Effects of nonredundance on a synthetic-apertureimaging system
Louis Sica
Naval Research Laboratory, Code 6530, Washington, D.C. 20375
Received April 23, 1992; accepted November 12, 1992
The analysis of signal-to-noise ratio (SNR) and associated imaging properties of an integrative synthetic-aperture imaging technique is extended to include the effects of an explicitly nonredundant aperture. This re-quires modification of the image estimator that has been used previously. In the case of a nonredundanttransmitter and point receiver, which corresponds to a coherently illuminated object in an analogous opticalsystem, no useful image is formed. As the receiver size increases, the system is capable of forming an image ofincreasing fidelity to an object that in the counterpart analogous system is illuminated with light of decreasingspatial coherence. As the receiver aperture enlarges, the image should become independent of coherentartifacts just as in the case of imaging with a redundant transmitter. Curves of SNR versus receiver-to-transmitter diameter ratio are given that compare the redundant and the nonredundant cases. The SNR issomewhat lower at all receiver sizes for the nonredundant transmitter aperture.
1. INTRODUCTION
An analysis of the signal-to-noise ratio (SNR) performanceof a synthetic-aperture imaging architecture was recentlyreported.",2 The system concept combines the ingredientsof active imaging and the nonredundant synthetic aper-ture3 to obtain a direct measurement of an object's Fouriertransform (see Fig. 1) by means of illumination with two-beam interference fringes. Such a concept can be realizedin angle-angle radar imaging, in laser radar, in sonar, orin the different size domain of assembly-line inspection.A general property of this variety of technique, in whichsignal intensity is integrated over receiver area, is that inthe case of a rough object noise in the Fourier-transformmeasurement decreases as receiver area increases. Thiswas noted by Ustinov et al.4 and Aleksoff5 and analyzed indetail in Ref. 2, where an expression for the SNR of theobject mutual-intensity function versus system size pa-rameters was derived.
Analysis of the SNR properties of the technique was ex-tended from the Fourier-transform plane to the computedimage of Ref. 1. It was shown there that the expressionfor the image estimate with respect to a statistical en-semble of ideally rough object surfaces was formally iden-tical to the image that was produced by a lens imaging thesame object that was illuminated with spatially partiallycoherent light. This formal identity implies that themethod that was studied should result, independently ofobject surface statistics, in images with fewer coherentartifacts than synthetic-aperture techniques that corre-spond to coherent imaging would have.
Although the system concept was originally envisagedas using a nonredundant aperture, the SNR calculationgiven in Ref. 1 was specialized, for the sake of simplicityin the initial study, to a transfer function correspondingto a redundant aperture. The purpose of this paper is toextend the treatment of Ref. 1 to an explicitly nonredun-dant aperture. Although the numerical results for theSNR turn out to be only a little less favorable than those
found in the redundant case, the analysis involves inter-esting system properties that were not completely exploredpreviously. The course of the paper is similar to that ofRef. 1. An image estimator for the nonredundant contri-bution in the Mills-cross geometry is first obtained from ageneral Fourier imaging relation. Appropriate Fouriertransformation shows that its behavior in the absence ofensemble averaging depends more critically on receiversize than in the case of a redundant aperture. The stan-dard deviation of the estimator with respect to an en-semble of ideally rough object surfaces is computed so thatthe SNR may be obtained. The SNR versus the ratio ofreceiver to transmitter diameters is compared with theprevious result for redundant apertures.
2. NONREDUNDANT TRANSMITTERRESPONSE FOR CROSS
A system architecture (see Fig. 1) and appropriate signalprocessing were described previously2 that result in ameasured estimate of the mutual intensity function for adelta-function-correlated rough object.6 The mutual in-tensity is given by the Fourier transform of the object in-tensity reflectivity according to the van Cittert-Zerniketheorem for speckle. The estimate, a spatial average [seeEqs. (17b) and (21) of Ref. 2] denoted by J, is an integralover fields in the receiver plane, a square of side 2L(Figs. 1 and 2):
A1 1 fL~u,+'L ( + V ( AR )
J~uu, =2L) ELAR AR / (1)
In this integral the tilde denotes the spatial Fouriertransform and A is the field just above the rough surfaceof an object, resulting from its illumination by a mono-chromatic transmitter source at ul, one among a numberof sources of an array in the far field at distance R. Posi-tion in the receiver plane is denoted by v = (vs,,vs), andvectors ul and u2 denote source positions in the transmit-
ing over a statistical ensemble of its surfaces, Eq. (1) hasan additional useful property: It is formally identical tothe mutual intensity (in the sense of partial coherencetheory) in the far field of the object when illuminated by asquare delta-function incoherent source of area (2L)2 lo-cated a distance R away. This may be shown by writingthe fields in Eq. (1) in terms of their values at the surfaceof the object in the form2
U+ A'xl"eA p [27rixl (i + v xAR ) J IeP[ AR 1dx
(i = 1, 2), (2)
where dx1 dx1 dy1 and a minus has been inserted in theexponential as compared with Ref. 1 for the sake of laterdevelopment. After performing the integral over dv, oneobtains
J(UU2) = 7 A(xl)A*(x2)
Xexp [-2vi(xi ul - X2 U2 ) 1x expLAR X sin 2rL(x - X2)
AR 1dx2 .
Transmitter ReceiverFig. 1. Pairs of apertures that are active simultaneously alongthe cross-shaped array illuminate the object with moving two-beam interference fringes. The time-dependent intensity inte-grated over the receiver area allows an estimate of the objectFourier transform to be measured.
V.
Hl
"2
X
Apertures
VY
y
Fig. 2. Coordinates used in Eq. (1) and those that follow.
ter plane (coincident with the receiver plane). The con-vention is used that dv = ddvy, and one integral sign iswritten for each boldface differential.
Whereas the architecture and the signal processing ofthe system as described previously were expressly chosento produce an unbiased estimate of the mutual-intensityfunction of an ideally rough object with respect to averag-
(3)
This result has the same general form as that given bypartial coherence theory.7 The sinc functions specify thedegree of coherence of quasi-monochromatic radiation atthe object surface because of a square source of side L inthe far field at distance R. It follows from this formalanalogy that one may adopt as the image estimator thecorresponding partial coherence relation for propagationfrom the u-coordinate plane to a computed image orx-coordinate plane:
(x) = 7 fP(u1)P*(2)j(u1,u 2)
x expL21ix.{1U - U2) duldu 2 , (4)
as done in Ref. 1. The estimator aspect of Eqs. (3) and(4), denoted by carets over appropriate symbols, is definedwith respect to an ensemble average over the A's and A's.
A. Specialization to a CrossTo specialize Eq. (4) to a case of relative nonredundancy,5
a pupil function that yields almost nonredundant measure-ments, a cross, may be substituted into Eq. (4), and theredundant terms caused by integration over each armalone may be removed from the resulting estimator for theintensity. The pupil function P for the cross may be rep-resented by
P(u) = wP(u")8(u,) + wP(u,)5(u.), (5a)
where w is the width of the narrow cross bars representedby the delta functions and P(u) is defined by
P(u) = 1 -L u L,
P(u) = 0 ul > L. (5b)
Object
I- - 1.
Louis Sica
Vol. 10, No. 4/April 1993/J. Opt. Soc. Am. A 569
After substituting Eqs. (5a) and (5b) into Eq. (4) and inte-grating the delta functions, one obtains
I(x) = 7 7P(l)P(Ux 2)J(uxi,0, fUx2, 0)
exp [27ri(uXi- Ux2 )X] duxdUx 2x exp[2 iAR fxu]d
+ W2 7 P(.1)P(Uy 2 )J(U1, 0l, 0, Uy2)
_ exp[2i(lx + UyY)]dU2P[ AR ] y
+ W2 77P P(Uyl)P(x 2)J(OUyl,0U2 2 0)
x exp[ 2i(-UJ UyY)]dU.2duy (
+ W2 P P(Uyl) P (y2) JAuyl Y0) UY2)
X exp[ i(u~ -A Uy2)Y duyldUy2 (6)
If the ensemble average of Eq. (6) is taken under the as-sumption of wide-sense statistical stationarity of the Afields in Eq. (1),
1(x) = w2f 7P(x1)P(x 2)J(Uxl - Ux2 0)
[2li(uxi - ux2)x 1x expL AR duxidux 2
+ w2 fJP(u. 1 )P(u, 2)J(uxb -Uy2)
* exp[2Fri(uxlx - UY2Y)ddX exp AR dxdy
+ w2 P(y1)P(Ux2 )J(UX 2 , uyl)
* exp[2ri(-Ux2X + Uy1Y)dfldfl+ exp f"fP AR u, 1 u
+ w2 f P(Uyl)P(Uy2)J(O, Uyl - Uy2)
X exp[27i(uY - Uy2)YldUdflx expL AR dyu2(7)
As a result of the averaging, the Js in Eq. (7) depend onlyon the difference of coordinates u - 2 in the u and uydirections. It should be noticed that the J's in the firstand last of the four integrals are the Fourier transformsof line scans through the object. Consequently these in-tegrals contribute to degradation of the image, as wasbeen noted previously by Reynolds et al.3 The middle twointegrals are identical, as can be seen by making the vari-able changes u,2 = - uy 2' and Ux2 = - x2' and taking intoaccount the evenness of the pupil functions. They arethen Fourier transforms of the truncated mutual-intensityfunction of the object. The sum of the middle two inte-grals of Eq. (6) may therefore be chosen as an image esti-mator. Since these two integrals are complex conjugatesof each other, the estimate is real.
B. Properties of the Image EstimatorBased on the above comments, an appropriate image esti-mator is
I(x) = -W2 P(UX1 )P(uy 2)J(U 1, 0, 0, uy2 )
2 Jx J E
Xexp [27ri(uxix - Uy2Y)dd+ exL 7 PAR) ux)(uy 2
+_W f P P(Uyl) P (U2) J( Uyl, .2, 0)2 x_
X exp 2ri(-U + lY)dU2duyl (8)
where a factor of 1/2 has been included before each inte-gral to compensate for the equality of the two terms whenthey are averaged. If follows from the above discussionthat the estimator is unbiased since in the spatial domain,after ensemble averaging, the image is generated by theconvolution of an object distribution and a space invariantpoint-spread function.9
Equation (8) may be transformed into an equivalent re-lation in the image plane by using Eq. (1) and performingthe pupil and receiver plane integrations. From Eq. (1),
J(uX, 0,0, u,2) = (2L)2 J) x J + VXl1 + vyi)
X A*(VXluy 2 + vyi)g(v)dvxidvyi, (9a)
J(0uyl, U 2 , 0) = (2L)2 J__ J A(vx2, uyl + uy2)
X A*(ux2 + vx2,vy 2 )g(v 2 )dvx2 dvy2, (9b)
where g(v) is a rectilinear function that equals zero whenv is outside the receiver area. After using Eqs. (9a), (9b),and (2) in Eq. (8) and performing integrations over thedu's and dv's, one obtains
Although the form of Eq. (10) is reminiscent of that forpartially coherent imaging, it behaves quite differently, asmay be seen by taking its coherent limit as the width of gspecified by L goes to zero. In this case, g = 1, andEq. (10) becomes
1x) = -J J A(xi yl)dys(x - x1)dx1
x 7 7 A*(x 2,y 2)dx 2s(y - Y2)dY2
2 J 77A*(X1, YI)dyiS(X x1)dx1
x 7 7A(x 2,Y 2)dx 2 s(y - Y2)dY2- (12)
In the coherent limit, the estimator does not form an im-age, at least not in any usual sense. The fact that anonredundant receiver does not yield an image in the caseof coherently irradiated objects was previously discussedby Reynolds. 0
The incoherent limit corresponds to a receiver of infi-nite size L -- -. Equation (11) shows that as g becomesnarrow Eq. (10) approaches
I(x)
= ( 2 w27 I IA(xi,yl)12s(x - xl)s(y - yl)dxldyl.
(13)
It is clearly desirable to choose a receiver size that yields aresponse close to Eq. (13) rather than Eq. (12), since thisresults in an image that is independent of object surfacephase information without coherent artifacts. We can in-tuitively sense that a critical point between the two limit-ing cases would appear to be reached when the width of gis roughly equal to that of s. The SNR of the image esti-mate of Eq. (10) computed below confirms this. Thechoice of SNR, through a choice in the magnitude of Lrelative to transmitter diameter, determines how closelythe behavior of the system conforms to Eq. (13) in a statis-tical sense.
3. SIGNAL-TO-NOISE RATIO
The SNR will be computed at the center of the image of alarge object as was previously done in the redundant caseby the author.' It is necessary to compute the averagemean and the average mean square of the image estimatorin the large-object limit so that the speckle scale at theequivalent receiver aperture P of the light scattered fromthe object is small compared with the length of P.
From Eq. (8), the average mean is
(IO,))= W2f JP(. 1 )P(, 2)J(U.1, -uy2)dux1duy2- x _x
0 82 (),
1
where the expression for the mutual intensity J employedin the evaluation was
J(Uxl, uy 2) = Io sinc( ) sinc( ) (14b)
corresponding to a square object of dimension 2a on a sideat a distance R from the receiver. This yields e8 =AR/2a.
The average value of the intensity squared must takeaccount of both terms of Eq. (8). Referring to these as Iand I2, we obtain
It follows from the fact that I2 = I,* as indicated abovethat I22 = J1*2 and (122) = (112)*.
Consequently only (I12) and (IJI2) must be computed.This is done in Appendix A, where it is shown that at
=
(Il2) = .4 I02,E4 + W Io E844 4(2L) 4
x J J P(rx~t)P(Vy2')G(V2)G(Vy2)dV2dVy2',
(17a)
where
G(v.2') = f g(vx1)g(vx2' + v)dvx 1, (17b)
(1I2)= 4 + 4(2L) J P2
(U. 1 )du.,j . (18)
By using Eqs. (17a) and (18) in Eq. (15) and noting fromEq. (14a) that
(I0, 0)) = w2 IOe.2 (19)
since P(0) = 1, one obtains
= (2) - (I)2)= {(2L)4 [ p 2( )G(V2)dV,]2
2 (20)
f2, , 2 P(V.2 ,dV.22 (20)
It follows from Eqs. (5b) that p 2= P and G(v) = 2L - vJ.
Then Eq. (20) becomes
X{(2L)4 [7 P(x 2')(2L - ItVx2 '1)dV.x2 ] + (2L)}
(20a)
Louis Sica
= W2J0,E"2p(0)2, (14a)
Vol. 10, No. 4/April 1993/J. Opt. Soc. Am. A 571
ST
15
Ic
7.5
5
2.5
Figrednor
qR requirements and no stringent phasing requirements onthe components making up this area.
- - - - - 4. CONCLUSION
The analysis of the SNR of a synthetic aperture imagingsystem has been extended to the case of an almost non-redundant transmitter based on a cross. Compared withthe case for a redundant transmitter, nonredundance in-creases the sensitivity of the technique to effects thatdepend on the area of the receiver. For a point receiver,which corresponds to coherence in the optical system ana-log, no image is in fact produced from the estimator. Asthe receiver size increases, the technique yields an image
2. 4 610 approaching that of an incoherent imaging system that isindependent of phase effects arising from the surface de-
Diameter Ratio tails of the object. It is found that the SNR for the non-
J.3. SNR for redundant and nonredundant geometry. The redundant system equals approximately 1 when r = 1 andlundant SNR is given by the dashed curves [Eq. (22)], and the rdnatsse qasapoiaey1we nredundant SNR is given by the solid curve [Eq. (21a)]. is somewhat less than that of the redundant system at all
receiver sizes.
For L Ž L/2, the function P(V.2') cuts off the integrationat LT. For this range of L, Eq. (20a) becomes
W 2i 0E.2 2 1 - 1 1/2
r = [ + J '1 , (20b)N/- r 2 2r7 416r 4
where r = L/L,. Finally, in view of Eq. (19), the SNRbecomes
S ((0, 0))
N ar(2 1 + 1 2\-r - +r 16r7
(21a)
which equals 1.13 when r = 1. Thus, for this value of r,the sparse aperture system has essentially the same SNRas a filled aperture in the case of coherent imaging. Forlarge values of r, Eq. (21a) approaches
N = r (for large r). (21b)
These results may be contrasted with the result previ-ously found for a redundant aperture:
r2
2 1-r --
S 3 6
N 1
1 - - r + - r2
3 6
(L 2 L,)
(L C L,)
APPENDIX A
By using Eqs. (16) and (9a), (Il2) may be computed at
x = 0. One obtains
(JI2) = P(Ux)P(Uy2)P(Ux3)P(uy4)(2L)4
x II7 (A(ux. + vxlvyl)A*(Vxliuy 2 + Vyl)
X A*(Vu2,u, 4 + V,2 )A(u. 3 + Vx2,Vy2))
x g(vl)g(v2)dvldv2 duxduy2dux3 duy4. (Al)
Applying the Gaussian moment theorem to the ensembleaverage inside the angle brackets yields
(I) = 7 I P(. 1 )P(y 2 )P(ux3)P(uy4)P2L4
X I~ f [J(uxl, -uy2 )J(ux3, -Uy4)
+ J(uxl + Vx1 - Vx2,-uy4 + vyl - vy2)
X J(ux 3 - VX1 + VX, -Uy 2 - Vyl + Vy2 )]
x g(vl)g(v2)dvldv2 dux1 duy2 dux3duy4.
(22)
For r = 1, the SNR = 2; when r = 0, the SNR = 1, corre-sponding to the usual situation of coherent imaging.
Equations (21a) and (22) are plotted in Fig. 3. It isclear that the use of a nonredundant aperture exacts aprice in terms of the receiver size that must be used toachieve a given SNR when one is using this technique.However, the trade-off of a large decrease in the numberof transmitter components for an increase in receiver col-lection area seems to be a favorable one. This is becausethe receiver area is an energy collector that may be seg-mented in any convenient way. There are no sampling
(A2)
Noting that the integral over the first J product equals(I,)2, and considering the large-object case for which the Jfunctions become delta functions with respect to the P's,one obtains
This result may now be substituted into Eq. (A5), withthe integration being performed over the first term inthe limit of a large object. The J's again act as deltafunctions compared with the P's and g's, and each two-dimensional integration over a J function yields IOeI2 as
Integration over the remaining delta functions produces
2. L. Sica, "Estimator and signal-to-noise ratio for an integra-tive synthetic aperture imaging technique," Appl. Opt. 30,206-213 (1991).
3. G. 0. Reynolds and D. J. Cronin, "Imaging with optical syn-thetic apertures (Mills-cross analog)," J. Opt. Soc. Am. 60,634-640 (1970).
4. 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).
5. C. C. Aleksoff, "Synthetic interferometric imaging techniquefor moving objects," Appl. Opt. 15, 1923-1929 (1976).
6. J. W Goodman, "Statistical properties of laser speckle pat-terns," in Laser Speckle and Related Phenomena, J. C.Dainty, ed. (Springer-Verlag, New York, 1984), p. 38.
7. M. Born and E. Wolf, Principles of Optics, 5th ed. (Perga-mon, Oxford, 1975), Chap. 10.
8. The T has even less redundancy than the cross. See A. R.Thompson, J. M. Moran, and G. W Swenson, Jr., Interfer-ometry and Synthesis in Radio Astronomy (Wiley-Inter-science, New York, 1986), p. 119.
9. This statement amounts to a definition of the term unbiasedin the context of this paper. It should be noted that an im-age generated by a sinc point-spread function may have nega-tive intensities and that therefore it may be desirable inpractice to use a tapered function, e.g., a triangle, for P(u).
10. G. 0. Reynolds, J. B. De Velis, G. B. Parrent, Jr., and B. J.Thompson, Physical Optics Notebook: Tutorials in FourierOptics (Optical Engineering Press, Bellingham, Wash., 1989),Chap. 37.
