knox-thompson and triple-correlation imaging through...

Vol. 5, No. 7/July 1988/J. Opt. Soc. Am. A 963

Knox-Thompson and triple-correlation imaging throughatmospheric turbulence

G. R. Ayers, M. J. Northcott, and J. C. Dainty

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

Received August 24, 1987; accepted December 7, 1987

A comparative study is made of the Knox-Thompson and triple-correlation techniques as applied to imagerestoration. Both photon-noise-degraded imaging and atmospheric-turbulence-degraded imaging are considered.The signal-to-noise ratios of the methods are studied analytically and with the aid of computer simulations. Theability to retain diffraction-limited information on imaging through turbulence is considered in terms of phase-closure relationships. On the basis of this work it is found that the two image-restoration techniques are effectivelyequivalent.

1. INTRODUCTION

The problem of imaging through turbulent media has beenstudied for many years. Of particular interest here is theimaging of objects through atmospheric turbulence. A de-tailed description of this effect is given by Roddier.'

As atmospheric conditions vary continuously, the ulti-mate resolution obtainable from conventional long-exposureimaging with a large telescope is severely limited by turbu-lence effects. However, if short-exposure, narrow-optical-bandwidth images are recorded, atmospherically inducedwave-front perturbations may be frozen. The turbulence-imposed resolution limit is therefore removed, permittingdiffraction-limited resolution to be obtained.

It is well known that this type of imaging of a point-sourceobject results in a speckled image because of interferenceeffects. This is a consequence of the complex amplitude inthe pupil of the imaging optics consisting of typically manyuncorrelated regions, or seeing cells; the size scale2 ro of theseregions increases as atmospheric conditions improve. Thelong-exposure angular resolution limit imposed is approxi-mately ro/X, which is typically much less than the diffractionlimit D/A of a telescope of diameter D.

In the technique of speckle interferometry3 the diffrac-tion-limited information present in short-exposure images isused to obtain the spatial autocorrelation of astronomicalobjects even in poor seeing conditions, i.e., at low light levelsand in the presence of telescope aberrations.4 During thepast 15 years, many speckle imaging techniques were pro-posed (see Ref. 5 for further details), although none of theseis in routine use. Two of the more promising techniques arethe method of Knox and Thompson6' 7 (KT) and a methodfirst suggested by Weigelt et al., 8-'0 which uses the conceptof triple correlation (TC).

Both techniques are analogous to speckle interferometry,as they involve the averaging of correlation functions ofmany short-exposure images. Additionally, as a conse-quence of phase closure," all three techniques possess trans-fer functions that permit diffraction-limited information tobe obtained even if ro is small. The KT and TC methodspermit direct recovery of the Fourier phase of the intensitydistribution of an object, as well as its Fourier modulus, from

their associated ensemble-averaged correlation functions.If the spatial autocorrelation of the object alone is known,then such phase information is potentially obtainable onlythrough indirect methods12 and by making assumptions con-cerning the intensity distribution of the object. For boththe KT and TC methods, rigorous analyses3" 4 show that theintensity distribution of an object is uniquely related, apartfrom trivial ambiguities, to the relevant averaged quantity ineach case (although there is no guarantee that current algo-rithms reconstruct this unique intensity distribution, partic-ularly in the presence of noise).

Although these correlation techniques make diffraction-limited information available, its quality must be assessedbefore it can be used effectively. A common measure ofquality used is the signal-to-noise ratio (SNR) at each spa-tial frequency u in the Fourier transform (S(u)) of theparticular average correlation being considered. The SNRmeasure can be defined as

SNR- I(S-(u))I,as

where ars is the standard deviation of the signal S(u). Thismeasure permits the comparison of similar techniques andadditionally gives the relative weight of the informationpresent so that it can be used optimally.

A consequence of using short exposure times and of thelow luminosity of many astronomical objects of interest isthat few photon events are recorded in individual speckleimages. Photon noise is therefore introduced that can seri-ously lower the obtainable SNR and as a result degrade thequality of the reconstructed image. Despite such noise,correlation techniques have been used successfully to studylow-luminosity astronomical objects (for example, see Ref.15).

The aim of this paper is to compare and relate the KT andTC methods with particular reference to the SNR's of theirassociated averaged correlations. The evaluation of theSNR's requires the ensemble averaging of many differentimage correlations' over both the Poisson statistics of thephoton events and the statistics of atmospheric turbulenceeffects. Fortunately, recent work16' 7 has associated the

0740-3232/88/070963-23$02.00 ©9 1988 Optical Society of America

Ayers et al.

964 J. Opt. Soc. Am. A/Vol. 5, No. 7/July 1988

much-used idea of phase closure in radio astronomy with thecomparatively new ideas of TC's and phase-closure pupil-plane interferometry as used in the optical and infraredwave bands. This association can be extended and general-ized to permit the approximation of the ensemble averagesof many different image correlations on the basis of phase-closure relationships, which was previously quite difficult.

The effect of photon noise on speckle interferometry wasinvestigated originally by Goodman and Belsher'8 ; theirwork was extended by Dainty and Greenaway1 9 with partic-ular reference to the problem of photon bias. Photon bias isa consequence of correlating photon events with themselvesand is present in both the KT and the TC average signals.Evaluation of the ensemble-average signals has been facili-tated by the development of computationally efficient pho-ton-coordinate-differencing algorithms,20 together withmodern photon detectors, such as the precision analog pho-ton-address (PAPA) detector," which returns a stream oftime-tagged photon coordinates. An increased SNR can beobtained by using the time-tag information." Previouswork23 -25 on the effects of photon noise and atmosphericturbulence was done for the KT and TC techniques. How-ever, only simplified approximations were obtained for en-semble averages of image correlations over the photon noiseand atmospheric statistics. Previous studies were con-cerned with only specific regions in Fourier space of theensemble-averaged signals; the work on the TC was con-cerned with only regions having the lowest SNR, which, infact, are not used in practice.26 Previous studies of the KTmethod involved only the basic two-dimensional correlationtechnique as originally proposed, 6 whereas the techniquecan be generalized completely to give a four-dimensionalaverage signal that contains considerably more informationand is analogous to the four-dimensional TC signal.

This paper is organized as follows. In Section 2 the math-ematical forms of the TC, the KT correlation, and the auto-correlation are defined in both image space and Fourierspace. A formalism for the TC is put forward that permits itto be related simply, in image space, to the KT correlation.The correlation techniques are shown diagrammatically andmathematically to be related intimately, each being simply acorrelation of an image with itself weighted by complexexponentials. Another technique,27"28 known as the phase-gradient method, is included and is shown to be a subset ofthe more general KT method.

In Section 3 the implementation of the correlation tech-niques at a low light level is described. Of particular impor-tance is the development of a differencing algorithm, whichpermits 'the evaluation of individual sections or subplanes ofthe Fourier transform of the 'l'C, known as the bispectruin;the full TC differencing algorithm is reduced to a weightedKW' implemnentation. This has the advantage that only thehigher SNR regions need be calculated, as opposed to thewhole of the bispectruin, which must be evaluated if a fulldifferencing algorithm is implemented. This problem canbe overcome by evaluating the required bispectrum planesdirectly in Fourier space. Such an evaluation, however,requires that each frame be Fourier transformed, a processthat becomes less efficient relative to photon differencing atlow light levels.

The effect of photon noise on SNR's is considered inSection 4. An estimate for the error of the phase of a com-

plex signal is proposed, and its dependency on photon noiseis studied for the KT and TC signals. The phase-errorestimate is shown to be related to the SNR. Averaging ofvarious image correlations over the photon statistics is facili-tated by a computer implementation of a generalization ofthe method of Goodman and Belsher,' 8 thereby making itpossible to extend the work on photon-noise dependency.

In Section 5 the imaging of a randomly translating objectis considered briefly. The necessity to centroid frames be-fore carrying out the KT processing is explained, as are theproblems associated with centroiding images in the presenceof photon noise.

In Section 6 the use of the KT and TC for imaging throughatmospheric turbulence is studied. The diffraction-limitedinformation present in the transfer functions of speckle in-terferometry, the KT, and the TC is shown to be a directconsequence of the concept of phase closure. Extensive useis made of a computer simulation of the imaging of objectsthrough atmospheric turbulence, using a large telescope.The dependence of SNR's and transfer functions on thenumber of photon events per image, N, and the mean num-ber of speckles per image, fi, is studied analytically and bymeans of Monte Carlo-type computer simulations.

Finally, in Section 7 results are compared, and conclusionsare drawn about the relative merits of the speckle imagingtechniques with respect to the areas studied.

2. DEFINITIONS AND RELATIONSHIPS

Consider a two-dimensional intensity distribution, i(x), andits Fourier transform, I(u), defined by

(u)== J i(x)exp(-27rjux)dx.

Both the TC8-10 and the KT correlation6' 7 may be definedin image space as correlations of i(x) or in Fourier space asproducts of I(u). The TC and its Fourier transform, thebispectrum, are defined by

iTC(Xl, X2) = J * x)i(x + xl)i(x + x2)dx,

ITC(U1, U2 ) = I(ul)I*(Ul + U2 )I(U2).

(2)

(3)

In an analogous way the KT double correlation and its Fou-rier transform are defined by

iKr(xl, Au) =| i*(x)i(x + xi)exp(27rjAux)dx, (4)

IPi(u,, Au) = I(u,)I*(ul + Au), (5)

where xl = xi, + xly and x2 = x2x + x2. are two-dimensionalspatial coordinate vectors; ul = ul, + uly, u2 = U2x + u2y, andAu = Au, + Auy are two-dimensional spatial-frequency vec-tors; and the superscript * indicates the complex conjugate.The x and y subscripts denote orthogonal component vec-tors.

It is apparent that Eqs. (1)-(5) are four dimensional.This fact causes implementation problems, extensive evalu-ation-time and data-storage requirements, particularly ifthe correlations are performed by using digitized images on acomputer. The basic KT technique does, however, consistonly of evaluating the three subplanes in Fourier space cor-

Ayers et al.

(1)


trum plane corresponds to a weighted version of the KTproduct in Fourier space. The complex weighting factor is,however, signal dependent and as such possesses a numberof important properties, which will become evident in latersections of this paper.

In order to understand the correlation techniques it isinstructive to consider the autocorrelation process. Thiscorresponds to the double correlation represented by Eq. (4)with the substitution Au = 0. The resulting representationsin the image and Fourier domains are

iA(xl) = iKT(Xl, 0) = i*(x)i(x + x1)dx,

IA(ul) = IKT(ul, 0) = I(ui)I*(U 1).

(8)

(9)

The autocorrelation of i(x) is the correlation of i(x) and i(x)multiplied by a complex exponential factor with zero spatialfrequency [see Fig. 2(a)]. Unfortunately the complete spa-tial symmetry of the operation prevents the preservation ofFourier-phase information. The argument of Eq. (9) is al-ways zero,

argIIA(uj)I = .A(Ul) = q(U1) -4(U 1 ) = 0, (10)

where 1(u) is the phase of 1(u) and arg$ I defines the phase ofthe complex number mod 27r.

Now consider the KT correlation of Eq. (4). This definesthe correlation of i(x) and i(x) multiplied by a complex

7: '<4�'

Fig. 1. Corresponding subplanes of (a) the bispectrum (u, = ul, U2= U2X) and (b) the KT (u, = ul,,, Au = Au.) four-dimensionaltransfer functions of a circular-aperture lens.

responding to Au = Aux, Au = Au,, and Au = Au, + Au.; the

Au vector components have constant values. If digitizedimages are used, then Au normally 7 corresponds to the fun-damental sampling vector interval. Further, different val-ues of Au may be used, and a number of subplanes can beevaluated. This is analogous to the many subplanes con-tained in the four-dimensional bispectrum (see Fig. 1).

The relationships between the two methods become moreapparent if Eq. (2) is rewritten by making the substitution U2

= Au. A single plane of the bispectrum can therefore bedescribed by

ITC(u 1, Au) = I(u 1)I*(ul + Au)I(Au). (6)

The corresponding image-space expression is obtained byinverse Fourier transforming, yielding the double correla-tion

iTC'(Xl, Au) = I(Au) J i*(x)i(x + xl)exp(2irjAux)dx

= I(Au)iKT(x 1, Au). (7)

On comparing Eqs. (5) and (6) it can be seen that for afixed spatial-frequency difference, Au, the defined bispec-

(a) iA(x)

I~~~~~~~~~~~~~~~~~~ cos(2itAu~xjl)

REAL

(b) P'<T(xlAul) sin(2,tAu~x,,)

........ i i*IMAG\ * / cos(2it2Aux,)

REAL

iTC41l2Au~)

( iI,(2Au ) sin(27t2Aux,1 )

IMAG

+h

iPG(X, +. ) I:y : X E _

(d)

t ~~~~~i PG (x 1,~aS)0000Gd0 ;0 sw-ax+ b

Fig. 2. Pictorial representation of correlation techniques in theimage plane: (a) the autocorrelation; (b) the KT correlation for asingle plane, Au = Au,; (c) a single bispectrum plane, u2 = 2Au =2Au,, represented as a double correlation in the image plane; and (d)correlations required for the phase-gradient technique, a = ax. Theasterisk indicates the correlation of the two-dimensional images,and multiplication by a function is indicated by the overlaid dashedaxes and one-dimensional function. The functions are constant inthe orthogonal direction. The real and imaginary parts of thecorrelations are shown separately.

(a)

uIx

u2 x

<i ....... I. .. :: m: t:: i:

(b)

Uix

Au,

Ayers et al.

.... 1.1; 11

...I


exponential factor with a spatial frequency greater than zero[see Fig. 2(b)]. For sampled images this generally corre-sponds to the fundamental spatial frequency. The KT cor-relation permits the retention of Fourier-phase-differenceinformation. The argument of Eq. (5),

arg1IKT(u1 , AU)} =- KT(U,, AU) = O(U1)- (u 1 + AU), (11)

gives the phase difference between two spatial frequenciesseparated by the vector Au.

The TC of Eq. (7) similarly permits the retention of phase-difference information. However, the absolute spatial posi-tion is indeterminable, owing to the property of the complexweighting factor that the linear phase is canceled. Addi-tionally, bispectrum planes contain phase information en-coded by correlating i(x) with i(x) multiplied by complexexponentials with higher spatial frequencies [for an exam-ple, see Fig. 2(c)]. The argument of Eq. (3),

arg$ITC(ul, U) = -ITC(Ul, U2) = (W) - O(Ul + U2) + 1(U2),

(12)

permits the determination of the phase difference betweentwo spatial frequencies separated by the vector Au, providedthat the phase at the frequency u2 = Au is already known.In particular, for u2 = Au and Au equal to the fundamentalsampling frequency in the case of discrete data, the TC givesphase-difference information identical to that given by theKT method, apart from the phase at Au. This is the linearphase, which cancels with the linear phase in the KT phasedifference to produce the shift-invariant property of thebispectrum,

DTC(U,, Au) = pKT(U1 , AU) + 0(AU). (13)

Another processing technique, the phase-gradient methodof Aitken et al.,'2728 is closely related to the correlations justdescribed. Phase information is again retained, but in theform of its derivative, not as explicit phase differences. Thephase-derivative information is contained in the imaginarypart of a double product in Fourier space,

IPG(ui, a) = [Im{II(ui)I*(u1)J] -7r

= (1(U 1)I2,0(U1)]- , (14)

where Iml I indicates the imaginary part, 0'(u) is the phasederivative, I'(u) is the derivative of I(u), and a is a vectorgradient in image space. The result can be achievedthrough image space in a number of ways. For example, thetwo simple correlations

iPG(Xl, +a) = i*(x)i(x + x1)(+a * x + b)dx,

iPG(Xl, -a) = i*(x)i(x + x1)(-a * x + b)dx (15)

may be performed, where b is a real constant, and thencombined and Fourier transformed (denoted by FTJ }) toyield the result of Eq. (14). This is similar to the opticalimplementation of the technique used by Aitken et al.,

IPG(ul, a) = [FT~iPG(x,, -a))]* - FT~iPG(xJ, +a))

= i[I'(u,)I*(u,)]* - I(ui)I*(u)J 2a27r

= [Im{IF(ui)I*(ui)1] a

= [II(U,)I'20/(U,)] a. (16)7r

The expressions in Eqs. (15) correspond to correlations ofi(x1) with i(x,) multiplied by linear ramp functions of oppo-site gradient [see Fig. 2(d)].

This technique is analogous to the KT method with a Auvalue such that there is only a linear fluctuation of phaseover that distance. This corresponds to the complex expo-nential in the KT correlation of Eq. (4) having a spatialfrequency, Au, such that the image being correlated extendsover only the linear region of the exponential. The imagi-nary part of the KT product in Fourier space therefore givesthe phase difference between spatial frequencies Au apart.Obviously, division by Au now yields, apart from a constantfactor, the weighted phase gradient contained in the result ofEq. (14):

Im1I1(u,)I*(u,)} =- if Im1IKT(uJ, Au) Lu-NoAu

- - 1 ImtI(u,)I*(u, + AU))AU oAu

-IA Im$I(u,)[I(u1 ) + I'(u,)Au]*)

-t II(u,)J2I'(u1). (17)

3. IMPLEMENTATION OF CORRELATIONTECHNIQUES AT A LOW LIGHT LEVEL

In a number of situations, for example, freezing out turbu-lence effects on imaging through the atmosphere, it is desir-able to record many short-exposure images. As a result, fewphoton events are detected in individual images. However,it can be shown that apart from removable photon biasterms, one of the properties common to all the correlationtechniques is that the ensemble average over many low-light-level frames is equivalent to the high-light-level corre-lation. At such low photon levels it is more efficient compu-tationally to perform the correlations in image space byusing photon-differencing algorithms rather than in Fourierspace by using fast-Fourier-transform routines and multipleproducts.

In the notation of Dainty and Greenaway,'9 the pth pho-ton-limited image is represented by a sum of Dirac deltafunctions positioned at the coordinates of the Np photonevents,

N,,

dp(x) = 3] (x - p)

k=1(18)

and its Fourier transform,

Ayers et al.


N,

Dp(u) = J| x - xpk)exp(-27rjux)dx, (19)k=1

where xpk is the position vector of the kth photon in the pthframe.

The photon-differencing algorithm for the KT methodcan be written as20

N,, N,,

iKT (x, Au) = 3 3 6(xp, - xp, + xj)exp(27rjAUXpk2)-k,=l k2=1

(20)

Thus the KT correlation for each image is obtained by plac-ing the complex number exp(27rj"uxPk 2) at position x = Xpkj

- Xpk2 whenever this vector coordinate difference occursbetween two photon events. The same algorithm can beused to obtain the autocorrelation but with the substitutionAu = 0.

There are many approaches that can be used to obtain theTC,22 but a differencing algorithm based on Eq. (7) is themost appropriate here; that is, each bispectrum plane corre-sponding to a different Au value is evaluated independentlyby using the following expression:

N.

iTC'(X1, AU) = 3 x - xpk)exp(-2ijAuxpk)dx

N, N,

X 3 E 6(X k - Xpkl + xl)exp(-27rijAuxpk). (21)

k,=l k,=l

Thus the correlation is reduced to the simpler KT differenc-ing algorithm folowed by a single complex multiplicationand hence requires little additional computation time.

The phase-gradient correlations of Eqs. (15) can be evalu-ated by using

N, N,,iPG(xl, a) = 3 3 2(x,-xv,, + xl)(axpk, + b). (22)

k,=1 k9=1

The ensemble averaging of photon-limited correlationsunfortunately introduces photon bias terms. Such termscorrespond to the correlation of photon events with them-selves and do not contribute any useful information to theaverage. The elimination of bias terms is achieved simplyby not including in the differencing algorithms any differ-ences between a photon event and itself. Therefore Eqs.(20) and (22) should include the proviso that k, #d k2, andEq. (21) is modified to read as

iTC'(X A&U) = f-

N,,

k= 1,k k, ,k ;k2

N,, A

X 3 k,=lk,,.k,, k,,-=,

6(x- xpk)exp(-2ijAuxpk)dx

3(Xpk, - Xpk, + X1),k.d ,

X exp(-27rjAuxpk,)- (23)

It is important to note, from the standpoint of photon-noiseanalysis, that the bias terms, not the ensemble-average biasterms, of each frame"9 are removed during processing.

The unbiased quantities averaged in each technique maybe stated in Fourier space as follows:

(a) For the autocorrelation technique,

Dp(u,)Dp*(ul) - Dp(O).

(b) For the KT technique,

DP(41)Dp*(ul + Au) - DP*(Au).

(c) For the TC for each bispectrum plane u2 = Au,

Dp(u,)Dp*(ul + Au)Dp(Au) - IDP(ul)I1

- ID P(U + AU)I2 - IDP (AU)12 + 2Np.

(d) For the phase-gradient correlation,

-j [D (u,)D *(u, + Au) - DP*(Au)2Au D +

- D (u)Dpul+ Au) + DP(Au)].

(24)

(25)

(26)

(27)

The variance of the above quantities with the number andthe distribution of photon events in each image describes theeffect photon noise on such correlation techniques.

4. PHOTON NOISE

With the exception of the autocorrelation technique, the aimof the techniques under study is primarily to preserve someestimate of the Fourier phase. As the averages are complexnumbers, both the variance of the real, UR', and imaginarypart, au2 of the Fourier transform are studied, together withthe covariance of the real and imaginary parts, Cov(I, R).The SNR measure used previously23"24 corresponds to

SNR,, = Imean signal|standard deviation of signal

= I(S)1 I

M

(28)

where am = (aR2 + a,2)1/2 and M is the number of indepen-dent realizations used. This measure provide a reasonableindication of the SNR; however, of concern here is the errorof the phase of the complex signal. An estimate of the phaseerror, OE, may be found by considering the variance of thesignal in a direction perpendicular to the mean signal,

[a,2 cos'2 0 + R 2sin'2 - Cov(I, R)sin 2]"'/2 1

tan OE -Il(S) '

(29)

where 0 is the argument of the mean complex signal (S).Provided that the number of realizations, M, used to obtainthe mean signal is large, then the small-angle approximationof the tangent may be used to yield the phase-error estimatedirectly:

[ar2 cos2 0 + IR2 sin2 0 - Cov(I, R)sin 20] '/2 1

=E l(S)I M

(30)

The am2, a,2, and aUR2 terms and the covariance Cov(I, R) ofall the unbiased estimated quantities [expressions (24)-(27)]

Ayers et al.


are evaluated where appropriate; see Appendix A. Themethod of evaluation is a computer implementation of ageneralization of the method described by Goodman andBelsher'8 for the power spectrum. The evaluated variancesare now used to compare the effects of photon noise on thecorrelation techniques.

The effect of photon noise on power-spectrum estimationhas been adequately studied by other authors.18"20 Of par-ticular interest in this work is the comparison of the photon-noise sensitivity of phase-difference estimates obtained byusing the KT and TC techniques. It is apparent from therelationships described in Section 2 that the KT and TCsubplanes may be compared directly, as they effectivelyyield the same information. The additional term present ineach bispectrum subplane is in general known from sub-planes defined by a smaller-magnitude frequency vector, u2.However, the bispectrum, being an average of a triple prod-uct, contains noise terms to the third power, causing thevariance of the observed signal to be greater than that of theKT double product. This is a basic property of high-ordercorrelations and suggests the use of correlations of the lowestorder possible, which permit the recovery of the same infor-mation. For example, Fig. 3(a) shows the theoretical errorOE, for a single image, of the phase-difference estimatesresulting from using the KT plane (ul, = 0, Au, = 0) and thebispectrum plane (u2x = Aux, uly = u2y = 0). The results arefor an asteroid-type object [see Fig. 3(c)], which is approxi-

4.0

3.1

I-0(a

!,

CD

0R

2.3

1.4

0.6

-0.3 ]--25.0 -20.0 -15.0 -10.0 -5.0 0.0

Spatial frequency in sampling intervals

(a)

0.3

-0.6

I-0(0)z

-1.4

-2.3

-3.1

-4.0 1 i-25.0 -20.0 -15.0 -10.0 -5.0 0.0

* 1; 1. a .. X/

;* _4*.C l,iia,

/Je m.C# St llll0A;0 i:: : .

10 50

7 /.-�.-<v-�

-6.0 0.0

(c)

0.0 7.0 0.0 2000

Fig. 3. (a) Plots of the theoretical error OE for a single image, of thephase-difference estimates resulting from using the bispectrumplane (u2x = Au., u1, = u2, = 0) and the KT plane (1, = 0, Au, = 0).(b) Plots of SNRm values for the same two planes. Plots (a) and (b)are on a natural logarithmic scale. (c) Asteroid-type object and itsFourier modulus.


(b)Fig. 4. (a) Comparison of the phase errors on corresponding crosssections of the bispectrum (u, = u,, u2 = u2.) and the KT (u, = ul",Au = Au.) subplanes for U2 = Au. = 4 frequency-sampling inter-vals. (b) Comparison of the SNRm values of the same two crosssections of the KT and bispectrum subplanes.

mately 10 X 20 pixels in a sampled region of 128 X 128 pixels,and an imaging system with a transfer function of valueunity for all spatial frequencies. The mean number of pho-tons per frame, N, is 10. Cross sections through these planesare shown as a graph in Fig. 4(a), corresponding to Au, = U2,= 4, i.e., 4 times the frequency sampling interval. TheSNRs, SNRm, for the same two planes are shown in Fig.3(b). If cross sections through these plots are compared [forexample, Fig. 4(b) shows the SNR of the KT section (Au, = 4and Auy, uly = 0) and the bispectrum (uW2 = 4 and u2y = uly= 0)], an additional indication of the advantage of the doubleproduct estimation is obtained.

As a second example, consider the estimation of the meannumber of photons per frame by the use of the bispectrumelement ul = u2 = 0 and the KT element ul = Au = 0. TheSNR for a single frame is

SNR(KT) = N2 (31)

(4N' + 2N') 1 '(2

for the KT method and

SNR(TC)= =3

(9N' + 18NV + 6NY) 1 2(32)

for the bispectrum method. It is apparent that the KTmethod has a greater SNR than does the bispectrum methodat this point for all values of N. In particular, for the limit-

(a)

(b)

Ayers et al.


ing cases of N - - and N << 1, the following results areobtained by using the expressions for the signal variancesgiven in Appendix A:

lim SNR(KT) - N"'Net-. 2

lim SNR(Tc)=N"TV-. 3

SNR(KT) = N.~ 2

SNR(TC) = N' ,6 C

N<< 1,

N<< 1.

(33)

(34)

(35)

(36)

(a)

U2x .1 S

The plots of OE and SNRm shown in Figs. 3 and 4 indicatea close relationship between these quantities, as expectedfrom Eqs. (28) and (29). Typically,

14Ez: 2S N Rm,

(37)

however, the only generalization that can be made providesan upper bound for the phase error in relation to SNRm:

(38)(1 + 2)1/2This inequality assumes nothing about the statistics of thefluctuating signal. Nevertheless it is a useful expression, asit permits the definition of a bound on the phase errorwithout calculation of OE, which, in general, is a far moredifficult task than the evaluation of SNRm. The additionalX arises in this expression because of a potential correlationbetween the real and the imaginary parts of the complexsignal, resulting in a nonzero covariance.

Nisenson and Papaliolios23 assumed that in the KT meth-od, for small Au values, am is the modulus of a complex errorvector that can possess any phase between ltr with equalprobability. This then gives, after averaging over all possi-ble phases, the expression for the phase error given by rela-tion (37). If the simple imaging case, as described for Fig. 3,is considered again, then the validity of these assumptionscan be investigated. Figure 5 shows plots of the ratio of OE,

defined by relation (30), to 1/(W2SNRm) for the KT andbispectrum planes used in Fig. 3. Both plots show the sameimportant features, and so only one of them, the KT plane, isstudied in detail. It should be noted that the result is unityin general with, in particular, the exception of the regionaround Au. = 0. This is an important exception, as thisregion is the main region used for the reconstruction of theobject's Fourier transform. These results are straightfor-ward to explain if the Fourier transform of a single realiza-tion is represented by D,(u) as defined by Eq. (19), which is asummation over a number of unit modulus complex expon-entials, resulting in a complex exponential at each spatialfrequency u that possesses a phase that is effectively uni-formly distributed mod 27r, correlated over a short range ofspatial frequencies and uncorrelated between realizations;the correlation over spatial frequencies increases if framescontain very few photon events. On this basis it is evidentthat away from the axes, where the KT signal is a product ofFourier-transform values for well-separated spatial frequen-cies (large Au), then the real and the imaginary parts of the

Au,

0.0 1.5

Fig. 5. Plots of the ratio OE, defined by relation (30), to 1(V2SNRm)for (a) the bispectrum plane defined by U2, = Au,, uly = U2, = 0, and(b) the KT plane defined by ul,, = 0, AU) = 0. Departure of the ratiofrom unity occurs only near the axes.

signal are essentially uncorrelated random variables withequal variances. This gives rise to the same result as thatobtained by Nisenson and Papaliolios.'23 However, the re-gions close to the axis (small Au) correspond to the regionsfor which the assumptions of Nisenson and Papaliolios aremade, whereas it is exactly these regions for which the as-sumptions break down. The phase of the signal in eachframe near the axis tends to be small [see Eq. (11)], givingrise to a large real part and a small imaginary part. The realpart and the imaginary part therefore tend to be correlated,and a nonzero covariance results. Additionally, the realvariance is large, and the imaginary variance is small. In

u 1x

e.

3,

dB . .Vs

0.0

t U1x(b)

1.5

Ayers et al.


general, the phase 0 of the averaged signal is also small, andso the following expression for the phase error results:

(E = tj 1-Ps)i Im112

(39)

As shown by Fig. 5(b), this error is smaller than that predict-ed by 1/0(SNRm), which contains the large real variancecontribution. Unfortunately, the nature of sampling leadsto occasional discontinuities in the Fourier phase of theobject being imaged. This can consequently give a largesignal phase 0, which increases the contributions of the realvariance and the covariance to the phase error [see relation(30)]. The result is that relation (38) tends to the equality.This is also shown by Fig. 5(b), in which large values for theplotted ratio are visible near the axis. The implication ofthis is that a larger number of frames is required in thesehigh-phase-error regions than that predicted by the SNRmvalue. The number of frames required could in fact be morethan twice the SNRm-based prediction. The latter does,however, appear to give a reasonable indication of the ex-pected error of the phase of the mean signal, again indicatingthe usefulness of evaluating the SNRm as opposed to themore complicated OE-

5. IMAGING A RANDOMLY TRANSLATINGOBJECT

The KT imaging technique is, unlike the TC, dependent onimage shifts resulting in signal degradation. Consider try-ing to image, using the KT technique, an object whose imageis randomly translating. Such translations may, for exam-ple, be a consequence of the first-order degrading effect ofatmospheric turbulence. The cause of the image shift is awave-front tilt relative to the optical axis of the imagingsystem, which, as in the case of atmospheric turbulence, iseffectively independent of other wave-front perturbations.Suppose that such tilts result in a Gaussian probability dis-tribution p(a) of image shifts a:

p(a) exp 1 ( _ (40)(27r)'/2ia ep -2

where ca is the standard deviation of the shifts. An ensem-ble-averaged transfer function may therefore be definedthat reduces the KT signal by an amount dependent on Auand cia:

(T8 (u)) = J exp(27rjua)exp[27rj(u + Au)ajp(a)da

= exp(-27r2Au2'ci2). (41)

As an example, consider a frequency sampling interval Au of0.5 arcsec-1 and a standard deviation of image shift of 0.5arcsec. Such numbers are reasonably typical in the astro-nomical case and result in the signal's being reduced byapproximately 0.75. This effectively reduces the improvedSNR obtained by averaging the KT double product as op-posed to the equivalent bispectrum plane.

In the absence of photon noise, centroiding images beforecorrelating makes the KT method a shift-invariant processand so removes the above effect, assuming that the sampled

data and Au are equal to the spatial-frequency samplinginterval. The Fourier spectrum of the centroided image,I(u), can be expressed by using the Fourier-shift theorem as

IIu(Au) ]U/AuI(U) = 1(u) L I(Au~ (42)

and the KT average obtained with centroided frames be-comes

I(u)I*(u + Au)= I(u) [fAAI lu]X I*(u + Au) JI(Au) J(U+&U)/&U

= I(u)I*(u + Au) I(Au)II(Au)I

(43)

The advantage of performing a double product for the KTmethod is lost, and the correlation is now a triple product inFourier space, equivalent to the bispectrum with respect tothird-order noise terms.

At low light levels the problem is complicated by thepresence of photon noise, which prevents the accurate deter-mination of the image centroid. The KT average obtainedby averaging a series of low-light-level randomly movingimages and centroiding each image before processing can beshown to give (see Appendix B)

(NpNP - ) (U + AU )I* (U + AU _ AU )I* (AU )Np2),

(44)

where N, is the number of photon events in the pth imageand I(u) is the Fourier transform of the object normalized byI(0). The ensemble average over the Poisson statistics ofthe number of photon events in each image is left unresolvedbecause of the intrinsic object dependency of the aboveresult. This result differs from the desired result,

(Np(Np - 1)(u)I*(u + Au)), (45)

because of the presence of the frequency-shift vector Au/Np.The latter is, however, negligibly small for all images exceptthose containing a few photons. In particular, for framescontaining two photons, the autocorrelation results, with nophase information, are preserved. The centroiding erroreffect obviously increases with Au. This result is important,as it suggests that the technique cannot be used in its presentform when only a few photon events per frame are recorded.This is in contrast to the shift-invariant TC technique, bywhich it is possible to image objects reasonably well at verylow photon levels.29'30

6. IMAGING THROUGH ATMOSPHERICTURBULENCE

Both the KT and TC techniques have been used successfullyfor imaging astronomical objects. Their success is based ontheir ability to overcome the degrading properties of atmo-spheric turbulence. Many images are recorded with expo-sure times short enough to freeze atmospheric perturbationsand then are processed to yield ensemble-average transferfunctions that contain diffraction-limited information. Ob-

Ayers et al.

Ayers et al. ~~~~~~~~~~~~~~~~Vol. 5, No. 7/July 1988/J. Opt. Soc. Am. A 971

tamnable resolutions are therefore ultimately limited by tele-

scope characteristics.The complex mathematical nature of the transfer func-

tions concerned makes a complete analytical study extreme-ly difficult. We therefore used Monte Carlo computer simu-

lations to obtain a reasonable idea of the effects of the statis-tical fluctuations of atmospheric turbulence. The wave-front phase perturbations induced by the turbulence areconsidered to have a Gaussian correlation function with a

conclusion length of approximately 0.5ro, ro being the Fried'parameter that describes the atmospheric conditions beingstudied. Wave-front amplitude perturbations are notpresent in the simulations. The simulations permit the useof any form of telescope pupil function and the imaging of

arbitrary object intensity distributions. The introductionof photon noise is also possible.

Initially, consider a further simplification of the problembased on the work of Roddier.1 6 Suppose that the atmo-

sphere is modeled by a distribution of seeing cells of diame-

ter ro in angular spatial-frequency space. The wave-frontphase is assumed to be constant across each seeing cell, and

the complex amplitude is assumed to be uncorrelated be-

tween cells. By using this basic model, the processing tech-

niques may be explained simply.Consider two seeing cells separated by a vector in the

telescope pupil, Xu, where X is the mean wavelength and u is

an angular spatial-frequency vector. If a point source is

imaged through the telescope by using a pupil function con-

sisting of two apertures, corresponding to the two seeingcells, then a fringe pattern is produced with a narrow spatial-

frequency bandwidth. The major component 1(u) at thefrequency u is produced by contributions from all pairs of

points with a separation Xu, with one point in each aperture

[see Fig. 6(a)]. If the major component is averaged over

many frames, then the result for frequencies greater thanro/X tends to zero because the phase difference, (P~A - 4B

mod 2ir, between the two apertures is distributed uniformlybetween ±ir with zero mean [see Fig. 7(a)]. As a result, the

Fourier component performs a random walk in the complexplane and averages to zero:

argII(u)) = 0(u) + 'DA - (B

(I(u)) =0, u > r/X,

(46)

(47)

where q5(u) is the Fourier phase at the frequency u and(indicates an ensemble average over many frames.

Consider now the autocorrelationl technique, which in

Fourier space corresponds to estimating the power spec-trum. The major Fourier component of the fringe pattern isaveraged as a product with its complex conjugate, and so the

atmospheric phase contribution is eliminated, and the aver-aged signal is nonzero [see Fig. 6(b)]:

argII(u)I(-u)) = qP(u) + 4)A - (DB + (-u) - 4)A + (PB = 0(48)

and

(I(u)I(-u)) = (II(U)I') # 0. (49)

Unfortunately phase information is not preserved. This isthe simplest form of the phase-closure idea"l used in radioastronomy.

(a)~~~~~U

(c)~~~~~

(d)~~~~U+S

Fig. 6. Diagrammatic representation of systems of pupil subaper-tures of diameter ro, showing how phase closure is realized. Thephase-closure requirement can be redefined in terms of spatial-frequency vectors forming a closed loop. (a) Vectors contributingto the Fourier spatial frequency u. (h) Phase closure is achieved inthe power spectrum by taking pairs of vectors of opposite sign, -uand +u, to form a closed loop. (c) Approximate phase closure isachieved in the KT method by taking two vectors, u and u + Au, andassuming that the pupil phase is constant over Au. The bispectrummethod adds a third vector, Au, to form phase closure explicitly.

(d) The third vector Au is essential for phase closure when XAu > ro.The KT method fails with this arrangement.

The KT technique is a small modification of the autocor-relation technique. The major Fourier component of thefringe pattern is averaged with a component at a frequencydisplaced by a small vector Au. Provided that the vectordisplacement Au does not force the vector difference -u -

Ayers et al.


PHASE OF FOURIER TRANSFORM {I(u)} PHASE DISTRIBUTION

> ~~~~~~~~~0CD co

FRAME NUMBER PHASE

KNOX-THOMPSON PHASE {I(u)I(-u-du)} PHASE DISTRIBUTION

0~~~~~t~~~~~~~~~~~~~~~~I 0j@rrM I

FRAME NUMBER PHASE

KNOX-THOMPSON PHASE {I(u)I(-u-Du)} PHASE DISTRIBUTION

> ~~~~~~~~~0_ _ _ _ _ _ _ _ _ _ _CD_ _I

FRAME NUMBER PHASE

BISPECTRUM PHASE {I(u)I(-u-du)I(du)} PHASE DISTRIBUTION

> 0

0~~~~~FRAME NUMBER PHASE

BISPECTRUM PHASE {I(u)I(-u-Du)I(Du)} PHASE DISTRIBUTION

> 0

0~~~~~FRAME NUMBER PHASE

Fig. 7. Graphs illustrating the retention of phase information in the KT and TC techniques. The results were obtained by using a computersimulation of imaging a point source through a 2-m telescope and approximately 0.7-arcsec seeing at a high light level. The point source is cen-troided and so has zero Fourier phase at all spatial frequencies. The phase-distribution statistics were obtained by using an ensemble of 5000different atmospheric realizations. The fluctuation of phase between frames is plotted for only the first 150 realizations. The telescope pupilconsists of three widely spaced subapertures of diameter ro. (a) The uniformly distributed Fourier phase associated with spatial frequenciesgreater than ro/X. (b), (d) Fourier-phase-difference information is retained in both the KT and the 'rC techniques for Au < ro/X. (c) Fourier-phase information is lost if the KT is used with Au > ro/X. (e) Phase-difference information is again retained, even though Au > ro/x, because ofthe phase-closure property of the bispectrum.

(a)

(b)

(c)

(d)

(e)

Ayers et al.

Ayers et al.

Au to be outside the spatial-frequency bandwidth of thefringe pattern, then Fourier-phase-difference information ispreserved in the averaged signal [see Fig. 7(b)]. Again, the

atmospheric phase effectively forms a closed loop [see Fig.

6(c)]:

argfI(u)(I(-u - Au)) = O(u) + 'DA - bB + P(-U - AU)- "A + 4B

= (u) - (u + Au) (50)


IP(u) = O(u)SW(u), (55)

where Sp(u) is the combined transfer function of the tele-scope-atmosphere imaging system. The ensemble-aver-aged transfer functions of the KT and TC techniques aredefined by

T(KT)(u,, AU) = (SP(U,)Sp*(Ul + AU)) (56)

and

and

(I(u)I(-u - Au)) # 0, Au < ro/X. (51) (a)

Suppose that this system of two apertures is extended tothree and that the Au value is made greater than ro/X [seeFig. 6(d)]. Now

arg{I(u)I(-u - Au) = k(u) + DA -B - (A-U - AU)

- (A + DC

= OM - O(U + )U) -uB + (4'C, (52)

(I(u)I(-u - Au)) = 0, Au < r0/X. (53)

It can be seen that the atmospheric phase contribution is notclosed. The resulting phase becomes uniformly distributed[see Fig. 7(c)], and the signal performs a random walk and

averages to zero. Thus the KT technique is limited to fre-quency differences Au < ro/X. If, however, the bispectrum

average is performed, then the phase is again closed andFourier-phase-difference information is preserved [see Figs.

7(d) and 7(e)]:

argfI(u)I(Au)I(-u - Au)} = 0(u) + ")A - (DB + O(AU)

+ DB -'[c+ 0(-U - AU)- (A + DC

= (U) - (U + Au) + O(Au).

(54)

Thus the bispectrum method, in contrast to the KT method,can obtain phase-difference information for phase differ-ences Au > ro/X. It is apparent that the phase-closure re-quirement can be restated in terms of the spatial-frequencyvectors that define a particular technique in Fourier space;the vector sum of the spatial-frequency vectors must be lessthan ro/X for the ensemble average of the signal to be non-

zero, the only exception being if frequency vectors can be

grouped together such that all the groups independentlyhave a sum that is less than ro/X.

By using this simplified model it can be seen how the

techniques are related through the phase-closure conceptand how they permit preservation of Fourier-phase informa-tion. However, the use of a telescope pupil consisting of afew subapertures permits the study of only a limited numberof spatial frequencies. This may be desirable in certain

circumstances, but in general the full telescope aperture isused to make use of all the available light flux and to permitthe study of many spatial frequencies.

Suppose that a series of short-exposure images is record-

ed. The pth image may be represented, by using the inco-

herent-imaging equation in Fourier space, as

(b)

julx

Au,

ga$:'''-,,..',:.':::'''.':';''''~~~~~~~~~~~~. .;.;.'... ... i ''i

-6.0 °°Fig. 8. Plots showing the moduli of corresponding subplanes of (a)

the bispectrum (u, = uix, U2 u 2) and (b) the KT (u, = ui,,, Au =Au,) four-dimensional transfer functions for imaging through a 2-mtelescope with approximately 0.7-arcsec seeing. The data wereobtained by using a Monte Carlo computer simulation, generating10,000 independent realizations of the atmosphere to obtain theensemble average. Both (a) and (b) are plotted with the same

natural logarithmic scale.

u2x

uix


0.0

-1.0

-2.0 -

-3.0 -

-4.0I

-5.0 I

0.0 10.0 20.0 30.0 40.0 50.0


(a)

0.0

-1.0

A. Additionally, both planes contain a high region aroundthe axis ulx = 0. This region permits the recovery of infor-mation about frequency differences of up to approximatelyro/X with a high signal-to-noise-ratio. It should be notedthat this region is wider in the KT case. This is due to thethird term present in the bispectrum triple product, whichreduces the bispectrum modulus, falling off in a mannersimilar to that of the short-exposure transfer function.

tu1x(a)

I-0(0

(I)

C).=3E

-2.0

-3.0

-4.0

-5.0

Au,

4-~

0.0 10.0 20.0 30.0 40.0 50.0


(b)Fig. 9. Graphs of cross sections through (a) the bispectrum (u, =U1., U2 = u2x) and (b) the KT (u, = u1I, Au = Au.) subplanes of theiratmosphere-telescope combination transfer functions. The crosssections correspond to frequency difference vectors U2x and Au. of 0,2, 4, 6, and 8 frequency sampling intervals. The telescope cutoffoccurs at 50 and with ro/X at approximately 4.5 frequency samplingintervals.

T(Tc)(u1, u2) = (S,(u 1)Sp*(ul + U0Sp(U2)), (57)

respectively.The effects of using the full aperture on these transfer

functions are, first, to introduce information on all spatialfrequencies out to the diffraction limit of the telescope and,second, to lower the transfer function for those frequenciesthat were present with the three-subaperture pupil function.The reduction of the average transfer functions is due tocontributions from combinations of subapertures, which arepresent in the full pupil, for which phase closure is notrealized. For example, consider the signal contributionsfrom the two frequency vectors, u and u + Au in Fig. 6(c),but with both of the vectors lying between different pairs ofsubapertures, thus preventing phase closure.

Although both transfer functions are four-dimensional, areasonable indication of their behavior may be obtained bystudying two-dimensional subplanes through them corre-sponding to ul = ulx, with Au = Au, for the KT correlationand u2 = u2X for the bispectrum. Figure 8 shows plots ofthese planes, obtained by using the Monte Carlo computersimulation, permitting a comparison of their basic proper-ties. Both possess a high central region for frequencies uix,Aux, and u2. less than the seeing cutoff of approximately rol

0.0 1.5Fig. 10. Plots of the ratio of the phase error OE to 1/('/2SNRm) forthe subplanes' of the KT and bispectrum results for the asteroid-type object shown in Fig. 3(c) and defined by ul = ulx, u2 = U2x, Aux.These plots are obtained by computationally simulating the effectof imaging the asteroid shown in Fig. 3(c) with a 2-m telescope and0.7-arcsec seeing. Departure from unity occurs only near the axes.

I-0Co

c.0o

0.0

(b)tu1x

Ayers et al.

Ayers et al.

Cross sections through these planes (see Fig. 9) show this

fact. However, the benefit of this additional term can alsobe seen. As the frequency-difference vectors Au, and u2X

increase, the KT signal, in contrast to the bispectrum, re-duces in value and becomes more random owing to the lack

of exact phase closure. The remaining high regions presentin the bispectrum are redundant because of the 12-fold sym-metry of the function.

Of particular interest is the number of different frequen-cy-difference vectors Au. and u2x by which high-SNR infor-

mation can be acquired. In general, the greater the number,the better the resulting reconstructed object intensity distri-bution is.14,2 4

Again, the problem arises of describing the statistics ofcomplex signals. The use of the phase-error measure OE

[relation (30)] poses many problems in an analytic studybecause of the complexity of the signal variances required.Consequently, the SNR measure SNRm [Eq. (28)] is used, as

in Section 4, to compare analytically the KT and TC tech-niques. The phase-error measure OE has, however, beenstudied computationally by using the Monte Carlo simula-tion previously described, and the following results compar-ing OE and SNRm were obtained for imaging through atmo-

spheric turbulence.Figure 10 shows plots of the ratio of the phase error OE to

1/(V/SNRm) for the subplanes of the KT signal and thebispectrum defined by ul = Ulx, u2 = u2., and Au = Aux.These plots are the result of computationally simulating theeffect of imaging the asteroid shown in Fig. 3(c) with a 2-m

telescope and 0.7-arcsec seeing. Five thousand independentrealizations of the atmosphere were used to obtain the en-semble averages. These plots should be compared directlywith those in Fig. 5. Although no photon noise is present,

the atmospheric-turbulence-induced signal fluctuations re-sult in the same overall effect; the variances of the real partand the imaginary part of the signal tend to be independentand equal, with zero covariance, away from the axes, whereasnear the axes the covariance tends to be larger with a smallimaginary variance and a large real variance. Again, theSNRm appears to give a reasonable indication of the phaseerror; however, the fluctuation of the ratio about unity nearthe axes is still present, a consequence of the nonzero covari-ance.

The SNR's of the KT and TC methods in Fourier spaceare defined by

(KT) ~~(I'(ui)Ip*(ui + Au))ISNR(KT)(u, Au) F M+ (58)

'(KT)

where

a(KT)= (IIP(U1)IP*(u1 + Au)12)

- I (IP(U)IP*(U + Au)I 2 ,

and

SNR(TC)(Ui, u2 ) - I(Ip(U,)Ip* (U + U2 )IP(u 2 )) I MU (TC)

where

(TC) = (IIP(U1)Ip*(U1 + U2)IP(U2)I2 )

-I (Ip(uI)Ip* (Ul + U2)Ip(U2) )12

(59)


and M is the number of independent images.In order for the SNR's to be compared, the averages must

be evaluated for a range of frequency vectors. By using the

model of Roddier,16 the following results are obtained. The

mean number of speckles per frame, ii, is assumed to bemuch greater than unity, and -ff = 2.0(D/ro)2 , where D is the

telescope diameter.The approximations that are required in order to estimate

the effect of atmospheric turbulence on ensemble averages

are repetitive and tedious. Consequently, a typical quanti-ty, the ensemble-average mean square of the bispectrum, istreated in detail as an example (see Appendix C). The

effect on imaging a point source at a high light level is

considered first. The Fourier spectrum of the pth image,I,(ui), therefore reduces to that of the point-spread functionof the telescope-atmosphere combination, Sp(u). For u1,

U2, AU > ro/X,

ii2(Sp(u1)Sp*(u1 + u2 )Sp(u2) ) _2T( 3)(u1, U2),

(ISp(U1)Sp*(U1 + u2 )Sp(u2)12 ) 1 I T(2)(u,)T(2)(U2)

X T(2)(ul + u2 ),

(SP(u1)SP*(u1 + Au)) ~ 0;

and, for u2 , Au < ro/X,

(Sp(U1)Sp*(u1 + u2 )Sp(u2 )) _ 1T(2)(U ) (Sp(U2)),ns

(60)

(61)

(62)

(63)

(64)(Sp(u 1)Sp*(u 1 + Au)) I T-(2)(Ul),ns

(ISp(U1)Sp*(U 1 + U2)Sp(U 2 )I2) 2_ 2 IT(2)(U1 )I21 (Sp(U 2 )) 12,

nis

(65)

(ISp(UI)Sp*(ul + Au)12) s 2 IT(2)(U )12lis

(66)

where T(2)(u) is the normalized overlap integral of two pupilsseparated by u and where T(3)(ul, u2 ) is the normalized

overlap integral of three pupils separated by u1 and u2.Consequently, the SNR's are

SNR (TC) ~~2T(3 )(ul, U2 )SNRc -n1/2[T(2)(ul)T(2) (u + u2)T 2)(u2)]

1 2 M,

U1, U2 > r0/X

U2 < r0/X (67)

and

SNR(KT) z 0, u1 , Au > r0/X

-z M, Au < r0 /X.

For ul, Au < ro/X at a high light level both techniques havestatistics that, within the approximation made here, reflectthe negative exponential behavior of the SNR of the powerspectrum.18 For frequencies outside this range, for which nopower-spectrum equivalent exists, the KT method has ap-

(68)


between the two plots. The width of this region gives anindication of the number of frequency-difference vectors forwhich high-SNR information is available for object recon-struction. When cross sections are compared for constantAuX and U2x values, it is found that both techniques havesimilar SNR's if the difference vectors are less than approxi-mately ro/X (see Fig. 12). Beyond this value the closure-phase property of the bispectrum maintains its SNR abovethat of the KT SNR, which tends to fluctuate wildly. Thehigh-SNR regions around the axes are constricted near theorigin for both techniques. This is particularly noticeable inthe bispectrum, where distinct regions of low SNR exist,corresponding to positions where two of the terms in thebispectrum triple product are identical, hence increasing thevariance of the bispectrum phase. These regions are clearlyvisible in Fig. 11: for example, around the line defined byUlx = u2x. In contrast to the transfer functions, the SNR'sfor the cross sections shown, with AuX, u2x < ro/X, maintain aroughly constant value for values of ul, up to approximatelyro/X of the telescope-imposed cutoff.

Unfortunately, the need to observe astronomical objectswith low luminosity and the short-exposure requirement of

-1.0xlO

10.0

6.0

I-0Co

z

2.0

-2.0

-6.0

-10.00.0 10.0 20.0 30.0 40.0 50.0

Spatial frequency in sampling intervals(a)

-1.5 0.0

Fig. 11. Plots of a single plane of (a) the KT and (b) the bispectrum(ul = u1 x, U2 = U2x, AU = Aux) SNR's at a high light level, using thecomputer simulation to obtain ensemble averages over 10,000 inde-pendent realizations of atmospheric turbulence.

proximately zero SNR, whereas the bispectrum SNR has a1/Vn1/2 dependency. This result disagrees with that ob-tained by Wirnitzer, 2 4 which shows no such dependency.The SNR of the information for u2 > ro/X is therefore typi-cally of the order of Yii,/2 lower than that for u2 < ro/X. Ifindividual images are considered to be independent, then inorder to obtain equivalent SNR values in both regions, amultiplicative factor of the order of !is more frames is re-quired when u 2 > ro/X.

Figure 11 shows plots of a single plane of the KT andbispectrum (u, = Ulx, u2 = u2x, Au = AuX) SNR's at a highlight level, obtained by using the computer simulation.Again, the high regions around ux = 0 should be compared

-1.0x10

10.0

6.0

I-0co

z2.0

-2.0

-6.0

-10.00.0 10.0 20.0 30.0 40.0 50.0


(b)Fig. 12. Graphs of cross sections through (a) the KT subplane (u,= u,,, Au = Au.) and (b) the bispectrum subplane (ul = ux, U2 =u2x) subplanes of the SNR's shown in Fig. 11. The cross sectionscorrespond to frequency-difference vectors U2x and Au. of 0, 2, 4, 6,and 8 frequency sampling intervals. The telescope cutoff occurs at50 and with ro/X at approximately 4.5 frequency sampling intervals.

(a)

00,~~~~~~~~~~~~~~~~~~~~~~~~~

-2.5

(b)

0.0

Ayers et al.

Ayers et al.

freezing the atmosphere result in low numbers of photonevents' being recorded in individual images. For such low-light-level images the effect of photon noise becomes impor-

tant. The photon-noise-dependent variances (see Appen-dix A) must therefore be averaged over the atmosphericstatistics in various regimes. These regimes are defined ingeneral by the ratio -f of the average number of photons, N,to the average number of speckles per frame, i,, and by thevalues of the frequency vectors, ul, u2, and Au. Again, with

the model of Roddier,16 averaging over the atmospheric sta-tistics yields the following approximations for the requiredstatistical moments:

For ul, u2, Au > ro/X,

6r(Tc)2 t N3T+ (U2) T + 2 (ud) 4+ T(2)(ul + U2) N4

T 2(U 1).() T(2)(Ul 2)(U+ 2 T 2 )(u2)N5 + T_ 2_ T 2)(u, + U2 )N 5

T (2 )(U, + U2)T(2) V5+ T 2)(u+ u(2)(U 2)N5

X T 2 )(u, + u2 )T 2 )(u 2 )N6 , N> 1, (69)


For ul, u2, Au > ro/X,

SNR(TC) ;Y N3/2T(TC)(U1, u 2)FM

-i << 1. (76)_ 2 N3/2T(3)(Ul, U2)s

Ens

The result is more complicated for larger ii, but the generalform corresponds to a gradual falling off of the dependenceon N, the variance becoming more dependent on N as

successive terms in relation (69) take precedence. Eventu-ally, for -i >> 1, the SNR is simply dependent on Th, as the N6-dependent term of relation (69) dominates the variance.When N is approximately equal to Th, then there is roughly alinear dependence of SNR(TC) on N when the N4 -dependentterm dominates the variance. The KT method has a SNR ofapproximately zero in the equivalent region, Au > ro/X.

For u2 , Au < ro/X and i << 1,

SN(TC) N'T (2 )(u,) (S (U2))S NR 3 FM = N4 ( " ) 2 (77)-n[X3 + NV41 ( S IWO)) 12]1/

and

2 "T(TC) 1:~ N1, N> m,<< 1,

2(KT) s: 0.

For u2 , Au < ro/X,

2 Nl(S(u 2))l2 + N TV 2 (u )UY(TC) 1:~N3 + (2 T (1

+N4 T(2) (U+ + N5 T (2) (U )I (S (U2) )1ins i~~~ns

inS nS

+ _~'-_ T +~u U21+ p(2

(70) SNR(KT) N T(2)( M.SR iinT ON

(78)

(71)

N> 1,+ N T (u,)TU2 S(u2 +)is2

2(Tc) N3 + N41(S (u2 ))12, N> 1, m << 1,

(KT) N2 + T(2)(u,) +N- T(2)(u + Au)

+ N T((u2)T(2)(ul + Au), N > 1,ins

(KT)2 N2 N>1, i<<1.

The mean signals are

N3T(TC)(ul + u2), N2 T( KT)(ul, Au)

for the bispectrum and KT methods, respectively.The SNR's of the two techniques may now be compared

for the various regimes.

In particular, relation (77) reduces to a linear dependence onii, equivalent to the power spectrum, for u2 = 0, as expected.Again, for larger -i, both SNR's are far more complicated,but, as before, their dependence on N gradually falls offwhen the terms in the variances that are dependent on high-er powers of N dominate. Both SNR's tend to the high-light-level value of unity, as for the power spectrum, for u2 =

Au = 0.

A Monte Carlo simulation was carried out (see Fig. 13), the

results of which reasonably confirm the N and ii, dependen-cies of the above SNR results.

The ensemble-averaged variances derived can be incorpo-rated easily into phase-reconstruction algorithms22 toweight optimally the information present in the averagesignals. The expression for the variance of the bispectrumgiven in relation (72) is particularly useful as a weighting

function, as it can be evaluated from knowledge of N and the

ensemble-average image spatial power spectrum at variousspatial frequencies. This weighting function will, however,suffer from the fact that it is independent of the phase of thebispectrum at each point. Two possible approaches to over-coming this are as follows. Firstly, a larger weighting func-

tion can be given to those points for which the bispectrumphase is not close to zero on the basis of the phase errortending to the equality in relation (38). Alternatively, thevariances and the covariance of the real and imaginary partsof the complex signal (see Appendix A) could be approxi-

mated in a manner similar to that used for the variance ofthe modulus, and the full expression for the phase errordefined in relation (30) could be evaluated.

978 J. Opt. Soc. Am. A/Vol. 5, No. 7/July 1988 Ayers et al.

-5.0xlO

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MEAN NUMBER OF PHOTONS PER FRAME

(b)

3.0x1 0

2.0

0.0

0.0 1.0 2.0 3.0 4.0 5.0

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x1 3.0

(e)

Fig. 13. Graphs of the results of a Monte Carlo computer simula-tion whereby ensemble averages over the atmospheric statistics aretaken for the photon-noise-dependent variances defined in Appen-dix A. (a) Dependency of the SNR of the bispectrum on 'n, and N.The relative SNR is the ratio of the SNR for the bispectrum pointdefined by (u1 = 0.4D/X, U2 = 0) at ro - 0.185D to that at ro - 0.095D.This ratio of ro corresponds to a ratio of 0.25 in the mean number ofspeckles per frame, -ns. The plot tends to 4 at very low light levels,corresponding to a 1/n-, dependency [see relations (67) and (77)],and tends to unity at high light levels, corresponding to a SNR thatis approximately independent of -n. [see relation (67)]. (b) A plotequivalent to (a) for the bispectrum point defined by (u1 = 0.6D/X,U2 = 0.3D/X). The plot tends to a value of 16 at very low light levels,corresponding to a 1/ii52 dependency [see relation (76)], and tends to2 at high light levels, corresponding to a //fl 5 dependency [seerelation 67)]. (c) The approximate linear dependence of the SNR ofthe same bispectrum point on N when the mean number of photonsper speckle is approximately 1. The mean number of speckles perimage, -n., in this example is approximately 1000. (d) The approxi-mate N3/2 dependency of the same bispectrum point when the meannumber of photons per speckle is much less than unity [see relation(76)]. The upper line is an TV3/2 -dependent reference. (e) Thelinear dependence of the SNR's of the KT and TC signals defined by(uI = 0.4D/X, u2 = 0, Au = 0) on N at low light levels [see relations(77) and (78)]. The upper line is a linear reference. The KT signalhas the slightly higher SNR; however, the lines appear to be super-imposed.

7. CONCLUSION

It is evident that the KT and TC methods of image process-ing are closely related from both the mathematics and im-plementation viewpoints. The extension of the KT methodto a more-general form, whereby the frequency-differencevector Au can take on a wide range of values, removes the TCadvantage of having a redundency of phase information thatcan increase the SNR. It is necessary to centroid images

before processing when using the KT method, and this re-moves the SNR gain over the TC that should result as aconsequence of being a lower-order correlation. Additional-ly, the need to centroid and the development of an efficientphoton-differencing TC algorithm result in implementationtimes' being equivalent, despite the KT method being alower-order correlation.

A more appropriate measure of the phase error, OE, of acomplex signal is suggested. This is compared with the

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traditional SNR-based measure whereby the phase error isapproximated by 1/(V2SNR). It is found that this measuregives a reasonable indication of the phase error; however, in

certain circumstances, for small-frequency-difference vec-tors, the value predicted by this method can be considerablyin error. It is seen that the SNR can be used to provide anupper bound on the phase error:

0 E SNRm (1 )Speckle interferometry, the KT correlation, and the TC

have been seen to be connected through phase-closure rela-

tionships. By considering phase closure in terms of spatial-frequency vectors forming closed loops in the telescope

pupil, it is possible to evaluate many ensemble averages ofimage correlations over the statistics of atmospheric turbu-lence. A computer implementation of a generalization of

the technique of Goodman and Belsher18 made possible aninvestigation of photon-noise-dependent SNR's. By usingthese developments, together with a computer simulation ofthe effects of imaging through turbulence, the SNR's of theTC and KT methods are compared with, in particular, theirdependence on n, and N.

It is found that the highest-SNR regions of both the TCand the KT average signals are of equivalent value andextent. They correspond to using frequency-difference vec-tors Au and u2 that have magnitudes that are less than or of

the order of ro/X. In this region both SNR's are linearlydependent on the number of photons per speckle at low lightlevels and approximately unity at a high light level. Theyreflect their close association with the negative exponentialstatistics of the power spectrum. Outside these high SNRregions, the KT signal is insignificant, and the bispectrumSNR shows an N312/-n2 dependency at low light levels and aI/in"/2 dependency at a high level. Again, this suggests the

effective equivalence of the two techniques, as for typical

values of in5, of the order of 1000, many more frames must be

processed for the additional information present in the bi-spectrum to be of any use. Results from a Monte Carlo-typecomputer simulation, which made possible the evaluation ofthe ensemble average over the atmospheric statistics withinthe constraints of the model used, agree well with the resultsobtained from the analytic study based on phase-closurerelations.

APPENDIX A

A computer implementation of a generalization of the meth-od of Goodman and Belsher1 8 for averaging signals in Fouri-

er space in the presence of photon noise was used to yield the

following results. Consider imaging, with a mean number Nof photon events per frame, an intensity distribution thathas a high-light-level image with the Fourier transform 1(u).

The KT signal of the pth frame is, in Fourier space,

D(KT)(ul, Au) = Dp(u,)D,*(u, + Au) - D,*(Au), (Al)

resulting in the following expressions for the second-orderstatistics. The second moment of the real part of the signalis given by

1/4 ([Dp(KT)(uo, Au) + D (KT)*(U,, AU)] 2 ), (A2)

and averaging over the photon-noise statistics gives

1/4[I(-u1)I(ui)I(U1 + Au)I(ul + Au)N4 + c.c.

+ 2I(-ul - Au)I(-u1)I(u1)I(ui + Au)N4

+ I(-u1 )I(-u1 )I(2u1 + 2Au)N3 + c.c.

+ 2I(-u1)I(u1)N3 + 2I(-u1)I(Au)I(u1 + Au)N3 + c.c.

+ 21(-u1 - Au)I(-u1)I(2u, + Au)N3 + c.c.

+ I(-ul - Au)I(-ul -Au)I(2u,)N3 + c.c.+ 2I(-u1 -_Au)I(u + Au)N3 + I(-Au)I(-Au)N2 + c.c.

+ I(-2u,)I(2u, + 2Au)N2 + c.c.

+ 21(-2u, - Au)I(2u1 + Au)N2 + 2NI,

where each c.c. denotes the complex conjugate of the whole

of the term to its left, including the sign.The second, moment of the imaginary part of the signal is

given by

-1/4 ([Dp(KT)(ul, Au) - DP(KT)*(U, AU)]2),


(A3)

-1/4[I(-u1)I(-u1)I(U1 + Au)I(ul + Au)N4 + c.c.

- 2I(-ui - Au)I(-u,)I(u,)I(u, + Au)N4

+ I(-u,)I(-u,)I(2u, + 2Au)N3 + c.c.- 2I(-u,)I(u,)N3 + 21(-u,)I(Au)I(u, + Au)N3 + c.c.

- 2I(-ul -Au)I(-u,)I(2u1 + Au)N3 + c.c.+ I(-ul - Au)I(-ul - Au)I(2u,)N3 + c.c.- 2I(-ul - Au)I(ul + Au)N3 + I(-Au)I(-Au)N2 + c.c.

+ I(-2u,)I(2u, + 2Au)N2 + c.c.

-2I(-2u, - Au)I(2u, + Au)N2 - 2N2].

The first moment of the real and imaginary parts of thesignal is given by

-j/4([Dp(KT)(u, AU) - Dp(KT)* (U, AU)]

X [Dp(KT)*(Ui, AU) + Dp(KT)*(Ui, Au)]), (A4)


-j/4[I(-ul_-Au)I(-ul - Au)I(u,)I(u,)N4 - c.c.

+ I(-ul - Au)I(-ul - Au)I(2u,)N3 - c.c. + 2I(-ul - Au)

x I(-Au)I(u1)N3 - c.c. + I(-2u, - 2Au)I(ui)I(u,)N3

- c.c. + I(-Au)I(-Au)N2 - c.c. + I(-2u, - 2Au)

x I(2u,)N2 - c.c.].

The TC signal of the pth frame is, in Fourier space,

Dp(TC)(ul, u2) = Dp(u,)Dp*(ul + u2 )Dp(u2 )

- IDP(U,)12 - IDP(ul + U2 )12

- IDp(U2)I2 + 2Np, (A5)

resulting in the following expressions for the second orderstatistics.

The second moment of the real part of the signal is givenby

Ayers et al.


1/4 ([D (TC)(Ul, U2) + DP(TC)* (U1, U2)] 2 ), (A6)

and averaging over the photon noise statistics gives

1/4 [I(-u 1)I(-ul)I(-u 2 )I(-u 2 )I(2uI + 2u2)N5 + c.c. + 21(-ul)

X I(-ul)I(-u2)I(ul)I(ul + u2)N5 + c.c. + I(-ul)I(-U1 )

X I(-2u2 )I(ul + u2)I(u1 + u2)N5 + c.c. + 2I(-ul)I(-u 2 )

X I(-u2)I(u2)I(ul + u2)N5 + c.c. + 2I(-ul)I(-u 2)I(u2)

X I(u 1)N5 + 21(-ul - u 2)I(-ul)I(-u 2 )I(ul)

X I(u, + 2u2)N5 + c.c. + 2I(-ul - u 2)I(-ul)I(-u 2 )

X I(u1 + u2)I(u1 + u2)N5 + c.c. + 2I(-ul - u 2)I(-ul)

X I(-u2)I(u2)I(2u, + u2)N5 + c.c. + 21(-ul - u 2)I(-ul)

X I(u1)I(u1 + u2)N5 + 2I(-u, - u 2 )I(-ul)I(u2 )I(ul - u2)

X I(U1 + U2)N5 + C-C- + I(-U 1 - U 2)I(-U1 - u 2)I(u2 )I(u2 )

X I(2u,)N5 + c.c. + 2I(-u, - u 2 )I(-u2 )I(u2 )I(ul + u2)N5

+ I(-u1)I(-u1)I(-2u2)I(2u, + 2u2)N4 + c.c. + 21(-u,)

X I(-ul)I(u1)I(u1)N4 + 4I(-ul)I(-u 2)I(u2)I(ul)N4

+ 2I(-u 1)I(-2u2 )I(u2 )I(ul + u2)N4 + c.c. + 21(-u,)I(u 1)N4

+ 21(-ul)I(u2)I(ul - u 2)N4 + c.c. + 2I(-ul - u 2)I(-ul)X I(-u2 )I(2u, + 2u2)N4 + c.c. + 41(-ul - u 2)I(-ul)I(ul)

X I(u1 + u2)N4 + 2I(-u, - u2)I(-ul)I(ul - u 2)

X I(u, + 2u2)N4 + c.c. + 2I(-ul - u2 )I(-ul)I(u 2 )

X I(2u,)N4 + c.c. + 2I(-ul - u 2)I(-U1 )

X I(2u, + u2)N4 + c.c. + 21(-ul - u 2)I(-ul + u2)I(-u2 )

X I(2u, + u2)N4 + c.c. + 2I(-ul - u 2)I(-ul + u2)

X I(u1 - u 2)I(u1 + u2)N4 + 21(-ul - u 2)I(-ul - u 2)

X I(U1 + U2)I(U1 + U2)N4 + I(-U 1 - u 2)I(-U1 - u 2)I(2u2)

X I(2u,)N4 + c.c. + 2I(-ul - u 2)I(-u2 )

X I(u, + 2u2)N4 + c.c. + 41(-ul-u91(-u 2 )I(u2 )

X I(u 1 + u2)N4 + 2I(-u 1 - U2)l(U1 + U2)N4

+ 2I(-ul - 2u2)I(-ul)I(u1)I(ul + 2u2)N4 + 2I(-ul - 2u2)

X I(-ul)I(u2 )I(2u, + u2)N4 + c.c. + 2I(-u2)I(-u2)I(u2)

X I(u2)N4 + 2I(-u2)I(u2)N4 + I(-2u 1)I(-u2 )I(-u2)

X I(2u1 + 2u2)N4 + c.c. + 2I(-2u,-u 2 )I(-u 2 )I(u2 )

X I(2u, + u2)N4 + I(-ul)I(-u 1)I(2u,)NY + c.c. + 2I(-ul)

X I(-u2 )I(ul + u2)N3 + c.c. + 2I(-ul + u2)I(u -U2)N3

+ I(-u 1 - u 2)I(-U1 - u 2)I(2u, + 2u2)N3 + c.c.

+ 2I(-u 1 + 2U2)I(-Ul + U2)I(2u 1 + U2)N3 + C-C.

+ 2I(-ul - 2u2)I(ul + 2u2)N3 + I(-u2 )I(-u 2 )

X I(2u2 )N3 + c.c. + I(-2u,)I(-2u 2 )I(2u1 + 2u2)N3 + c.c.

+ 21(-2u, - u2)I(2u, + u2)N3 + 2N3

+ I(-ul)I(-ul)I(-u2)I(-u2)I(ul + u2)X I(u1 + u2)NT6 + c.c. + 2I(-ul - u 2)I(-ul)I(-u 2)I(u2)

X I(u 1 )I(uI + U2)N 6 ].

The second moment of the imaginary part of the signal isgiven by

_1/4 (D (TC)(ull U2 ) - DP(TC)*(ul, U2 )]2), (A7)


-1/4[I(-ul)I(-ul)I(-U2)I(-U2)I[2u1 + 2U2)N5 + C-C.

+ 2 I(-uI)I(-ulI(-u2 )I(u1 )I(u, + u2)N5 + c.c. + I(-ul)

X I(-u1 )I(-2u2 )I(ul + u2)I(u1 + u2)N5 + c.c. + 2I(-ul)

X I(-u2)I(-u2 )I(u2)I(ul + u2)N5 + c.c. - 2I(-ul)I(-u 2 )

X I(u2)I(ul)N5- 2I(-ul - u 2)I(-ul)I(-u 2)I(ul)

X I(u, + 2u2)N5 + c.c. + 2I(-ul - u 2)I(-ul)I(-u 2 )

X I(u1 + u2)I(u1 + u2)N5 + c.c. - 2I(-ul - u 2)I(-ul)

X I(-u2)1(u2)I(2u, + u2)N5 + c.c.-2I(-u -u 2)I(-ul)

X I(ud)I(u1 + u2)N5 - 2I(-ul - u 2)I(-ul)I(u 2 )I(ul - u2)

X I(ul + u2)N5 + c.c. + I(-u1 - u 2)I(-ul -u 2)I(u2)I(u2 )

X I(2u,)N5 + c.c. - 21(-ul -u 2 )I(-u2 )I(u2 )I(ul + u2)N5

+ I(-ul)I(-u1 )I(-2u2)I(2u, + 2u2)N4 + c.c. + 2I(-ul)

X I(-ul)I(u1 )I(u1 )N4 + 4I(-ul)I(-u2)I(u2)I(ul)N4

+ 2I(-u1)I(-2u2)I(u2)I(ul + u2)N4 + c.c. - 21(-ul)

X I(u1 )N4 - 21(-ul)I(u2 )I(ul - uN 4 + c.c.

+ 21(-ul - u 2)I(-ul)I(-u 2)I(2u, + 2u2)N4 + c.c.

+ 4I(-ul - u 2)I(-ul)I(ul)I(ul + u2)N4 + c.c.- 2I(-u1 -u 2)I(-ul)I(ul- u 2)I(u1 + 2u2)N4 + c.c.+ 2I(-ul - u 2)I(-ul)I(u2)I(2u,)N4 + c.c. - 2I(-u, - u2)

X I(-u1)I(2u1 + u2 )N4 + c.c.-21(-u -u 2)I(-uI + u2 )

X I(-u2 )I(2u, + u2)N4 + c.c. - 21(-ul - u2)I(-ul + u2)

x I(ul - u2)I(u1 + u2)N4 + 2I(-ul - u 2)I(-ul - u2)

X I(ul + u2)I(ul + u2)N4 + I(-u 1 - u 2)I(-U1 - u 2)I(2u2 )

X I(2u,)N4 + c.c. - 21(-u1 - U 2)I(-U2 )

X I(uj + 2u2)N4 + c.c. + 4I(-ul - u 2)I(-u2)I(u2 )

X I(uI + u2)N4 - 2I(-u, - u 2)I(u1 + u2)N4 + c.c.

- 2I(-ul - 2U2)I(-Ul)I(ul)I(ul + 2U2)N4

- 2I(-ul - 2u2)I(-ul)I(u2)I(2u, + u2)N4 + c.c.

Ayers et al.

Ayers et al.

+ 2I(-u2)I(-u2)I(u2)I(u2)N4 - 2l(-u2)I(u2)N4

+ I(-2u,)I(-u2 )I(-u2 )I(2u, + 2u2)N4 + c.c.

- 2I(-2u, -u 2)I(-u2)I(u2)I(2u, + u2)N4 + I(-u1)

X I(-u1)I(2u1)N3 + c.c. + 2I(-ul)I(-u 2 )

X I(u1 + u2 )N3 + c.c. - 2I(-ul + u2)I(u1 -U2)3

+ I(-u 1 - u 2)I(-ul - u2)I(2u, + 2u2)N3 + c.c.

- 2I(-ul - 2u2)I(-ul + u2)I(2u1 + u2)N3 + c.c.

- 21(-ul - 2u2)I(u1 + 2u2 )N3 + I(-u2 )I(-u2 )

X I(2u 2)N3 + c.c. + I(-2u 1 )I(-2u 2 )I(2u, + 2u2 )N3 + c.c.

- 2I(-2u, - u2)I(2u1 + u2)N3 - 2N3 + I(-u,)I(-ul)

X I(-u 2 )I(-u 2 )I(ul + u2)I(u1 + u2)N6 + c.c.

- 2I(-ul -u 2)I(-ul)I(-u2)I(u2)I(ul)I(ul + u2)N6].

The first joint moment of the real and imaginary parts ofthe signal is given by

-j/4([DP(TC)(ul, U2) - DP (TC)*(U, U2)

X [Dp(TC)(ul, u2 ) + DP(Tc)*(ui, u 2 ]), (A8)


-j/4[-I(-ul)I(-ul)I(-u2)I(-u2)I(2u1 + 2u2)N5 - c.c.

- 2I(-u1)I(-ul)I(-u2 )I(ul)I(Ul + u2)N5- c.c. - I(-ul)

x I(-ul)I(-2u2 )I(ul + u2)I(u1 + u2)N5 - c.c. - 2I(-ul)

x I(-u2 )I(-u 2)I(U2)I(ul + u2)N5 - c.c. - 21(-u1 - u2)

x I(-u)I(-u 2 )I(ul + u2)I(u1 + u2)N5 - c.c.

+ I(-u1 - u 2)I(-u1 -u 2)I(u2)I(u2)I(2ul)N5 - c.c.

- I(-u1)I(-u1)I(-2u2)I(2u, + 2u2)N4 - c.c. - 2I(-ul)

x I(-2u2 )I(u2 )I(ul + u2 )N4 - c.c. - 2I(-u, - u2 )I(-ul)

x I(-u2 )I(2u, + 2u2)N4 - c.c. + 21(-ul - u 2)I(-ul)I(u2 )

X I(2u,)N 4 - c.c. + I(-U 1 - u 2)I(-U1 - u 2)I(2u2 )

X I(2u,)N 4 - c.c. - 1(-2ul)I(-u2)I(-u2)

X I(2u, + 2u2)N4 - c.c. + I(-u 1)I(-u 1)I(2u1)N3 - c.c.

+ 21(-ul)I(-u2 )I(ul + u2)N3 - c.c. - I(-u1 - u 2)

x I(-U1 - u2 )I(2u, + 2u2)N3 - c.c. + I(-u2 )I(-u2 )

x I(2u2)N3 - c.c. - I(-2u,)I(-2u 2)I(2u, + 2u2)N3 - c.c.

- I(-ul)I(-ul)I(-u 2)I(-U2 )I(U1 + U2)

X I(u 1 + u2)N 6- c.c.]

APPENDIX B

Centroiding frames before performing the KT techniqueresults in the introduction of errors into the ensemble aver-age when photon noise is present.


The pth low-light-level image is described by Eq. (18), andits Fourier transform is described by Eq. (19). The follow-ing expression may be written, concerning the shift, Xp, fromthe image centroid of the photon-event distribution, theimage centroid being the center of gravity of the image pho-ton-event distribution:

NP(Xp - Xg) = 1 (Xk - Xg),k=l

(Bi)

where xg is any point in the frame used as a reference.Hence

Np

Z Xk

X k= ~N(B2)Np

The KT signal in Fourier space is

Dp(u,)Dp*(ul + Au) -DP*(Au),

and if DW(u) is centroided, using the Fourier-shift theorem,to give

Dp(u)exp(-27rjXPu), (B4)

then the resulting KT signal formed is

Dp(ul)exp(-2irjXpuo)Dp*(u, + Au)exp[27rjXp(ul + Au)]

- Dp*(Au)exp(27rjXpAu). (B5)

Taken by the method of Goodman and Belsher,18 the ensem-ble average over the photon-event distribution yields

(NPNP - 1)i(u + Au U + Au - Au)*(Au)NP- 2)

(- Np) ( Np)

+ PAu -N) (N ) (B6)

where r(u) is the object Fourier transform I(u) normalizedby I(0) and ( ) indicates an ensemble average over thePoisson statistics of the photon events. Obviously the biasterms cancel, as is desired. However, the resulting signaldiffers from the required result,

(Np(Np - 1)(u)I*(ul + Au)), (B7)

obtained from a series of photon-limited images of a station-ary object that is not centroided before the KT average istaken. The average over the bias term alone yields a resultthat is completely general for any frequency Au. Conse-quently the same result may be used to describe the result-ing image obtained by summing a series of centroided imagesthat contain photon noise:

(E[DP(u)exp(-27rjXPu)]) = KNI(u - N_ )P

(B8)

where E[ ] indicates the expectation value of the signal thatis due to the photon-event distribution.

(B3)


APPENDIX C

Roddier 16 developed a model that may be extended, in con-junction with some simple assumptions concerning the sta-tistics of atmosphere-induced wave-front perturbations, toapproximate the effect of atmospheric turbulence on ensem-ble averages of intensity distribution correlations in general.

Imaging through atmospheric turbulence with a large sin-gle-aperture telescope is modeled by describing the tele-scope pupil function Ho(x) as a distribution of seeing cells ofdiameter ro. The wave-front phase is assumed to be con-stant across each cell, and the complex amplitude in thepupil, A(x), is assumed to be uncorrelated between cells.The complex amplitude modulus is assumed to be unity.The Fourier transform of the intensity distribution of animage of a point source is therefore described by

Ho(x)Ho*(x + Xu)A(x)A*(x + Xu)dxS,(u) = pupil

Jpupil

This integral may be approximated, using the simple model,by a sum of complex exponentials corresponding to the com-plex amplitude difference between the cells in pairs of seeingcells:

IHO(X) 12dx

(Cl)

occur in more than one pair, result in phase closure and anonzero ensemble-average contribution. Such combina-tions all result in the phase-closure relationship

(a q -Iq') + ('Ir -(Ir') + ('ps -)

+ ( Dq - q) + (or"- Or') + (bst - 4s') = 0, (C4)

which may be realized in a number of different ways. To seehow these arise and to obtain their contribution to the en-semble average, it is instructive to consider imaging througha telescope pupil with an atmosphere-induced complex am-plitude distribution that is delta correlated:

CA(X) = (A(x')A*(x' + x)) = 5(x). (C5)

This corresponds to making N(O) approach infinity, so thatthe summation approximation of Eq. (C3) again becomes anintegral expression:

(a) U1 +U5UI

-U1

- U2

- Ut

¾u2

1N(u)

SV(u) = 1 N(u) expU( 8 -N(O) Z=l (C2) (b) U1 - U2

u+ U2

where 4'b - 4t' is the wave-front phase difference betweenthe seeing cells in the sth pair, separated by Au. There areapproximately N(u) such pairs in the pupil.

In order to study the SNR's of the KT and TC techniques,a number of different correlations must be averaged over theatmospheric statistics. By using the above model, a particu-lar correlation is treated to determine the form of the ap-proximations required in general. These and similar ap-proximations are used to average all the correlations neededfor the investigation so that the forms of the dominant termsmay be determined.

Consider, as an example, averaging the squared modulusof the bispectrum of a point source, for frequencies ul, u 2 >ro/X. This term is of particular importance, as it permits theevaluation of the variance of the bispectrum:

(ISp(u1)SP*(u1 + U2)SP(u2)12)

I N(ul) N(u,+U 2) N(U2)

kN(O)' q1 r=1 E _E

U, + _-U2

(c)

U,

- Ut

Ut

(d) U.l + X - U2 U2 11 - U1 - U2

-U,

U2

-U2

UI

-Ut

U,

(e) u,+u2

2

-U2

expUQ(4q - (Iq') + i(yr - (Dr')

expUj(Qqt' - 4Dq )

+ j(],./ - (lr) + j(iQsI - (P')]j- (C3)

In goneral, this expression corresponds to suniming overcontributions from six different pairs of seeing cells, and forsuch combinations the ensemble average tends to zero.However, certain other combinations, for which seeing cells

-U UU U-2 Ut

Fig. 14. Phase-closure relationships can be viewed in terms ofspatial-frequency vectors forming closed loops in the telescope pu-pil. (a) Representation of the six vectors that must be consideredwhen analyzing the modulus squared of the bispectrum at a pointdefined by (U1,1U2). (b) How the vectors can be selected in conjugatepairs to cancel the phase, each pair being analogous to the powerspectrum at that particular frequency. (c), (d) Other possible com-binations of vectors that result in phase closure. (e), (f) Redundantconfigurations between those represented in (b)-(d).

W -U2

U2

U

-Ut

+ j(ps - )]N(u,) N(u,+u2 ) N(U2)

q'= r=l s'=

Ayers et al.


(ISp(u1 )Sp*(U1 + u2)Sp(u2)I2)

= f | f f f HO(x1 )Ho*(x 1 + XU1 )HO(X2 )

X HO* [X2 - X(U1 + U2)lHo(X 3)Ho*(x 3 + XU2 )Ho*(X 4 )

X HO(x4 + Xul)Ho*(x5)Ho[x 5 - X(U1 + U2)]Ho*(X 6)

X HO(x6 + Xu2 )(A(xo)A*(xl + Xul)A(x2)

X A*[x 2 - X(u1 + u2 )]A(x 3 )A*(x 3 + Xu2)A*(x 4 )

X A(x 4 + Xul)A*(x5)A[x5 - X(u1 + U2 )]A*(X 6 )

X A(x6 + XU2))dxldx 2dx3 dx4 dx5dx6 ,

where the normalization factor Y is defined by

Y = [f f f IHo(xO)I2IHo(x 2 )12IHo(x 3 )I2 lddx2dX 3 ]. (C6)

By averaging over the complex amplitude statistics, thephase-closure nonzero ensemble-average contribution termscan be approximated. Since the complex amplitude is as-sumed to possess a unit modulus, its higher moments cannotbe generated by using Reed's theorem,30 which is applicableonly if the complex amplitude is a Gaussian random vari-able. Instead, the high-order moment is broken down intocombinations of lower-order moments on the basis of phase-closure considerations. This breakdown of the higher-ordermoments is perhaps seen more easily if the phase-closurerelationship is thought of in terms of spatial-frequency vec-tors forming closed loops in the telescope pupil (see Fig. 14).

(a) Consider terms for which

(Nq - Dq') + (0q -s 'q ) = 0,

(0rbr + (0t r)°(4 -D ) + (Os. -'Iv) = 0

These correspond to the following nonzero contribution tothe ensemble average over the complex amplitude statistics[see Fig. 14(b)]:

(A(xi)A*(x 4)) (A(x 4 + Xul)A*(x1 + Xul)) (A(x 2)A*(x 5))

X (A[x 5 - X(u1 + u 2 )]A*[x 2 - X(u1 + U2)])

X (A(x 3 )A*(x 6 ))(A(x 6 + XU2 )A*(x 3 + Xu2 ))

= (X4 - X1 )(X 5 - X2)6(X6 - X3). (C7)

This results in the following contribution to the averagemodulus squared of the bispectrum, which is nonzero for allspatial frequencies out to the diffraction limit of the tele-scope [for an unapodized aperture this is simplified to aproduct of terms dependent on the normalized diffraction-limited transfer function T(2)(u)]:

|fpupi f| IHo(x)I 2IHo(X1 + Xu1 )I2 IHo(x 2 )I2

X IHO(X1 - X(u1 + u2 )]I 2 IHO(x3 )12 IHO(x3 + Xu2 )I2

X dxldx2dx3

- 1 T(2)(u 1)T(2)(u2 )T(2)(u1 + U2), (C8)n3

where the mean number of speckles per frame, ins, is definedby

| IHo(x)l2dx

i l ICA(x)I2dx

(b) Consider terms for which

(Rq - 'Iq') + ((qP" -(q') = 0(41r -r') + (4r 'r') + ('I's -(') + (4)"- 4!s ) 0,

These correspond to two terms [see Fig. 14(c)] that haveidentical nonzero contributions to the ensemble averageover the complex amplitude statistics, provided that thepupil is symmetric. For example,

(A(x1 )A*(x 4)) (A(x4 + Xul)A*(xl + Xul)) (A(x2)A* (x6))

X (A(x 3)A*[x 2 - X(U1 + U2 )])(A[X 5 - X(U1 + U2)]

X A*(x3 + Xu2)) (A(X6 + XU2)A*(X5))

= 6(X4 - X1 (X6 - X2 )6(X 5 - X6 - XU2 )

X 6[x2 - X3 - X(u1 + u2 )]6[x 3 - X5 + X(u1 + u 2 ) - XU2]

(C9)

represents the contribution [shown in relation (C10)] to theaverage squared modulus of the bispectrum, which is non-zero for all spatial frequencies out to the diffraction limit ofthe telescope. This simplifies for the case of an unapodizedaperture to a product of terms dependent on the normalizeddiffraction-limited transfer function, T(2)(u), and the nor-malized overlap integral, T(4)(u2, U1 + U2), of four pupils withseparations from an arbitrary origin of zero, Xu2, X(u1 + U2),

and X(ul + U2) + Xu2:

If IHO(X1)I2IHO(X1 + Xu1)12dxf| IHO(X2)I2

Y pupil Jpupil

X IH0(x2 - Xu )121 HO[x -_ X(U1 + U2)11

X 1H0(X2 + XU2 )12 dX2

__ 1 T1(2)(u)T(4)1(U2 U1 + U2).-S4

(C10)

The symmetry of the bispectrum modulus squared resultsin additional terms for the same phase-closure realization,namely,

2 T(2)(u2 )T(4)(ul, U1 + U2),ns

2 T(2)(ul + u2 )T(4)(ul, U2).ns

(c) Consider terms for which

('Iq - 4') + ((rb 4) + OD' s t) °(-tq/ q ) + ((¢rt - rt1) + ((.s" 4)s') =O.

(C11)

These correspond to four terms that have identical nonzerocontributions to the ensemble average over the complexamplitude statistics. For example,

Ayers et al.


(A(x 3)A*(xl + Xul)) (A(x,)A*fX 2 - X(U1 + U2)])

X (A(x2)A*(x 3 + Xu2 )) (A(x 4 + Xul)A*(x 6))

X (A[x 5 - X(u 1 + U2 )]A*(x 4 ))(A(x 6 + X1U2 )A*(x 5))

= (x1 - x3 + u)6[x2 - X1 -x(u + u2)]6(x3 - x2 + "12)

X O(x6 - X4- "u)[x 4 - x5 + X(u1 + U2)]

X O(x5 - - X1U2). (C12)

Another example is shown diagrammatically in Fig. 14(d).Again, a diffraction-limited contribution results, which, af-ter all terms are combined, is simplified by considering anunapodized aperture, yielding

IHO(X1)I2 1HO(X1 + XU1)1

21HO[X1 + X(U1 + U2)II2dxi}

#_4 T (3)(Ul, U2) T(3) (U , U2). (C13)ns

The normalized diffraction-limited bispectral transfer func-tion T(3)(ul, U2) is the overlap integral of three pupils withseparations from an arbitrary origin of zero, Xul, and X(ul +U2)-

The phase-closure realization being considered here hassome redundancy with the preceding realizations, (a) and(b). For example, in relation (CIO), if x1 + Xu1 = x2, thenthe phase-closure conditions of (c) are realized. The redun-dancy must therefore be eliminated by subtracting the rele-vant duplicated terms. In this example this corresponds tosubtracting

2 T( 4 )(u2 , Ul + U2).

Compensating for all the redundancy requires subtraction of

45 [T (4) (U2 Ul + U2) + TV4)(ul, ul + U2) + T(4)(ul, U2)]

+ 25 T(3)(ul, U2). (C14)ins

Figures 14(e) and 14(f) show examples of phase-closure rela-tions that have redundancy.

(d) Finally, consider realizing the closure phase of Eq.(C4) without any of the equality conditions of considerations(a)-(c) being true. Averaging over the complex amplitudesyields 40 terms that have the following identical, nonzerocontributions to the ensemble average, provided that thepupil is symmetric:

Y| f IHo(X1)I4IH0(X 1 + XU1 )14IHO[x1 + M(U1 + U2)]14dx 1YPupil

T(Nul, U2)

n.,5(C15)

The ensemble-averaged squared modulus of the bispec-trum therefore consists of a sum of many terms with, inparticular, differing dependences on ii. However, it is as-sumed that in5 is much greater than unity, and so, in thislimit, terms that are inversely proportional to the lowestpower of ini will dominate easily. In this example the contri-

bution of the term defined by Eq. (C8) will dominate allothers.

This approach has been applied to all the ensemble aver-ages over the atmospheric statistics required for the SNRstudy.

ACKNOWLEDGMENTS

This study was supported by the U.K. Science and Engineer-ing Research Council (grant GR/D 92332) and the U.S.Army (grant DAJA 45-85-C-0028).

Note added in proof: Some of the results in this paper arestated by Chelli.3 2

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