degradation dus aux mouvements erratiques et au recentrage des images, et qu'elle est plus simple et plus rapide, du point de vue calcul, que la methode Knox-Thompson. Des exemples de reconstruction a partir d'images speckle rCelles et simulCes sont prCsentCs.

[Traduit par la revue] Can. 1. Phys. 66, 847 (1988)

1. Introduction Atmospheric turbulence is the primary obstacle to high-

resolution optical imaging of astronomical objects from the ground (1). Wavefronts arriving at a ground-based telescope &e severely distorted by the spatially and temporally varying refractive index of the atmosphere. Consequently, the instanta- neous point-spread function (PSF) of the telescope-atmosphere combination has an envelope of angular size A/ro, where A is the wavelength and ro is the coherence distance on the wave- front, typically about 10cm at a good site. When the telescope diameter D is much larger than ro, the PSF contains a fine structure of bright that are the size of the diffraction- limited PSF, -X/D, within the broader envelope. This pattern of bright patches, called speckles, changes within the envelope about every 10 ms. Conventional long-exposure images have a smooth PSF of width -A/ro and contain no finer angular infor- mation about the object's structure. Thus, a 4 m telescope like the Canada-France-Hawaii telescope has a very large light- collecting area but a resolving capability 40 times less than its theoretical diffraction limit.

In 1970, Labeyrie (2) demonstrated that diffraction-limited information about the complex spatial spectrum of an object could be recovered from a series of short-exposure speckle images, whose exposure times were short enough, - 10 ms, to freeze the motion of the atmosphere. His speckle interferome- try (SI) measured the diffraction-limited modulus of the spatial spectrum and has been successfully applied to astronomical measurements where modulus alone is sufficient; for example, the case of binary or simple symmetric objects.

The reconstruction of the image of an arbitrary object from its measured spectrum requires an estimate of phase as well as modulus. Phase-retrieval algorithms, such as those developed by Fienup (3), make use of other a priori information about the source, such as the positivity of an intensity distribution and the support of the image autocorrelation function, the latter being the Fourier transform of the modulus squared. While goodresults have been obtained with these techniques, particu- larly in two-dimensional (2-D) imaging, as yet there is no guarantee that the solutions are unique. The direct measure- ment of phase by phase tracking is quite impractical because of the rate and magnitude of the phase fluctuations, which have a standard deviation of several 2~ revolutions. A much more

phase from a set of speckle images and then integrate to obtain the phase. The Knox-Thompson (KT) algorithm (4), intro- duced in 1974, finds a 2-D array of the phase differences between adjacent spectral components. This is equivalent to approximating the gradient of the phase function. More recently we have developed the phase-gradient (PG) algorithm (5, 6\, which estimates the true spatial derivatives of the phase from the speckle images. One advantage of the PG process is that it does not experience the degradation from image wandering that affects the KT process.

Because speckle images must be recorded in very narrow band of light, AA/X < 0.05, with exposure times of 10 ms, most stellar objects of interest produce a relatively small num- ber of photon events per image. At these low photon rates, it is advantageous to use detectors that deliver the addresses of the individual photon events instead of intensity, and to execute the necessary computations in the image domain. In this paper the properties of the photon-address implementations of the PG and KT algorithms are compared in low-photon applications. The PG process is more computationally efficient than the KT process and gives less noisy images in the photon-address mode.

2. Phase-gradient processes The basic principles of SI, KT, and PG have been described

elsewhere (2, 4 , 5); therefore, only a summary is given here as a background to the comparison of properties presented later. As is customary, it is assumed that the object lies within the 2"-4" isoplanatic patch of the atmosphere, so the convolution relationship in = o * s, applies, where in is the nth image in a set of N speckle images, o is the object-intensity distribution, and s, is the PSF of the telescope and time-varying atmosphere combined. In the Fourier domain, I, = 0 .S, , where I,, 0, and S, are the Fourier transforms of i,, , o, and s,, respectively. The object spectrum has magnitude 101 and phase a.

The SI, KT, and PG can be implemented as products of spectra or image cross-correlations on the individual speckle images, the former being most appropriate for strong sources while the latter has advantages at low photon fluxes.

The SI estimates the average power spectrum of the speckle images, (11,12), by computing either the average of the modu- lus squared,

successful approach has been to estimate the gradient of the [ I ] M = ( 1 1 ~ 1 ~ ) ~

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Page 2: High-resolution reconstruction of photon-limited stellar images using phase gradients

848 CAN. J. PHYS. VOL. 66, 1988

or the Fourier transform (FT) of the average image autocorrela- tion function

where (.) denotes the true expected value, (.)N is the average over N images, and O is the cross-correlation of two functions. A set of speckle images made with an unresolved, neighbouring source provides an estimate, M,, of the speckle transfer func- tion ( 1 ~ ~ 1 ~ ) . The reference star may be observed immediately before and after the source under study. For a classical intensity distribution, the object modulus is given by

and an estimate of 101 is (M/M,)'I2. The KT algorithm computes the spectral-domain average

autocorrelation product

either from I,, or from a Fourier transform of the image cross- correlations as described by

where Aft is a small spatial-frequency shift such that Aft < ro/X, and 6 is x or y depending on whether the shift is parallel to f, or f,. The key to the KT process is that the angle of (PnS) is

Computation of (PIu)N and (P,ly)N generates a 2-D array of phase differences, Aa, and A%, in directions parallel to the coordinate axes. In general, Af can be in any direction and the corresponding Aa is evaluated.

The PG algorithm is based on the Fourier differentiation theorem, which is used to calculate

where [ takes the values x or y depending on whether differ- entiation is with respect to f, or fy. The desired quantity is the average product

which can be evaluated with I, and [7] in the spectral domain or with the corresponding image-domain correlations in the expression

It can be shown (5) that the derivative of the object phase is given by

The denominator in [ lo] is the image-power spectrum given by the SI process. Equation 1101 implies that the expected value of the phase of S, is zero, which is a valid assumption in practice.

In practical situations where classical intensity distributions cannot be assumed, photon noise introduces bias terms into (I1,l2)~, (Pne)N, and (D,[)N. Equations P I , 161, and [lo1 are valid estimators provided the bias terms are accurately esti- mated and subtracted.

It was mentioned earlier that a neighbouring, unresolved

reference source could be used to calibrate the measured power spectrum, (11,12)N, for the severe attenuation introduced by the speckle transfer function at f > ro/X. The PG and KT phase- measuring processes offer a further calibration possibility with the reference-source speckle data. The phase function obtained by processing the reference-source speckle images is a map of the aberrations in the telescope's transfer function, and can be used to compensate the phase computed for the resolved source. In other words, the PG and KT processes have the potential to restore images that have been degraded by telescope aberra- tions as well as atmospheric turbulence.

3. Photon-address mode At light levels of less than lo3 photons/image, the detectors

are designed to give the addresses of the individual photon events instead of the intensity distribution. The likelihood of two photon events occurring simultaneously at the same loca- tion in the detector is very small, although photon events may occur at the same address at different times during the expo- sure interval. In terms of individual photon events, the nth image can be described by

N"

[I 11 i,,(r) = 1 6 ( r - rk) k = l

where N, is the number of photons detected in the nth image and rk is the position of the kth photon event. Substitution of [I I] into [2], [5], and [9] gives the expressions defining the image-domain, photon-address mode of operation for SI, KT, and PG processes, respectively, as follows:

and

The noise bias terms at low photon levels are generated by the self-products of photon events, which occur when k = 1. Therefore, the bias terms can be eliminated by omitting all such contributions in 1121-[14]. Implementation of the cross- correlation computation is simply a matter of generating the 2-D histograms of address differences in the case of [12] and of weighted address differences in [ 131 and [ 141. Thus, provided only the k f 1 terms are computed, the quantities evaluated by [12]-[14] are free of noise bias and can be used directly to find 101, Aa, and a ' . Although terms arising from k = 1 are elimin- ated, cross-correlations between photons that have arrived at the same address at different times during the exposure interval are included.

4. Properties of KT and PG processes Spectral-domain processing with PG requires three Fourier

transforms per speckle image, whereas the KT process requires only one. However, in the image-domain mode, both pro- cesses require three correlation operations per image. It is in this mode that PG has some clear advantages. An inspection of [ 5 ] and [9] reveals that the image correlation in KT processes requires complex arithmetic and trigonometric functions, while

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Page 3: High-resolution reconstruction of photon-limited stellar images using phase gradients

AITKEN AND JOHNSON 849

the corresponding correlations in PG can be canied out com- pletely with integer arithmetic on real numbers. This means that PG can be executed with half the number of computational operations and half the memory size needed by the KT algorithm.

The KT process is based on the premise that for sufficiently small A f , the atmospherically induced phase fluctuations will be essentially the same at f and f + A f . Therefore, the phase difference between spectral components at f and f + Af can be measured despite the large absolute phase fluctuations they experience. As Af increases, these phase fluctuations become gradually uncorrelated until by Af - ro /A they are quite un- correlated, in which case (PI IS ) would be zero. Thus, a practi- cal compromise must be made between the sampling interval and the degradation in (PI IS ) that can be tolerated. The quantity (DI l t ) in PG has no such dependence on the degree of turbu- lence and hence ro. In fact, the spatial-frequency sampling interval used with PG could, in principle, be larger than ro /X, provided it is adequately sampling the object spectrum. This gives a greater degree of flexibility in choosing the number of samples and (or) the field of view required in a given observation.

In addition to the fine structure found in the speckle images, the images also appear to wander. If a centered image is defined as i , (r) and the shift due to wandering is w,, the spectrum of the shifted image is In,( f ) = In( f ) exp (- j 2 n f . w,,). Let Pg = (Pnt) to simplify the notation. Now the KT autocorrelation product takes the form

= Pg(exp ( - j 2 n A f t w,,t))

= PC exp (- 2n2A f ; ~ : )

where a& = (w;) = (w&) since the x and y components of wn can be considered to be statistically independent. Typical val- ues of u, would be about L / 4 , where L is the width of the image field, and Aft is 1/L, for which the estimate of (Pnt ) would be reduced by a factor of 0 .74 . If ( A f t u w ) were doubled by either increased wandering, an increase in the frequency shift, or both, the attenuation factor would be 0 . 2 9 . Because of this effect, it is desirable to shift the centroid of the speckle images to the centre of the field to preserve the contrast of P t .

Ayers et al. ( 7 ) have shown that at low photon levels, shift- ing the centroid of the individual speckle images before pro- cessing yields

where I( f ) = I( f ) / I ( O ) . This result differs from the desired low-level results,

[I71 Pg = (Nn(Nn - l ) f ( f + A f s > f * ( f >)

in that there is an additional noisy term and the shifts Afg/N,, produce errors in the phase-difference estimates. The defects become significant only when N,, is small, but this suggests that the technique cannot be used in its present form for images containing a small number of photons.

When the PG process is applied to the wandering image defined above, the derivative is

[18] ILSt = [I;( - j2nwngI, , ] exp (- j 2 n f .wn)

Let Dt = (Dnt) and the average product arising from the shifted quantities be Dwt , then

[I91 DwS = (Re {- jIhwgI,*,))

= Dg - 2 ~ ( w , , s ) ( 1 ~ l 1 1 2 )

where ( W , , ~ ) is the x or y component of the centroid of the long exposure image, and the second term in [19] is simply the phase derivative produced by a nonzero centroid, multiplied by ( ( 1 ~ ~ 1 ~ ) . Note that there is no degradation of the desired quantity Dt , just an added term that gives the true position of the source in the coordinate system if ( w , ) is not zero.

The effect of shifting the centroid, if used in the case of the process PG, can be determined at low photon levels using the Fourier transform of the image defined in [ l 1 1 ; that is,

N"

In = k = l exp (- j 2 n f .rk)

For the nth image, the centroid can be written as

Thus, for Nn # 0

The expected value of [22] is found by the method of Good- intensity, man and Belcher (8) in which the photon and atmospheric statistics are treated as being independent. Averages over the [23] P(rk) = d r photon statistics are taken first, followed by averages over the atmospheric fluctuations. The probability density of the photon position rk is the classical intensity distribution (i.e., Nn infinite)

I = in(rk)/In ( 0 )

of the nth image, normalized with respect to the integrated Since the photon positions rk and rl are statistically indepen-

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Page 4: High-resolution reconstruction of photon-limited stellar images using phase gradients

CAN. 1. PHYS. VOL.. 66, 1988

FIG. 1. Examples of image reconstructions from simulated speckle images: (a) PG and (b ) KT with N = 400; ( c ) PG and ( d ) KT with N = 200. Reprinted from ref. 6 with permission.

dent, averaging over photon statistics on an individual frame [24] and [25] should be compared with [16] and [17], respec- yields tively. In summary, the PG process is not degraded by image

wandering and does not require shifting the image's centroid. [24] DC< = ( ( N ~ - - 2, Re {-j[i;tii;< - i~,1(0)11~~/2~})rn In fact, at low light levels, shifting the is detrimental

where represents the average over the atmospheric fluctu- ations for all frames where Nn # 0. If N,, is independent of the atmospheric fluctuations, ((N, - l)(N,, - 2)),, can be reduced to {N2 - 2N + 2[1- exp (-N)]), where N is the average number of photons per image and the exponential factor takes into account the fact that shifting the centroid has no physical mean- ing when N, = 0. The first term gives the desired Dt, while the second term divided by (11~1~) is just the constant phase derivative, 2n(cnS), associated with the 6 component of the centroid of the average image. Thus, the effect of shifting the centroid is to remove the tilt due to the image offset in the coordinate system and to scale the desired result,

by the factor ((N, - l)(Nn - 2)),/(Nn(Nn - I)). Equations

and should not be used when applying PG processing. How- ever, if for other reasons the data is recentered, the PG process does not suffer the distortions inherent in the KT process at low levels, as indicated by the A fs/Nn shifts in [17].

Because speckle images must be recorded with a very nar- row bandwidth and short exposure times, most objects of inter- est deliver a relatively small number of photons to each individual image. Therefore, the speckle images are extremely noisy. It has been mentioned earlier that the noise bias terms of the desired quantities can be removed by omitting the self-correla- tion of the photons during the cross-correlation operations. Here, the variances of the fluctuating, bias-free estimates of (Ill2), Ps, and Dg are given for the case of low photon levels.

The variance of (11~1~) is

[26I u; = (11n14) - (11n12)2

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Page 5: High-resolution reconstruction of photon-limited stellar images using phase gradients

AITKEN AND JOHNSON

Equation [26] , and the expressions for the variances of Pg and Dg to follow, can be evaluated in the manner described by Nisenson and Papaliolios ( 9 ) and Ayers et al. (7). First, [20] is substituted into [26] . Then, averages are taken over the photon statistics with k # 1 to eliminate the bias terms, and the prob- ability distribution of [23] is applied independently to each rk. The process is straightforward but tedious, especially in the case of Pg and D g . Six distinct terms are produced in a:; how- ever, when N is small and f > ro /X, the dominant term is N 2 . All other terms are smaller by at least ( ro /D)2 .

The variance of Pg is

which reduces in a similar manner to eight terms, the dominant term being N 2 .

Finally, the variance of Dg is

Of the 14 different terms occurring in a&, the most important ones at low levels are

Note that in contrast to a: and a & , the dominant terms of a: depend on the derivatives of the long-exposure spectrum. In all cases the dominant, noise spectral-density functions are con- stants; that is, the photon noise is white. Expressions [26]-1281 give the variance for one image frame. When N speckle images are processed, the variances are proportional to 1 IN .

An estimate of the variance of the phase difference given by the KT process can be obtained by using the fact that the variance of a phasor's angle is approximately the fractional variance of the modulus when the fractional variance is less than 0.2. Thus, for N images

[30I ":,KT = a ~ g l l ~ g 1 ~ ~ For small A f , Ipgl = (111 2) , which can be expressed as (1)

when the object is an unresolved point source and f > ( ro /X ) . T( f ) is the optical tra.nsfer function (OTF) of the telescope. Applying [31] and a;< = N 2 to [30] gives

1321 a,,,, = N ( r O / ~ 1 2 T ( IN-'/^

in which N,, is the number of photons per speckle since ( ~ / r ~ ) ~ is approximately the number of speckles in an image (1 ) .

The variance of a; depends on the noise in both D,,( and 11,12 and is

[331 u2.g = + ( ( D ~ ) / ( ~ ~ ~ ~ ) ) * U : I N -

= ( 1 1 1 ~ ) - ~ [ u ; ~ + (a ; )2a: ]~- '

assuming that the fluctuations of D,,< and 11,,12 are uncorrelated. To examine a& further, one needs a model for the long- exposure or average spectrum, ( i ( f ) ) . The usual theoretical form is exp(-3.44( f / f I ) 5 / 3 ) , where f I = ro /X; however, this model is not valid at low spatial frequencies and has an un- defined second derivative at f = 0 . A Gaussian model that is a good approximation to the observed spectra is

[341 ( f ( f ) ) = exp (-4.4(f l f I i )2)

which has the same width as the theoretical model ate- ' of the

FIG. 2. Typical speckle image with 600 photons in a 256 X 256 pixel format. The image field is 2.48 X 2.48 arcsec.

peak value and is well behaved at f = 0 (10) . With the deriva- tives of [34] applied to [29] and the resulting expression for a& substituted into [33] along with [ 3 1 ] , the standard devia- tion of a; becomes

[35] = I f / ) + l ~ ; I l l ( N p s T ( f )N112)

A further simplification is to assume a point source so that a; = 0 . To facilitate comparison with the KT result, we can approximate the difference angle by CtjA f , where Af is the sampling interval. Thus,

[361 a,,,, = 2 . l ( A f l f / ~ l l ~ ~ ~ ~ ( f )N lI2l

In practice, the ratio ( f l / A f ) is usually in the range of 3-5. A comparison of [36] with [32] indicates that the dominant noise terms are essentially the same for the two processes at low photon levels. The PG process should have lower noise when ( f I / A f ) > 2. The KT noise given by [32] is really a lower bound, because the small amount of noise added by shifting the centroid, or the contrast reduction occurring if the images are not shifted to the centroid, has not been included in the analysis.

The phase function is found by means of a modified version of the relaxation algorithm described by Hudgin ( l l ) , which adjusts the phase values to conform to the array of phase differ- ences. The phase differences estimated by the KT process can be integrated directly. However, because the PG algorithm measures phase derivatives, it is necessary to determine the phase differences between sample points. If the object phase does not vary rapidly with spatial frequency, a simple linear estimation is adequate. A more accurate estimate of phase differences is obtained by integrating polynomials that have been fitted to the derivatives at up to six neighbouring sample points. These extra steps in the process add very little to the

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Page 6: High-resolution reconstruction of photon-limited stellar images using phase gradients

CAN. J . PHYS. VOL. 66, 1988

FIG. 3. Reconstructions of the binary star P-Del: ( a ) PG and ( b ) KT with = 600; (c) PG and (d) KT with N = 150. The field shown is 0.62 x 0.62 arcsec.

overall processing time because they are done only once, at the end of the process.

A source of error in the PG process would be inaccuracies in the estimate of the denominator of [ lo] from which a[ is computed. The problem is not a matter of random noise, which affects the KT and PG process in much the same way, but one of incorrect photon-noise bias removal. An incor- rect bias on the modulus estimate directly alters the phase gradients. Fortunately, in the photon-addressmode, the photon- noise bias can be removed exactly on a frame-by-frame basis by omitting the photon self-products. No problems of this sort have been experienced on our applications of the photon-address PG process to simulated and real data.

5. Image-reconstruction examples The PG and KT algorithms have been implemented in the

photon-address mode and applied to the reconstruction of simu- lated and real speckle images. Simulated images were gener- ated on a FPS 5 100 array processor hosted by a VAX 1 11750 computer. The speed of the array processor was necessary to produce large numbers of images for various object and atmo- spheric parameters in a reasonable time. However, to avoid the delays introduced by using virtual memory in our multi-user facility, we had to limit the image size to 64 x 64 pixels, the

capacity of the array processor's memory. Atmospheric effects were simulated by generating a random phase at each point in the aperture and then applying a ~ a u s s i a n filter to obtain a specific ro.

The two reconstruction processes are installed on the array processor and on a SUN 31160. In the latter case, the image cross-correlations are executed in integer arithmetic with the result that the PG algorithm runs two times faster than the KT one when processing real, 256 x 256 pixel, speckle images.

Figures 1 a and 1 b show the images reconstructed by the PG and KT algorithms, respectively, from 1000, 64 X 64 pixel simulated, photon-limited, speckle images. The source is a pair of compact unresolved bright objects separated by 10 pixels and having peaks in the ratio 2 : l . The average number of photons is N = 400, and the number of speckles in an image is about 40. In Figs. 1 c and 1 d, N has been reduced to 200. Since the estimated modulus is the same in both processes, it is evident that the phase reconstruction by the KT algorithm is noisier.

The two processes have been applied to 4500 frames of photon-address data from Beta Delphinus (P-Del), acquired with a multi-anode, multichannel analyzer (MAMA) detector on the Steward Observatory's 2.3 m telescope. N is 600 photons/ image, and the reference-point source is Zeta ~ e l ~ h i n u s for

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Page 7: High-resolution reconstruction of photon-limited stellar images using phase gradients

AlTKEN AND JOHNSON 853

which 10 000 frames at & = 300 are processed. Figure 2 shows a typical speckle image containing 600 photons in a 256 X 256 pixel format. The occasional extra high event represents two photons that arrived at the same address at different times. Figures 3a and 3b show contour plots of the images reconstructed by the PG and KT processes respectively. Only the central region corresponding to 64 x 64 pixels of the original image field is shown. The larger component of P-Del is visible at the center of the field; just below it is the smaller component, whose height is about 0.3 of the larger peak. Above the large peak and placed symmetrically with respect to the smaller peak is a ghost artifact, typical of double-star reconstructions with noisy phase information. The contours are in steps of 0.05 with the peak at 1 .O, and the field shown is 0.3 1 X 0.3 1 arcsec.

To explore the performance of the two processes at lower light levels, we have repeated the reconstructions using every fourth photon in the lists. The results for 150 photons/image are shown in Figs. 3c and 3d. Note that the ghost in the PG reconstruction is less prominent than in the KT reconstruction, where it has risen to the 0.2 contour. The bridge that has appeared between the two components indicates that the noise at 150 photons/image is just beginning to degrade the resolv- ing capability. The P-Del speckle images had been shifted to the centroid as part of the data pre-processing at Steward Ob- servatory. Therefore, the wandering degradation that affects the KT process has already been eliminated. At the same time, the recentering has removed the absolute position of the object within the observed field. Noise arising from this recentering in these examples is negligible because of the relatively large number of photons.

wandering does not degrade the PG results, while it reduces the KT autocorrelation product if it is not compensated by recentering of the individual images. Recentering generates its own kind of noise, which becomes significant for &, in the order of 10 photons/image. In this operating range, the PG process does not suffer the distortion affecting the KT one, although for other reasons the data must be recentered. The primary advantage of the PG process over the KT process is its higher computational speed and lower memory requirement when implemented in the photon-address mode. The PG method is expected to have a noise performance comparable to, or slightly better than, that of the KT algorithm; this is confirmed in the computer simulations and real data examples presented here.

Acknowledgements The authors wish to thank E. K. Hege, Steward Observatory,

Tucson, AZ, for the P-Del speckle images, and to acknowl- edge the crucial role of the MAMA image detector provided by J. Morgan, Centre for Space and Astrophysics, Stanford Uni- versity, Stanford CA, in the data acquisition. This work is sup- ported by grants from the Natural Sciences and Engineering Research Council of Canada.

1. F. RODDIER. Prog. Opt. 19, 281 (1981). 2. A. LABEYRIE. Astron. Astrophys. 6, 85 (1970). 3. J. R. FIENUP. Appl. Opt. 21, 2758 (1982). 4. K. T. KNOX. J. Opt. SOC. Am. 66, 1236 (1976). 5. G. J. M. AITKEN, R. JOHNSON, and R. HOUTMAN. Opt. Com-

mun. 56, 379 (1986). 6. G. J. M. AITKEN and R. JOHNSON. Appl. Opt. 26,4246 (1987).

6. Conclusion 7. G. R. AYERS, M. J. NORTHCOTT, and-J. C. DAINTY. J. Opt. SOC. Am. A, 5, 963 (1988).

The PG and KT are two methods of estimating 8. J. W. GOODMAN and J. F. BELSHER. Proc. SPIE Int. Soc. Opt. the gradients of spectral phase from a set of stellar speckle Eng. 75, 141 (1976). images. At low light levels where the image detector can deliver 9. P. NISENSON and C. PAPALIOLIOS. Opt. Commun. 47,91 (1983). the addresses of the individual photon events, the PG process 10. N. J. WOOLF. Annu. Rev. Astron. Astrophys. 20, 367 (1982). appears to have several advantages over the KT one. Image 11. R. H. HUDGIN. J. Opt. SOC. Am. 67, 375 (1977).

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