Reduction of image speckle noise with the use of an integrative synthetic aperture imaging technique isstudied. It is found that the Fourier inversion of the mutual intensity estimate [Appl. Opt. 30, 206-213(1991)] yields an image intensity that corresponds exactly to the illumination of the object with partiallycoherent light from source optics imaging a delta-function incoherent source. An expression for thesignal-to-noise ratio at an image point is derived for a large rough object with delta-function correlatedamplitude reflection. A synthetic aperture system receiver of sufficiently small diameter yields imagespeckle with a signal-to-noise ratio (SNR) equal to 1. When the receiver and the transmitter diameters areequal, the SNR is 2 for linearly polarized speckle. The SNR continues to increase with receiver size and islinear in the diameter for large receiver-to-transmitter diameter ratios.
1. IntroductionIn a recent paper,' an active pupil plane imagingsystem was described that consists of an array ofilluminating beams and an integrating receiver aper-ture. The system's architecture and processing pro-vide an estimate of the Fourier transform of theintensity reflectivity of a laser-illuminated object, aswould ordinarily be obtained only from a system thatuses incoherent light. The estimated transform isFourier inverted, and a corresponding image estimateis computed. The basic mechanism used by thesystem is to illuminate an object with interferencefringes at different spatial frequencies resulting frompairs of differently spaced apertures in the transmit-ter such that each spatial frequency is identified witha unique temporal beat frequency at the detector.Various aspects of similar concepts have been studiedby Ustinov et al.' and Aleksoff,' who recognized thatsuch systems would have the effect of reducingspeckle if the receiver were sufficiently large. Refer-ence 1 gave an expression for the standard deviationof the estimate of the object Fourier transform result-ing from use of the integrative synthetic aperture(ISA) technique in the presence of speckle and shotnoise. The magnitude of the standard deviation canbe made small as the receiver size increases, indepen-
The author is with the Naval Research Laboratory, Code 6530,Washington, D.C., 20375.
dent of the system resolution, implying that specklecontrast in the image should be reduced.
The purpose of this paper is to compute the effect oftransmitter and receiver aperture sizes on imagespeckle contrast directly for the case of a continuousaperture corresponding to the ISA transmitter array,because for the purpose of target identification,4
speckle contrast is an important indicator of imagedegradation in coherent systems. The calculation isdiscussed in Sections II and III. Section II shows thatthe image resulting from the system-provided Four-ier transform estimate, followed by its inversion, isformally identical to the image that would result froman equivalent system in which the imaged target isilluminated with spatially partially coherent light of agiven correlation scale. This identification with aconventional optical system provides an intuitiveunderstanding of the speckle reduction effects of thesystem. It also shows that the method describedimplies the reduction of artifacts because of coherentimaging in general, independently of the precisestatistics of object surfaces. In Section III, the stan-dard deviation of the image intensity is computed atthe center of a large uniform square image, and theratio of mean to standard deviation, or reciprocal ofthe speckle contrast, is obtained for an ideally roughobject surface. It is found that for a small receivercorresponding to coherent illumination in the equiva-lent imaging system, the speckle contrast equals onefor a range of transmitter configurations. To providenumerical values for other receiver sizes, the generaltransmitter aperture is specialized to a filled square
aperture. As the receiver size increases, the specklecontrast decreases corresponding to the illuminatinglight of a smaller spatial coherence function widthcompared with the impulse response width in theequivalent imaging system.
II. Equivalent Partially Coherent ImagingIn Ref. 1, a system architecture (see Fig. 1) andappropriate signal processing are described that re-sult in an estimate of the mutual intensity functionfor a delta-function correlated rough object. Theestimate [see Eqs. (17b) and (21) of Ref. 1] denoted byJ, is an integral over fields in the receiver plane, asquare of side 2L:
J(U1, U2) = 1 XLRA ( A R). (
Object
In this integral, the tilde denotes the Fourier trans-form, andA is the field just above the rough surface ofan object, resulting from its illumination by a trans-mitter source at ui, one among a number of sources ofan array in the far field at distance R. Position in thereceiver plane (Fig. 2) is denoted by v = (v., v ), andvectors ul and 2 denote source positions in thetransmitter plane (coincident with the receiver plane).The convention is used that dv = dvdvy, and oneintegral sign is used for each boldface differential.
If ensemble averaging (over object surface struc-ture) of both sides of Eq. (1) is carried out for the caseof statistically stationary fields with circular complexGaussian statistics corresponding to ideal surfaceroughness, then
(1) (J(u, U2,) = (2L ( +) A*(u2 + ))dV,
or, as a result of statistical stationarity,
(J(U 1 , U2)) = () 2fL d ( V = j ( XR -)
(2a)
(2b)
Thus, the estimate of Eq. (1) is unbiased.. It should be noted that it has been shown in Ref. 1,
that for fully developed speckle with Gaussian statis-tics, the estimate of mutual intensity by the spatialaverage of Eq. (1) approaches the ensemble average,Eq. (2b) as L becomes large. Further, by the vanCittert-Zernike theorem for speckle,5 the Fouriertransform of J with respect to u - 2 gives theintensity reflectivity, i.e., the image, of the object.
Corresponding to the estimate for the mutualintensity in Eq. (1), an estimate for the image inten-sity may be provisionally adopted by analogy withpartial coherence theory6 :
Transmitter ReceiverFig. 1. Example of a possible system architecture. A transmitterof the sparse aperture type, a Mills-Cross, is embedded in thereceiver plane. Mutually coherent beams illuminating the objectfrom any two points on the cross result in interference fringesacross the object.
where the function P(u) specifies the range of sourcelocations on the transmitter. While for a real trans-
mitter of the kind envisaged, aperture location isordinarily a discrete variable, the use of continuousvariables permits system function to be computedand interpreted more easily and, following Reynoldsand Cronin7 as well as others, will be adopted here.
The choice of Eq. (3a) for the intensity estimatorwill be justified first on the grounds that it is unbi-ased, but it will further be shown below that it may beinterpreted as an exact relation for a certain equiva-lent partially coherent imaging system. Because Eq.(1) provides an unbiased estimate for the mutualintensity arising from an ensemble of ideally roughobjects, it follows that the ensemble average of Eq.(3a) provides an unbiased estimate of the correspond-ing image. If P(u) is interpreted as the imagingsystem pupil function and the expression of Eq. (2b)is used for the mutual intensity, the ensemble aver-age of Eq. (3a) may be interpreted as the image-planeintensity that is due to a pupil plane mutual intensityJ(ul - u) arising from the object to be imaged actingas a delta-function spatially incoherent source8 :
I(x) = JJ P(u,)P*(u2)J(u1 - u2 )exp[ (AR 2j duidu2.
(3b)
If Eq. (3b) is transformed by the change of variables
Interchanging the order of integration results in thefamiliar representation of incoherent imaging
1(X) = f T(u)J(u1)expf Ri K du, (4b)
where r(ul) represents the usual transfer functioncorresponding to the pupil P and J is the Fouriertransform of the object intensity reflectivity. Conse-
(n
S L1,M1 R'-Plane L, x-PlaneFig. 3. Equivalent optical system for the interpretation of Eq. (7).S is a delta-function incoherent source. The source optics, consist-ing of lens L, and square mask Ml, form an image on the x' plane,where a partially transparent object is located. The intensity imageof this object formed by lens L2 is observed in the x plane.
quently, the estimate of Eq. (3a) is seen to beunbiased for the image of a rough object.
Equation (3a) gives the exact image intensity thatwould result from illuminating any object indepen-dently of its surface statistics with light from adelta-function incoherent source by using appropri-ate source optics. If the fields in Eq. (1) are expressedin terms of their Fourier transforms in the form
- (u + v r 2Tix 1(ui + v)1A R ) = JA(x)exp Idx (i = 1, 2) (5)
and the estimate of Eq. (1) is used in Eq. (3a), one has
i(X) = ff, f' P(ul)P(u 2)exp[IU_ +R ]
[ 2'TriU 2 (X2 + x)dl ARp -duldu,
x A(x )A*(x2) 1 ,( exp[2r (A dvdxldx,. (6)
After performing the indicated operations followed bythe change of variables xi = -xl and x2 = -x2 anddenoting the Fourier transform of P by K, thisbecomes
Equation (7) represents (up to a constant) imaging atone-to-one magnification by an imaging system withamplitude spread function K of an object that trans-mits or reflects complex amplitude A(-xl) after illu-mination by light from a square delta-function spa-tially incoherent source of width 2L at distance R.The situation is represented by the optical systemshown in Fig. 3. In this equivalent system, the sincfunctions represent the mutual intensity of the illumi-nating light from the square source optics pupil, theeffective source of Hopkins.9 The imaging equation inthis general form with spatially partially coherentlight is described by Born and Wolf.6 In the limit of alarge source size, such as L -> c, Eq. (7) describesincoherent imaging for an object independent of itssurface statistics. However, because source size isordinarily limited, statistical fluctuations of the im-age that are due to object surface roughness will occurin general. These fluctuations are considered below.
III. Computation of Speckle ContrastSpeckle contrast is defined as the standard deviationof the intensity divided by the mean intensity, or thereciprocal of the SNR. To compute the standarddeviation, one must first compute the mean square ofthe estimate given by Eq. (3a). One immediately
where it is implicit that the ensemble average arisesfrom an ensemble of rough object surfaces. From Eq.(1), the average over the J's may be written as
J(u,, u2) J(u3 , U4 )) = f L At UR A ( A R Ul(2L)4 EJ- XR )(K
x A* U, + u KlRA u + ul,,du'du`. (9)K R ' R
Equation (9), when we use the Gaussian momenttheorem, becomes
(Ul, U2 ) (u3, U4)) = J - U2\jU (u- U + 1 fLK (R ) (R ) (2L) -L
x uU + U 1 5U3 -U2 + u-Ul d
x J - KR AR Idu'du (10)
Inserting this result into Eq. (8) and noting that theintegral involving each J factor in the first term of Eq.(10) has the same form as those in Eq. (3b), one findsthat
u2 (I2) -()
= ffff0 P(l)P*(u2)P(u3)P*(u 4)
12 LX (2L 4fL J(u1 -u 4 + u1
- U")
x J(u, - u + ull - ul)du1'dull
exp[2rix(ul + u3 - U2 - U4 )] (11x expt AR Id~ud~u. (1
For the purpose of evaluating the SNR, it suffices tocompute (x) and I(x) in the middle of a centeredimage at x = 0. Making the change of variables
after performing the integrations over U4 and u2 toobtain the transfer functions corresponding to thepupil function P. The integral over duldull is in the
0.2 = fr T(Ul)T(U3) (2L)2 f2L ( 2L )
x (1 - I2YI) J(ul + V) J(ul - V)dVduldu3. (13a)
After rearrangement, Eq. (13a) becomes
(2 L) 2 :L I2 L1 2 L f -
X T(Ul) J( 1 dUl T(U3) J( du3dV. (13b)
To proceed further, one must specialize the calcula-tion. This may be done by realizing that the impor-tant statistical characteristics of fully developed im-age speckle are independent of object size and shapefor objects that are large compared with the systemresolution. These characteristics depend only on theimaging lens aperture." Consequently, J may bechosen to correspond to a uniform square object that,by the van Cittert-Zernike theorem for speckle re-ferred to above,5 implies that J has the form
J(U. - U0 2, Uyl - Uy2) = I sinc[ 2ra(uRi- u0
x sinct [2Tauyl -y2)] (14)
In Eq. (14), I, is the intensity at the receiver aperture,2a is the width of the object at distance R from thereceiver, and the speckle scale E in Eq. (13b) may beidentified as = R/2a. As the object size becomeslarge, the sinc function becomes narrow comparedwith the system transfer function and acts like a deltafunction. Consequently, o2becomes
2L)3 f2L (i - iI)(1 - T(-V)T(V)dV
X r J (u~ + Vdul f J 3 du, (15a)
(15b)
after integrating the J's by using the function givenby Eq. (14) and assuming that T is aberrationless.Evaluating the image intensity given by Eq. (4b) atx = 0 for infinitesimal speckle correlation length, wefind that E yields
I(0) = fY t(ul) J (e dul = T(0) rYJ (tul = T(O)Ihe,'. (16)
U = I,,2E.,4 2L 1 - IV-1 I - T2 (V)dVT2 _LY f-.'2 L 2L� V'�
and (16) as
I(0)=r
2L
[f2L - 2L \/ 2L 2(0) 1/
(17)
This result has a form analogous to that obtained byDainty and others'2 considering speckle reduction byscanning with finite apertures. However, the specificformula for that case is different from Eq. (17) andcannot be used for a numerical description of thissystem.
It is to be noted from Eq. (17) that in general theSNR depends on the shape of the system transferfunction but is independent of E, as expected from thefact that a limiting condition has been used. However,when the receiver size is small compared with theextent of the transfer function, Eq. (17) can beevaluated independently of the shape of 7(v). In thiscase, corresponding to coherent imaging for the equiv-alent optical system of Fig. 2,
I(0) 2L 2L
b'J-2L ( 2L -dVdVj (4L 1
which is to be expected.It is also possible to ascertain the behavior of the
speckle contrast as L becomes large compared withthe resolution limit of the transfer function, becausethe integration is halted by the edge of the transferfunction rather than the limits 2L because ofreceiver size. In this case,
I(0) 2L
= rnre2L V01 2(\ ~V) 1~~y1/22L 2L )( 2L T
2(0) |
2L
2(r: dV V] (19)
Thus the behavior for a large L depends only on thevolume under the normalized r2 surface.
To evaluate Eq. (17) for intermediate values of L,we must assume some specific transfer function. Thesimplest case to evaluate is that of the filled squareaperture.' 3 The behavior in this case should indicatethat expected from more general transfer functionsbecause the limiting behaviors of all must be given byEqs. (18) and (19). For a square aperture,
T(V) = (24 )2 (1 II)
Vl 2LT
•IVYI 2L'
otherwise,
-V (2L 0 2L! 2L
T(V) =
where 2L, is the diameter of the aperture. Using Eq.(20) in Eq. (17), we find that the integral in thedenominator is the product of two equal integrals.Hence, Eq. (17) becomes
I(0) 2L
U UL (lI 1 (IVI dV (1f- 2 L 2L)\ 2)(
where the x and y subscripts on V are no longerneeded. There are now two cases to consider. In thefirst case, L > L, whereas in the second case, L < LTWhen the case first holds, the second factor in theintegrand of the denominator of Eq. (21) cuts off (at2L) before the first so that the SNR becomes
I(0) L (L,)2
r ' (L( 2L).(22a)
When the second case holds, the first factor in theintegrands of Eq. (21) cuts off first and the SNRbecomes
I(0) L
0U f2L( _ )(1 _ )2 dV1
2L 1L 2
1 3L + 6L 234 642
(L L4). (22b)
From either relation one obtains
(0)-=2, L =L,.(U
(23a)
Equation (22b) shows that I/r = 1 when L becomessmall, as before. In addition, from Eq. (22a) one findsthat as L becomes large compared with LT,
I(°)=2L (L>>L,). (23b)
It is interesting to attempt to account for thesimple dependence shown in Eq. (23b) by the follow-ing naive reasoning. If the number of coherence areaswithin the area of the point spread function of theequivalent system is taken to be the number ofstatistically independent contributions to the specklefluctuation in the image at a point, the SNR shouldincrease with the square root of this number. Fromthe quantities defined, the coherence area equals(XR/2L)2 and the point spread function area equals(XR/2L,) 2. Consequently, the square root of theirratio is L/L, which gives the general dependence ofEq. (23b).
Using both Eqs. (22a) and (22b), I/(J versus LIL isplotted 4 in Fig. 4. Figure 4 indicates that a largeincrease in the SNR can be obtained, but at the cost ofa large receiver aperture size compared with the
Diameter RatioFig. 4. Plot of the SNR (inverse speckle contrast) versus thenumber of coherence patches within the point-spread-functionarea, LIL,.
transmitter aperture. However, even at equal aper-tures, an enhancement of 2 is obtained, which isuseful when coupled with the fact that only onepolarization has been considered. A second specklepattern realization arising from the other polariza-tion would increase the SNR by an additional factor of+/i leading to its overall increase by a factor of 2.8.
IV. Summary and Conclusion
This research has treated the reduction in imagespeckle noise caused by the use of the ISA imagingtechnique discussed in Ref. 1 in the approximation ofa continuous redundant aperture transmitter. It hasbeen shown that the overall imaging system, includ-ing Fourier inversion of the mutual intensity esti-mate, is formally identical to an imaging system thatuses partially coherent light derived from a delta-function incoherent source. This indicates qualita-tively that image artifacts that are due to coherentimaging should be reduced regardless of the statisticsof object surfaces. The SNR, or reciprocal specklecontrast, has been computed in the image of anideally rough object that is large compared with theresolution limit of the system. This SNR depends only
on the receiver size and the system transfer functionshape. For a receiver of large diameter compared withthe transmitter, the SNR increases linearly with thereceiver dimension. For a receiver and a transmitterof equal diameter, the SNR is 2 for speckle of a givenlinear polarization component.
The author thanks W. H. Carter for a usefulconversion.
References and Notes1. L. Sica, "Estimator and signal-to-noise ratio for an integrative
2. N. D. Ustinov, A. V. Anufriev, A. L. Vol'pov, Yu. A. Zimin, andA. I. Tolmachev, "Active aperture synthesis in observation ofobjects via distorting media," Sov. J. Quantum Electron. 17,108-110 (1987).
3. C. C. Aleksoff, "Synthetic interferometric imaging techniquefor moving objects," Appl. Opt. 15, 1923-1929 (1976).
4. A. Kozma and C. R. Christensen, "Effects of speckle onresolution," J. Opt. Soc. Am. 66, 1257-1260 (1976).
5. J. W. Goodman, "Statistical properties of laser specklepatterns," in Laser Speckle and Related Phenomena, J. C.Dainty, ed. (Springer-Verlag, New York, 1984), p. 38.
6. M. Born and E. Wolf, Principles of Optics, 5th ed., (Pergamon,Oxford, 1975) p. 529.
7. G. 0. Reynolds and D. J. Cronin, "Imaging with opticalsynthetic apertures (Mills-Cross analog)," J. Opt. Soc. Am.60, 634-640 (1970).
8. M. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence(Prentice-Hall, Englewood Cliffs, N.J., 1964), p. 35.
9. H. H. Hopkins, "The concept of partial coherence in optics,"Proc. R. Soc. London Ser. A 208, 263-277 (1951).
10. A. Papoulis, Probability Random Variables and StochasticProcesses (McGraw-Hill, New York, 1965), p. 325.
11. Ref. 5, p. 41. Analogously, the far-field statistics of speckle atthe receiver depend on the intensity reflectivity and shape ofthe object. See L. I. Goldfischer, "Autocorrelation function andpower spectral density of laser-produced speckle patterns," J.Opt. Soc. Am. 55, 247-253 (1965).
12. J. C. Dainty, "Some statistical properties of random specklepatterns in coherent and partially coherent illumination,"Opt. Acta 17, 761-772 (1970). For reviews of various specklereduction methods see G. Parry, "Speckle patterns in partiallycoherent light," and T. S. McKechnie, "Speckle reduction,"both in Laser Speckle and Related Phenomena, J. C. Dainty,ed. (Springer-Verlag, New York, 1984).
13. The square aperture is also of interest because it is advanta-geous to know the object Fourier transform at points on asquare grid for purposes of eventual Fourier inversion by theFFT algorithm.
14. This plot has the same qualitative character as that given inDainty's paper in Ref. 12.
