bispectral analysis at low light levels and astronomical speckle masking

14 J. Opt. Soc. Am. A/Vol. 2, No. 1/January 1985

Bispectral analysis at low light levels and astronomicalspeckle masking

B. Wirnitzer

Physikalisches Institut, Erwin-Rommel-Strasse 1, 8520 Erlangen, Federal Republic of Germany

Received February 26, 1984; accepted September 10, 1984

The bispectrum is the Fourier transform of the triple correlation, sometimes also referred to as the triple-productintegral. The influence of photon noise on the bispectrum of an image-intensity distribution is discussed. As anexample, the astronomical speckle-masking method is considered. Speckle masking is a method to overcomeimage degradation that is due to the turbulent atmosphere. It is shown theoretically that bispectral analysis inspeckle masking should yield true, diffraction-limited images in all those cases in which the speckle-interferometryprocess has been successful in reconstructing the object autocorrelation.

1. INTRODUCTION

An image that is detected at low light intensity consists of onlya few isolated photon events. If such photon-limited imagesare the raw data for image-processing algorithms, the influ-ence of photon noise must be considered. This influence wasstudied by Goodman and Belsher for the case of power-spectrum analysis.1'2 An interesting application of this workis the astronomical speckle-interferometry method3 at lowlight levels.2' 4

The so-called bispectrum is a valuable generalization of the(second-order) power spectrum to a spectrum of third order.The bispectrum contains both modulus and phase informa-tion, and the bispectrum is, under certain conditions, insen-sitive to noise.

During the last few years bispectra and their Fouriertransforms, the triple correlations, have proved to be usefultools in image processing. An example is the astronomicalspeckle-masking method,5 -7 which overcomes the imagedegradation that is due to both the turbulent atmosphere ofthe earth and telescope aberrations. Speckle masking yieldstrue diffraction-limited images of (general) astronomicalobjects. 7 Since the most interesting astronomical objects areusually faint, it is desirable to extend speckle masking to rawdata, which are affected by photon noise.

Some more examples of triple correlation and bispectrumanalysis are given in Refs. 8-13 (for a summary, see Ref. 13).It is desirable to perform many of these analyses at low lightlevels.

In Section 2 of this paper, the influence of photon noise onbispectral analysis is discussed from a general point of view.The modeling and the principal method of solution parallelthose of Goodman and Belsherl'2 given for power-spectrumanalysis. We found that at low light levels the bispectrum ofthe detected image is identical with the bispectrum of theclassical intensity falling on the detector plus some fre-quency-dependent bias terms.

In Section 3, we apply bispectral analysis to photon-counting speckle interferograms of faint astronomical objects.We discuss the signal-to-noise ratio (SNR) in the averaged

bispectrum of many speckle interferograms. From the av-eraged bispectrum true diffraction-limited images of generalastronomical objects can be reconstructed. As a fundamentalresult, we found that image reconstruction is possible in allpractical cases in which the speckle-interferometry processcan be used to provide the object autocorrelation. The SNRof speckle interferometry is discussed in Refs. 2, 4, and 14-16,for example.

2. BISPECTRAL ANALYSIS AT LOW LIGHTLEVELS

A. Bispectra and Triple CorrelationsBefore we consider the influence of photon noise, we start withthe definitions of the triple correlation I(3)(x, x') and the bi-spectrum I(3)(u, v) of an image-intensity distribution I(x):

I(3)(x, x') = JfI(x")I(x" + x)I(x" + x')dx",

I(3)(u, v) = f I(3)(X, x')exp[-27ri(ux + vx')Idxdx'

= I(u)I(v)I(-u -v),

(1)

(2)

where I(u), I(v), and IF-u - v) are spectra (or, equivalently,the Fourier transforms) of 1(x).

Since the intensity distribution I(x) is a real function, itsspectrum is Hermitian, i.e., I*(u) = I(-u). As a conse-quence,

I(3)(u, v = 0) = I(u)I(0)I(-u)= const.Ii(u)12, (3)

with

const. = 1(0).

Therefore the bispectrum contains the power spectrum as aspecial case. Bispectrum analysis goes beyond power-spec-trum analysis since the bispectrum contains almost completeand highly redundant information about both the Fouriermodulus and the Fourier phase of the image-intensity dis-

0740-3232/85/010014-08$02.00 © 1985 Optical Society of America

B. Wirnitzer

Vol. 2, No. 1/January 1985/J. Opt. Soc. Am. A 15

tribution. Therefore image reconstruction is possible withhigh SNR, as is discussed in detail in Ref. 11.

B. Influence of Photon NoiseWe now discuss how photon noise influences the bispectrumof an image. The modeling and the principal method of so-lution will follow those given by Goodman and Belsher1' 2 forthe case of power spectra.

At low light levels, we cannot record a continuous intensitydistribution I(x), but we can use an image intensifier to collectraw data consisting of isolated photoevents. These raw dataare described as

ND(x) = E 5(x -xj),

j=1(4)

where 5(0) is the two-dimensional Dirac delta function, xj isthe location of the jth photoevent, and N is the total numberof detected photons. If the photoevents in the detected imagedo not have a 6-like structure, the center of gravity of eachphoton dot is determined in a preprocessing step.

In our processing, the photon-counting images are firstFourier transformed:

D (u) = 6(x - xj)exp(-2iriux)dxfj=1

N= _ exp(-2iriuxj). (5)

j=1

Evaluation of the bispectrum yields

D(3)(U, V) = D(U)D(V)Df(-U - V)

= Z expj-2iri[u(xj - xj) + v(xk - xI)]I. (6)j,k,l

We are interested in the expected value E[D(3)(u, v)] ofD)(3 )(U, v). The expected value must be taken with respectto three statistical processes:

(1) The photon-detection process, the probability densityp(xj) of a photon's being detected at position xj, is propor-tional to the classical intensity

p(xj) = CI(xA), (7)

where the constant c depends on the properties of the de-tector.

(2) The total number N of photons detected in the imageis described by a Poisson distribution.

(3) Finally, we average over the statistics of I(x) [moreexactly, the average is over the rate function X(x, t) that isproportional to I(x, t); see Ref. 1].

We first take the expected value [statistical process (1)] withrespect to the photon coordinates

Ejkl[D( 3)(u, v)]

= Z EjkL(expt-27ri[u(xj - XI) + V(Xk - X)]). (8)j,k,l

In order to evaluate Eq. (8), we collect

case (a):case (b):case (c):

N terms in which j = k = 1,N(N- 1) terms in which j #d h = 1,N(N-1) terms in which j = k /- 1,

case (d): N(N - 1) terms in which j 5d I = k,case (e): N(N-1)(N-2) terms in whichj w- k #d 1.

We first discuss the terms in case (a):

NE EjkI(expj-27ri[u(xj - xi) + v(Xk - xI)]}) = N. (9)

j=k=1=1

For the terms in case (b) we obtain

NE Ejkl(expj-27ri[u(xj - xI) + v(Xk - xA)]D)

js^k=1=1

= N(N - 1) L Ejk (exp{-2iri[u(xj -X)])j k

= N(N - )c 21 1(u) 2, (10)

where we used the fact that, for i #d k, xj and xi are inde-pendent random variables, and therefore the probabilitydensity p(xj, Xk) is [see Eq. (7)]

p(xi, xk) = p(xj)p(xk)= c2I(xj)I(xk). (11)

The terms in cases (c) and (d) for Eq. (9) are evaluated in asimilar way. Collecting all terms yields

EjkL[D( 3 )(u, v)]

= N+ c2N(N - 1)[II(u)12 + II(v)12 + jI(-u - v)12]+ c3 N(N - 1)(N - 2)I(3)(U, V). (12)

What follows is [statistical process (2)] the average EN[.] overthe Poisson statistics of the number N of detected photons[for given I(x)]. For a Poisson process,

(13)

where N1 denotes the conditional mean of N [I(x) known].Continuing the averaging, we find that

EjklN[D( 3 )(U, V)]

+ NV2[I`(u)12 + I '(V)12 + `(-U - V)12]

+ N1 T3 (3)(U, V), (14)

where i(u) = cI(u) denotes the normalized spectrum. Theconstant c is determined by c fI(x)dx = fp(x)dx = 1.

Finally, we evaluate the average [statistical process (3)] overthe statistics of I(x) and obtain

E[D( 3 )(U, V)]

= TV +N2[(jI(U)j2) + (Ii(v)12 ) + (Ii(-U - V)12)]+ NT3([(3) (U, V)), (15)

where N denotes the unconditional mean of N and the anglebrackets denote the average over the statistics of I(x).

From Eq. (15) we see that photon noise in the detectedimages causes an undesired, frequency-dependent photonbias. This photon bias contains terms that are proportionalto the spectral density (IT(u)I 2) of the classical image intensity.For high light levels (N V O) the term N 3(i (3)(U, v)) domi-nates, and the bispectrum of the detected image is identicalwith the bispectrum of the intensity distribution at the de-tector. The influence of photon noise is illustrated by theexamples in Figs. 1 and 2.

B. Wirnitzer

ENMN - 1) ... (N - r + 1)] = TV,,r,


(a)

(b) (c)

Fig. 1. One-dimensional example for a triple correlation and a bispectrum. (a) Double pulse, (b) its triple correlation, (c) its bispectrum.

(a) (b)

Fig. 2. Illustration of the influence of photon noise. (a) Expected value of the bispectrum of the pulse in Fig. l(a) in the case of photon-limitedraw data, (b) Fourier transform of the bispectrum in (a). We assume that the raw data contained five photons, on average.

C. Unbiased EstimateWe are interested in an unbiased estimate of (r(3)(u, v)). Asolution to the problem will be obvious after we write the re-sult of Goodman and Belsher for the spectral density of aphoton-limited image':

E[ID(u)j 2] = N + N 2 (V (u) 2) (16)

Equation (16) could be used to compensate for the bias terms

in the bispectrum in Eq. (15). An alternative is to evaluatefrom the raw data D(x) a new quantity Q(3)(u, v):

Q(3)(u, v) = fi(3)(u, v) -

[JD(u)12 + ID(v)12 + ID(-u - v)2- 2N]. (17)

Now we evaluate the expected value of Q(3)(u, v) instead ofthe expected value of D( 3)(u, v). Using Eqs. (15) and (16),we obtain

B. Wirnitzer


E[Q(3)(u, v)] = N3(T(3)(u, v)), (18)

i.e., we obtain an unbiased estimate of the bispectrum of I(x),although the initial data were photon limited. Fouriertransformation of Eq. (18) yields the triple correlation of theimage.

3. BISPECTRUM ANALYSIS OF PHOTON-COUNTING SPECKLE INTERFEROGRAMS

We want to compare the SNR in the power spectrum with thatin the bispectrum of photon-counting speckle interferograms.For that purpose, we repeat briefly in Subsection 3.A the es-sentials for speckle interferometry and speckle masking. InSubsection 3.B, we expand speckle masking to photoncounting of raw data.

A. Speckle Interferometry and Speckle MaskingAstronomical speckle methods are used to overcome imagedegradation that is due to the turbulent atmosphere of theearth. The fascinating resolution gain of these techniquesis up to a factor of 50.

In astronomical speckle methods many short-exposurephotographs, so-called speckle interferograms, are evaluated.For high light levels, the nth speckle interferogram can bedescribed by the following incoherent, quasi-monochromatic,and space-invariant imaging equation:

In (x) = 0(x) * Pn (x), (19)

where O(x) is the two-dimensional object intensity, the as-terisk denotes convolution, and Pn (x) denotes the combinedpoint-spread function of the telescope and the atmosphere.The aim of all speckle methods is to overcome the imagedegradation Pn (x).

First we review speckle interferometry, which is a power-spectrum analysis of many speckle interferograms. 3 Theensemble average (.) over the power spectrum of many framesof data yields

(I In (u)l2 ) = I O (U)J 2(I Pn (U)J 2). (20)

The speckle-interferometry transfer function (IPn(u)12 ) isknown from theory or from the measurement of an astro-nomical point object. Compensation of (lPn(u)12) providesthe object power spectrum I O(u)2 or, after a Fourier trans-formation, the object autocorrelation. The object autocor-relation has diffraction-limited resolution since (JPn (u)12) is

positive and nonzero for all frequencies up to the diffractionlimit of the telescope.

A speckle method that provides true images instead of au-tocorrelations is speckle masking. In speckle masking,third-order correlations are used instead of second-ordercorrelations. Image reconstruction in speckle masking canbe performed in the correlation domain 5,6 or in the Fourierdomain. 7 Here we concentrate on image reconstruction inthe Fourier domain, i.e., we reconstruct the true image fromthe average bispectrum of many speckle interferograms:

(In( 3 )(u, V)) = C(3)(U, V)(Pf(3)(U, V)). (21)

From Eq. (21) the complex spectrum O(u) of the object canbe reconstructed in modulus and phase, as discussed in detailin Refs. 7 and 11. The basis for the reconstruction is the fact

that the transfer function (Pn (3)(u, v)) is positive and nonzerofor all frequencies below the diffraction limit of the telescope,providing a diffraction-limited object bispectrum.

For phase recovery of the object Fourier transform, a re-cursive algorithm is applied to the bispectrum in Eq. (21).Since the four-dimensional bispectrum contains highly re-dundant information about the two-dimensional Fourierphase of the object, phase recovery is possible with a highSNR. Furthermore, since (

3)(u, v)) is found to be

real, 7

phase I(In(3)(u, v))I = phase I{(3)(u, v)}, (22)

and therefore, since (In (3)(u, v) ) can directly serve as raw datafor the phase recursion, no transfer function has to be com-pensated for.

In order to reconstruct the Fourier modulus it is possible,for example, to produce speckle-interferometry data or,equivalently, power-spectrum data. This is achieved bysetting u = 0 in Eq. (21), as shown in Eq. (3). Bolstering ofthe SNR is possible by averaging redundant modulus infor-mation, which is contained in the bispectrum. Modulus re-covery from bispectra is discussed in Ref. 11, but it has notbeen fully examined until now.

B. Photon-Counting Speckle MaskingWe now apply the results of Subsection 2.C to generalizespeckle masking to the case of low light levels.

Speckle interferograms of faint astronomical objects con-sists of a few isolated photoevents. Our aim is to gain fromthese photon-counting raw data the ensemble-average bi-spectrum (I, (3)(u, v)) of the classical intensity arriving at thedetector.

A key to the problem is to calculate from each photon-counting speckle interferogram Dn (x) the quantity

Q. (3)(U, V)

= D&(3)(u, v)

- [JDn(u)J2 + IJ)n(v)12 + IDn(-u - v)12 - 2N]

and to evaluate the ensemble average over many frames ofdata. Applying the results of Subsection 2.C, we find for theexpected value of (Q,( 3)(u, v))

E[(Qn(3 )(u, v))] = N 3(in( 3 )(u, v)), (24)

where (In( 3)(U, V)) is the desired (normalized) bispectrum.

From (i3 (3)(u, v)) true images can be reconstructed, as brieflydescribed above.

C. Signal-to-Noise Ratio in the BispectrumIn this subsection, an expression for the rms SNR for bi-spectrum analysis in speckle masking is presented and com-pared with the SNR of power-spectrum analysis in speckleinterferometry. We consider first the SNR in the bispectrumas obtained from Z frames (or realizations) of data:

(SNRBs)Z = EyQ/)

7N3(7 (3)(u, V)) (5UQ/J

where uQ denotes the rms variance of the quantity Qn(3) (u

v) defined in Eq. (23).

(23)

B. Wirnitzer


The evaluation of o-Q in Eq. (25) is given in Appendix A.We note without proof that aQ contains terms that dependon the frequencies (u), (2u), (u + v), (u + 2v), etc. as a gen-eralization of the half-frequency noise, known from the powerspectrum of a photon-limited image.

In Appendix A, it is shown that for bright objects with anaverage number of photons per speckle nI much larger thanone,

AOBI < arctan UQ/1\/z = arctanj 1/(SNRBI)zl

(29)

where oTQ/lv/§Z is the (rms) variance in the bispectrum if Zspeckle interferograms are evaluated. We have assumed thatZ is large; therefore (SNRBI)z >> 1; hence

(26)

In the case of low light levels, we found for frequencies u, v $,O ul 0 1 vJ that

for i << 1, (27)

where 6(3)(u, v) = 6(3)(u, v)/3(3)(0, 0) is the normalized, dif-fraction-limited, and aberration-free bispectral transferfunction of the telescope (see Ref. 7 of App. B).

For comparison, we give the SNR for the power spectrumas exploited in speckle interferometry 4 :

(SNRpow)z -\a(SNRpow)z j (u)1o(u)l2

for nT >> 1,

for n << 1, (28)

where 6 (u) is the normalized, diffraction-limited, and aber-ration-free transfer function of the telescope. Figure 3 showsthat the SNR in the power spectrum exceeds the SNR in thebispectrum. However, the bispectrum contains redundantinformation about the object's Fourier transform. Averagingthis redundant information yields a reconstruction with im-proved SNR, as illustrated by curve (c) in Fig. 3 and as dis-cussed below.

We want 1to estimate the SNR of speckle masking. The newaspect of speckle making is that one can gain the Fourier phaseof the object, not only its modulus. Therefore we restrict ourconsideration to the phase-recovery algorithm7 "'1 that is ap-plied to the bispectrum. The phase error in the bispectrumcan be estimated by

Mz

0

0

- 1

-2

-3

photons per speckleFig. 3. SNR for various speckle methods: (a) SNR in the bispec-trum, (b) in the power spectrum, (c) in the Fourier phase recoveredfrom the bispectrum of Z frames of data. We assume that the speckleinterferograms contain 2500 speckles on average and that o(u) = 1.

arctanl1/(SNRBi)z - 1/(SNRBI)Z- (30)

Now we take into consideration that in phase recovery frombispectra redundant information about the object Fourierphase is averaged. A detailed discussion about the influenceof noise on the phase-retrieval algorithm is given in Ref. 11.Here we give only a crude estimation.

If we assume that the speckle interferograms contain -n,speckles on average, then the Fourier spectrum I(u) containsapproximately nl statistically independent correlation areas.The bispectrum I(3)(u, v) = I(u)I(v)!(-u - v) will containapproximately g,2/4 statistically independent correlationvolumes [the factor 1/4 is due to an inherent symmetry ofbispectra: I(3)(u, v) = I(3)(v, u) = I(3)(-v,-u-v)]. If nSphase values are reconstructed from n 2/4 complex bispec-trum values, then the phase error is reduced by a factor ofV/4Y'1/ 2 (see also Ref. 11). Using expression (29), we obtain

for the error in the reconstructed phase:

A0PH - 27 -1/21 1/(SNRBI)ZI- (31)

Next we define a measure for the SNR in the reconstructedphase:

SNRPHASE = '-L kPH

(32)

According to Eq. (32), SNRPHASE = 1 means that the meanphase error AOPH is equal to r/2.

Using expressions (27), (31), and (32), we find for the SNRin the reconstructed phase that

(SNRPHASE)Z Z X '/2 X 7n 1/2 for i >> 1,

(SNRPHASE)Z - N/Z X '/2 X N1/2 X n X b(u) X o(u)for n << 1, (32')

where we have used n X ni = N. For amplitude recovery fromthe bispectrum, divisions are involved, and the SNR in ex-pressions (32) may be regarded as an upper limit. Since thebispectrum contains the power spectrum in it; central planes,u = 0, v = 0, and u = -v, modulus recovery is at least possiblewith the SNR known from speckle interferometry. '

Comparison of the SNR in expressions (28) and (32) showsthat for N > 4, the Fourier phase recovered from the bispec-trum has a higher SNR than the Fourier modulus (or powerspectrum) as obtained in speckle interferometry. Ini allspeckle-interferometry experiments performed until now thespeckle interferograms contained more than four photons, onaverage. Therefore it should be possible to use the raw dataof all these experiments to recover the Fourier phase of theobject and to reconstruct true images.

B. Wirnitzer

- I 1/(SNRBi)zl,

(SNRBI)Z - N/Z for 7T >> 1.

(SNRBI)Z - A/Z_ jE3/26 (3) (U, V) 6 (3) (U, V)

1


CONCLUSIONS

We have discussed the influence of photon noise on bispectralanalysis of images. As a result, we found that for low lightlevels a constant and a frequency-dependent photon biascontribute to the bispectrum. A method was suggested foravoiding this undesired photon bias.

As an example, we expanded the astronomical speckle-masking method to the case of extremely low light levels. Thecalculation of the SNR in the bispectrum showed that imagereconstruction is possible in all cases in which the speckle-interferometry process is successful, provided that the averagenumber of photons per speckle interferogram exceeds four.

APPENDIX A

We want to calculate the SNR of bispectral analysis in specklemasking for Z frames (or realizations) of data. The quantityQn (3 )(u, v) defined in Eq. (23) is an unbiased estimator of thedesired bispectrum (in(3)(u, v)). We found that

E[Qn( 3)(U, V)I = N3 (T (3)(u, V)). (Al)

In further calculations, we use the one-dimensional variablesu and v instead of the two-dimensional vectors u and v forsimplicity without sacrificing generality. In order to estimatethe SNR, we must calculate the rms variance aQ of Qn (3)(u,v) for a single frame of data:

UQ2 = E[IQn( 3 )12I - IE[Qn(3 )]12

= ElIDn( 3 )(u, v)12 + ID&(u)14 + ID&(v)I4

+ If5n(-u - v)I4 + 2[jDn ()D(V)I 2

+ ID&(u)JJn(-u - v)12 + IDn(v)Dn(-u - v)12]+ 4N2 - 4N[IDn(U)12 +I Dn (V)12 + IDn(-u - v)12]

- [Dn(3)(u, V) + Dn(3)*(u v)]EDtn (u)I2 + bn(v)I2

+ IDn(-u - v)12 - 2N]l - IE[Qn(3)]12 .

First we evaluate the expected value with respect to thephoton coordinates. To this end, the summations in terms(a)-(o) of Eq. (A3) are placed in different classes, similar tothe calculation in Section 2. We are interested ultimately inthe case of very high or very low light intensities. Thereforewe may restrict our computation to those index classes thatproduce terms depending on N 6 or that are frequency inde-pendent and depend on N, N2, or N3, as discussed below.

We consider the evaluation of term (a) of Eq. (A3) as anexample. First, we select those contributions that are fre-quency independent. The conditions for u independence andv independence are the index combinations (j = 1, m = o) or(J = m, 1 = o) and (k = 1, n = o) or (k = n, 1 = o), respectively.Combination of both conditions yields for term (a) of Eq.A(3)

index class

j=k=l=m=n=oj = l = k FZ m = n = o

j = l = o = m y- k = n

j = m 5- l = o = n = k

j = m = k = n mk -l = o

j=m ilo~ = ho F k =n

obtain

N

N(N- 1)

N(N- 1)

N(N- 1)

N(N- 1)

N(N- 1)(N-2).

We also need the index class

j F4 k F 1 F m ,# u 5- 0,

yielding N(N - 1)(N - 2)(N - 3)(N - 4)(N - 5)1T (3)(u, v)12.

(A2)

Using Eq. (5) and Eq. (6) yields

rYQ2 = F_ E(expj-27riju(xj - x- xm + x0)j,k lm,n,o

+ V(xk - Xi - Xn + XO)]})

+ _ E(exp{-2zri[u(xj - Xk- X + Xm]})j,k,l,m

+ F_ E(expj-27ri[v(xj - Xk - x1 + Xm]1)j,k,l,m

(a)

(b)

(c)

+ F_ E(exp-27ri[(-u - v)(x1 - Xk - xi + xm)]1) (d)j,k,l,m

+ 2 _ E(expl-2iri[u(xj - Xk) + v(x1 - Xm)]1) (e)j,k,l,m

+ 2 Z E(expl-27ri[u(xj - Xk - xj + Xm)j,k,l,m

+ V(-Xl + Xm)]1)

+ 2 F E(expl-27ri[u(-xj + xk)j,k,l,m

+ V(-Xl + Xm - Xi + XkI1)

+ 4N 2

(g)

- 4y E(exp{-27ri[u(xj - x)]j)j,k

- 4N F_ E(expl-2ri[v(xj - xk)]1)j,k

- 4N _ E(exp{-27ri[(-u - v)(x1 - xA)I1)jk

- Y E(expf-2-ri[u(xj - x1 + xm - xn)j,k,l,m,n

+ V(Xk -k)]j)

- E E(expj-2ri[u(xj-xx)j,k,l,m,n

+ V(Xk - Xj + Xm -Xn)]})

- F E(expl-22ri[u(xj - XI + Xm - Xn)j,k,l,m,n

+ V(Xk - Xi + Xmn -XAD

+ 2N Y E(expj-27ri[u(xj - xI) + V(xk - xI)]1)j,k,1

+ c.c. of terms (l)-(o)

(h) - IE(Qn(3 )12.

(j)

(k)

(1)

(m)

(n)

(o)

(p)

(A3)

B. Wirnitzer


Finally, we average over the statistics of N; using Eq. (13) weobtain

NI + 4NI2 + NI3 + ... + Ni61in(3)(U, V)12. (a)

In a similar way, the other terms in Eq. (A3) are evaluated:

where E(3)(u, v) is the bispectrum of the aberration-free anddiffraction-limited transfer function of the telescope. Wenote that E(3)(u, v) and 6 (3)(u, v) are similar in the first ap-proximation.

Using expressions (A8)-(A1O) in expression (A7) yields

(b) + (c) + (d)

(e) + (f) + (g)

(h)

(i) + (g) + (k)

(1) + (m) + (n)

(o)

3(N1 + 2Ni2) +

3(2N + 2N, 2) +...

4N 12 + 4NV,

-3(4N2+ 4Ni) +...

-3(NI + 2N12) +.. ..

+2NI2 + 2Nr + . . . ,

-3(NI + 2NI2 ) + 2NI2 + 2NI + . . ..

Collecting terms (a)-(p), we obtain

NI-3 + . .. + NI19' (3)(U, V)12 (a) -(p). (A4)

Averaging over the statistics of I(x) yields

N3 + . .. .N6(IT'(3)(U, V)l2) (a) - (p). (A5)

Now we evaluate the term JE[Qn3]I2 in Eq. (A). In Section2 we found that

Ejkz,,[Q 3)] = N313n(3)(U, v)

= NI3Tn(U)'n(V)Tn(-u - V).(A6)

The task remains to average over the statistics of a singleframe of data In(x). If the image-intensity distribution ex-tends over a region L X L, then T(u) is approximately a circularcomplex Gaussian random variable for u >> 1/L.2 For u, v -0 and Jul 5: lvl, i(u), T(v), and -(_u - v) are approximatelyindependent, and therefore Ei[T(u)u(v)T(-u - v)] - 0.Therefore we find that, for a single frame of data,

IE[In( 3 )]120 o for u, v 5, 0, Jul s6 Ivl.

Using Eqs. (A3), (A5), and (A7) yields

o-Q2 = N3 + ... + N6(I4 (3)(U, V)12).

(A7)

(A8)

Now we are ready to determine the rmis SNR associated withZ frames of data. Equations (Al) and (A8) yield

(SNRBI)Z = (In+ . + N6(I( 3)(u v)2))"2 . (A7T)

We can express (T (3)(u, v)) as

(T (3)(u, v)) = (Pn( 3)(U, v))( 3)(u, v). (A8')

The normalized optical transfer function (Pn (3 )(U, v)) of theatmosphere-telescope combination can be shown to be

(p (3) (U, v)) _ i 3/25 (3)(U V)

u, v >> ro/f, n7 >> 1, (A9)

where 6(0) is the normalized, aberration-free, and diffrac-tion-limited bispectral transfer function of the telescope,7 Xis the wavelength, ro is Fried's coherence parameter, and ifis the average number of speckles per frame. Similarly, wefind that

( u1n(3)(U, v)»2) - lS-31T(3)(» V)12

u, v >> ro/;\f, iiS >> 1, (A10)

(SNRBI)Z = ~ ~ ~Tn-3/26(3)(u, v)o(u, v)

(N 3 +... n 3JT(3 )(u, v)1216(u, v)12)1/2

(All)

We now consider the case of very high limit levels. For N- (IT - -) we obtain

(SNRBI)z - \, 7n - , (A12)

where we used the approximation 6(3)(u, v)I/JI()(u, v)12 - I

and where we assumed a pointlike object, causing (3)(u, v)= 1.

For low light levels

(SNRBI)Z - jj3/26(3)(U, V)5(3)(u, v)NZ, n << 1,(A13)

where in = NV/-n is the average number of photons perspeckle.

ACKNOWLEDGMENTS

I want to thank A. W. Lohmann and G. P. Weigelt for stimu-lating remarks and the Deutsche Forschungsgemeinschaft forfinancial support.

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B. Wirnitzer

bispectral analysis at low light levels and astronomical speckle masking

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