Equalizing and coherence measure correlators

David Casasent and Alan Furman

A correlator is described that combines optical and electronic processing. The optical section is the firstFourier transform taking stage of a joint transform correlator; its output is electronically processed by amodified spectrum analyzer. Increased flexibility and various processing operations beyond those normallypossible in an optical system result.

1. Introduction

The importance of the correlation operation in imageand signal processing is well accepted. Many diverseoptical correlator systems have been devised,1-7 be-ginning with the holographic matched filter correla-tor,1,2 that enable this correlation operation to be per-formed in parallel on 2-D imagery or multichannel 1-Ddata and at high throughput rates. The basic opticalsystem which we chose to modify is the joint transformcorrelator (JTC).3-5 Rau6 had previously suggesteddetection of the joint transform of two functions (placedside by side in the input plane) by a vidicon camera. Aspectrum analyzer was applied to the camera's videosignal, yielding the correlation of the two inputs in ademonstration of optical character recognition. Intemporal offset correlation, Macovski and Ramsey7

devised a method whereby a video signal equivalent tothe above one was produced by transforming the co-herent superposition of the two images (fully overlap-ping) after having frequency-shifted the light used toilluminate one of the images. The transform planeintensity then contains a temporal ac term composedof a carrier (at the shift frequency) modulated by thecomplex transform product. Since the width of theinput format is reduced typically by a factor of 3, thetransform plane detector spatial resolution required isreduced proportionally; however, because an ac (intime) intensity distribution must be detected, a non-integrating (e.g., image dissector) detector must beused.

In the system to be described herein, the jointtransform detection system is modified, and an equal-izing correlator produced that enables compensationof various types of signal distortions to be realized. A

The authors are with Carnegie-Mellon University, Department ofElectrical Engineering, Pittsburgh, Pennsylvania 15213.

Received 30 January 1978.0003-6935/78/1101-3418$0.50/0.

) 1978 Optical Society of America.

theoretical description (Sec. II) of the electronic readoutjoint transform correlator and experimental demon-strations of it (Sec. III) are followed by similar sectionsfor the equalizing correlator (Secs. IV and V). Weconclude with a discussion of other extensions of thissystem including a coherence measure operation (Sec.VI).

II. Electronic JTC Readout; Theoretical

The conventional joint transform correlator3-5 is re-viewed first. In this system, the two functions f and gto be correlated, both assumed to have a width of b, areplaced side by side in the input plane Po with a center-to-center spacing 2b.8 We represent the transmittanceof Po by

ti(x) =f(x+b) +g(x -b). (1)

We use 1-D notation both for simplicity and becausewe will deal mainly with 1-D processing in this paper.The Fourier transform plane intensity distribution(modulus squared of the transform of ti) is

Il(u) = IF(u)12 + IG(u)12 + 21F1 IGI cos[47rub + +p(u)], (2)

where F and G are the Fourier transforms of f and g,and the coordinate x1 of the transform plane P, is re-lated to the spatial frequency u by u = xl/XJL. Thefocal length of the transform lens is fL, and the phaseterm k(u) = arg G (u) - arg F(u) equals the differencebetween the phases of the transforms of the two func-tions.

This Fourier transform distribution is recorded onan optically addressed spatial light modulator (SLM),9and the SLM is now read by illuminating it with a planewave. Assuming that the SLM's amplitude transmit-tance is approximately a linear function of writing in-tensity II, we obtain at the output plane P2, uponforming the Fourier transform of this transmittance.

u(x2) =fsf+gg+ftg * (x2+ 2b) +g of* b (x2- 2b),(3)

where denotes correlation, and * denotes convolution.

3418 APPLIED OPTICS / Vol. 17, No. 21 / 1 November 1978

2b

C

P.

0Lc Es P,

Fig. 1. Schematic of the basic electronic JTC system.

MULTIPLIERS INTEGRATORS SOUARERS

Fig. 2. Block diagram of the electronic section of the basic electronicJTC system.

This is the desired correlation of the input functionscentered at x2 = 12b (assuming equal focal lengths forboth transform lenses).

In the electronic readout JTC system, the intensitydistribution in Eq. (2) is converted to an electrical timefunction (video signal) by a scanning photodetector asshown in Fig. 1, and the second transform is producedin modified form electronically as shown in Fig. 2. Werepresent the output electronic signal from the detectorat P1 in 1-D by

v(t) = R1,(u), (4)

where R is the detector's responsivity in volts/unit in-tensity, s is the detector's scanning speed, and hence xi= st, and u = st/XfL. This video signal v(t) is multi-plied by quadrature sinusoids sin(cwot) and cos(wot),these two separate products integrated, and the inte-grals squared and summed. We then obtain a newoutput:

) = [ J v(t) coswotdt] + [f X v(t) sinwotdt]

= (RL) 1 uo Il(U) exp(j2rut)du|

(RXfL)21 9-l[Il(u) rect(u/2uo)] 12

= (RXfL) 2 sinc(2uot) * [ff + gg + gf * 6( - 2b)

+f g * 5Q + 20]2, (5)

where T is the time constant of the integrator (theproduct of a resistance and capacitance in the typicalop amp integrator), 2to is the line scan time, o = sto!XfL, and = woXfL/27rs. The sinc function in Eq. (5)represents the finite frequency bandwidth of the cor-relator system. If the transform plane P1 is repetitivelyscanned, the integrators in Fig. 2 are discharged to zero

initial value after each line, and the oscillator's fre-quency wo (and hence correlation shift parameter t) are

~ incremented between scans, the desired output corre-y V"DEO lation of the two input functions results.

A. Anatomy of the Joint Transform Pattern

We now examine the joint transform intensitywaveform at P1 of Fig. 1 as the input functions f and gare changed as an aid in visualizing the examples andsystem extensions to follow. From Eq. (2), we see thatIF 12 + IG 12 is the slowly varying part of the outputvideo signal. The important part of this signal is a co-sine function at a frequency 2b, phase modulated by thedifference between the phases of the transforms of thetwo functions and its strength proportional to theproduct of their magnitudes.

We assume approximately equal power spectra IF II G I in all cases to simplify the analysis. For the case

of equal inputs f = g, the expected output is the auto-correlation, 0 = 0, and Eq. (2) becomes

Il(u) = 21G12(1 + cos4irub). (6)

The recorded fringe pattern or cos[ ] term thus has100% amplitude modulation. If one input function isdisplaced away from the other in P(, i.e., the separationbetween f and g is increased (say to 2.5b), then the fre-quency of the fringes increases to 2.5b. With f = g, Eq.(2) now becomes

Il(u) = 2 G 2(1 + cos57rub) = 21 G 2[1 + cos(4irub + rub)]. (7)

From the second formulation in Eq. (7), we see that wecan also view this as the addition of a linear phase termwhose slope, or derivative with respect to a, is propor-tional to the shift in the location of g in Po and thuscontains data on the location of g in f.

Iff $P4- g, the fringe pattern is no longer a pure cosine.Equation (2) describes the resultant electronic outputfrom the plane P1 detector; phase modulation 0(u) #0. The power spectral density at the carrier frequency2b is now decreased. This spectral component is thecorrelation, and hence the loss in this component dueto q5 5d 0 causes an associated loss in signal-to-noise ratio(SNR) of the correlation output.

If f (x) = ag(x), f and g are identical except for a de-creased strength for f by a factor a. The detectedpattern is now

I1(u) = GI2 (1 + a + 2a cos47rub). (8)

As seen, the modulation is reduced with the amplitudeof the information-bearing ac term reduced propor-tional to the decreased strength of f.

By electronically introducing phase modulation intov(t), one can compensate for phase differences betweenF and G such as those due to signal propagation dif-ferences and P1 detector scanning errors. By sensingthe level of the ac signal in Eq. (8), the presence of dif-ferences in the strengths of the input functions can befound and compensated for. We discuss such exten-sions of this electronic readout JTC system later and fornow only note that considerable information on the

1 November 1978 / Vol. 17, No. 21 / APPLIED OPTICS 3419

(a) (b)

Fig. 3. Example of pulse-coded waveform correlation:transparency; (b) correlation output.

(a) input

similarity as well as the relative positions and strengthsof the two input functions may be extracted from theamplitude and phase of the ac fringe signal.

11l. Autocorrelation Experiments

The optical system of Fig. 1 was used. A 50-mWHe-Ne laser source at A = 633 nm and equal focallengths fLs = fLc = 381 mm for the spherical and cy-lindrical lenses provide unity vertical image magnifi-cation (fLs/fLc = 1) from Po to P1 with the horizontaltransform of the input pattern on each line appearingat the corresponding horizontal output line (with aneffective lens focal length fL = 381 mm). A linearself-scanned CCD-addressed photodiode array (ReticonCCPD-1728) with 1728 16-gim square photodiode ele-ments on 16-gm centers was used as the output detector.This circumvented the geometric distortion and MTFproblems associated with vidicon and similar detectors.The array's 27.6-mm length covered L114 cycles/mmof spatial frequency.

To simplify the output electronic system, a Tektronix3L5 spectrum analyzer with 571B mainframe was usedto convert the video signal into the desired correlation.In this system, the video signal v(t) is heterodyned withan oscillator, and the product is bandpass filtered andpeak detected. For a fixed oscillator frequency, thefinal output is the amplitude of the input frequencycomponent at the difference of the oscillator frequencyand the center frequency of the bandpass filter. As theoscillator frequency is swept, the output is the spectralamplitude density of the input in time. While thissystem does not explicitly realize Eq. (5), it provides anequivalent output.

Since transform lens focal length and detector reso-lution were fixed, the input size had to be adjusted toaccommodate the latter. If the full width of a jointtransform input is 3b (where the physical size of eachinput is b with a center-to-center input plane separation2b), then one must use a transform plane sampling in-terval of u, < 1/6b. For various practical reasons we

used us = 1/12b, oversampling by a factor of 2. Thedetector sampling interval of 16 gm results in us = 0.132cycles/mm and a nominal input width of 3b = 3.77mm.

To demonstrate the use of this system in the corre-lation of pulse coded signals, a pseudorandom,twenty-pulse, feedback shift register code was generatedby a computer program. The bit sequence used was1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,1. Bar pattern inputimages in the JTC input format (see left side of Fig. 3)were recorded with a Xerox Graphics Printer andphotoreduced 200:1. Actual full widths ranged from3.3 mm (top) to 4 mm (bottom). Input patterns withthree different separations corresponding to three dif-ferent signal range delays and the corresponding elec-tronic analysis system outputs are shown in Fig. 3. Aspredicted by Eq. (5), the frequency of the cosine carrierincreases as the separation between the inputs (or ef-fectively the target range delay) increases. The corre-sponding frequency (horizontal axis in Fig. 3) or hori-zontal displacement at which the output correlationpeak occurs is analogous to the target range, time ofarrival r, or shift necessary between the two inputfunctions to achieve maximum correlation

p(r) = fg(t)f(t + r)dt. (9)

As shown in the right side of Fig. 3, the location of theoutput autocorrelation peak shifts to the right as theseparation or range delay between the input signalsincreases. The middle peak represents a temporalcarrier frequency of 150 kHz in v (t).

Similar experimental demonstrations of this corre-lator were obtained for text and aerial imagery inputs.The results for the aerial input case are shown in Fig.4. The objective here was to determine the location ofthe reference object g (the airfield) in various inputpatterns f. This pattern recognition operation is ap-propriate for a missile guidance application. 1 0 Theshifted location of g in f in the sequence of three ground

(a) (b)

Fig. 4. Example of pattern recognition on aerial imagery: (a) inputtransparency; (b) correlation output.

3420 APPLIED OPTICS / Vol. 17, No. 21 / 1 November 1978

image sensor data frames shown in Fig. 4 is intended torepresent the drift of the projectile or missile off course.As shown in Fig. 4, the coordinate of the output corre-lation peak is proportional to the location of the refer-ence object in the input data and hence provides thedata necessary to correct the flight path of the projectileor missile.

IV. Equalizing Correlator; Theory

In the conventional JTC system, the output is thecorrelation of the two input functions f and g (with noconvenient way to modify the operation performed). Ifthese functions are the same except that one of them,say f(x), has been acted upon by a linear space-invariantsystem, described by an impulse response h(x), thenf (x) = g(x) * h(x), and the correlation output from theJTC becomes

f(x) g(x) = [g(x) * h(x)] g(x) = h(x) * [g(x) ®)g(x)]. (10)

The effect of the convolution with h (x) shown in Eq.(10) is usually a degradation of the output correlation.The correlation peak is weaker and less sharply defined,and the output SNR is less than that of the autocorre-lation case g(x) ® g(x). By suitably modifying thebasic optical/electronic correlator, one can neutralizethe effect of this type of signal distortion when thedistorting transfer function is known a priori or can beestimated. This capability follows from the access, forelectronic modification, to the joint transform patternvideo signal afforded by the hybrid correlator system.Compensation for linear time-invariant distortion isknown as equalization in the communications field;'1

hence we refer to our new system as an equalizing cor-relator.

To see how this equalization can be accomplished, weexamine the video signal (ac term of interest only) thatresults when g(x) and g(x) * h(x) are the inputs:

/St 2 Ist ) l ,(2sb)tv(t) 2RIG(- IH( -)Cos 2r t\XfL/ \ XfL I I Xfz

+ argH (SL) ] + etc.,

used in the processing of radar or sonar signals degradedby the propagation path or medium12 and for the com-pensation of imperfections in optical processor systemelements. In the latter category are nonuniformtransform plane scanning speed and a restricted classof transform plane lens errors, namely, those equivalentto applying different space-invariant filters to the twoinputs f and g. If the distorting function is known apriori, direct correction is possible. In more realisticsituations, some knowledge is available of the expecteddegradations, or various degradations can be assumed,and the system's performance improvement can be usedto judge the appropriateness of the assumed degrada-tion on an a posteriori basis.

V. Equalizing Correlator; Experiment

To demonstrate the equalization concept, we corre-lated a distorted signal against the original signal andapplied equalization to recover the correlation peak.The formatted pair of inputs is shown in Fig. 5(a). Theoriginal signal [left side of Fig. 5(a)] was the pseudo-random pulse train used in the autocorrelation experi-ments. The distorting transfer function H was of unit

- Eu - - *3 *� EmE. - WE(a)

(11)

where H(u) = Y [h(t)] is the distorting transfer function.If this signal is fed to a phase modulator while a tem-poral voltage waveform -arg H(st/XfL) is applied to itsphase control input, the distorting term arg H(st/XfL)is removed from the cos factor. Equivalently we canphase-modulate the heterodyne oscillator; this lattermethod is easier to implement electronically. Similarly,if v(t) and another signal IH(st/XfL)I-' are electroni-cally multiplied, the product will be free of the multi-plicative distorting factor HI in Eq. (11). Fourier-transforming the resulting signal will yield the auto-correlation of g.

Arbitrary linear space-invariant filtering can beperformed on the correlation output by such a system;equalization of input distortion is a special case of this.Since the system's filtering function is determined bya pair of voltage waveforms, it can be changed with theease and speed with which a signal generator can bereprogrammed. The equalizing correlator can thus be

(b)

(c)

Fig. 5. Example of equalizing correlation: (a) phase-distorted inputand reference signals; (b) conventional correlation output; (c) equa-

lizing correlator output.

1 November 1978 / Vol. 17, No. 21 / APPLIED OPTICS 3421

V/\/-

magnitude, while its phase part, shown in Fig. 6, con-sisted of five rectangular pulses of magnitude 7r with abaseline of zero. The distorted input pattern wasgenerated by numerically convolving the undistortedinput with h (x), thresholding the resulting signal, andcomputer-graphically printing and photoreducing thishard-clipped waveform [right side of Fig. 5(a)]. Theconventional correlation of these two patterns is shownin Fig. 5(b). Positional resolution and SNR are bothseen to be poor. The correlation output with equali-zation circuitry in operation is shown in Fig. 5(c)-aclear correlation, with greatly enhanced SNR and po-sition-measurement accuracy.

The particular transfer function H used in thisdemonstration was chosen to simplify the video signalprocessing circuitry. The distorted video output signalwas fed to a phase modulator, which reversed the po-larity of the video signal at appropriate times to realizemodulation by -arg H. A pulse generator was used toproduce the five-pulse sequence (of the same form asFig. 6) that was used to control the modulator.

This equalizing correlator system is another methodby which the flexibility and repertoire of operationsachievable on an optical processor can be increased.Matched spatial filter weighting and synthesis con-trol,2"13 hybrid optical/digital processing,14 and space-variant processing'5"16 are other methods by which theflexibility of an optical correlator can be extended.

VI. Extensions and Discussion

Another use of the general optical/electronic JTC isin the normalization of correlations. Correlation deg-radation can arise when one of the input signals to becorrelated is reduced in strength by a factor a from thatof the other input function. The degraded jointtransform pattern for this case is described by Eq. (8).The amplitude 2a of the cos term in the video outputsignal can easily be measured and used to control thevertical scale in the output patterns [Figs. 3(b), 4(b), and5(c)]. This results in a normalized output correlationpattern independent of changes in the strength of theinput signal. Other nonlinear processing operationssuch as generating the logarithm of the spectrum (asrequired in cepstral processing) can also be realizedusing such a system.

A. Carrier Period Variance and Coherence Measure

The presence or absence of a correlation and itspositional coordinate are often of primary concernrather than the actual shape of the correlation function.We therefore now consider another new optical/elec-tronic system, which we shall call a coherence measurecorrelator, that greatly improves the speed at which thecorrelation/no correlation decision can be made alongwith ways of implementing it. We conclude this sectionwith a discussion of how it can be applied in range/Doppler multichannel signal processing. This systemuses the configuration of optics and scanning detectorshown in Fig. 1. However, the video signal is now op-erated on by a different kind of electronic processorthan that discussed so far.

H H,9gH(u)

rnH RHFig. 6. Distorting phase transfer function used in the equalizing

correlator experiment.

As was pointed out in Sec. II, phase modulation of thefringe pattern causes a correlation loss, whereas a co-herent uniform fringe frequency (identical inputs)produces a large output correlation. Thus, by mea-suring the coherence, or spectral purity, of the videosignal, the electronics can detect an autocorrelation, ormatch of the input functions, or compute the degree ofsimilarity between them. Several qualitatively relatedproperties of the video signal may be used as coherencemeasures, namely, carrier period variance, carrier phasemodulation or phase variance, and instantaneous fre-quency variance. The smaller any of these quantitiesare, the greater the video signal coherence and hence thegreater the degree of similarity.

We now describe how the carrier phase variancemight be determined by electronic postprocessing of thevideo output described by Eq. (2). Bandpass-filteringEq. (2) removes the first two terms and leaves 21F l G Icos[47rub + 0(u)]. The amplitude modulation presenton this cosine fringe carrier can be removed by a limiterleaving as the output the phase-modulated carriercos[47rub + 0(u)]. If this signal is fed to a phase de-modulator, we obtain a voltage output proportional tothe phase modulation 0(u) on the carrier. Squaringthis quantity and integrating it electronically over a linescan time yield a final output voltage proportional to thecarrier's phase variance

var[(u) = 5 02(U)du-to

(12)

If a frequency demodulator is used instead of a phasedemodulator, we obtain the instantaneous frequencyvariance

var[o'(u)] = to [0'(u)]2

du.-to

(13)

To compute the carrier (fringe) period variance overone detector line scan, we threshold the video signal intoa digital pulse sequence we represent by (u, U2, * , UN)in which these u, are the u values at which positive-going zero crossings of the phase-modulated carrieroccur [i.e., where cos(47rub + k) = 0, and its derivativewith respect to u is positive]. If this pulse sequence isthen fed to a digital or combination analog/digitalprocessor, the mean carrier period

_ 1 N UN

UkNk=1 N

and the carrier period variance

(U- U)2 = (.\Uk - U)2 N Y_ uN )Nk=1 Nk=1 N

(14)

(15)

3422 APPLIED OPTICS / Vol. 17, No. 21 / 1 November 1978

. .u

can be calculated, where AUk = uk - Uk 1 is the carrierperiod at the kth cycle.

This electronic processor can take many forms. Themean and mean square pulse width over one line scantime can be calculated simultaneously. At the con-clusion of the scan we subtract the square of the meanfrom the mean square pulse width. This yields thevariance. An alternate simpler circuit to measure co-herence is the phase-locked loop (PLL).17 A PLLconsists of a voltage-controlled oscillator (VCO) and aphase comparator. The latter compares the VCO'soutput with the input signal presented to the loop, andthe phase comparator output is connected via a low passfilter to the control input of the VCD. When the loopis in lock, the VCD is forced to track the input signal,and its control input voltage indicates the instantaneousfrequency of the input signal. The PLL thus behavesas an FM demodulator. In our case, the ac content(variance) of the VCO control voltage can serve as acoherence measure when the video signal is applied tothe PLL's input.

Coherence measure processing requires a single linescan rather than repetitive scans, as do the system ofFig. 2 and its relative, the spectrum analyzer. Theprocessor's output is computed almost immediately atthe end of each scan. This improvement in efficiencyis especially important when the 2-D nature of opticalprocessing is exploited in performing multichannel 1-Dsignal processing.

An example of such processing is the correlation oftwo signals differing in Doppler. The first processingstep in this case is to correlate one signal against anumber of different doppler versions of the other signal,seeking the best match. The second step is to extractthe range from the correlation of the optimally similarpair found in step 1. This processing operation can beimplemented with the aid of the coherence measurecorrelator as follows: On successive horizontal lines inthe input plane, signals are written in pairs, with one ofthe signals repeated and the other signal written in

different doppler versions on each line. This array ofspatial signals is Fourier transformed in the horizontaldirection and imaged in the vertical direction by aspherical-cylindrical lens system. The horizontal jointtransform fringe patterns on successive lines are sensedby a 2-D raster scanning detector. The coherencemeasure parameter is computed on-line for each scanline from the detector's video signal. The line whichyields the greatest coherence indicates the optimalDoppler match; to determine range, only a 1-D trans-form (as in Fig. 2) of the one line with the largest co-herence need be performed. Thus, range/Dopplerprocessing can be performed in one scanner frame timeby a relatively simple configuration of components.

The support of the Air Force Office of Scientific Re-search on contract AFOSR 75-2851 administered by theAir Systems Command and the Office of Naval Re-search on contract NR 350-011 for the work reported onherein is gratefully acknowledged.References1. A. Vander Lugt, IEEE Trans. Inf. Theory IT-10, 139 (1964).2. A. Vander Lugt and F. B. Rotz, Appl. Opt. 9, 215 (1970).3. C. S. Weaver and J. W. Goodman, Appl. Opt. 5, 1248 (1966).4. J. E. Rau, J. Opt. Soc. Am. 56, 1490 (1966).5. C. S. Weaver et al., Appl. Opt. 9, 1672 (1970).6. J. E. Rau, J. Opt. Soc. Am. 57, 798 (1967).7. A. Macovski and S. D. Ramsey, Opt. Commun. 4, 319 (1972).8. D. Casasent and A. Furman, Appl. Opt. 16, 285 (1977).9. D. Casasent, Proc. IEEE 65, 143 (1977).

10. D. Casasent and M. Saverino, SPIE Proc. 118, 11 (1977).11. W. D. Gregg, Analog and Digital Communications (Wiley, New

York, 1977).12. Topical issue on Adaptive Optics, J. Opt. Soc. Am. 67, 269-409

(1977).13. D. Casasent and A. Furman, Appl. Opt. 16, 1662 (1977).14. D. Casasent and W. Sterling, IEEE Trans. Compt. C-24, 318

(1975).15. J. W. Goodman, Proc. IEEE 65, 29 (1977).16. D. Casasent and D. Psaltis, Proc. IEEE 65, 77 (1977).17. F. M. Gardner, Phaselock Techniques (Wiley, New York,

1966).

William R. Hunter (left) and Richard Tousey photographedon the occasion of Dr. Tousey's retirement from the U. S. Naval Re-search Laboratory. Mr. Hunter is now on a sabbatical at Imperial

College, London.

1 November 1978 / Vol. 17, No. 21 / APPLIED OPTICS 3423