1. INTRODUCTIONIn optical pattern recognition two essentially differenttypes of task are distinguished: detection of a target andestimation of its exact position.1–3 When correlation fil-ters are used, these problems can be solved in two steps.First, the detection is carried out by searching for corre-lation peaks in the filter output, and then coordinates ofthese peaks are taken as position estimations. The qual-ity of both procedures is limited by the presence of somerandom noise in an observed scene. The detection capa-bilities of correlation filters can be quantitatively ex-pressed in terms of probability of detection errors (falsealarms),1,4,5 signal-to-noise ratio,6 discrimination capa-bility,7 peak-to-output energy ratio,8,9 etc. Optimizationof these criteria leads to reducing false recognition errors,which can occur everywhere in the correlation plane. Af-ter the detection task has been solved, we still are facedwith small errors of target position estimations that aredue to distortion of the object by noise. The coordinateestimations lie in the vicinity of their actual values. Sys-tematic biases in the target location may be taken into ac-count and hence can easily be compensated. Thereforeaccuracy of the target location can be characterized onlyby means of the variance of measurement errors along co-ordinates. Solution of the variance minimization prob-lem depends on a mathematical model of an input scene.A number of solutions have been proposed when the inputscene contains a reference object corrupted by additive,stationary random noise. It has been shown1,2 that theclassical matched filter6 (CMF) is optimal with respect tolocation errors. However, it appears that the problemhas not been solved when the input scene contains both atarget embedded on a nonoverlapping background andadditive, random noise. This model arises frequently inpattern recognition. So solution of the detectionproblem8–11 has been a subject of intensive investigation.It is also important, perhaps, to exploit this model tominimize the variance of location errors because the loca-tion errors depend very much on a background neighbor-hood of the target.The purpose of this paper is to investigate the perfor-
mance of correlation filters in terms of accuracy of loca-
tion measurement when an input scene contains a noisytarget embedded in nonoverlapping noise. The presenta-tion is organized as follows. Section 2 introduces somepreliminary concepts and definitions; for a special case italso provides a derivation of an explicit filter formulaminimizing location errors. In Section 3, using computersimulation, we discuss and compare the error variance ofthe derived filter with the variances obtained by a phase-only filter12 (POF), a CMF, and a minimum mean squareerror filter10 (MMSE). Section 4 summarizes our conclu-sions.
2. LOCATION ERROR ESTIMATION ANDFILTER DESIGNLet us consider the nonoverlapping model of an inputscene. We use one-dimensional notation for simplicity.In this section integrals are to be taken between infinitelimits. Let t(x) denote a reference object, and let T(f )denote its Fourier transform:
T~f ! 5 E t~x !exp~2j2pfx !dx. (1)
We assume that an input scene s(x) contains the targett(x) having unknown coordinate x0 , nonoverlappingbackground b(x), and additive noise n(x):
s~x, x0! 5 t~x 2 x0! 1 b~x, x0! 1 n~x !. (2)
The coordinate x0 is considered as a random variable withan arbitrary probability-density function. The nonover-lapping background signal b(x, x0) is regarded as a prod-uct of a realization b(x) from a stationary random process[with expected value mb and power spectral density B(f )]and an inverse support function of the target w(x) [withFourier transformW(f )] defined as zero within the targetarea and unity elsewhere:
b~x, x0! 5 b~x !w~x 2 x0!. (3)
Throughout this paper we will use the same notation for arandom process and its realization. An example of thesignal b(x) is a spatial homogeneous texture that can bedescribed by a stationary random process. The noise
1996 Optical Society of America
1654 J. Opt. Soc. Am. A/Vol. 13, No. 8 /August 1996 V. Kober and J. Campos
n(x) is assumed to be a realization from a zero-mean, sta-tionary process with power spectral density N(f ). Thismodel can describe well the sensor’s noise. It is also as-sumed that the stationary processes and the random tar-get location x0 are statistically independent of each other.To be specific, we base our analysis on a VanderLugt
correlator.6 Let h(x) and H(f ) be the real impulse re-sponse and the transfer function of the filter to be de-signed, respectively. The correlation output signal canbe written as follows:
c~x, x0! 5 ct~x 2 x0! 1 cb~x, x0! 1 cn~x !, (4)
where
ct~x ! 5 E t~j!h~x 2 j!dj (5)
is the deterministic correlation output of the referencesignal;
cb~x, x0! 5 E b~j, x0!h~x 2 j!dj, (6)
cn~x ! 5 E n~j!h~x 2 j!dj (7)
are the random correlation outputs of the random signalsb(x, x0) and n(x), respectively. We note that the latteris a stationary random process with zero mean and withpower spectral density Nc(f ) 5 N(f )uH(f )u2.It is obvious from the model of the input scene in Eq. (2)
that detection and localization problems must be solvedby matching with the reference object t(x) as well as withthe weighted inverse support function mbw(x). We nowsuppose that the signals ct(x 2 x0) and E[cb(x, x0)]P C2 [i.e., continuous functions with derivatives up to or-der 2; E(•) is the expected value], the maximum ofE[c(x, x0)], and the random correlation intensity peakuc(x, x0)u
2 lie in the close vicinity of the coordinate x0 .Thus we can represent the output of a linear system inEq. (4) as a sum of deterministic and zero-mean, randomparts:
c~x, x0! 5 E@c~x, x0!# 1 cb0~x, x0! 1 cn~x !, (8)
where c b0(x, x0) 5 cb(x, x0) 2 E[cb(x, x0)] is a zero-
mean, random process.It is clear that c b
0(x, x0) can be regarded as the corre-lation output of the zero-mean, stationary random processb0(x) 5 b(x) 2 mb . Let us use a Taylor-series expan-sion of the signal E[c(x, x0)] at the coordinate x0 :
E@c(x, x0)#9ux5x0denote the first- and second-order de-
rivatives of the expected value of the filter output c(x, x0)taken at the coordinate x0 . The second derivative in Eq.(9) may be computed as follows:
d2 [ 2 E@c~x, x0!#9ux5x05 4p2E f 2H~f !
3 @T~f ! 1 mbW~f !#df. (10)
We also use the symbol d1 to denote the first derivative ofE[c(x, x0)] evaluated at the coordinate x0 :
d1 [ E@c~x, x0!#8ux5x05 2pjE fH~f !
3 @T~f ! 1 mbW~f !#df. (11)
It is clear that if the maximum of E[c(x, x0)] lies at thecoordinate x0 , then d1 is equal to zero. The location D0 ofthe intensity maximum of c(x, x0) can be found by solv-ing the following equation:
]c~x, x0!
]x Ux5D01x0
5 0. (12)
Using Eqs. (8)–(12), we find an expression for D0:
D0 5d1 1 rb 1 rn
d2, (13)
where
rb []cb
0~x, x0!
]x Ux5D01x0
, rn []cn~x !
]x Ux5D01x0
are the derivatives of the zero-mean random processesc b0(x, x0) and cn(x) with respect to x evaluated at
x 5 D0 1 x0 . The derivative is a linear operation, andthus rb and rn are also zero-mean random processes.Strictly speaking, since the random processes rb and rndepend on the location D0, the latter should be defined asthe solution to Eq. (13). However, by the assumptionthat the values of D0 are small, the mean and the varianceof D0 can be approximated by the mean and the varianceof the right-hand side of Eq. (13) evaluated in the vicinityof x0 . The expected value of the location D0 is computedas
E~D0! 5d1
d25
jE fH~f !@T~f ! 1 mbW~f !#df
2pE f 2H~f !@T~f ! 1 mbW~f !#df.
(14)
If the spectrum H(f )[T(f ) 1 mbW(f )] is an even func-tion, i.e., H(f )@T(f ) 1 mbW(f )# 5 H(2f )@T(2f )1mbW(2f )], then E(D0) is equal to zero (unbiased positionestimation). This condition is satisfied when the fre-quency response H(f ) 5 @T(f ) 1 mbW(f )#*A(f ). HereA(f ) is an even function, and the asterisk denotes thecomplex conjugate.As was stated above, the accuracy of target location is
defined as the variance of measurement errors along co-ordinates. The variance of D0 can be expressed as
Var~D0! 5Var~rb! 1 Var~rn!
d22
, (15)
where Var(rb) and Var(rn) are the variances of rb and rn ,respectively.
V. Kober and J. Campos Vol. 13, No. 8 /August 1996 /J. Opt. Soc. Am. A 1655
Since the derivative preserves the stationary properties ofa random process,13 the latter term, rn , is a stationaryrandom process with zero mean and with variance
Var~rn! 5 4p2E f 2N~f !uH~f !u2 df. (16)
In Appendix A we show that the expression for the vari-ance of the nonstationary random process rb is given by
Var~rb! 5 4p2E E E f1f2H~f1!H* ~f2!B0~f !
3 W~f1 2 f !W* ~f2 2 f !exp@ j2pD0
3 ~f1 2 f2!#dfdf1df2 , (17)
where B0(f ) is the power spectrum of the zero-mean, sta-tionary random process b0(x). If the intensity maximumof c(x, x0) is in the vicinity of x0 , then the variance in Eq.(17) is approximated by
Var~rb! ' 4p2E E E f1f2H~f1!H* ~f2!B0~f !
3 W~f1 2 f !W* ~f2 2 f !dfdf1df2 . (18)
It is evident that if the background noise is a zero-mean,overlapping random process, then W(f ) 5 d(f ) is theDirac delta function, and Eq. (18) can be further simpli-fied to
Var~rb! ' 4p2E f 2B0~f !uH~f !u2 df. (19)
We note that this equation is the same as Eq. (16) ob-tained for the random process rn . Substituting Eq. (16)and relation (19) into Eq. (15), we arrive at the known re-sults for the case of the CMF when the target is in thepresence of additive, stationary noise.1,2,14
Using Eqs. (10), (15), and (16) and relation (18), we canobtain an estimation for the variance of location errors asfollows:
Var~D0! 'E E E @f1f2H~f1!H* ~f2!B0~f !W~f1 2 f !W* ~f2 2 f !#dfdf1df2 1 E f 2N~f !uH~f !u2 df
4p2U E f 2H~f !@T~f ! 1 mbW~f !#dfU2. (20)
Next it is useful to make some simplifying assump-tions. Let us suppose that the characteristic function ofthe random variable D0 on the right-hand side of Eq. (17)is close to the Dirac delta function, say
E@exp~ j2pfD0!# ' ad~f !, (21)
where a is a normalizing constant. This situation aris-esif D0 has approximately the uniform distribution with awide range. Then, by applying the statistical averagingto the right-hand side of Eq. (17) with respect to D0, weobtain the variance of rb and consequently the variance ofD0 from Eqs. (10) and (15)–(17) as follows:
Var~rb! ' 4p2aE f 2uH~f !u2@B0~f !+uW~f !u2#df, (22)
Var~D0! 'E f 2uH~f !u2$@aB0~f !+uW~f !u2# 1 N~f !%df
4p2U E f 2H~f !@T~f ! 1 mbW~f !#dfU2,
(23)
where the symbol + denotes the convolution operation.Use of the Schwarz inequality in the last relation leads
to the following expression for a filter that minimizes thevariance of location errors:
Hopt~f ! 5T* ~f ! 1 mbW* ~f !
@aB0~f !+uW~f !u2# 1 N~f !. (24)
The above filter is the generalized matched filter9 (GMF)derived for the input scene in Eq. (2). The filter yieldsthe following variance of location errors:
Varopt~D0!
5 H 4p2E F f 2uT~f ! 1 mbW~f !u2
@aB0~f !+uW~f !u2# 1 N~f !dfG J 21
. (25)
As was pointed out by a reviewer, the same formulas asthose in Eqs. (24) and (25) can be obtained in the follow-ing way (see Appendix B): (1) Reduce the nonoverlap-ping model of the input scene in Eq. (2) to the classicaladditive model by applying an average over the randomvariable x0 . (2) Use the known formulas for the additivemodel1,2,14 to compute the variance of location errors andthe transfer function of a filter minimizing the variance.In Section 3 we discuss and illustrate by computer simu-lation the difference between location accuracy estima-tions given in relation (20) and Eq. (25).It can be seen from Eq. (25) that spectral coefficients of
the target and the weighted inverse support function atlower frequencies give a larger contribution to location er-rors than those at higher frequencies. It is easy to showthat if the background noise is a zero-mean, overlappingrandom process, and a is quite small, then Eqs. (24) and
(25) are simplified to expressions known for the classicalmatched filter1,2,14:
Hopt~f ! 'T* ~f !
B0~f ! 1 N~f !, (26)
Varopt~D0! ' H 4p2E F f 2uT~f !u2
B0~f ! 1 N~f !dfG J 21
. (27)
The above results can be easily extended to the two-dimensional case by using either the partial derivativesalong both coordinates or the directional derivatives in
1656 J. Opt. Soc. Am. A/Vol. 13, No. 8 /August 1996 V. Kober and J. Campos
Eqs. (9)–(13). So, for the case of directional derivatives,the expression for Varopt(D
0) in Eq. (25) becomes (see Ap-pendix C)
Varopt~D0! 5 X4p2E E H ~fx cos f 1 fy sin f!2uT~fx , fy! 1 mbW~fx , fy!u2
where fx and fy are the coordinates in the Fourier plane,f is a direction in the correlation plane chosen to mini-mize the error variance, T(fx , fy) and W(fx , fy) are two-dimensional spectra of the target and the inverse supportfunction, respectively, and N(fx , fy) and B0(fx , fy) aretwo-dimensional power spectral densities of additivenoise and zero-mean nonoverlapping noise, respectively.Here D0 is an estimation of the intensity peak location ofthe filter output in the direction at angle f.Next we consider a relationship between two methods
of the variance estimation in Eq. (17) with use of theGMF. The first method is based on the assumption thatlocation errors are small. This leads to the expression forthe variance in relation (18). On the other hand, the sec-ond method of the variance estimation is based on the as-sumption that location errors are uniformly distributed ina wide range. Strictly speaking, the latter assumption isnot valid, because when the intensity peak of c(x, x0) isfar from x0 , the Taylor-series expansion of the signalE[c(x, x0)] in Eq. (9) requires the terms with powershigher than 2, and, moreover, the random variable D0
should be computed as the solution to Eq. (13). However,it very much simplifies the problem, and the explicit for-mula for the filter that minimizes location errors and theexpression for the minimal variance are easily derived.With the second method we arrive at Eqs. (24), (25), and(28). Finally, we consider the relation between the vari-ances of location errors in relation (20) and Eq. (25).From now on, we take into account the frequency band-width of a linear system. So let F be the frequency band-width of the filter H(f ). Then the variance of the non-stationary random process rb in relation (18) can berewritten as follows:
Var~rb! ' 4p2E E EFf1f2H~f1!H* ~f2!B0~f !W~f1 2 f !
3 W* ~f2 2 f !dfdf1df2
5 4p2EFB0~f !U E
Ff1H~f1!W~f1 2 f !df1U2 df.
(29)
Using the Schwarz inequality, we have
Var~rb! < 4p2EFB0~f !F E
Ff1
2uH~f1!u2
3 uW~f1 2 f !u2df1EFdf1Gdf
5 4p2FEFf 2uH~f !u2@B0~f !+uW~f !u2#df. (30)
The latter term is rather similar to the expression in re-lation (22), which leads to the variance of location errorsin Eq. (25) by using the GMF in Eq. (24). The only
(slight) difference is the normalizing constant F. It isknown from the bandwidth theorem of signal theory thatthe bandwidth–duration product of a signal cannot beless than a certain minimum value.15 This fact preventsarbitrary specification of signals on the time–frequencyplane. Let T and F be widths of the impulse responseand the transfer function of the filter. Several definitionsof the widths are used in signal processing, such asequivalent widths and autocorrelation widths.15 For ex-ample, a convenient measure of the width of a signal g(x)is called the equivalent width and defined as the area ofthe signal divided by its central ordinate,
T 5
E2`
1`
g~x !dx
g~0 !. (31)
The physical meaning of the equivalent width of a func-tion is the following: it is the width of a rectangle whoseheight is equal to the central ordinate and whose area isthe same as that of the function. The correspondingmeasure of the width of its Fourier transform is
F 5
E2`
1`
G~f !df
G~0 !. (32)
According to the reciprocal property of equivalent widths,it follows that the product of T and F is equal to unity.Returning to relation (30), we can substitute the band-width F of the filter by T21. Here T is the width of theimpulse response of the filter. Since the normalizing con-stant a in relation (21) is approximately equal to T21, itfollows that the variance estimations of rb in Eq. (17) andrelation (18) are bounded above by the expression in rela-tion (22). Hence the variance in relation (20) is alsobounded above by the variance in Eq. (25). Equivalentwidth is not always the best measure of the width. Forexample, if g(0) 5 0 or G(0) 5 0, then the equivalentwidths in Eqs. (31) and (32) do not exist. However, otherwidth definitions and corresponding bandwidth–durationratios can be used as well.15
In Section 3, with the help of computer simulation, weillustrate the relationship between the variance estima-tions in relation (20) and Eq. (25) by using the GMF. Wewill see a similar behavior of both variance estimations.
3. DISCUSSION AND COMPUTERSIMULATIONSIn this section we analyze the performance of the GMF,the POF, the CMF, and the MMSE filter in terms of loca-tion accuracy for spatially nonoverlapping target and in-put scene noise in Eq. (2). The CMF is chosen because it
V. Kober and J. Campos Vol. 13, No. 8 /August 1996 /J. Opt. Soc. Am. A 1657
is optimal for the case of an overlapping model of additivenoise. Recently, the POF has been compared with theCMF for the overlapping model with respect to locationerrors.2 We test the performance of the POF for the non-overlapping model. It was shown that, for spatially non-overlapping target and background noise, the MMSE fil-ter provides a good detection performance,10 and,moreover, it coincides with the filter optimizing the peak-to-output energy ratio.8,9
In Section 2 it is assumed that location errors aresmall. With the help of computer simulation we illus-trate how small they are. Test target and correspondinginverse support function are one-dimensional signals,shown in Fig. 1. The target has a mean value of 0.51.The size of the input signal is 256 samples. The inversesupport function w(x) is equal to zero in the intervalx1 < w(x) < x2 , where the target function is nonzero.We use the model of white noise for nonoverlapping back-ground and for overlapping scene noise. Also, colorednoise with the exponential correlation function is consid-ered as spatially nonoverlapping noise. In order to com-pute a spectrum, we use a one-dimensional fast Fouriertransform.In this section we deal with discontinuous functions.
There is some mathematical convenience to using theStieltjes integral, but we prefer a mathematical represen-tation by using the Dirac delta function d (x), because
Var~D0! '
VarbE UE f1H~f1!W~f1 2 f !df1U2 df 1 VarnE f 2uH~f !2 df
4p2U E f 2H~f !@T~f ! 1 mbW~f !#dfU2 . (35)
sometimes it has a simple physical interpretation. Letus define the generalized sifting property of d (x) for a dis-continuous function g(x) as follows15:
Fig. 1. Test reference object t(x), inverse support function w(x),and first-order derivative w(x) of inverse support function.
d~x !+g~x ! 5 limD→0
g~x 1 D! 1 g~x 2 D!
2
5g~x 1 0 ! 1 g~x 2 0 !
2, (33)
where g(x 2 0) and g(x 1 0) are limits of g(x) on theleft-hand and right-hand sides, respectively. It is as-sumed that the limits exist. In a similar way we can de-fine generalized differentiation of g(x):
g8~x ! 5 limD→0
g~x 1 D/2! 2 g~x 2 D/2!
D. (34)
It is easy to show that if g(x) is a continuous function,then the above operations produce the same results asthose with regular operations. On the other hand, atpoints of discontinuity of the function the operations inEqs. (33) and (34) take into account a symmetrical neigh-borhood and thus yield unbiased results.
A. Location Errors for White NonoverlappingBackground and White Input Scene NoiseFirst, let us consider the variance of location errors in re-lation (20) when nonoverlapping and overlapping noisesare white, with the variances Varb and Varn , respec-tively; that is,
For a given target and correlation filter the variance de-pends on three parameters of the input scene model inEq. (2): Varb , Varn , and mb . Now we investigate thevariance behavior as a function of mb for various correla-tion filters. Let Varb and Varn be fixed and nonzero.First, we determine the frequency response of the CMF asfollows:
HCMF~f ! 5 T* ~f !/~Varb 1 Varn!. (36)
For the case of the nonoverlapping white-noise model theanalysis of location errors will be done by using severalsimple theorems. So we state the following theorem forthe CMF:
Theorem 1. Suppose that s(x, x0) in Eq. (2) is areal, positive function, and suppose also that both noisecomponents in Eq. (2) are white; if s(x, x0) is the input toa linear system with the impulse response of the CMF,then (1) the variance of location errors in relation (35) willdiverge at a real, positive value mb
` of nonoverlappingnoise mean, where
mb` 5 2E f 2uT~f !u2 df YE f 2T* ~f !W~f !df, (37)
1658 J. Opt. Soc. Am. A/Vol. 13, No. 8 /August 1996 V. Kober and J. Campos
and (2) the expected value of the system output,E[c(x, x0)], has no local maxima when the nonoverlap-ping noise mean is equal to mb
`.
Proof. The first part of the theorem has a simpleproof. After Eq. (37) is substituted into relation (35), itfollows that for fixed values of the variances Varb andVarn the variance of location errors as a function of mbconverges to infinity at mb
`. Now we show that mb` is a
real, positive value. According to Parseval’s theorem, thedenominator in Eq. (37) is the integral of a product be-tween the first-order derivatives of the target and the in-verse support function. Let us refer to the first deriva-tive of the inverse support function as w8(x). It is easilyshown that w8(x) can be represented as a sum of twoDirac delta functions. Figure 1 demonstrates w8(x) forthe test signal. The inverse support function w(x) isequal to zero in the interval x1 < w(x) < x2 , where thetarget function is positive. With the aid of Eq. (34) thedelta functions are determined as follows:
w8~x1! 5 limD→0
w~x1 1 D/2! 2 w~x1 2 D/2!
D5 2d~x1!,
w8~x2! 5 limD→0
w~x2 1 D/2! 2 w~x2 2 D/2!
D5 d~x2!.
(38)
The first derivative of the target, denoted as t8(x), has thesame interval. It is obvious from the model of the inputscene in Eq. (2) that t8(x1) is positive but that t8(x2) isnegative. Taking into account the sifting property in Eq.(33), we arrive at the conclusion that the denominator inEq. (37) is negative. In contrast, since the numerator inEq. (37) is the energy of t8(x), then it is always positive.Therefore mb
` is a real, positive value.Next we show that the condition in Eq. (37) leads to the
second statement of the theorem. If the nonoverlappingnoise mean is equal to mb
`, then the second-order deriva-tive d2 in Eq. (10) is equal to zero. Hence the expectedvalue E[c(x0 1 D, x0)] in Eq. (9) is a linear function ver-sus D. This means that E[c(x, x0)] has no local maxima.
It is important to take into account the sign of mb`.
Since signals in optical systems are positive, it followsthat a situation in which it is theoretically impossible tolocalize a target by using the CMF is realistic. Slightvariation of the mean mb from the critical value mb
` re-duces very much the variation errors. This fact is illus-trated by computer simulation in Fig. 2(a) with use of Eq.(37) and the test signal in Fig. 1. In this case the valuesof the noise variances are Varb 5 1 and Varn 5 1. Forthe case of the CMF the value of mb
` is equal to 0.48. Itis interesting to note that the variance of location errorsis a symmetrical function. In Section 2, to make theFourier-series expansion of E[c(x, x0)] in Eq. (9), we as-sumed that the maximum of E[c(x, x0)] lies in the vicin-ity of the coordinate x0 and that the localization proce-dure is carried out by taking the coordinate of themaximum in the correlation plane as a target coordinateestimation. This is true while mb < mb
`. However,when mb . mb
`, both of the above operations should be
performed with the minimum values; i.e., for the Fourier-series expansion in Eq. (9) we suppose that the minimumof E[c(x, x0)] lies in the vicinity of x0 , and a target coor-dinate estimation is computed as the coordinate of theminimum in the correlation plane. In Section 2 we de-rived all formulas without checking out the sufficient con-dition of the maximum of a function. Therefore they arevalid for the case of mb . mb
`, and we observe the sym-metrical behavior of the variance of location errors as afunction of mb .We now formulate conditions when the CMF yields an
unbiased position estimation.
Theorem 2. The CMF provides an unbiased posi-tion estimation if at least one of the following conditionsis satisfied: (1) the nonoverlapping noise mean is equalto zero; (2) the target function has the same boundary val-ues, i.e., t(x1) is equal to t(x2).
Fig. 2. Standard deviation (St.D) of location errors (in percent)as a function of nonoverlapping noise mean for the set of corre-lation filters (CMF, POF, GMF, MMSE, GMF–OPT). Standarddeviations of white overlapping and nonoverlapping noises areequal to unity: (a) nonoverlapping noise is white, (b) nonover-lapping noise is colored with correlation coefficient of 0.95.
V. Kober and J. Campos Vol. 13, No. 8 /August 1996 /J. Opt. Soc. Am. A 1659
Proof. Let us evaluate the numerator d1 of the meanof the location errors in Eq. (14) by using the CMF. Wecan rewrite the first derivative d1 , defined in Eq. (11), asfollows:
d1 5 2pjF E fuT~f !u2 df 1 mbE f T* ~f !W~f !df G .(39)
Obviously, the former addendum in Eq. (39) is equal tozero, because uT(f )u2 is an even function. Using Parse-val’s theorem, we can represent the latter term in the spa-tial domain as the integral of a product of w8(x) and thetarget t(x). It was shown above that w8(x) is a sum oftwo Dirac delta functions located at the positions x1 andx2 . Therefore d1 may be computed as a product betweenmb and the difference of target values taken at x2 and x1 .This product is equal to zero when either mb 5 0 ort(x2) 2 t(x1) 5 0.
The fact that mb 5 0 leads to unbiased position esti-
mation is the known result for the CMF. On the otherhand, for the nonoverlapping model a symmetrical behav-ior of the target function at the boundary points also leadsto unbiased position estimation.The theorems can be illustrated by a simple example.
Let a test target be a rectangular unity impulse. Thenthe inverse support function is the complementary unityimpulse. These functions have the same first derivativesbut with opposite signs. It follows from Eq. (37) thatmb
`
is equal to unity. We see that for arbitrary values ofVarb and Varn the expected value of the linear systemoutput is a constant (unity) everywhere in the correlationplane, i.e., E[c(x, x0)] 5 1, and hence it is impossible tolocalize the target.It follows from relation (35) that the variance of loca-
tion errors is linearly proportional to both Varb and Varn .Figure 3 illustrates the standard deviation of location er-rors normalized to the size of the test signal in Fig. 1 ver-sus the standard deviation of additive, overlapping noisewhile the nonoverlapping background is various constantbiases. Here the overlapping noise is zero mean. It can
Fig. 3. Standard deviation of location errors (in percent) as a function of standard deviation of additive white noise for the set of cor-relation filters (CMF, POF, GMF, MMSE, GMF–OPT). The nonoverlapping background is a constant bias mb with values (a) mb 5 0,(b) mb 5 0.1, (c) mb 5 0.3, and (d) mb 5 0.5.
1660 J. Opt. Soc. Am. A/Vol. 13, No. 8 /August 1996 V. Kober and J. Campos
be seen that the CMF is optimal for additive noise in Fig.3(a) and that it loses its optimality when the backgroundbias goes to the critical value mb
` in Figs. 3(b)–3(d). InFig. 4 the ratio between the normalized standard devia-tion of location errors and the standard deviation of non-overlapping white noise is shown as a function of the ratiobetween the standard deviations of overlapping white andnonoverlapping white noises. The mean of nonoverlap-ping noise is varied. It may be seen from Figs. 2–4 thatthe performance of the CMF in terms of location errorsdepends very much on the expected value of nonoverlap-ping noise and is linearly proportional to the variances ofboth noises.It is easy to extend the above theorems to other corre-
lation filters that do not take into account explicitly or im-plicitly the mean of nonoverlapping noise. We now con-sider the performance of the POF with the frequencyresponse defined as12
HPOF~f ! 5 T* ~f !/uT~f !u. (40)
In a similar manner as that for the CMF, by inserting Eq.(40) into relation (35), we can find the critical value mb
`
for the POF:
mb` 5 2E f 2uT~f !udf YE f 2T* ~f !W~f !/uT~f !udf.
(41)
For the test signal used in computer simulation this valueis equal to 0.42. It can also be shown that the POF pro-vides an unbiased position estimation if either the meanof nonoverlapping noise is equal to zero or the whitenedtarget function tw(x), i.e., with the spectrum T(f )/uT(f )u,has the same values at points x1 and x2 , that is,tw(x1) 5 tw(x2). The influence of the noise variances onthe variance of location errors is the same as that for theCMF. A comparison of location errors for the POF andthe CMF is presented in Figs. 2–4. We see that the CMFis essentially better than the POF when the mean of non-
Fig. 4. Ratio between standard deviation of location errors and standard deviation of nonoverlapping white noise (in percent) versusratio between standard deviations of additive white noise and nonoverlapping white noise for the set of correlation filters (CMF, POF,GMF, MMSE, GMF–OPT). The mean of the nonoverlapping background, mb , is (a) mb 5 0, (b) mb 5 0.1, (c) mb 5 0.3, and (d)mb 5 0.5.
V. Kober and J. Campos Vol. 13, No. 8 /August 1996 /J. Opt. Soc. Am. A 1661
overlapping noise is far from its critical value mb`. Both
filters fail to localize the target at their values of mb`.
Next we consider correlation filters whose frequency re-sponses are complex conjugate to T(f ) 1 mbW(f ).These are the GMF and the MMSE filter. As was shownin the Section 2, these filters have an unbiased positionestimation in Eq. (14). To analyze the behavior of thevariance of location errors as a function of mb , we statethe following theorem for the GMF:
Theorem 3. Suppose that s(x, x0) in Eq. (2) is areal, positive function, and suppose also that both noisecomponents in Eq. (2) are white; if s(x, x0) is the input toa linear system with the impulse response of the GMF,then (1) the variance of location errors in relation (35) willdiverge at a real, positive value mb
` of nonoverlappingnoise mean if and only if the target function is a rectan-gular impulse mb
` is determined as the height of the tar-get impulse and (2) the expected value of the system out-put, E[c(x, x0)], is a constant when the nonoverlappingnoise mean is equal to mb
`.
Proof. Substituting the frequency response of the GMFin Eq. (24) into relation (35), we see that the denominatorbecomes
d2 5 4p2E f 2uT~f ! 1 mbW~f !u2 df. (42)
It is clear that d2 > 0. In contrast, by applying Parse-val’s theorem, we can represent d2 in the spatial domainas follows:
d2 5 E ut8~x ! 1 mbw8~x !u2 dx. (43)
Solving the equation
t8~x ! 1 mb`w8~x ! 5 0 (44)
with respect tomb`, we see that the equality in Eq. (44) is
valid at all x if and only if the target function is a rectan-gular impulse. In this case mb
` is determined as theheight of the target impulse. Since the input signal is areal, positive value, then mb
` is also a real, positivevalue.Since the second-order derivative d2 in Eq. (42) is equal
to zero when the mean of nonoverlapping noise is equal tomb
` and since the first-order derivative d1 in Eq. (11) isalways equal to zero for the GMF, then from Eq. (9) it fol-lows that the expected value E[c(x0 1 D, x0)] is a con-stant value. This means that it is impossible to localizethe target.
The transfer function of the MMSE filter for the inputmodel of signal in Eq. (2) can be written as10
HMMSE~f !
5T* ~f ! 1 mbW* ~f !
uT~f ! 1 mbW~f !u2 1 @aB0~f !+uW~f !u2# 1 N~f !.
(45)
In a similar way, by inserting Eq. (45) into relation (35)and taking into account that both noises are white, we
can show that Theorem 3 is also valid for the MMSE fil-ter.Since the test signal in Fig. 1 is not a rectangular im-
pulse, the GMF and the MMSE filter have no criticalpoints mb
`. On the other hand, since the target functionis quite similar to a rectangular impulse, the filters pro-duce maxima in Fig. 2(a) at mb 5 0.34 (GMF) andmb 5 0.30 (MMSE). The variance of location errors forthe GMF is linearly proportional to Varb and Varn . Incontrast, for the case of the MMSE filter the influence ofnoise variances on the variance of location errors is morecomplicated. Figures 2–4 illustrate computer simulationresults for all used filters in terms of location errors. Wealso show on the test signal how well the variance of lo-cation errors in relation (35) can be evaluated by usingEq. (25). This estimation is called GMF–OPT. We seethat the plots for the GMF are bounded above by curvesfor the GMF–OPT estimation. For the case of the MMSEfilter we show curves with Varb 5 1. Comparing theCMF, the POF, the GMF, and the MMSE filter, we notethat the GMF has the best performance in terms of loca-tion accuracy. The GMF coincides with the CMF for theadditive model of input scene in Fig. 3a, and it is essen-tially better than other filters for the nonoverlappingmodel in Figs. 3(b)–3(d) and 4.Finally, we note that the GMF and the MMSE filter
provide better results in terms of location accuracy incomparison with those of the CMF and the POF becausethe GMF and the MMSE filter match to t(x) 1 w(x)mb .In contrast, the CMF and the POF match only to t(x).
B. Location Errors for Colored NonoverlappingBackground and White Input Scene NoiseTo investigate the performance of correlation filters forthe nonoverlapping colored-noise model, we use the col-ored noise as a realization of a zero-mean, stationary pro-cess b0(x) with an exponential correlation function; thatis,
R~t! 5 Varb exp~2lutu!, (46)
where l is a correlation parameter.To obtain a simple physical interpretation of the noise
model, let us discretize the correlation function R(t) ac-cording to a sampling rate 1/T, where T is the samplinginterval. Then the parameter r 5 exp(2lT) can be con-sidered as the correlation coefficient13 of random valuesb0(kT) and b0[(k 1 1)T], that is, always less than unity.Here k is an integer index of the discretized sequenceb0(kT). Thus, for r → 0, we arrive at the white-noisemodel investigated above. On the other hand, the condi-tion r → 1 leads to a strong correlated background signal.The spectral density function of the colored noise b0(x)
is computed as follows:
B0~f ! 5 Varb2l
l2 1 ~2pf !2. (47)
Finally, the variance of location errors in relation (20)takes the following form:
1662 J. Opt. Soc. Am. A/Vol. 13, No. 8 /August 1996 V. Kober and J. Campos
Var~D0! '
VarbE 2l@l2 1 ~2pf !2#21U E f1H~f1!W~f1 2 f !df1U2 df 1 VarnE f 2uH~f !u2 df
4p2U E f 2H~f !@T~f ! 1 mbW~f !#dfU2. (48)
It can be seen that the variance of location errors dependson four parameters of the input model (Varn , Varb , mb ,and r). Comparing relations (35) and (48), we concludethat the above results describing the behavior of Var(D0)versus mb can easily be extended to the colored-noisemodel. Let us summarize some of them for the case ofcolored nonoverlapping noise:
1. The CMF determined in relation (26) fails to localizethe reference object when the mean of nonoverlappingnoise is
mb` 5
2 E f 2uT~f !u2@l2 1 ~2pf !2# Y$Varb2l 1 Varn@l2 1 ~2pf !2#%df
E f 2T* ~f !W~f !@l2 1 ~2pf !2# Y$Varb2l 1 Varn@l2 1 ~2pf !2#%df. (49)
2. For the case of the POF the critical value mb is thesame as that of the white-noise model, i.e., determinedfrom Eq. (41).3. The GMF and the MMSE filter cannot localize the
reference object if the target function is a rectangular im-pulse, and the critical value mb
` is equal to the height ofthe target impulse.
Figures 2(a) and 2(b) illustrate the variance of locationerrors as a function of mb for two noise models. The cor-relation coefficient r is chosen as 0.95, and the variancesof overlapping and nonoverlapping noises are equal tounity. It is clear that the variance of location errors hasa similar behavior when the nonoverlapping noise iswhite or colored. We see that each filter has an extremevalue (infinity or maximum) at a critical point of mb .Table 1 shows these critical values and corresponding val-ues of the standard deviation of location errors for theCMF, POF, GMF, MMSE, and GMF–OPT [estimation byEq. (25)] cases, while the correlation coefficient r is var-ied. Since the transfer function of the POF does not de-pend on the correlation coefficient, the POF yields a con-stant of mb . In contrast, the position of the extremevalue for other filters changes when the correlation coef-ficient is varied. It is interesting to note that increasingthe correlation coefficient leads to better approximation ofthe variance in relation (48) by the variance in Eq. (25).At the same time nonoverlapping colored noise withhigher correlation coefficient has the effect of increasingthe variance of location errors.It follows from relation (48) that the variance of loca-
tion errors is linearly proportional to both Varb and Varnonly for the case of the POF. Other filters have nonlineardependencies on the noise variances. Figure 5 illustratesthe ratio between the normalized standard deviation of lo-cation errors and the standard deviation of nonoverlap-ping colored noise as a function of the ratio between the
standard deviations of overlapping white and nonoverlap-ping colored noises. The mean and the correlation coef-ficient of nonoverlapping noise are varied. For the CMF,GMF, MMSE, and GMF –OPT cases the variance Varb ischosen as unity. Comparing Fig. 5 with Figs. 3 and 4, wecan observe how the variance of location errors changeswhen the nonoverlapping noise becomes colored. Wenote that the influence of the correlation coefficient isstronger when the ratio between the variances of bothnoises is larger. It is also important to take into accountpossible changes in the extreme point positions in Table 1
while the correlation coefficient is varied. Similarly tothe white-noise model, the graphs for the GMF arebounded above by curves for the GMF–OPT estimation.Finally, we note that for the colored-noise model the GMFalso has the best performance in terms of location accu-racy.
4. CONCLUSIONSWe have considered the problem of the accuracy of loca-tion measurement of an input scene containing a noisytarget in nonoverlapping background noise. In particu-lar, we have estimated the variance of location errors and,with some simplifying assumptions, derived the filterminimizing this variance. The filter coincides with theGMF. We have extended the obtained results from theone-dimensional case to the two-dimensional case. Forthe case of white and colored nonoverlapping noise, com-puter simulations have been provided to compare the per-formance of the GMF with those of the POF, the CMF,and the MMSE filter. We have also made the estimationof location errors for the GMF by use of Eq. (25). The re-sults have shown that the GMF has the best performancein terms of location accuracy.Since the normalized correlation function16 provides in-
tensity invariance for the discrimination problem, thismethod can also be investigated with respect to locationerrors for a noisy target embedded on a nonoverlappingbackground by using the proposed approach. In this casethe difficulty of the analysis greatly increases.Further investigation can be done to verify how close
the experimental localization errors are to their theoreti-cal evaluations given in the paper for various kinds of fil-ters and input noisy scenes.
V. Kober and J. Campos Vol. 13, No. 8 /August 1996 /J. Opt. Soc. Am. A 1663
Table 1. Critical Values of mb and Corresponding Standard Deviations of Location Errors for CorrelationFilters with Different Correlation Coefficients While the Variances of Overlapping and Nonoverlapping
APPENDIX AIn this appendix we derive an expression for the varianceof the derivative of the zero-mean random processesc b0(x, x0) with respect to x evaluated at x 5 D0 1 x0 ,
that is, rb [ @]cb0(x, x0)/]x#ux5D01x0
. Because rb is azero-mean, random process, its variance may be com-puted as follows:
Var~rb! 5 E~rbrb* !
5 E H E E b0~D0 1 x0 2 j!
3 w~D0 2 j!h8~j!@b0~D0 1 x0 2 b!
3 w~D0 2 b!h8~b!#* djdbJ , (A1)
where h8(x) 5 ]h(x)/]x is the derivative (it is assumed toexist) of the impulse response of the filter. Let Rb(x) bethe autocorrelation function of the zero-mean, stationaryrandom process b0(x). We can rewrite Eq. (A1) in thefollowing way:
Var~rb! 5 E E Rb~b 2 j!w~D0 2 j!h8~j!
3 @w~D0 2 b!h8~b!#* djdb
5 $Rb~g!+@h8~g!w~D0 2 g!#+@h8~ 2 g!
3 w~D0 1 g!#* %g50 . (A2)
Taking Fourier transforms from the relevant quantitiesin Eq. (A2), we obtain
Var~rb! 5 4p2E B0~f !@fH~f !+W~f !exp~ j2pD0f !#
3 @fH~f !+W~f !exp~ j2pD0f !#* df, (A3)
where B0(f ) is the power spectrum of the random processb0(x).Using Eq. (A3), we obtain the final expression for the
variance of rb in Eq. (17) as follows:
Var~rb! 5 4p2E E E f1f2H~f1!H* ~f2!B0~f !
3 W~f1 2 f !W* ~f2 2 f !
3 exp@ j2pD0~f1 2 f2!#dfdf1df2 . (A4)
APPENDIX BIn this appendix we reduce the nonoverlapping model tothe additive one and then deduce formulas for the vari-ance of the location errors and a filter minimizing thevariance. The nonoverlapping model in Eq. (2) can berepresented as follows:
s~x, x0! 5 t~x 2 x0! 1 w~x 2 x0!b~x ! 1 n~x !
5 @t~x 2 x0! 1 w~x 2 x0!mb#
1 @w~x 2 x0!b0~x ! 1 n~x !#. (B1)
In this additive representation the target signal ist(x 2 x0) 1 w(x 2 x0)mb , and the additive noise isw(x 2 x0)b
0(x) 1 n(x). For a fixed x0 the componentw(x 2 x0)b
0(x) of this noise is a nonstationary zero-mean random process. If x0 is unknown, it may be re-garded as random, and properties of the random processw(x 2 x0)b
0(x) have to be evaluated as an average overthe random variable x0 . For instance, its autocorrelationfunction averaged over x0 is
Rwb~x1 , x2! 5 Rb0~x1 2 x2!E
2`
1`E2`
1`
W~f1!W~f2!
3 exp@2i2p~f1x1 1 f2x2!#
3 CFx0~f1 1 f2!df1df2 , (B2)
where CFx0(f1 1 f2) is the characteristic function of thedistribution density of x0 . When x0 is distributed uni-formly over an area S, with S → `,
CFx0~f1 1 f2! → ad~f1 1 f2!, (B3)
Rwb~x1 , x2! → aRb0~x1 2 x2!E
2`
1`
uW~f !u2
3 exp@2i2pf~x1 2 x2!#df, (B4)
where a is a normalizing constant.Thus, on average over x0 , the random process
w(x 2 x0)b0(x) is, in the limit, a stationary process with
the spectral density
Hwb~f ! 5 aB0~f !+uW~f !u2. (B5)
Now one can apply formulas for the classical additivemodel1 to obtain the optimal filter and the variance of thelocation errors as in Eqs. (24) and (25).
1664 J. Opt. Soc. Am. A/Vol. 13, No. 8 /August 1996 V. Kober and J. Campos
Fig. 5. Ratio between standard deviation of location errors and standard deviation of nonoverlapping colored noise (in percent) versusratio between standard deviations of additive white noise and nonoverlapping colored noise for the set of correlation filters (CMF, POF,GMF, MMSE, GMF–OPT). The mean mb and the correlation coefficient r of the nonoverlapping background are (a) mb 5 0, r 5 0.5;(b) mb 5 0, r 5 0.95; (c) mb 5 0.1, r 5 0.5; (d) mb 5 0.1, r 5 0.95; (e) mb 5 0.3, r 5 0.5; (f ) mb 5 0.3, r 5 0.95; (g) mb 5 0.5,r 5 0.5; and (h) mb 5 0.5, r 5 0.95.
V. Kober and J. Campos Vol. 13, No. 8 /August 1996 /J. Opt. Soc. Am. A 1665
APPENDIX CIn this appendix we consider an extension of the resultsobtained for one-dimensional signals to the two-dimensional case. Let c(x, y; x0 , y0) 5 c(ru; x0 , y0) bethe filter output. Here u 5 (cos f, sin f) is a vector,r 5 (x2 1 y2)1/2 and f 5 tan21(y/x) represent a signalin polar coordinates, and (x0 , y0) are unknown coordi-nates of the target. Now we suppose that the maximumof E[c(ru; x0 , y0)] lies at the coordinates (x0 , y0) andthat the correlation intensity peak uc(x, x0)u
2 occurs in
Var~D0! 'E E ~fx cos f 1 fy sin f!2uH~fx , fy!u2@aB0~fx , fy!+uW~fx , fy!u2 1 N~fx , fy!#dfxdfy
4p2E E ~fx cos f 1 fy sin f!2H~fx , fy!@T~fx , fy! 1 mbW~fx , fy!#dfxdfy
. (C6)
the vicinity of the coordinates (x0 , y0). Let us expandthe signal E[c(ru; x0 , y0)] in the Fourier series at the co-ordinates (x0 , y0) in direction f:
2)1/2, D 5 r 2 r0, and (u¹)[ (cos f)]/x 1 (sin f)]/y is the differential operator in di-rection f.Our objective is to minimize the variance of the location
error in the correlation plane at angle f. In a similarfashion to that done in Section 2 for the one-dimensionalcase we derive an expression for d2 [see Eq. (10)] as fol-lows:
d2 5 4p2E E ~fx cos f 1 fy sin f!2H~fx ,fy!@T~fx , fy!
1 mbW~fx ,fy!#dfxdfy , (C2)
where fx and fy are the coordinates in the Fourier planeand T(fx , fy), W(fx , fy), and H(fx , fy) are two-dimensional spectra of the target, the inverse supportfunction, and the transfer function of the filter, respec-tively. The directional derivative of the additive, station-ary process cn(ru) is a zero-mean, stationary random pro-cess with the power spectral density
Nn8~fx , fy! 5 4p2~fx cos f 1 fy sin f!2N~fx , fy!,(C3)
where N(fx , fy) is the two-dimensional power spectraldensity of the additive noise n(x, y). Using Eq. (16) andrelation (B3), we can express the variance of rn as follows:
Var~rn! 5 4p2E E ~fx cos f 1 fy sin f!2
3 uH~fx , fy!u2N~fx , fy!dfxdfy . (C4)
In a similar manner it can be shown that the variance ofthe nonstationary, random process rb in relation (22) canbe estimated as
where B0(fx , fy) is the two-dimensional power spectraldensity of the zero-mean, stationary noise b0(x, y). Fi-nally, from Eqs. (B2) and (B5) and relations (23), (B3),and (B4) we obtain an expression for the variance of loca-tion errors,
Applying the Schwarz inequality to the last relation, wearrive at the variance expression in Eq. (28). Here D0 isan estimation of the intensity peak location of the filteroutput in direction f.
ACKNOWLEDGMENTSThe authors thank B. Javidi and M. J. Yzuel for helpfuldiscussions. We also thank the reviewers for useful com-ments. This research was supported financially by theComision Interministerial de Ciencia y Tecnologıa underproject TAP93-0667-C03-01. V. Kober acknowledges fi-nancial support from Generalitat de Catalunya.
*Correspondence should be addressed to Vitaly Kober.Tel: (343) 581-2602; e-mail: [email protected]. Perma-nent address, Institute of Information TransmissionProblems RAS, 19 Yermolovoy str., 101 447 Moscow,Russia.
REFERENCES1. L. P. Yaroslavsky, ‘‘The theory of optimal methods for local-
ization of objects in pictures,’’ in Progress in Optics XXXII,E. Wolf, ed. (Elsevier, Amsterdam, 1993), pp. 145–201.
2. B. V. K. Vijaya Kumar, F. M. Dickey, and J. M. DeLau-rentis, ‘‘Correlation filters minimizing peak location er-rors,’’ J. Opt. Soc. Am. A 9, 678–682 (1992).
3. M. Fleisher, U. Mahlab, and J. Shamir, ‘‘Target locationmeasurement by optical correlators: performance crite-rion,’’ Appl. Opt. 31, 230–235 (1992).
4. G. Zalman and J. Shamir, ‘‘Reducing probability in opticalpattern recognition,’’ J. Opt. Soc. Am. A 8, 1866–1873(1991).
5. U. Mahlab, M. Fleisher, and J. Shamir, ‘‘Error probabilityin optical pattern recognition,’’ Opt. Commun. 77, 415–422(1990).
6. A. VanderLugt, ‘‘Signal detection by complex filters,’’ IEEETrans. Inf. Theory IT-10, 139–145 (1964).
7. G. Zalman and J. Shamir, ‘‘Maximum discrimination filter,’’J. Opt. Soc. Am. A 8, 814–821 (1991).
8. B. Javidi and J. Wang, ‘‘Optimum filter for detection of atarget in nonoverlapping scene noise,’’ Appl. Opt. 33, 4454–4458 (1994).
9. B. Javidi and J. Wang, ‘‘Design of filters to detect a noisytarget in nonoverlapping background noise,’’ J. Opt. Soc.
1666 J. Opt. Soc. Am. A/Vol. 13, No. 8 /August 1996 V. Kober and J. Campos
Am. A 11, 2604–2612 (1994).10. B. Javidi, G. Zhang, F. Parchekani, and P. Refregier, ‘‘Per-
formance of minimum-mean-square-error filters for spa-tially nonoverlapping target and input-scene noise,’’ Appl.Opt. 33, 8197–8207 (1994).
11. B. Javidi, P. Refregier, and P. K. Willet, ‘‘Optimum receiverdesign for pattern recognition with nonoverlapping targetand scene noise,’’ Opt. Lett. 18, 1660–1662 (1993).
12. J. L. Horner and P. D. Gianino, ‘‘Phase-only matched filter-
ing,’’ Appl. Opt. 23, 812–816 (1984).13. A. Papoulis, Probability, Random Variables and Stochastic
Processes (McGraw-Hill, New York, 1991).14. J. M. Wozencraft and I. M. Jacobs, Principles of Communi-
cation Engineering (Wiley, New York, 1965).15. R. N. Bracewell, The Fourier Transform and Its Applica-
tions (McGraw-Hill, New York, 1986).16. F. M. Dickey and L. A. Romero, ‘‘Normalized correlation for
pattern recognition,’’ Opt. Lett. 16, 1186–1188 (1991).
