The advantages of negative feedback are well known in electronics and extensively used in the operationalamplifier. The properties of such a system are nearly independent of the parameters in the forward branchof the system; they are only determined by external elements in the backward branch. An optical analogof such an operational amplifier is reported. The essential operations, amplifications, and inversion of thecirculating signals are carried out using a TV system. The capability of the system to compensate for spatialinhomogeneities and for nonlinearities is demonstrated. In addition, the system is able to create the inverseof a transfer function located in the feedback branch.
1. Introduction
The principle of negative feedback is commonlyused in electronics for improvement of unstable, non-linear, and noisy amplifiers.' If the amplification in theforward branch of the circuit is large, the properties ofthe total system are essentially determined by externalelements in the feedback branch.
To construct an optical analog of such an operationalamplifier, we, consequently, have to amplify the circu-lating image signal and to invert it for negative feed-back.
This can be implemented with a TV system in thecircuit,2 '3 since electrical signals may be amplified andinverted (unlike incoherent light signals). Althoughsome authors have proposed coherent light amplifica-tion in feedback systems,4'5 we prefer (incoherent) TVsystems for their insensitivity to noise and their prac-ticality. A passive feedback system with TV has beenused by Sato et al. 6 to reduce ghost images and forspatial filtering.
We shall demonstrate some useful capabilities of anactive system with negative feedback, which was pro-posed earlier,7 such as compensation of a spatial inho-mogeneous field and of nonlinearities.
Thereby the compensation of nonlinearities isequivalent to that achieved in electronics. But becauseof its 3-D character (two space coordinates and onetemporal coordinate) the optical operational amplifieris more than a pure translation from electronics. It is,for example, able to generate oscillations7 and to solve
The authors are with Universitit Erlangen-Niurnberg, Physikal-isches Institut, 8520-Erlangen, West Germany.
linear differential equations in space and time. Thismay be either the diffusion equation8 or even the waveequation. One possibility, which will be demonstratedin this paper, is the capability of creating the inverse ofthe transfer function in the feedback branch of thesystem.
11. Some Capabilities of an Optical OperationalAmplifier
In this section we want to derive some of the capa-bilities of an optical operational amplifier; its practicalrealization will be discussed in Sec. III. According toFig. 1, the circulating signal is first operated on by it, isamplified by the factor , is operated on by X, and fi-nally inverted.
The operators and XY may depend on the spacecoordinates x and y, which might represent an inhom-ogeneous field sensitivity. They may depend on theintensity itself, which might represent a nonlinearsystem, or they may depend on spatial and temporalfrequencies 1u,v,f, which might represent a linear filter.Indeed, they may depend on all these parameters at thesame time.
A. Operational Amplifier with General Operators
We shall discuss the general case before treating someuseful special cases. The feedback system is assumedto be stable, that is, after an initial period, the outputsignal will be stationary in time, if the input signal is alsostationary. Consequently, we have only to take intoaccount the behavior of the operators at f = 0. In otherwords, .? and K may be considered as depending onlyon spatial variables not on temporal variables. Usingthe notation of Fig. 1, we get:
Now dividing by fi and making fi very large, we find fora finite output O' that
g[ -11(0')] 0. (4)
Application of the inverse operator -1 to (4) yields
0 - 11(O') 9-10o).
OBJECT SIGNALO Xyt)
FORWARD BRANCH AMPLIFIER OUTPUT SIGNALOPERATOR P >>1 O' (xyt)
Fig. 1. Block diagram of an optical operational amplifier.
(5)
If the system contains no sources, 9-1(0) = 0, and Eqs.(6) and (7) follow:
0 = (0'),
o = i-'(O).
(6)
(7)
These indicate that the action of the whole system isequivalent to that of an operator I- 1 . The operatorin the forward branch has been effectively canceled.This will be true even if, for example, the operators arespace variant or nonlinear, within the limit that thesystem remains stable.
If the input is a distorted signal 0 = Xf(IO) the outputsignal will be a corrected version O' = Io. This will bedemonstrated later for some special operators.
B. Inhomogeneous Field
It is well known that TV cameras generally have anonuniform sensitivity in the object field or that theintensity of images produced by lenses decreases withthe fourth power of the cosine of the field angle.
If, for example, those elements are introduced intothe operational amplifier, the operators ? and arerepresented by space variant factors SG (x,y) andSH(X,Y).
With the notation of Fig. 1, for a stable system weget
Fig. 2. Compensation for nonuniform camera sensitivity; (a) uniformtriangular input signal is distorted by the camera and (b) is com-
pensated by inserting the same camera into the feedback branch.
O'(X,Y) = [O(X,Y) - OF(X,Y)I * SG(X,Y) * 3,
OF(X,Y) = 0'(X,Y) SH(X,Y),
(8)
(9)
0'(XY) = O(X,Y) + SG(XY) (10)Ifl. -SG (X,Y) SH (X,Y)
If the product - SG- SH is large compared with 1, then(11) follows:
1O'(xy) O(xy)S1
SH (X,Y)(11)
Fig. 3. (a) Oscillogram of one line from a distorted video signal [Fig.2(a)]. (b) One line from a compensated signal [Fig. 2(b)].
The system has ignored the inhomogeneous sensitivityin the forward branch, while the inhomogeneity in thebackward branch has been inverted.
Lee9 has already used Eq. (10) to create an outputdepending nonlinearly on an input function SG. Butsince he uses only passive feedback, the range of possiblenonlinearities is restricted.
In Figs. 2 and 3, the effect of inhomogeneity com-pensation is experimentally demonstrated. The object,a triangular grating with uniform amplitude, was dis-turbed by a transparency SH(X,Y). Fig. 2(a) shows theresulting nonuniform input picture.
In the backward branch of the amplifier, the sametransparency SH(X,Y) has been introduced. According
Fig. 4. Compensation for nonlinear distortion: (a) video oscillogramof square type distortion of a triangular object grating, and (b) partialcompensation by introducing the same nonlinearity into the feedback
branch.
to (4) we get a nearly complete compensation of thedisturbed input, which is also demonstrated in Fig.2(b).
Figs. 3(a) and 3(b) show the corresponding oscillo-grams of one TV line.
C. Compensation of Nonlinearities
The conversion of optical signals into electrical signalsand back to optical signals is frequently nonlinear.Vidicon type cameras have a y value of about 0.5 mon-itoring a y value of 2. So the operators 9 and If arenonlinear functions NG (I) and NH(1) of the intensityI.
After the system has reached stability, the followingrelations hold:
0' = NG(O - OF)M3
OF = NH(O ),
(12)
(13)
0' = NG[O - NH (°')] * A- (14)
If we consider only strictly monotonic nonlinearities,the inverse functions N- 1and N-l are unique. So wefind from Eq. (14)
N-1(') = O -NH(O').
Equation (16) demonstrates the capability of the op-erational amplifier to compensate for nonlinearly dis-torted inputs, by applying exactly the same nonlinearityin the feedback loop. This is demonstrated in two ex-periments represented by Figs. 4 and 5. A triangular-object grating has been distorted by an 12 type nonlin-earity [Fig. 4(a)] and by an ()1/2 type nonlinearity [Fig.5(a)]. Exactly the same nonlinearity has been appliedin the feedback branch, and the results of the compen-sation are shown in Figs. 4(b) and 5(b). The compen-sation is not complete, since in the experiments thevalue of is not very large. Details of the experimentsare described in Sec. III.
Before we proceed to spatial filtering, we shall discussa special nonlinearity-the addition of a bias, a fre-quently used method in incoherent processing to avoidnegative signals. If NG (I) = I + BG, we shall find thatNC'(I) = I - BG. From Eq. (16), it follows that, if NH= I, the output signal O' will be 0 + BG. This is auseful result. A pure bias nonlinearity in the forwardbranch will occur unchanged in the output signal.
D. Spatial FilteringSo far, we have discussed operations without spread;
we can consider the optical loop consisting of about 105channels (pixels) in parallel. Spread is introduced byaberrations of the lenses and by optical or electronicfilters. (This is discussed later. If the operators 9 andIf depend on the spatial frequencies and the temporalbehavior leads to stability, we can find equations similarto Eqs. (8)-(10) in the frequency space denoted by thetilde:
O'( U,v) = O(,v)- G(A,v)
1 + -G(U) H(,Iv)(17)
Again, if we proceed to the ideal operational amplifier,that is, ( 3 G * H >> 1, we find
1Z = *O 'U, ) = A, v) - gv (18)
If, for example, an aberrated lens has a transfer functionH, the same aberrated lens introduced into the back-ward branch of an optical operational amplifier may
(15)
In the operational amplifier is very large, so the leftside of Eq. (15) will degenerate to Njl (0) if O' is finite.N 1(0) exists and is zero, for example, for passive sys-tems. If it exists for active systems, NG'(0) will be aconstant bias term -BG. From the above Eq. (16)follows:
0'= Nj1(O + BG). (16)
If the bias BG is zero, the output signal undergoes asecond nonlinearity, which is the inverse of that insertedinto the feedback branch.
One can easily convince oneself that Eq. (16) is trueby setting NG(I) = I and NH(I) = 12 and solving Eq.(14). The output signal is found to be O' = (0)1/2.
Fig. 5. Compensation for nonlinear distortion: (a) video oscillogramof square root type distortion of a triangular object grating, and (b)partial compensation by introducing the same distortion into the
create its own inverse filter. Note that, if H becomesnegative for certain ,v, this may cause oscillations.This problem can be avoided, if we can choose G = He(as suggested by Lohmann3 ), where the asterisk denotesthe complex conjugate.
Figures 6-8 demonstrate the results of some filteringexperiments. For the experiment in Fig. 6 the objectwas a sine grating with linearly increasing frequency.
The TV camera, which converted the object intensitydistribution into an electrical signal, was slightly defo-cused. The corresponding video oscillogram of thedefocused grating is shown in Fig. 6(a). This videosignal has been stored on a TV storage tube, whichserves as an input device. Now the defocused camerahas been introduced into the feedback branch of theoperational amplifier without changing the focus. The
Fig. 6. Inverse filtering. (a) Oscillogram of a distorted input signalproduced by a defocused camera. The object is a sine grating withlinearly increasing frequency. The envelope is the transfer functionof the defocused camera. (b) Partial transfer function compensationproduced by inserting the same defocused camera into the feedback
branch.
Fig. 7. Experiments are similar to those in Fig. 6, except that theamount of defocus in the feedback loop is different from that presentin the input signal: (a) input signal; (b) restored signal produced byadditional defocusing in the feedback branch; and (c) overcompen-sated signal produced by excessive defocusing in the feedback
branch.
Fig. 8. Image restoration by autocompensation: (a) input imageproduced by a defocused camera; (b) autocompensation using thesame defocused camera in the feedback branch; and (c) image ob-
slight improvement of high frequency contrast, ac-cording to Eq. (10) or (11), is shown in Fig. 6(b). Thecompensation can be improved by applying a higheramplification: in the circuit for a better approximationof the ideal operational amplifier. Another possibleimprovement for this kind of blurring is to increase thedefocusing in the feedback part of the loop, which wasdone in the experiments in Fig. 7. In Fig. 7(b) the re-stored signal is shown. By further increase of defo-cusing we can even achieve a high pass character of theoutput [Fig. 7(c)]. The quantitative experiments ofFigs. 6 and 7 should be complimented with qualitativeexperiments [Figs. 8(a) and 8(b)]. To improve the de-focused picture of Fig. 8(a), the defocused camera wasinserted into the feedback branch to create its own in-verse filter. The result is shown in Fig. 8(b). Thecontrol experiment of Fig. 8(c) demonstrates the max-imum obtainable image quality of the TV components,without feedback and without defocusing.
E. Combination of Variables
Generally the operators and depend on all vari-ables at the same time-space, time, intensity, spatialand temporal frequencies. This interdependencemakes TV optical feedback versatile and powerful. Asshown in Sec. II.A, the mechanism of autocompensationworks with general operators within the limits of sta-bility. Indeed, if we have a camera with a nonuniformfield, followed by a nonlinear amplifier, and then aspatial filter, all inserted into the feedback branch, theoutput signal does not depend on the properties of theforward branch. The action of the total system is thereverse of that of the operator inserted into the feedbackbranch.
For the example mentioned above, we obtain Eq. (19),where indicates a Fourier transformation:
0' = - N_' S 0)SH I i~
We see that the order of restoration is the reverse of theorder of input degradation. The system not onlycompensates for pictures, it can also be used for mani-pulating pictures, for example, division of (x,y) bySH(X,Y), where the transparency SH would be insertedinto the feedback loop. One property may be of par-ticular importance. Light to voltage convertors in-variably contain spatial inhomogeneities, which largely
(19)
reduce the range of possible filter operations. Thesecan be compensated by inserting the image convertorinto the forward branch of an operational amplifier. In
or addition, after averaging over temporally varying noise,we have a signal available with extremely high SNR. Inprinciple, the dynamic range of the signal could be-MON similar to that of a computer, so that subsequent digital
OTPU' processing would not be limited by the input signal butby the computer.
111. TV Optical Operational Amplifier
So far, we have discussed the system from thestandpoint of black box theory. Proceeding in aphysically realizable system, two interesting problemsarise in addition to technical problems such as geo-metrical distortion.
(1) In the loop, we seomtimes convert incoherentlight signals into electirical signals and vice versa. Wemust take care that electrical signals, which may bebipolar, correspond only to positive intensities, beforethey are fed into a monitor.
(2) In the discussion in Sec. II, we assumed that thesystem would be stable. Since there is a temporal delayin the loop, 7 the negative feedback changes to positivefeedback for temporal frequencies in the range of f 10Hz. We must take care that the total gain in that fre-quency range is less than unity to avoid oscillations,although the gain at f = 0 Hz should be very large.Consequently, we have to use a very slow TV system.We have used a specially selected vidicon camera withan afterimage of 30% after 400 msec. With these con-siderations in mind, the device shown in Fig. 9 was de-signed. As an input device, we use a vidicon TV storagetube. This is charged with an input image by the sameTV camera used later in the feedback branch, thusavoiding distortion problems connected with two dif-ferent cameras.
According to Fig. 1, the feedback signal OF has to besubtracted from the input 0 in the device shown in Fig.9. This can be easily performed, since 0 and OF are
INITIAL OUTPUT
I O/
.5
0 TIME TV-CYCLES)v40ms
Fig. 10. Initial behavior of a TV optical operational amplifier fordifferent amplification factor fl. The time constant of the camera is
electrical signals. The difference will then be amplifiedand fed into the feedback branch, where it is convertedinto an optical signal. After performing an opticaloperation, such as 2-D convolution, the signal is againconverted into an electrical signal. In this device, thetwo requirements demanded in the beginning of thissection can be satisfied without introduction of a bias.The system performed a positive-stable output, evenwith an amplification ( 3 5.5.
In Fig. 10, the calculated behavior, immediately afterthe system is turned on, is shown for different (3. Thenormalized initial output 0/3 is plotted vs time for aconstant input signal. For small (, the output reachesits stable value without going negative. For large d thesignal would become negative, which is forbidden. Themaximum leading to signals >0 is about 5.5, whichcoincides with our experiments. Of course, (3 = 5.5 isfar from the ideal operational amplifier, but the maxi-mum obtainable depends directly on the slowness ofthe TV system. The theory of temporal behavior,which will be dealt with in a subsequent paper, showsthat a camera, which is ten times slower, will make itpossible to apply an amplification that is about tentimes larger.
IV. Conclusions
The principle of active-negative feedback can beapplied for incoherent optical systems. To achieveamplification and inversion, TV systems are used inaddition to optical systems. If the amplification in theforward branch of the system is large, it may be calledan optical operational amplifier, in that its propertiesare determined only by the properties of the feedbackbranch. The action or the total system is the reverseof that of the operator inserted into the feedbackbranch. Consequently, the system has the capabilityof autocompensation of image degradations, which maybe nonlinear distortions, multiplicative errors, or lensaberrations. The degradations may be space variant.
We are grateful for helpful discussions with A. W.Lohmann and S. K. Case. This paper was presented atthe ICO-11 Madrid meeting in September 1978.
References1. H. J. Reich, J. G. Skalnik, and H. L. Krauss, Theory and Appli-
cations of Active Devices (Van Nostrand, Princeton, 1966); U.Tietze and C. Schenk, Halbleiter Schaltungstechnik (SpringerVerlag, New York, 1976).
2. G. Hdusler and A. W. Lohmann, Opt. Commun. 21, 365 (1977).3. A. W. Lohmann, Opt. Commun. 22, 165 (1977).4. M. 0. Hagler, Appl. Opt. 10, 2783 (1971).5. G. Crosta, Opt. Quant. Elect. 10, 9 (1978).6. K. Sato, K. Sasaki, and R. Yamamoto, Appl. Opt. 17, 717
(1978).7. G. Hdusler and M. Simon, Opt. Acta 25, 327 (1978).8. G. Hausler, A. W. Lohmann, and M. Simon, Opt. Acta to be
published.9. S. H. Lee, Opt. Eng. 13, 198 (1974).
10. G. L. Rogers, Noncoherent Optical Processing (Wiley, New York,
1977).
Optical SignalProcessing for C3 1
October 29-30, 1979Marriott Hotel * Boston, Massachusetts
Chairman. William J. Miceli, Rome Air DevelopmentCenter. Seminar Committee. Norman J. Berg, HarryDiamond Laboratories; John Burgess, Rome AirDevelopment Center; H. John Caulfield, AerodyneResearch Corp.; David Casasent, Carnegie-MellonUniversity; Michael Hamilton, Air Force AvionicsLaboratory; Sam Horvitz, Naval Underwater SystemsCenter; John A. Neff, Air Force Office of ScientificResearch; Richard Picard, Rome Air DevelopmentCenter; William T. Rhodes, Georgia Institute ofTechnology; Gerhard Sauermann, MITRE Corp.
The growth of modern Command, Control, Com-munications and Intelligence 1C311 systems continues tostrain the state of the art of signal processing tech-nology. High-resolution imaging radar spread spectrumcommunications, adaptive array processing and wide-band spectrum analysis are examples of applicationswhose requirements magnify the limitations of existingapproaches and thus promote new technologies.
These difficult requirements can be approached withoptical signal processing, where additional benefits ac-crue in reduced costs and complexity because of simplercomponents. Optical processing techniques appear to bequite compatible with SAW/CCD/digital technologies,thus permitting the configuration of versatile combina-tions. Recent developments obtained with such hybridarchitecture include scale, rotationally and Doppler in-variant processing, antenna beam steering, optical logic,and high-resolution spectrum analysis.
The purpose of this seminar is to re-examine thepromise of real-time optical signal processing in the con-text of C31. Systems users will describe their needs andconstraints, and the (hybrid) optical implementation ofvarious processing algorithms will be discussed. Devicetechnologists will review state-of-the-art hardware, andinnovative solutions to old problems will be presented.
Tutorial reviews, plus original papers describingnovel C31 processing applications are solicited. Topicssuitable for presentation include, but are not limited to:
The location for this seminar was selected becauseof its proximity to the Electronic System Division(ESD), the Rome Air Development Center Deputy forElectronic Technology (RADC/ET), and many engineer-ing installations, permitting broad participation from allsectors, providing a much-needed forum for the dis-cussion of current and future requirements/solutions.
Proceedings of this seminar will be published bySPIE.
