# Analysis of the dual discrimination ability of the two-port photorefractive joint transform correlator.

**ABSTRACT** An all-optical joint transform correlator featuring two operative correlation planes(ports) with complementary performance is presented. We present the theory of operation, derive the input-output characteristics, and demonstrate computer simulations and experimental results. The two-port joint transform correlator is based on simultaneous use of two photorefractive wave-mixing architectures. The first port uses two-beam coupling, and the second port uses four-wave mixing. The performance of the two ports depends on an experimentally controlled beam intensity ratio and the photorefractive coupling coefficient. With appropriate selection of these parameters, the first port is capable of high discrimination, while simultaneously the second offers a low discrimination output. Our results show that the two-beam coupling port can achieve peak-to-noise and signal-to-noise ratio values better than the phase-only correlator, whereas the four-wave-mixing port performs similarly to the classical joint transform correlator. This leads to a potential application in which the correlator could be set up so that in one port a general class is detected (interclass) and, in the other, the specific item in a class is detected (intraclass).

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**ABSTRACT:**A photorefractive nonlinear joint-transform correlator based on the incoherent-to-coherent conversion is presented and analyzed. The nonlinearity of this incoherent-erasure joint-transform correlator (IEJTC) is tunable from the classical-matched to the phase-extraction limit. Correlation peak intensity, sharpness, and discrimination ability increase with the incoherent beam intensity. At easily achievable incoherent-to-coherent beam intensity ratios the IEJTC has its optimal performance, at which the IEJTC approaches the performance of the inverse filter for clean inputs and surpasses the inverse filter performance for noisy inputs. We examine this nonlinearity by using the transform method of analysis and computer simulations. Our study focuses on the effect of saturation on the correlation ability. Our results provide an explanation of why extending the severity of saturation by increasing the incoherent-to-coherent intensity ratio beyond a turning point results in lower optical efficiency, degraded correlation peak, and increased higher-order harmonics. © 1996 Optical Society of AmericaJournal of the Optical Society of America A 02/1996; 13(7):1345-1356. · 1.67 Impact Factor - SourceAvailable from: George Asimellis[Show abstract] [Hide abstract]

**ABSTRACT:**Compansive diffraction nonlinearities reduce the noise and improve the performance of joint transform correlators. The compression and expansion of the photorefractive two-beam coupling parallel optical device is similar to that of the limiting square-law serial electronic re-ceiver. Computer simulations of this device indicate superior perfor-mance over a wide range of noise levels. © 1998 Society of Photo-Optical Instrumentation Engineers. [S0091-3286(98)01901-1]Optical Engineering 02/1998; 37(1):66–74. · 0.96 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the recent trends and advancements in real time pattern recognition with special reference to joint transform correlator (JTC) architectures and algorithms. Techniques for achieving both single and multiple target detection in noise-free, noisy and cluttered input scenes using the classical, binary and fringe-adjusted JTCs are discussed. Distortion invariant target recognition using tools such as the synthetic discriminant function is also incorporated. Various optoelectronic and all-optical architectures and implementation issues for effectively implementing the various types of JTC techniques are discussed. Extensive simulation results are presented to investigate the effectiveness of the various JTC techniques, especially the fringe-adjusted JTC.Proc SPIE 01/2001;

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Analysis of the dual discrimination ability of the

two-port photorefractive joint transform correlator

GeorgeAsimellis, Mark Cronin-Golomb, Jehad Khoury,

Jonathan Kane, and Charles Woods

An all-optical joint transformcorrelator featuring twooperativecorrelation planes1ports2with complemen-

tary performanceis presented.Wepresent thetheory of operation, derivetheinput–output characteris-

tics, and demonstrate computer simulations and experimental results.

correlator is based on simultaneous use of two photorefractive wave-mixing architectures.

The first port uses two-beam coupling, and the second port uses four-wave mixing.

the two ports depends on an experimentally controlled beam intensity ratio and the photorefractive

coupling coefficient.With appropriate selection of these parameters, the first port is capable of high

discrimination, while simultaneously the second offers a low discrimination output.

that the two-beam coupling port can achieve peak-to-noise and signal-to-noise ratio values better than

the phase-only correlator, whereas the four-wave-mixing port performs similarly to the classical joint

transform correlator.This leads toa potential application in which thecorrelator could beset up sothat

in one port a general class is detected 1interclass2 and, in the other, the specific item in a class is detected

1intraclass2.

Key words:J oint transform correlator, photorefractive pattern recognition, discrimination ability.

r1995 Optical Society ofAmerica

The two-port joint transform

The performance of

Our results show

1.

Experimental demonstration and modeling of the

first two-port joint transform correlator 1TPJ TC2 was

introduced in Ref. 1. In this paper we develop the

device theory, analyze an interesting aspect of this

correlator, namely its dual discrimination ability, and

verify the conclusions and results with computer

simulation and experiment.

The TPJ TC is an all-optical photorefractive joint

transform correlator 1J TC2 with two output correla-

tion planes, called ports, that operate simultaneously

and have different discrimination characteristics.

Discrimination is defined as the ability to reject

1discriminate against2 any other signal not identical

to the reference signal. No optical pattern recogni-

tion scheme developed today satisfies together low

Introduction

and high discrimination requirements.

cal J TC is known for its low discrimination ability,2

whereas the binary J TC offers high discrimination.3

Tworecently proposedall-optical photorefractiveJ TC’s

have a tunable discrimination ability,4,5which de-

pends on a beam intensity ratio.

beam ratio, the discrimination can be either low or

high.The TPJ TC attempts to satisfy both low and

high discrimination requirements.

The model presented in this paper uses the physics

of photorefractive four-wave mixing and self-pumped

phase conjugation6,7together with principles of non-

linear signal processing.

forward pump, modified by the signal Fourier trans-

form, and a plane-wave reference beam.

viewed as an application of the two-beam-coupling

J TC proposedin Ref. 4.Thefour-wave-mixing geom-

etry employs the same forward pump and the phase

conjugate of the outgoing signal beam as the back-

ward pump beam.The Fourier transform of the

diffracted beam generates the second J TC port.

This port is an application of the four-wave-mixing

J TC.5

The performance of the two ports is shown to be

dependent on the photorefractive gain and the inten-

sity ratio of the signal-to-probe beams.

The classi-

Depending on the

The first port utilizes a

It can be

When the

G. Asimellis, M. Cronin-Golomb, and J . Khoury are with the

Electro-Optics Technology Center, Tufts University, Medford, Mas-

sachusetts 02115.J . Kane and C. Woods are with the Rome

Laboratories, Hanscom Air Force Base, Massachusetts 01731.

Khoury is alsoa visiting scientist at RomeLaboratories.

Received 10 April 1995; revised manuscript received 27 J uly

1995.

0003-6935@95@358154-13$06.00@0.

r1995 Optical Society ofAmerica.

J .

8154 APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 2

beamratiois much smaller than theexponential gain,

both ports behave similarly to the classical J TC,

whereas at very large beam ratios both ports exceed

the performance of the phase-only correlator in terms

of peak intensity, peak-to-noise ratio 1PNR2, and

signal-to-noiseratio1SNR2.

pare with the gain, the two ports have different

characteristics.In particular, the two-beam-cou-

pling port has high discrimination, and the second

port has low discrimination.

preferable when an image has to be selected from

among others of the same class, but tends to reject a

correct object if the object is somewhat rotated or

scaled.In such cases a low discrimination scheme

works better.

At beam ratios that com-

High discrimination is

2.

The advent of charged-coupled-device 1CCD2 arrays

and spatial light modulators 1SLM’s2 made possible

real-time optical correlation systems based on the

J TC.8–11

Yet these innovations did not change the

fundamental concept of the classical J TC, namely the

use of quadratic processing in a square-law receiver,

be it a photographic film or an intensity-responding

camera.The quadratic processing of the spectral

amplitudeisequivalent tothat oftheclassical matched

filter.It is well known that the matched filter is

characterized by poor performance.12

A variety of nonlinear operations have been pro-

posed toimprovetheperformanceof theJ TC in terms

of correlation peak intensity, noise robustness, dis-

crimination ability, SNR, andPNR.

mention the binarization and other fractional power

laws studied by J avidi.13

evaluate the binary J TC thresholding level are cur-

rently under study.14

Other J TC variations include

fringe adjustment15and image preprocessing.16

thesemodifications arerealizableby serial operations

that include a digital processing unit and additional

CCD camerasandSLM’s.

mance improvements, however, have come at the

expense of the advantages that the optical parallel

processing is best known for, namely processing speed

and operational simplicity.

The holographic properties of photorefractive crys-

tals can be the nucleus of a fast, all-optical J TC

architecture. In particular, a photorefractive crystal

placed at theFourier-transform planecan providethe

appropriatespatial frequency mixing neededfor corre-

lation becausetheinterferencegrating can bewritten

on the intensity-responding real-time holographic

crystal.The use of photorefractives in an all-optical

J TC instead of an electronically assisted J TC offers

thefollowing advantages:

Background

For example, we

Advanced techniques to

All

Theunquestionableperfor-

x

A single photorefractive crystal can be substi-

tuted for a CCD camera, a digital processing unit, and

a SLM display.Only basic optic elements are neces-

sary for J TC operation, and processing speed is no

longer compromised.This results in a compact, fast,

low-energy budget and low-cost design.

x

The photorefractive crystal can achieve much

larger resolution compared with the current SLM

displays and CCD cameras.

x

The inherent photorefractive nonlinearities of

the various beam-coupling energy-transfer mecha-

nisms provide a wide variety of spatial frequency

mixing tools.

Photorefractives have already been introduced by a

number of researchers in linear correlation filter

architectures.17–19

The nonlinear aspects of photore-

fractivemixing havebeen considered tobesideeffects

and, until recently, they have not been actively uti-

lized and investigated.The two recently proposed

photorefractive nonlinear J TC’s actively use these

nonlinearities to achieve significant performance im-

provementsovertheprevious

schemes.4,5

These performance improvements can

be assessed by many metrics such as peak intensity,

PNR, SNR, and discrimination ability.

A setup of thetwo-port correlator is shown in Fig. 1.

Interference by coherent beams 1 1signal beam2 and 4

1probe beam2 establishes a grating inside the photore-

fractive crystal.The readout beam is a plane wave,

and the signal beam carries the joint Fourier trans-

form of the scene 1s2 and reference 1r2 signals, which

are placed alongside the input plane.

crystal, the probe beam is selectively amplified or

deamplified by the joint power spectrum.

quent Fourier transformation of this beam yields

correlation plane 1, which is henceforth called port 1.

Correlation plane 2 is obtained as follows:

pumped phase-conjugate mirror is placed in the prop-

agation path of beam 1, so that a replica of the

outgoing signal beam reenters the crystal.

pumped phase conjugator provides self-alignment

and aberration corrections, sothat the entering beam

is Bragg matched with the grating.

theexisting grating insidethecrystal generates beam

3, which lies along the propagation path of the

reference beam.A beam splitter is used to separate

photorefractive

On exiting the

Subse-

a self-

The self-

Diffraction from

Fig. 1.Experimental arrangement.

10 December 1995 @ Vol. 34, No. 35 @ APPLIED OPTICS8155

Page 3

the diffracted beam, and a transform lens produces

output correlation plane 2, which is henceforth called

port 2.

A common element in the photorefractive J TC’s

discussed in Refs. 4 and 5 and theTPJ TC is that their

input–output curve in the spatial frequency plane is

similar to that of the limiter-quadratic detector20

1LQD2 but with a saturative soft clipping instead of

hard clipping. Such nonlinearities amplify only the

weak portions of the signal 1in the quadratic region2

and deamplify the intense portions 1in the limiting

region2.The LQD is a proven robust alternative to

the classical quadratic detector 1QD2 for the problem

of a signal embedded in non-Gaussian noise.

theoretical investigation21it has been shown that the

relative efficiency of the LQD with respect to the QD

is independent of the noise variance 1s22 if the limiter

value is proportional to s.

selection of the limiter value, the LQD has consider-

ably more false-alarm stability and better perfor-

mancethan theQD for non-Gaussian noise.

A purely quadratic J TC produces broad and dim

correlation peaks.It is typically a scheme with low

discrimination and is preferable when the inputs

have a relative scale and rotation difference or the

noise has Gaussian characteristics.

purely limiting 1alsoknown as hard clipping2 produces

thin and intense peaks when the signals are identical

and no peak if they are different.

preferable when maximum discrimination is desired,

which reduces, for example, the false-alarm probabil-

ity.

In the input–output curves of the photorefractive

J TC’s thesignal spectral intensity appears multiplied

by the signal-to-probe beam ratio.

parameter is controlled by the experimental settings

and substantially affects the crystal response to the

input signal intensity.When this parameter is small,

so that most of the modified spectral signal intensity

is below the limiter level, the crystal response is

predominantly quadratic, whereas for a large param-

eter the crystal response is mainly limiting.

tion from the one to the other mode is possible by

simply increasing the beam ratio.

Section 3, the transition occurs at different values of

beam ratio for the two ports of the TPJ TC.

thebasis of thedifferent performanceof thetwoports.

In Section 3 we also present the expressions that

describe the operational modes, analyze the effect of

these modes on the relative performance of the two

ports, and discuss the conditions under which one

port may operate near the limiting region while the

other port operates near thequadratic region.

In a

With an appropriate

A J TC that is

A limiting J TC is

The value of this

Transi-

As is shown in

This is

3.

In deriving the TPJ TC input-output characteristics

we consider the transmission grating geometry, with

the notation of Fig. 1.We use the method of grating

integral, which is fully presented elsewhere,22,23with

Theory

the following assumptions:

same wavelength l, all wave vectors are in the plane

of the paper, all waves are polarized perpendicular to

this plane, and the grating wave vector is parallel to

the crystal optical axis.The absorption coefficients

are assumed to be small and are omitted.

intensity I0and input intensity ratiom are defined as

all beams are of the

The total

I05o

i51

4

Ai2,m 5

A11022

A41022.

112

The coupled-wave equations can be written as

1boldfaced characters indicatecomplex quantities2

d

dzA11z2 5 2

1

2GC1z2A41z2,

d

dzA41z2 5

1

2G*C*1z2A11z2,

d

dzA21z2 5

1

2GC1z2A31z2,

d

dzA31z2 5 2

1

2G*C*1z2A21z2,

122

where G is the amplitude complex coupling constant

and C1z2 is thegrating interferencefunction23:

C1z2 5

A11z2A4*1z2 1 A21z2A3*1z2

I0

.

132

Of special interest is thegrating action integral, u1z2

which is defined as

u1z2 5e

0

z1

2g0C1z20dz,

142

where g is the real part of G.

externally superimposed electric fields on the crystal

1diffusion limit2, the coupling constant is real and no

phase coupling occurs between the mutually interact-

ing beams. If L is the crystal thickness along the

direction of propagation and U 5 u1L2, the solutions of

Eqs. 122 can be expressed in terms of wave amplitudes

only as

In the absence of

A11z2 5 A1102cos3u1z24 2 A4102sin3u1z24,

A41z2 5 A4102cos3u1z24 1 A1102sin3u1z24,

A21z25A21L2cos3u1z22U41A31L2sin3u1z22U4,

A31z25A31L2cos3u1z22U42A21L2sin3u1z22U4.

15a2

15b2

15c2

15d2

The wave amplitudes as expressed above offer a

clean representation of the energy-transfer mecha-

nisms andthegrating diffraction efficiency.

more, the validity of these solutions is not confined

for thin interaction lengths only or for small coupling

constants. To evaluate the value of the grating

action integral, we consider the boundary conditions,

Further-

8156APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 4

which are

A1102 5 A10,A4102 5 A40,A21L2 5 A2L,A31L2 5 0.

162

Because A31L2 5 0, the grating can be considered as

exclusively written by beams 1 and 4 if A2is small

compared with A1 and A4.

mixing approximation.In such a case the following

known solutions for A11z2 and A41z2 from the two-wave-

mixing theory,7

This is the two-wave-

A11z2 5 A103

1 1 m21

1 1 m21exp1gz24

1@2

,

A41z2 5 A403

1 1 m

1 1 m exp12gz24

1@2,

172

must satisfy thegeneral solutions of Eqs. 15a2 and 15b2.

By incorporating the conditions 162 and expressions 172

into Eq. 142, we obtain the solution for the grating

action integral:

u1gz, m2 5 tan213

exp1

gz

222 1

1

Œmexp1

Œm 1

gz

224

.

182

The grating diffraction efficiency h is defined by the

portion of beam A2at z 5 L that is deflected in the

direction of beam 4 at z 5 0.

easy toseefrom Eq. 15d2 that

Because A31L2 5 0, it is

h 5

I30

I2L

5 sin23u1gL, m24.

192

Beam 1 illuminates a transparency that contains

the two input signals, whose transmissivity is de-

noted by r1x, y 1 y02 and s1x, y 2 y02, and is subse-

quently Fourier transformed by a lens with focal

length f1.As a result, beam A10incident upon the

crystal 1signal beam2 is modulated by the sum of the

individual complex Fourier transforms of the scene

and thereferencesignals, R 1nx, ny2 1 S1nx, ny2.

and ny indicate the spatial frequency coordinates.

If A˜10is the signal beam amplitude before the input

transparency, then

Herenx

A105 A˜10

0R 1nx, ny2 1 S1nx, ny20

lf1

.

1102

Themodified input intensity ratiois written as

m 5

A˜102

A402

0R 1nx, ny2 1 S1nx, ny202

1lf122

5 meffE1nx, ny2,

1112

where meffis the effective beam ratio at the Fourier-

transform plane and E1nx, ny2 is the normalized signal

intensity spectrum.

follows:

We define meffand E1nx, ny2 as

meff5

A˜102

A402

E02

1lf122,

1122

E1nx, ny2 5

0R 1nx, ny2 1 S1nx, ny202

E02

,

1132

where E0is the integrated transmissivity of the input

signal r 1 s. Thesedefinitions aresuch that both the

effective beam ratio and the normalized signal inten-

sity are dimensionless. In addition, any changes in

thesignal beam intensity by alterations in thearea of

the input signals are accounted for:

A˜102E02can beviewedas theintegratedintensity after

passage of beam 1 through the transparency.

effective beam ratio 1simply called beam ratio2 is

controlled by the experimental settings and the nor-

malized signal intensity 1alternatively referred to as

energy2 depends on theinput characteristics.

By the substitution of Eqs. 1112 and 182 into Eqs. 192,

15a2, and 15b2, the grating diffraction efficiency h and

thewaveamplitudes A1Land A2Ltaketheform

the product

The

h 5 sin25

tan213

exp1

gL

222 1

1

1meffE 21@2exp1

1meffE 21@21

gL

2246

,

1142

A1L5 A4053meffE11 2 h241@22 1h21@26,

1152

A4L5 A405311 2 h241@21 3meffEh41@26.

1162

The phase-conjugate mirror creates a replica of

beam A1L.Under certain conditions, including one

in which thephase-conjugatereflectivity b is indepen-

dent of the intensity spatial fluctuations, one can

express A2Las A2L5 bA1L.

sions for A2Land A3LintoEq. 15d2 weobtain

Substituting the expres-

A305 2bA4053meffE11 2 h241@22 1h21@261h21@2.

1172

Correlation planes 1 and 2 are produced after

inverse Fourier transformation of beams A4Land A30,

respectively.By setting A4L5 A40f11nx, ny2 and A305

2bA40f21nx, ny2, we derive the input–output functions

f11nx, ny2 and f21nx, ny2 for thetwoports:

f11nx, ny2 5 31 2 h1nx, ny241@21 3meffE1nx, ny2h1nx, ny241@2,

1182

f21nx, ny2 5 5meffE1nx, ny231 2 h1nx, ny24h1nx, ny261@2

2 h1nx, ny2.

1192

4.

The two input–output operators f11nx, ny2 and f21nx, ny2

describe nonlinear processing of the joint intensity

spectrum.The characteristics of these nonlineari-

Analysis

10 December 1995 @ Vol. 34, No. 35 @ APPLIED OPTICS8157

Page 5

ties are subject to the values of the experiment-

controlled beam ratio meff and the photorefractive

gain factor gL.The nature of these characteristics

and the way that they can be used for performance

optimization areinvestigated in this analysis.

There are twofundamental aspects of this process-

ing: the mixing of the spectra, which is responsible

for the correlation properties of the J TC, and the

selectiveamplification of thejoint intensity spectrum,

which is responsible for the different characteristics

of the two ports. The crystal responds to the inten-

sity of the signal, not to the amplitude.

accountablefor themixing of thespectra, and wenote

that both input–output operators are functions of the

signal energy E1nx, ny2.Thecrystal responseis differ-

ent for different signal energy values, as is demon-

strated by the energy dependence of the diffraction

efficiency h1nx, ny2, as shown in Eq. 1142.

denceresults in theselectiveamplification.

It is well understood that the diffraction efficiency

is negligible for any of the limits meffE 9 exp1gL2 and

meffE : exp1gL2.Wecan examinethebehavior of the

input–output response curves of the two ports at

these limits by using series expansion and keeping

only the first-order terms.

thefollowing twocases of approximations:

This is

This depen-

By doing so we arrive at

1a2

Approximations at small beam ratio values:

A beam ratio is small when, for most of the spatial

frequencies 1nx, ny2, the condition meffE 1nx, ny2 9

exp1gL2 is true. Such low beam ratio values are

obtainedwith a strong readout beam,A4.

tion efficiency and the input–output operators are

expressed as

Thediffrac-

h1gL, meffE 2

mE9exp1gL2

5 meffE exp12

gL

224 sinh21

gL

42

1202

f11nx, ny2

mE9exp1gL2

5 1 1 meffE exp12

3

4gL22 sinh1

gL

42,

1212

f21nx, ny2

mE9exp1gL2

5 meffE exp12

3

4gL22 sinh1

gL

42,

1222

Both input–output operators are quadratic with

respect tothe signal amplitude 1linear with the signal

energy2. Usually the low spatial frequencies, which

correspond to the dc autocorrelation terms, are very

intense.Thusat small beamratiostheTPJ TC ampli-

fies the low frequencies.This is typical of the classi-

cal J TC quadratic receiver, which has a very low

discrimination ability.

1b2

Approximations at largebeam ratios:

ratiois largewhen, for most of thespatial frequencies

1nx, ny2, the condition meffE1nx, ny2 : exp1gL2 holds.

This is accomplished with a strong signal beam, A1.

Thediffraction efficiency and theinput–output opera-

A beam

tors are

h1gL, meffE 2

mE:exp1gL2

5

exp1

gL

224 sinh21

meffE

gL

42

,

1232

f11nx, ny2

mE:exp1gL2

5 exp1

gL

222

1

meffEexp1

gL

224 sinh21

gL

42,

1242

f21nx, ny2

mE:exp1gL2

5 exp1

gL

222 1 2

1

meffEexp1gL2

3 4 sinh21

gL

42.

1252

In this region both operators have an inverse

relation with the input intensity.

high frequencies, which correspond to the cross-

correlation grating, that are of low intensity.

fore at large beam ratios the TPJ TC is amplifying the

high frequencies of the spectra.

meffE = ` the operators are purely limiting, and their

output is independent of the signal amplitude.

J TC with amplitude removal is equivalent to the

phase-extraction correlator, which is characterized by

a very high discrimination ability.

Typically it is the

There-

At the limit of

A

Figure 2 displays two sets of the input–output

operators f11nx, ny2 of the two-beam-coupling port 1solid

curves2 and f21nx, ny2 of the four-wave-mixing port

1dashed curves2 versus the product meffE.

11 has been added tof2sothat at the low beam ratio

limit all the curves converge to11.2

curves reach a plateau when meffE : exp1gL2, and

they are linear with the signal energy when meffE 9

exp1gL2.

The curves are different at intermediate beam ra-

tios.For such beam ratios, the product meffE1nx, ny2

for small E satisfies the quadratic approximation

1A bias of

As expected, all

Fig. 2.

port 1solid curves2 and the four-wave-mixing port 1dashed curves2

for gain ranging from 26 to16.

TPJ TC input-output curves for the two-beam-coupling

8158APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 6

3,exp1gL24, whereas for large E it satisfies the limit-

ing condition 3.exp1gL24.

sponse has mixed characteristics, namely quadratic

for the weak points of the intensity spectrum and

saturating 1limiting2 for the intense points, with a

smooth transition for the intermediate intensities.

Such a soft-clipping detector has an improved perfor-

mance compared with both the hard-clipping detec-

tors andtheQD’s.This is attributedtothereduction

of thefrequency-planecontrast by selectiveamplifica-

tion of the weak parts of the signal.

plane in a J TC is dominated by the intense on-axis dc

term; yet the important information lays at the

interference grating written by the cross-correlation

term, which is normally weak.

tively amplified by a soft-clipping detector, while the

intense dc term is relatively deamplified.

QD amplifies thewholespectrum, and predominantly

the dc terms, whereas a purely hard-clipping detector

does not amplify the weak portion of the signal.

Photorefractive selective amplification at the spatial

frequency plane has been proposed for contrast ma-

nipulation,24image enhancement,25and noise reduc-

tion.26

For both ports of the TPJ TC, the limiter level,

wherethetransition from thequadratictothethresh-

olding region occurs, depends on the gain gL.

beam ratio at which transition occurs is approxi-

mately exp1gL2, but it is different for the two ports.

We note from Fig. 2 that for positive gain the

two-beam-coupling port saturates earlier than the

four-wave-mixing port.At negative gain, the four-

wave-mixing port saturates first.

sition is smooth and not abrupt, we attempt to

estimate the meffE value at the limiter level by the

point of inflection of the input–output curve, which

can be found analytically.

values for the point of inflection of the two input–

output curves.

We note that 1i2 the meffE values at the points of

inflection are approximately related to a geometrical

succession, whose step is different for the two ports;

1ii2 both ports require larger signal intensities for

increasing photorefractive gain to saturate, and 1iii2

the separation of the two curves is emphasized for

largeand positivephotorefractivegains.

Figure 3 displays the TPJ TC input-output curves

for a gain of 18 to show in detail the different

responses of the two ports.

sponds to the two-beam-coupling port, and the four-

wave-mixing port is indicated by the dashed curve.

Therefore the output re-

The frequency

This grating is selec-

A purely

The

Because the tran-

Table 1 presents the meffE

The solid curve corre-

At very large beam ratios 3meff. 1044, both curves

have a limiting response, which is identical tothat of

theinversefilter and is noted by thehorizontal dotted

linein thefigure. At small beam ratios, theresponse

is linear in thesignal energy for thefour-wave-mixing

port.The similarity with the matched-filter re-

sponse curve 1a line with a slope of 12 is clearly

identified. There is a relatively broad range of beam

ratiovalues for which thetworesponsecurves display

different characteristics, namely the two-beam-cou-

pling port has a response close to that of the phase-

only correlator 1a line with a slope of 0.52, whereas the

four-wave-mixing port performs like the classical

matched filter.

Similar separation occurs for negative photorefrac-

tivegain values, which areachieved simply by revers-

ing the crystal orientation.

values necessary for separation are of the order of

1024, which means that thesignal beam has tobevery

weak.With a very weak beam, the self-pumped

conjugator does not operate, and the very weak beam

limit is impractical from the experimental point of

view.

We summarize our theoretical predictions regard-

ing the relative performance of the two ports with

positivephotorefractivegain in Table2.

However, the beam ratio

5.

The objective of this analysis was to examine the

performance of the TPJ TC by comparison of the

Computer Simulation

Table 1.Point of Inflection for Various Photorefractive Gains

Photorefractive

Gain gL

meffE at the Point of Inflection

Port 1 Port 2

8

6

4

2

0.2

1085.7

203.2

28.7

8782.5

1040.1

120.4

11.74.8

1.2 1.3

Table 2.TPJTC Performance at Different Operating Points

Beam RatioPort 1Port 2

meffE 9 exp1gL2

Quadratic—low dis-

crimination

Nearly saturating—

high discrimina-

tion

Saturating—very

high discrimina-

tion

Quadratic—low dis-

crimination

Nearly qua-

dratic—low dis-

crimination

Saturating—high

discrimination

meffE < exp1gL2

meffE : exp1gL2

Fig. 3.

curve2 and the four-wave-mixing port 1dashed curve2 for a gain of

8. The slopes of three known filters 1classical matched, phase

only, and inverse2 aredrawn for comparativeanalysis.

Input–output curves of the two-beam-coupling port 1solid

10 December 1995 @ Vol. 34, No. 35 @ APPLIED OPTICS8159

Page 7

correlation peaks with known J TC schemes, namely

theclassical J TC andthephase-only J TC.

plane was an array of 512 3 1024 pixels, in which the

scene and the reference signals, combined in an array

of 128 3 256 pixels, were centered on the optical axis.

We used the following sets of scene and reference:

1a2 a set of two identical large disks, and 1b2 a set of a

largeand small disks. Thelargeand thesmall disks

had radii of 28 and 14 pixels, respectively, and their

center-to-center separation was 128 pixels.

Thenonlinear functions f11nx, ny2 and f21nx, ny2 in Eqs.

1182 and 1192 were applied in the spatial frequency

plane with the values for the beam ratio1meff2 of 1021,

102, and 104, typical values for small, intermediate,

andlargebeamratios, respectively.

tive gain 1gL2 was 18.The fast Fourier transforma-

tions werecorrected sothat theenergy was preserved

in all planes.

In the output plane only the 160 3 160 pixel region

area around the first-order correlation peak was

selected for display. Figure 4 shows the computer

simulation results with the classical J TC and the

phase-only J TC for the similar 1top row2 and the

dissimilar 1bottom row2 disks.

the frequency-plane processing was quadratic in the

spectral amplitude, whereas for the phase-only27J TC

the absolute value of the spectral amplitude was

considered.The z-axis ratio between Figs. 41b2, 41c2,

and 41e2, 41f2 is 5:1 because of the relative decrease in

peak intensity.This axis convention is held for all

Theinput

Thephotorefrac-

For the classical J TC,

cases when similar-disk and dissimilar-disk correla-

tion peak intensities arecompared.

Figure5 displays theTPJ TC correlation peak inten-

sity plots with the similar disks for different beam

ratios.Figure6 repeats this for thedissimilar disks.

SNR and PNR measurements for the corresponding

amplitude arrays surrounding the correlation peaks

areshown in Table3 for thesimilar disks and in Table

4 for the dissimilar disks.

PNR definitions as described in Ref. 28.

As expected, the shape of the peaks and the values

of intensity, SNR, and PNR depend on theoperational

point, which is controlled by the beam ratio.

beam ratio values 3Figs. 51a2, 51b2, 61a2, 61b24 give both

ports characteristics of the classical J TC.

the similar shapes of the correlation peaks between

both ports of the TPJ TC with meff 5 0.1 and the

classical J TC.These peaks are typically broad and

not intense. Such a correlator sometimes fails to

detect objects in the presence of nonoverlapping non-

Gaussian noise, but performs well with clean inputs.

Its low discrimination is somewhat tolerable to rota-

tion and scaling of the twoinputs, as indicated by the

sizablecorrelation peak with thedissimilar disks.

Large beam ratio values produce peaks with a

much smaller full-width at half-maximum 1FWHM2,

as well as larger PNR and intensity values than the

correlation peak of the classical J TC or the phase-

only J TC.Both ports outperform both the classical

We used the SNR and

Small

We note

Fig. 4.

classical J TC correlation peak intensity, 1c2 the phase-only J TC correlation peak intensity, 1d2 the input plane with dissimilar disks, 1e2 the

classical J TC correlation peak intensity, 1f2 thephase-only J TC correlation peak intensity.

Computer simulations with similar 1top row2 and dissimilar 1bottom row2 disks:

1a2 the input plane with similar disks, 1b2 the

8160APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 8

and the phase-only J TC’s:

port shown in Fig. 51e2, with beam ratiomeff5 104, has

PNR and SNR values three times larger than the

corresponding peak produced by the phase-only J TC,

as shown in Fig. 41c2.

Such a correlator successfully detects an object in

the presence of nonoverlapping non-Gaussian noise

and performs excellently with clean inputs.

characterized by a very high discrimination ability,

the peak from the first

It is

but is very sensitive to input rotation and scaling.

This is demonstrated by the fact that the output with

the dissimilar disks is seriously degraded for both

ports.

A disadvantage in operating well past the transi-

tion point is the increased presence of higher har-

monic terms. The severity of the nonlinearity is

such that most of the signal is hard clipped.

output of a hard-clipping nonlinear processor can be

The

Fig. 5.

and the four-wave-mixing port 2 1bottom row2.

1a2 Port 1, beam ratioat 0.1; 1b2 port 2, beam ratioat 0.1; 1c2 port 1, beam ratioat 102; 1d2 port 2, beam ratioat 102; 1e2 port 1, beam ratioat 104;

1f2 port 2, beam ratioat 104.

TPJ TC computer simulations with thesimilar disks:intensity plots of correlation peaks of thetwo-beam-coupling port 1 1toprow2

The same columns correspond to the same beam ratio.The photorefractive gain was 8.

Fig. 6.

row2 and thefour-wave-mixing port 2 1bottom row2.

8.

1a2 Port 1, beam ratioat 0.1; 1b2 port 2, beam ratioat 0.1; 1c2 port 1, beam ratioat 102; 1d2 port 2, beam ratioat 102; 1e2 port 1, beam ratioat

104; 1f2 port 2, beam ratioat 104.

TPJ TC computer simulations with the dissimilar disks:intensity plots of correlation peaks of the two-beam-coupling port 1 1top

Thesamecolumns correspond tothesamebeam ratio.Thephotorefractivegain was

10 December 1995 @ Vol. 34, No. 35 @ APPLIED OPTICS8161

Page 9

expressed in an infinite series of harmonic terms,

whose weight is such that they can no longer be

neglected.As a result of light diffraction into the

higher harmonics the intensity at both ports is re-

duced.

Intermediate beam ratio values 3<exp1gL24 offer a

wide range of beam ratio values at which the two

ports of the TPJ TC operate in a different mode.

two-wave-mixing port operates closer to the flat sec-

tion, whereas thesecondport is closer tothequadratic.

In the case of the similar disks, the first port output

compared with that of the second port has a thinner

and more intense peak.

inputs, the first port output is drastically degraded,

whereas the second port output is less affected and

turns out to offer a brighter spot.

when Figs. 51c2 and 51d2 are compared with Figs. 61c2

and 61d2.

Thus the TPJ TC at intermediate beam ratios offers

a dual performance pattern, with port 2 fitted for the

low discrimination interclass identification and port 1

suitable for high discrimination intraclass identifica-

tion.At this range the performance of port 1 com-

pares in shape with that of the phase-only J TC, and

port 2 compares with the classical J TC.

compare Figs. 51c2 and 61c2 with Figs. 41c2 and 41f2 and

thePNR and theSNR values in Tables 3 and 4.

We conducted comparative simulations with the

median-thresholded14J TC and the TPJ TC with

meff5 102and gL 5 8. The numerical results for the

PNR and the SNR and the shape of the correlation

peaks were similar to that of the phase-only J TC.

This result was expected, given the similar perfor-

manceof thebinary J TC and thephase-only J TC with

similar inputs.

The

In the case of dissimilar

This is indicated

One can

6.

The experimental setup is shown in Fig. 1.

coherent illumination was provided by an argon laser

Experimental

The

at l 5 0.514 µm, split intotwobeams 11 and42 toallow

for different beam intensities.

on a barium titanate crystal with an angle of 20° and

a beam waist of 4 mm in diameter.

the reference signals were placed in the path of beam

1 andpassedthrough lens L11f15 16 cm2 sothat their

Fourier transforms appeared at the barium titanate

crystal. Exiting beam 1 was phase conjugated by a

second barium titanate crystal, placed in a self-

pumped configuration.This phase-conjugate beam

was used as backward pump beam 2.

Port 1, the two-wave-mixing output, was recovered

when beam 4 was allowed to propagate through

Fourier-transform lens L21f25 13.5 cm2.

four-wave-mixing output, was recoveredwhen a beam

splitter was placed in the path of beam 4 so that

diffracted phase-conjugate beam 3 was reflected off of

the beam splitter and passed through Fourier-

transform lens L31f35 13.5 cm2.

There were two different types of input pairs that

weexamined, namely a set of similar objects and a set

of dissimilar objects. Physically the objects were

disks cut intoa piece of sheet metal with a separation

of 1.5 mm. In the case in which the disks were

similar their radii were 0.3875 mm, versus that in

which they weredissimilar, wheredisk 1 had a radius

of 0.3875 mm and disk 2 had a radius of 0.175 mm.

The same analogies at the input planes have been

applied in thecomputer simulation.

We have covered two different beam intensity ra-

tios, one small and one large.

positiveand negativephotorefractivegains by revers-

ing the orientation of the crystal C axis.

positive gain, beam 1 pumps beam 4, and with a

negativegain, beam 4 pumps beam 1.

The beam intensities were 1i2 at the small beam

ratio, I1 5 2.51 3 104µW@mm2and I4 5 1.885 3

104µW@mm2; and 1ii2 at thelargebeam ratio, I15 2.51

3 104µW@mm2and I45 4.12 3 102µW@mm2.

intensity for beam 1 was measured after the passage

through the metal sheet.

definition given in Eq. 1112, we calculated that the

effective beam ratio for the first case 1small beam

ratio2 was meff5 197 and for the second 1large beam

ratio2 meff5 9007.Thesmall beam ratiosatisfies the

condition meff< exp1gL2 while the large beam ratio

satisfies meff: exp1gL2.

The experimental results are shown for the similar

images in Fig. 7 and for the dissimilar images in Fig.

8.On the left-hand side of the figures is the two-

wave-mixing output 1port 12, and on the right-hand

side of the figures is the four-wave-mixing output

1port 22.In both figures, cases A–D correspond tothe

large beam ratio, while cases E–H correspond to the

small beamratio.ThecasesA, B andG, H arefor the

positive gains, and C, D and E, F are for the negative

gains.

At the large beam ratio at either port and for both

positiveand negativegains, thecorrelator operates in

the saturation region, as we can see from Fig. 3.

The beams impinged

The scene and

Port 2, the

We applied both

With a

The

Using these data and our

As

Table 3. SNR and PNR for the TPJTC and the Classical and the

Phase-Only JTC’s with Similar Disks

J TCSNRPNR

Two-Port

meff5 0.1

meff5 102

meff5 104

Classical

Phase-only

Port 1

6.69

17.71

58.36

Port 2

6.05

7.64

29.48

Port 1

4.72

17.01

54.26

Port 2

3.87

5.82

25.12

6.64

17.12

3.84

18.92

Table 4. SNR and PNR for TPJTC and the Classical and the Phase-only

JTC’s with Dissimilar Disks

J TCSNRPNR

Two-Port

meff5 0.1

meff5 102

meff5 104

Classical

Phase-only

Port 1

9.16

6.16

8.07

Port 2

8.72

8.35

6.56

Port 1

3.09

4.49

6.88

Port 2

2.92

3.21

4.28

8.72

7.13

2.92

5.33

8162APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 10

Fig. 7.

gain; C, port 1, largebeam ratio, negativegain; D, port 2, largebeam ratio, negativegain; E, port 1, small beam ratio, negativegain; F, port

2, small beam ratio, negativegain; G, port 1, small beam ratio, positivegain; H, port 2, small beam ratio, positivegain.

TPJ TC experimental results with similar disks:A, port 1, large beam ratio, positive gain; B, port 2, large beam ratio, positive

10 December 1995 @ Vol. 34, No. 35 @ APPLIED OPTICS8163

Page 11

a result, the dissimilar disks 1Figs. 8A and 8B2 did not

display a distinct correlation peak but only a pair of

rings, while the similar disks 1Figs. 7A and 7B2

showed distinct, intense and sharp correlation peaks,

with the rings having a much smaller intensity than

the peaks.

the computer simulation results, in which one should

compare Figs. 51e2 and 51f2 with 7A and 7B and Figs.

61e2 and 61f2 with 8A and 8B.

lies in the fact that we do not actually observe full

These results are in full agreement with

The notable difference

Fig. 8.

gain; C, port 1, largebeam ratio, negativegain; D, port 2, largebeam ratio, negativegain; E, port 1, small beam ratio, negativegain; F, port

2, small beam ratio, negativegain; G, port 1, small beam ratio, positivegain; H, port 2, small beam ratio, positivegain.

TPJ TC experimental results with dissimilar disks:A, port 1, large beam ratio, positive gain; B, port 2, large beam ratio, positive

8164 APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 12

rings but rather crescents.

discrepancy can beunderstood if it is realized that the

resolution of the crystal has been assumed to be

isotropic and that the recording process is highly

nonlinear.

At the small beam ratiothe performance of the two

ports changes significantly on change of the beam

ratioand between the twoports.

Fig. 2, at a gain of 16 and beam ratio100, two-wave-

mixing port 1 is closetothelimiting region, whileport

2 is at the quadratic region.

port 1 at the dissimilar disks 1Fig. 8G2 did not display

any correlation peak, and we observed only the dc

spot.The absence of a correlation peak is indicated

by the computer simulation result shown in Fig. 61c2.

At the same time, port 2 showed very broad peaks,

which also agree with the simulation 3compare Fig.

8H with Fig. 61d24.This clearly demonstrates that

port 1 has high discrimination and port 2 has low

discrimination. The results with the similar disks

1Figs. 7G and 7H2 showed distinct, intense and sharp

correlation peaks, with the rings having a much

smaller intensity than the peaks for port 1 only, while

port 2 had correlation peaks similar to the cone.

Theseresults arein full agreement with thecomputer

simulation results, for which one should compare

Figs. 51e2 and 51f2 with 7G and 7H.

At a negative photorefractive gain, the small beam

ratio corresponds to the limiting region.

expect tosee results similar tothe ones obtained with

thelargebeam ratio.This is demonstrated in Fig. 7,

where the shapes of Figs. 7E and 7F are similar to

thoseof Figs. 7C and 7D.

The explanation of this

As we can see from

The results show that

Thus we

7.

A new all-optical TPJ TC based on four-wave mixing

in photorefractive media has been presented and

analyzed.It is, to our knowledge, the first J TC that

permits two simultaneously performing ports that

feature complementary discrimination ability, en-

abling both intraclass and interclass discrimination.

TheTPJ TC architecturemaintains parallel process-

ing at all planes, and bypasses the need for a serial

digital processor and additional CCD’s and SLM’s at

the spatial frequency plane.

with a soft-clipping quadratic-limiter nonlinearity.

The characteristics of this nonlinearity can be con-

trolled by the ratio between the incident intensities

parameter and the photorefractive gain 1gL2.

analysis has found that when the intensity ratio is

much smaller than exp1gL2 both ports behave like the

QD, whereas at intensity ratios much larger than

exp1gL2 both ports have a thresholding behavior.

the quadratic region both PNR and SNR values

indicate a similar performance to the classical J TC,

whereas in the limiting region the performance of

both ports in terms of PNR and SNR exceed that of

thephase-only J TC.

With approximately equal intensities of the signal

and the readout beams and positive photorefractive

gain we achieved limiting behavior in the two-wave-

Conclusions

The TPJ TC operates

Our

In

mixing port and quadratic behavior in the four-wave-

mixing port. At these settings the discrimination of

the first port was very high.

able when false warnings are to be eliminated.

second port had a classical J TC performance, with

broad but dim correlation peaks.

native scheme is preferable when the inputs have a

relativescaleand rotation difference.

This scheme is prefer-

The

This less discrimi-

References and Notes

1. J . Khoury, M. Cronin-Golomb, J . Kane, and C. Woods, ‘‘Two-

port photorefractive correlator,’’ in OSA Annual Meeting, Vol.

16 of 1993 OSA Technical Digest Series 1Optical Society of

America, Washington, D.C., 19932, paper ThI8, p. 174.

2. C. S. Weaver and J . W. Goodman, ‘‘Technique for optically

convolving twofunctions,’’Appl. Opt. 5, 1248–1249 119662.

3. B. J avidi and C. J . Kuo, ‘‘J oint transform image correlation

using a binary spatial light modulator at the Fourier plane,’’

Appl. Opt. 27, 663–665 119882.

4. J . Khoury, M. Cronin-Golomb, P. Gianino, and C. Woods,

‘‘Photorefractivetwo-beam-coupling nonlinear joint-transform

correlator,’’ J . Opt. Soc.Am. B 11, 2167–2174 119942.

5. J . Khoury, J . Kane, G. Asimellis, M. Cronin-Golomb, and C.

Woods, ‘‘All-optical nonlinear joint Fourier transform correla-

tor,’’Appl. Opt. 33, 8216–8225 119942.

6. M. Cronin-Golomb, B. Fischer, J . O. White, and A. Yariv,

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7. J . Feinberg, ‘‘Self-pumped continuous-wave conjugator using

internal reflections,’’ Opt. Lett. 7, 486–488 119822.

8. D.A. Gregory, ‘‘Real-time pattern recognition using a modified

LCTV in a coherent optical correlator,’’Appl. Opt. 25, 467–468

119862.

9. B. J avidi and J . L. Horner, ‘‘Single SLM joint transform

correlator,’’ Opt. Eng. 28, 1027–1032 119892.

10. K. H. Fielding and J . L. Horner, ‘‘1-f joint transform correla-

tor,’’ Opt. Eng. 29, 1081–1087 119902.

11. D. L. Flannery and J . L. Horner, ‘‘Fourier optical signal

processors,’’ Proc. IEEE 77, 1511–1527 119892.

12. B. J avidi and J . Wang, ‘‘Limitation of theclassical definition of

the correlation signal-to-noise ratioin optical pattern recogni-

tion with disjoint signal and scene noise,’’ Appl. Opt. 31,

6826–6829 119922.

13. B. J avidi, ‘‘Nonlinear joint power spectrum based optical

correlation,’’Appl. Opt. 28, 2358–2367 119892.

14. T. J . Grycewicz, ‘‘Fourier-plane windowing in the binary joint

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34, 3933–3941 119952.

15. F. Cheng, P. Andres, F. T. S. Yu, and D. Gregory, ‘‘Intensity

compensation fiber for joint transformcorrelator peak enhance-

ment,’’Appl. Opt. 32, 4357–4364 119932.

16. M. S.Alam, O. Perez, and M.A. Karim, ‘‘Preprocessed multiob-

ject joint transform correlator,’’ Appl. Opt. 32, 3102–3107

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17. J . O. White and A. Yariv, ‘‘Real time image processing via

four-wavemixing,’’Appl. Phys. Lett. 37, 5–7 119802.

18. M. G. Nicholson, I. R. Cooper, M. W. McCall, and C. R. Petts,

‘‘Simple computational model of image correlation by four-

wavemixing in photorefractivemedia,’’Appl. Opt. 26, 278–286

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19. M. R. Weiss and A. Siahmakoun, ‘‘Autocorrelation via two-

wavemixing in barium titanate,’’Opt. Eng. 30, 403–406 119912.

20. R. D. Martin and S. C. Schwartz, ‘‘Robust detection of a known

signal in nearly Gaussian noise,’’ IEEE Trans. Inf. Theory

IT-17, 50–56 119712.

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21. R. D. Martin and C. P. McGath, ‘‘Robust detection of stochastic

signals,’’ IEEE Trans. Inf. Theory IT-20, 37–41 119742.

22. J . Khoury, J . Fu, and M. Cronin-Golomb, ‘‘Quadratic process-

ing and nonlinear optical phase rectification in noise reduc-

tion,’’ J . Opt. Soc.Am. B 11, 10 119942.

23. P.Yeh, Introduction toPhotorefractiveNonlinear Optics,1Wiley,

New York, 19932, Chap. 4.

24. J . A. Khoury, G. Hussain, and R. W. Eason, ‘‘Contrast manipu-

lation and controllable spatial filtering via photorefractive

two-beam coupling,’’ Opt. Commun. 70, 272–276 119892.

25. T. Y. Chang, J . H. Hong, and P. Yeh, ‘‘Spatial amplification:

an image-processing technique using the selective amplifica-

tion of spatial frequencies,’’ Opt. Lett. 15, 743–745 119902.

26. J . A. K houry, C. Woods, and M. Cronin-Golomb, ‘‘Noise

reduction using adaptive spatial filtering in photorefractive

two-beam coupling,’’ Opt. Lett. 16, 747–749 119912.

27. The phase-only filter, introduced by J . L. Horner and P. D.

Gianino in ‘‘Phase-only matched filtering,’’ Appl. Opt. 23,

812–816 119842, has as output fromthefilter planetheabsolute

value of the reference spectral amplitude.

between thephase-only filter andthephase-only J TC is that in

theJ TC thespectral amplitudeis madeof thejoint spectrumof

thesceneand thereference.

The difference

28. Our definition for the SNR can be found in J . L. Horner and P.

D. Gianino, ‘‘Signal-dependent phase distortion in optical

correlators,’’ Appl. Opt. 26, 2484–2487 119872.

tion for the PNR was originally called peak-to-correlation

energy, and the PNR we used is equivalent to the peak-to-

correlation energy introduced in Eq. 182 of J . L. Horner,

‘‘Metrics for assessing pattern-recognition performance,’’Appl.

Opt. 31, 165–166 119922.

Our defini-

8166APPLIED OPTICS @ Vol. 34, No. 35 @ 10 December 1995

Page 14

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