Estimator and signal-to-noise ratio for an integrative synthetic aperture imaging technique Louis Sica An imaging system is described which uses the following concepts: laser illumination of objects, nonredun- dant apertures, and phase closure. A sparse transmitter array is envisioned,each aperture of which emits at a different laser frequency such that any pair of beams givesrise to a unique beat signal. The light reflected by an object thus irradiated is sensed by a spatially integrating detector array. An estimator is given for the Fourier components of the object at spatial frequencies corresponding to the unique temporal beats sensed by the receiver array. The standard deviation of the estimator is computed taking both shot noise and laser speckle into account. It is'found that the signal-to-noise ratio for both kinds of noise increases with the square root of the area of the detector array. This allows the signal-to-noise ratio of the system to be increased independent of the resolution. I. Introduction A number of techniques have been developed which allow measurements performed in the entrance pupil of an optical system to be used to compute an optical image, thus bypassing the role of conventional imaging optics. Such techniques, which have been referred to loosely as pupil plane imaging techniques, encompass the use of such historically famous devices as the Mi- chelson stellar interferometer,l the intensity interfer- ometer 2 of Brown-Twiss, and more recently the inter- ferometers used by radio astronomers. 3 The advantage of pupil plane techniques lies in achieving the high resolution of a large collecting aperture while avoiding the difficulty of fabricating one. Formerly, the objects imaged by such methods have had the property of self-radiance. However, with the advent of the laser, new versions of pupil plane imaging are evolving based on the active illumination of objects with laser light. Given coherent illumination of an object by one or more lasers, pupil plane measure- ments together with appropriate data processing can result in a computed image equivalent to that formed by imaging optics. 4 Under the proper circumstances, the image may even be similar to that which might be The author is with U.S. Naval Research Laboratory, Washington, DC 20375-5000. Received 22 December 1989. produced by incoherent illumination. Examples of some of the techniques being explored in this area are the phase retrieval technique of Fienup, 5 and the FOCI technique of Ebstein and Korff. 6 In these examples, the incoherent image may be achieved by repeating the measurements and processing for different laser wave- lengths or different realizations of the scattered field corresponding to slightly different object aspect an- gles, and then properly combining the results. The present work analyzes a new system architec- ture and processing scheme for pupil plane imaging which has some interesting and even surprising prop- erties. It combines several well-known conceptual in- gredients in a new way: active imaging, the nonredun- dant/synthetic aperture, 7 phase closure, and the goal of direct observation of the incoherent source Fourier transform as is achieved (in principle) in the Michel- son stellar interferometer. The system concept is re- lated to those considered by Aleksoff 9 "1 0 and Ustinov et al. 1 1 Aleksoff explored synthetic imaging techniques in which part of the spatial frequency information was generated by large area detector reception of object scattered radiation during translational 9 or rotation- al' 0 motion through a two-source fringe field. He also assessed the importance of receiver size in reducing undesirable coherence effects (speckle in the context of the present work) for multiple point targets. In the present work, spatial frequency information is also obtained by illuminating an object with interference fringes, but all spatial frequency information is gener- ated from multiple antenna spacings independent of object motion. In the limit of an infinite aperture, the system is closely related to that treated theoretically by Ustinov et al.1 for looking through the turbulent 206 APPLIED OPTICS / Vol. 30, No. 2 / 10 January 1991
Transcript
Estimator and signal-to-noise ratio for an integrativesynthetic aperture imaging technique
Louis Sica
An imaging system is described which uses the following concepts: laser illumination of objects, nonredun-dant apertures, and phase closure. A sparse transmitter array is envisioned, each aperture of which emits at adifferent laser frequency such that any pair of beams gives rise to a unique beat signal. The light reflected byan object thus irradiated is sensed by a spatially integrating detector array. An estimator is given for theFourier components of the object at spatial frequencies corresponding to the unique temporal beats sensed bythe receiver array. The standard deviation of the estimator is computed taking both shot noise and laserspeckle into account. It is'found that the signal-to-noise ratio for both kinds of noise increases with thesquare root of the area of the detector array. This allows the signal-to-noise ratio of the system to be increasedindependent of the resolution.
I. Introduction
A number of techniques have been developed whichallow measurements performed in the entrance pupilof an optical system to be used to compute an opticalimage, thus bypassing the role of conventional imagingoptics. Such techniques, which have been referred toloosely as pupil plane imaging techniques, encompassthe use of such historically famous devices as the Mi-chelson stellar interferometer,l the intensity interfer-ometer2 of Brown-Twiss, and more recently the inter-ferometers used by radio astronomers. 3 Theadvantage of pupil plane techniques lies in achievingthe high resolution of a large collecting aperture whileavoiding the difficulty of fabricating one. Formerly,the objects imaged by such methods have had theproperty of self-radiance. However, with the adventof the laser, new versions of pupil plane imaging areevolving based on the active illumination of objectswith laser light. Given coherent illumination of anobject by one or more lasers, pupil plane measure-ments together with appropriate data processing canresult in a computed image equivalent to that formedby imaging optics.4 Under the proper circumstances,the image may even be similar to that which might be
The author is with U.S. Naval Research Laboratory, Washington,DC 20375-5000.
Received 22 December 1989.
produced by incoherent illumination. Examples ofsome of the techniques being explored in this area arethe phase retrieval technique of Fienup,5 and the FOCItechnique of Ebstein and Korff.6 In these examples,the incoherent image may be achieved by repeating themeasurements and processing for different laser wave-lengths or different realizations of the scattered fieldcorresponding to slightly different object aspect an-gles, and then properly combining the results.
The present work analyzes a new system architec-ture and processing scheme for pupil plane imagingwhich has some interesting and even surprising prop-erties. It combines several well-known conceptual in-gredients in a new way: active imaging, the nonredun-dant/synthetic aperture,7 phase closure, and the goalof direct observation of the incoherent source Fouriertransform as is achieved (in principle) in the Michel-son stellar interferometer. The system concept is re-lated to those considered by Aleksoff9"10 and Ustinov etal.11 Aleksoff explored synthetic imaging techniquesin which part of the spatial frequency information wasgenerated by large area detector reception of objectscattered radiation during translational9 or rotation-al'0 motion through a two-source fringe field. He alsoassessed the importance of receiver size in reducingundesirable coherence effects (speckle in the contextof the present work) for multiple point targets. In thepresent work, spatial frequency information is alsoobtained by illuminating an object with interferencefringes, but all spatial frequency information is gener-ated from multiple antenna spacings independent ofobject motion. In the limit of an infinite aperture, thesystem is closely related to that treated theoreticallyby Ustinov et al.1 for looking through the turbulent
atmosphere. However, the present system employs afinite receiver and a source giving rise to nonredundantspatial frequencies. It is primarily concerned with theshot and speckle noise contingent on the architectureand processing. Advantages not noted previously arefound to characterize the modified system: it pro-duces a better estimate of the Fourier transform of theobject intensity reflectivity than coherent imaging, it isintegrative with respect to both shot noise and specklenoise (a fact that can be used to improve signal-to-noise ratio), and speckle motion due to object rotationmay improve the signal-to-noise ratio for Gaussiannoise while not affecting shot noise.12
The architecture of the system is briefly described,and its function in the limit of an infinite receivingaperture is examined. This limiting behavior is simi-lar to the average behavior of a finite receiver andallows its properties to be understood in a simple way.Then, a multiaperture system in two dimensions isanalyzed with regard to both its deterministic andstatistical properties. Portions of the statistical calcu-lations are relegated to the Appendices. The effects ofboth well-developed speckle (Gaussian statistics) andshot noise (Poisson statistics) are taken into account.Finally, a summary of the chief properties of the sys-tem which have been established is given.
II. System Description and Operation
An overall diagram of the system architecture isshown in Fig. 1. Light derived from a laser is split intogroups of beams which emanate from unequallyspaced apertures along the three arms of an equiangu-lar Y, a nonredundant aperture arrangement. Thebeams from several apertures at a time illuminate theplanar object and the interference of any two beamstaken along produces a set of straight line Young inter-ference fringes. The orientation of the fringes andtheir spacings depend on the locations of the corre-sponding emitting apertures on the Y, and overall, alarge variety of spacings and orientations may be pro-duced.7 The frequency of the light emitted from eachaperture of the transmitter is frequency shifted so thateach pair of Young's interference fringes moves overthe object surface at a characteristic rate. The fre-quencies are chosen such that the difference frequencyof any pair of beams is different from that of any otherpair. Some of the radiation scattered by the objectpropagates to the (segmented) receiver which acts likea single, large square law detector, since its output isthe instantaneous sum of the outputs of any individualsegments. The image of the object may be computedfrom the receiver output as described below. Beforegiving a detailed Fourier optics account of the opera-tion of the system in the next section, however, a sim-pler semiquantitative description is given.
Consider waves from two transmitter apertures atfrequencies v and v which are locally plane in theregion of a distant object. Let their direction cosinesbe +a and -a with respect to the x-axis, and y withrespect to the z-axis. They superpose to produce afield U given by
Fig. 1. Overall imaging system. The small circles indicate laserapertures of the sparse transmitter array. The receiver may be
segmented.
U = cos2rv1 - ax + ) + cos27rv(t + ax - z (1)
Since harmonics of v and Pi are beyond the detectorfrequency response, intensity U2 is
= 1 + co{2ir(vl + v)(t - 0) - 2r(v1 - v)
c
+ co¶2x(vl - v)(t - 0) - 27r(v1 + v) -Jx (2)
where the z dependence is hidden in 0. The secondterm in Eq. (2) may be neglected since it is also beyondthe detector frequency response. As a consequence,U2 becomes
where Av = - . This relation describes an interfer-ence fringe moving from left to right along the positivex-axis. For a frequency offset Av = 109 Hz and afrequency v = 1015 Hz, Av/2v = 0.5 X 10-6, which has anegligible effect on the fringe spacing. Consequently,Eq. (3) may be rewritten as
LP = 1 + cos[2rAv(t - 0)] cos(27r2a )
+ sin[2irv(t - 0)] sin(27r2a A). (4)
If the reflectivity of a planar object is given by O(x,y),the power reflected is the integral of the product of U2and O(x,y) over the object area:
This equation gives the sine and cosine Fourier trans-forms of the object in the x-direction as coefficients oftime oscillating sine and cosine terms. Due to theconservation of energy, Eq. (5) would give the signal vstime seen by a (wraparound) detector subtending 2-ir srat the object except for a phase lag due to the lighttravel time from object to detector. Note that thisnecessarily follows in spite of the randomness of theintensity of the reflected light (a speckle pattern) inthe case of a rough object.
For the system in Fig. 1, each pair of apertures wouldgive rise to a different spatial frequency which wouldbe functionally related to the difference between thebroadcast frequencies of the two apertures. As shownbelow, appropriate processing of the signal makes itpossible to compute the magnitude and phase of theFourier transform of the object at the spatial frequen-cies determined by the location of each pair of aper-tures. More importantly, however, it is shown that adetector aperture of finite size can produce a signalthat is a good approximation to that produced by aninfinite or wraparound detector. This means thatproperties exhibited by the idealized system are quali-tatively exhibited by a sufficiently large finite system.The analysis necessary to demonstrate these proper-ties clearly requires a treatment based on diffractionrather than intensity. This is given below.
11. Finite Aperture System
The coordinate geometry is shown in Fig. 2. The xy-coordinates denote the object plane, uv-coordinatesdenote the detector-transmitter plane at a distance R,and active transmitter aperture positions are denotedby ui and uj. For R sufficiently large, the wave fromeach transmitter aperture is locally plane over theobject (the Fraunhofer approximation). After (dif-fuse) reflection from the object, the fields at detectorplanes U and Uj due to the ith and jth transmittersare, up to an unimportant constant,13
Ui = exp(-27rivit + io) A(xl) exP(27rixi
X exp(2lrixi *) dx1 ,
U = exp(-27rivjt + i!jo) j A(X2)
X exp(27rix 2 * A ) exp(27rix 2 A dX2
(6)
(7)
where dx dxdy, A(xi) denotes the complex amplitudeof the field from us in the object plane just after reflec-tion from the object, vi and j are the frequencies of the
UX
,.-Transrnitter7/ Apertures
R W
V
Fig. 2. Coordinates used in Eqs. (6) and (7).
light emitted from transmitter apertures i and j, andTio and Tjo denote their respective phase errors withrespect to a reference aperture. As mentioned follow-ing Eq. (3), the frequency shifts are chosen to be suffi-ciently small that the corresponding changes in wave-length may be neglected. Therefore both vi and Vj areassociated with the same wavelength X.
One may evidently write
Ui = exp(-2rivit + i'io)A Ui +)
(8a)
Uj = exp(-2rivjt + io)A( )'
where the tilde indicates the Fourier transform withrespect to spatial variables. The total field at thereceiver from N active transmitter apertures is
NU=zUi,
and the corresponding instantaneous power collectedis
ILj UU*dV =ZE Uij dv, (8b)
where L1 and L2 denote the edge coordinates of thereceiver, not necessarily centered, in the uv-plane. In-serting the relations of Eq. (8a) into Eq. (8b) results in
The signal output by the receiver is the spatial integralof the instantaneous intensity. It depends on thetransmitter aperture frequency differences vi - vj andthe phase error differences Jio - jo.
An estimate of the Fourier cosine (or sine) transformof the object may be computed from the receiver signalby an estimator formula. The frequency of the cosineused in the estimator for a given spatial frequency ischosen to be equal to that of the beat frequency of thepair of active apertures giving rise to that spatial fre-quency. For the cosine transform the estimator is
SCnm = fT IL(t) cos(27rVnnt)dt. (12)
The use of the caret labels a quantity an estimate, T isthe time during which the intensity is recorded, (2L)2 isthe area of the square receiver, and Vnm = Vn-Vm-Equation (12) indicates that the receiver output signalis (numerically) time integrated with the appropriatecosine function and divided by the product of T and(2L)2 to compute the estimate. The estimator for thesine transform is essentially similar:
SSnm = 1T IL(t) sin(27rVnrt)dt. (13)
The estimator may be evaluated by inserting the rela-tion of Eq. (10) into Eq. (12). This yields
T 1 N N N
SCnm = J (2L)2 T Gii + 2Gij
X cos(-2rvijt + 'Fij + jij)] coS(2lrVnmt)dt. (14a)
In this initial deterministic treatment, only static ob-jects are considered so that only the cosine terms de-pend on time. In this case, the integral of the sum ofthe Gii with the cosine in Eq. (14a) reduces to theproduct of the average light intensity over the receiveraperture and the time average of the cosine over obser-vation time T. The average value of the cosine de-creases as one over the number of beat periods in timeT, so that this term becomes negligible for appropri-ately chosen observation times and beat frequencies.Consequently, Eq. (14a) becomes
N N rT
SCnm = (2L)2 T E 2G J [Cos(* + dij)
X cos(2rvijt) + sin(q- + jij) sin(27rvijt)] CoS(27rVnmt)dt. (14b)
If the cosine is multiplied through the square brack-ets, the first term yields a cosine squared when vij =Vnm. This term has a dc component of 1/2. It is theonly term whose time average values does not decreasewith increasing T. Consequently, Eq. (14b) becomes
SCn0 h 1 Gnm COS('nn + 6nm)- (15)(2L)a m
Similarly,
S9Snm = (2L)2 Gn sin(1nm + °nm). (16)
Alternatively, one may write1 4
'9Cnm = (2L)2IL A (u+ V) A- (um ) dv COS(4'nr + Onm)X / 1XR/a
(17a)
or
S~nrn 2(L 2 (U,+ V) (m V)"CnM= 2 A ( R)A ( XR )dv
X exp(iI,,tm) + c.c. (17b)
The connection between the result of Eq. (17b) andthat of Eq. (5) may be demonstrated by inserting intoEq. (17b) the explicit expressions for the fields fromEqs. (6) and (7):
SCnm = 2(2L) 2 exp(iInm) T JA(xl)
X exp 2rix, -UAR exp 2rix, * ,R dxl
X A*(x 2 ) exp(-27riX 2 R)
X exp(-27rix2 - ) dx2dv + c.c., (18a)
or after rearrangement,
SCnm = 2(2L) 2 exp(inm) J J A(x1 )A*(X2 )
X ex{27ri ( 1 n2 Um)]
X J exp[27ri (x1 x 2 dvdxdX2 + C C (18b)
For the following development the receiver is consid-ered to be considerably larger than the transmitter andto be symmetrically located with respect to the origin.The integral in dudv approaches a delta function as L'- (L is kept constant), and Eq. (18b) approaches
(XR)2 1
S9Cnm = (2L)2 exp(itnm) f IA(x1 )2
X exi{2irix. (Un Urn) ]dxl + c.c. (19)
The result in Eq. (19) is equal to the coefficient of thetemporal cosine oscillation in Eq. (5) except for a nu-merical coefficient and systematic error in phase angle.The (2L)2 was put in the denominator of Eq. (12)because it is an appropriate normalization for the fol-lowing statistical treatment of scattered fields due torough objects in the case of a finite receiver aperture(although not for an infinite aperture). It should benoted that the choice of normalization does not affectthe signal-to-noise ratio computed later.
The analysis from this point on makes use of ensem-ble averaging techniques that make it far easier toanalyze the behavior of a system with a finite aperturereceiver. The technique is first applied in this sectionto derive a fundamental property of the estimator.Later it is used to obtain the signal-to-noise ratio of thesystem.
If the ensemble average of the estimator given by Eq.(17b) is computed for Gaussian distributed fields asare produced by scattering from a rough surface in thefar field, one has .
(SCnm)G = 2(2L)2 exp(iInm)
XK ((Un+ V) A (Um + v)) + c.c., (20)
where the subscript G indicates Gaussian averaging.If the field is also statistically stationary in the widesense,1 5 one obtains
(SCnm)G = U]| COS(nm + *nm), (21)
where J is the mutual intensity function, 6Onm is its
phase, and 2L = L2-L 1. The integral over dudv is justthe receiver area (2L)2 after the ensemble average al-lows J to be factored out. Since the mutual intensityfunction in the far field of an incoherent source isproportional to the Fourier transform of the objectintensity, the estimator given by Eq. (12) is unbiased,although it contains a systematic phase error due tothe transmitter. It follows that, provided the stan-dard deviation of the estimator is small, the techniquegives results similar to imaging with spatially incoher-ent light even though the object is illuminated withlaser light. It should be noted that phase errors 11nmmay be eliminated by use of the phase closure princi-ple.8 Application of this principle results in an imagefree of the degrading effects of phase errors, but uncer-tain as to absolute position.
IV. Signal-to-Noise Ratio
It is necessary to compute the mean and meansquare of SCnm with respect to Poisson and Gaussianprocesses in turn to obtain the standard deviation ofthe estimate. First Eq. (12) must be put in a formappropriate for the calculation of Poisson moments 7 :
SCnm = | IL(t) _ dt = im E IL(tk)dtkCnm(tk),
(22a)
where
COS(27vrnmt)Cnm(t) = (2L) 2 T
The quantity ILdt is intensity integrated over areatimes time or energy. If the intensity is expressed asthe number of photoelectric events per unit area perunit time by taking the number of photons and thedetector quantum efficiency into account, ILdt is the
(22b)
number of photoelectric events in time dt and is Pois-son distributed. The sum in Eq. (22a) is an infinitesum of statistically independent Poisson variables, onefor each value of tk in interval dtk, and each multipliedby a constant Cnm. The average value of powers ofrandom variables such as Scnm may be computed bythe use of characteristic functions using the theory ofstochastic processes.17 In Appendix A it is shown that,for a Poisson process,
T(9Cnm)P = IL(t)Cnm(t)dt,
( Cnm)P = I IL(t)Cnm(t)dt + [Jf L(t)Cnm(t)dt],
(23)
(24)
where the P subscript on the angular brackets in Eqs.(23) and (24) indicates that the average is taken withrespect to the Poisson process only. The variance ofthe estimator is by definition
(Cnm)P - (SCnm)P = | IL(t)Cnm(t)dt.fo n
(25)
Inserting Eqs. (10) and (22b) into Eq. (23) producesthe result in Eq. (14a) which leads to Eq. (17b) restatedhere as the Poisson average:
(SCnm)P = 2(2L)2 j ( V)
x A*(um + V)dv exp(iTnm) + C.C., (26)
(SCnm)P = lJnml COS(Tnm + 6nm)- (27a)
The carets over both J and 0 in Eq. (27a) refer to thefact that the integral over dv is an estimator for themutual intensity function of the field due to a roughobject for the spatial frequency indicated by the sub-script nm, since, as shown before, the Gaussian averageof Eq. (26) or Eq. (27a) produces the real part of themutual intensity function exactly:
((-SCnm)P)G = IJnml COS(i1nm + Onm)- (27b)
Note that here the carets on Jnm and nm have beenremoved by the Gaussian ensemble averaging.
Next, it is useful to evaluate Eq. (25) for the varianceof the estimator due to the Poisson process alone sincethe result will be used later. Inserting Eq. (10) anddefinition (22b) into Eq. (25) produces
Only the first sum over Gii has a longtime averagewhich does not approach zero after multiplication bythe dc value of the squared cosine factor to the right ofthe curly bracket. The harmonic componentof thesquared cosine term produces a nonzero time averageonly if its harmonic equals one of the other frequencies.
There is at most one such frequency18 and the corre-sponding term is considerably smaller than the sumover Gii. Consequently, one may write
N
2cnm - Gii (29)
Therefore, using Eqs. (27a) and (29), the signal-to-noise ratio for the Poisson contribution alone is
(SCnm)P I17nmI COS('nm + 6nm)AL(2T)" (30)
17Cnm (N 12)EGii,
where AL = (2L)2, the area of the receiver. Two impor-tant characteristics of the method are indicated in Eq.(30). First, the sum under the radical in the denomi-nator indicates that the beams from all the activeapertures combine to produce the shot noise at eachspatial frequency. Second, the signal-to-noise ratio isintegrative with respect to both the receiver area andthe observation time.
Equation (30) only takes into account the effect ofPoisson noise, the first term in Eq. (24), on the signal-to-noise ratio. This is appropriate only for the case ofan object with a relatively smooth or a specified roughsurface. To complete the noise calculation for the caseof rough objects with unspecified surfaces, the Gauss-ian contribution due to speckle must also be taken intoaccount. To accomplish this, the Gaussian averagingof Eq. (24) must be carried out, and then the standarddeviation may be computed.
This operation may be indicated by
((SCnm)P)G = (Cnm)G + ((SCnm)PJ)G, (31)
where the first term on the right may be obtained bytaking a Gaussian average of Eq. (29). The variance ofthe estimator is then
(C(Snm)P)G - ((SCnm)P)G) = ( Gnm)C
+ [((9Cnm)P)G -JnmI COS (Inm + 0Jnm)] (32)
where use has been made of Eq. (27b). Using Eq. (29),the first term on the right-hand side of Eq. (32) is easilyevaluated to be
2 1 N
(aCn.)G (2L)4 2TE (Gii)G = (2L) 2 T IT, (33)
where IT = NapLo is the total speckle averaged intensityvia all active transmitter apertures Nap, and I, is thespeckle averaged power per unit area at the receiverdue to one transmitter aperture. A factor of (2L)2 hasbeen eliminated from the denominator as a result ofthe evaluation of (Gii)G- The terms inside the squarebracket on the right-hand side of Eq. (32), denoted by
2'JRCnmG, are evaluated in Appendix B where it is shownthat
IYRCnmG s Ins, (34)
where ns is the number of speckles in the receiveraperture. The variance therefore becomes
S 12 ITI+ It 1cIC - (2L)22T T +N.,~ ns; (35)
From Eqs. (27b) and (35), the signal-to-noise ratio maybe put in the form
((SCnm)P)G I/in. COS(Tnm + Ornm)
aSC NQP +±11/2 (2L)22TIT nS
(36)
where the relation (IT/Nap) Anm = Jnm has been used.In the limit of high overall transmitter power the sig-nal-to-noise ratio becomes
The total number of photoelectric events at the receiv-er is ARTIT-
V. Conclusion
It may be useful to list the properties derived abovealong with the properties that follow immediately fromthem:
(1) The signal-to-noise ratio with respect to bothshot noise and speckle noise increases with the squareroot of the receiver area, since ns increases with receiv-er area.
(2) The resolution is determined by the size of thetransmitter aperture and the range rather than thereceiver which determines the signal-to-noise ratio.
(3) The signal-to-noise ratio for speckle may be in-creased by increasing the integrating area of the receiv-er without affecting the resolution.
(4) There is no threshold energy per unit area of thereceiver necessary to achieve a good signal-to-noiseratio since the total energy collected during the obser-vation time over the whole receiver determines theshot noise. The most economical compromise be-tween a large number of small detector concentratorunits and a small number of large detector concentra-tor units can be chosen to achieve the necessary receiv-er area.
(5) The image computed from the collected datacan have a higher signal-to-noise ratio with respect tospeckle noise than a coherent image at the same resolu-tion due to the decoupling of resolution and signal-to-noise ratio.19
The above properties would evidently characterizethe synthetic aperture device described under the con-dition that phase errors were eliminated through theuse of the phase closure technique. Especially inter-esting is the fact that speckle in the resultant imagecould have reduced visibility compared with coherentimagery even though objects are laser illuminated.
The author would like to thank one of the refereesfor bringing Refs. 4 and 10 to his attention.
Appendix AIn this Appendix, Eqs. (23) and (24) are derived by
means of the characteristic function theory as devel-oped in Papoulis.17 The estimate whose first two mo-ments are desired is given by Eq. (22a) as a linearcombination of statistically independent Poisson dis-tributed random variables. The characteristic func-tion for a given value of k is the expectation
Ok = E[expjwIL(tk)dtkC(tk)I,
which may be shown to be equal took = exp(IL(tk)dtkexpUwC(tk)] - ).
(Al)
(A2)
Consequently, characteristic function for all k is
0 J 1 kk = IJ exp(IL(tk)dtktexpUWC(tk)I - 1),k k
and the second characteristic function T = ln is
Ink = E IL(tk)dtkexpUWC(tk) - 1.k
Then T may be written asT
* = no = J IL(t)JexpwC(t)] - ldt.
The moments to be computed are given in terms of the'derivatives of ; namely,
1 d0(0) (Sn), (A6)jfl dwn
where it is necessary to compute these derivatives interms of the derivatives of T evaluated at co = 0. Theneeded expressions are easily computed to be
a= exp'I,
ao= exp'I'HI (A7)O9 w
(A3)
(A4)
(SCnm)p = /2 *nm expiInm + 1/2Xnm exp - n, (Bi)2
URCnmG may be written as= ((S0nrn)2)C - jJ2n COS2('nn + OJnm)
Using the Gaussian moment theorem, this becomes(n . = 1 = fJL2
[ (U)12 + J(V2-V)J(VXR)]dV 1dV2, (B4)
or(3 J(Un~Urn)2 + 1 fL2 / ' - 2dv 1d 2.
(A5) =Jm~n / J(2L) 4JL L VII)2
(B5)
Similarly, one finds that(l 2 ) (Un-s 1 fL2 L2
U Rn) + (2L)4fL1 I
x (Un - Un+ V - 2)j(U - Um+ V2 - v)dvdV2, (B6)
and a similar result for its complex conjugate. WhenEqs. (B5) and (B6) are inserted into Eq. (B2) oneobtains
2 1 rL 2 L vj-v 2 \ 2IR~mG~_ 4I I dlR )Idv~d" 2RnrG2(2L)4JfL, J 1 \ R
1 rL2 L2 |jU -U. + V1 V2\ fu- U, -(V V2)1+ 2(2L)4 fL, fI XR SL X I CoS[O(UnmV12) + 2'Inn]dVidV222)LIl R L AI j
2 r = a2q, /a)2]
If derivatives of Eq. (A5), evaluated at = 0, are usedto evaluate Eq. (A7), use of Eq. (A6) leads to Eqs. (23)and (24).
Because the coordinate dependencies on v and 2 aredifferences, the limits of integration may be changedfrom -L to +L in Eq. (B7) where L = (L2 - L1 )/2.Making this change and taking absolute values of bothsides of Eq. (B7) one obtains
IGCnmGI 2(2L4 L j(V )| dV1dv2 + 1 [ J(U. - U+ V1 V2 [Un Um -(V 1 - V2)]2(2L) JL JL \XR/ 2(2)4 J-L R L dvRvI
Appendix B
In this Appendix, the square bracket on the right-hand side of Eq. (32) denoted by RCnmG is evaluated.Using the result of Eq. (27a) in the form
after bringing the absolute value inside the secondintegral and noting that Icos(e + 2nm) S 1. Equa-tion (B8) has the form of a well-known integral due tothe integration over difference coordinates. It may beput in the form20
For the cases of interest here, the receiver width ismuch larger than the speckle width. Then the limitsof integration +2L in the two integrals of Eq. (B10)may be replaced by i- with only a small error. Itfollows from application of the convolution theoremthat the second integral of Eq. (B10) is smaller thanthe same integral evaluated at (u, - un) = 0. Conse-quently,
Iacn G < 1 2 _J J( T) dT,,dT,. (B11)
For definiteness, let J be given by the peaked function2rT,, 2rTy
J = I sinc -sinc -, (B12)
corresponding to a square object, where the specklewidth is a = XR/D, and D is the object diameter. If theintegral of Eq. (Bli) is evaluated using the function inEq. (B12),
acn.G < - (B13)ns
where ns = (2L)2/(2o) 2, the number of speckles in thereceiver aperture.
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12. This point will be dealt with in a future paper.13. For example, the 1hR dependence has been ignored to simplify
the notation. However, since the final derived formula dependon field quantities evaluated at the receiver only, it is unneces-sary to carry along such constants.
14. From this point on, the analysis will be given for Sc only, with theunderstanding that a similar analysis could be carried out for Ss.
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18. This depends on the generation scheme for the frequency off-sets. Schemes can be created such that none of the differencefrequencies equals any of the difference frequency harmonics.
19. This topic will be treated more fully in a future paper where itwill be shown that the architecture described plus processing isequivalent to a conventional imaging system using partiallycoherent light with the coherence scale parameter depending onthe receiver size.
