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Comparison on nonlinear joint transform correlator and nonlinear matched filter based correlator
Volume 75, number
I
OPTICS
COMMUNICATIONS
I February
1990
COMPARISON OF NONLINEAR JOINT TRANSFORM CORRELATOR AND NONLINEAR MATCHED FILTER BASED CORRELATOR Bahram JAVIDI Department qfElectr:cal Engineering, Unrversity qf Connecticut, Storrs. CT 06269. USA Received
19 July 1989
We compare the performance of the nonlinear joint transform correlator, the nonlinearly transformed filter based correlator and the conventional linear correlator. The pattern recognition systems will be compared in the areas of correlation peak, peak to sidelobe ratio and the correlation width. We show that compared to the linear correlator, both types of nonlinear correlators produce better correlation performance
1. Introduction
Recently, we introduced two types of nonlinear image correlators with performance substantially superior to that of the conventional linear optical correlators [ l-41. The first type [ 1,2] is joint Fourier transform based [ 51 which uses nonlinearity at the Fourier plane to threshold the joint power spectrum. The performance of the nonlinear joint transform optical correlator (JTC) has been compared with the classic joint transform image correlator. We have shown that the nonlinear correlator can provide a much higher autocorrelation peak intensity, smaller correlation sidelobes, narrower correlation width and better discrimination sensitivity. The nonlinear JTC concept has been extended to nonlinearly transformed matched filter based optical correlator [ 3,4]. The nonlinearly transformed matched filter is computed electronically by expressing the linear matched filter as an amplitude and phase modulated bandpass waveform using the amplitllde and phase of the Fourier transform of the refcrcnce function. The modulated bandpass real function is then transformed using a general type of nonlinearity. The nonlinear filter is displayed on an electrically addressed SLM located at the Fourier plane of an optical correlator. The nonlinear matched filter can provide high peak intensity, small sidelobes and well-defined correlation spots. The non8
linear filter can improve the performance of the filter based associative processors [ 61. In this paper, we compare the performance of the nonlinear JTC and the nonlinearly transformed matched filter based correlators. The nonlinear correlators will be compared in the areas of correlation peak, correlation peak to sidelobe ratio, correlation width and discrimination sensitivity. The nonlinear correlators will be investigated for kth law device nonlinearity. The effects of the various degrees of the nonlinearity on the performance of the nonlinear correlators will be determined. We show that in nonlinear JTC, the nonlinearity affects the Fourier magnitude of both the input signal and the reference sigthe nal. In nonlinear filter based correlator, nonlinearity affects the Fourier magnitude of the reference signal only.
2. Analysis In this section. we provide mathematical expressions for the two types of nonlinear correlators. The correlation functions will be determined in terms of the transfer characteristics used for the nonlinear transformation. The nonlinear transformed filter based correlator is shown in fig. 1. Plane P, is the input plane that contains the input signal displayed on SLM,. Plane Pz is the Fourier plane that contains the nonlinear
0030-4018/90/$03.50
0 Elsevier Science Publishers
B.V. (North-Holland)
Volume 75, number
OPTICS
1
I February
COMMUNICATIONS
1990
k&Law Nonllnearltv
a
Nonlinear Flltergtz) Correlation Input
Plane
"
-1
Electrically Addressed
Input SLM
SLM
Filter plane
<
f
f
f
b
f
*
FTL, Linearly
--+
Polarized Coherent Light Plane Klh
Law
Nonlinearity
Linearly Polarlied Coherent Llght
SLM
Fig.
1. (a) Nonlinear
matched
filter based optical correlator.
filter displayed on SLM2. The output functions produced at plane P,. The input signal and the erence signal are denoted by s(x, y) and r(x, y), spectively. The Fourier transforms of the signals represented by
are refreare
(la)
(b) Nonlinear
joint transform
FT[r(x,y)l=R(a,P)
correlator.
exp[ih(wP)l
7
(lb)
where (a, /3) are the angular spatial frequency coordinates and S( (Y,8) exp [ i&( cr, /I) ] and R(cu, P) exp[ iq3R(Ly, p) ] correspond to the Fourier transforms of the input signals s(x, y) and T(X, y ), respectively. Here, R(a, p) and S(a, /3) are the am9
Volume 75, number
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COMMUNICATIONS
plitude spectrum and QR( (Y,,!?) and &( a!, /J’) are the phases of the Fourier transforms. We express the linear matched filter as an amplitude and phase modulated bandpass function E(CX./J)=R(a,B)
.
cos[S(jcX-d,(cu,p)]
(2)
where _Y,,is the central frequency of the bandpass function. It can be seen from the above equation that E( cy. p) is a real function and can be displayed on an electrically addressed spatial light modulator located at the Fourier plane of an optical correlator. In the linear case. the inverse Fourier transform of the product of the input signal Fourier transform and the above equation can produce the linear correlation signals at the output plane. The nonlinearly transformed matched filter is obtained by thresholding the modulated bandpass function with a nonlinear mapping and displaying the thresholded result at the Fourier plane on an electrically addressed SLM. The thresholded function can be considered as the output of a nonlinear system. The nonlinear characteristics of the thrcsholding network is denoted by g(E) where E is the modulated bandpass function given by eq. (3). An expression for the thresholded filter can be obtained by a similar approach as the analysis of nonlinear systems using the transform method. The amplitude modulation of the nonlinear matched filter depends on the type of the nonlinearity that is used. To obtain a specific expression for the nonlinear matched filter. we consider full wave (odd ) lith law device nonlinearity g(E) =f?” . =-
E>O.
/fYIA,
Ii‘tO.
(3)
where g(E) is the transfer characteristic of the nonlinearity, E is the linear matched filter input of the nonlinearity and k is a nonnegative real number. The nonlinear matched filter produced at the output of the h-th law device is given by [4,5] gk(E)=
5 ,,=I
Wk+
1) [R(w/j)
2Af( I - (zb--!i)/z)r(
I + (z,+li)/2)
, odd
xcos[z’~~~s,cu-L’~~(~.p)]
where r is the gamma function stands for the lith law device. IO
(4)
.
and the subscript
I\
1990
It can be seen from this equation that each harmonic term is phase modulated by E’times the phase of the matched filter function and the higher order filters are diffracted to 11x,,.The envelope of each harmonic term is proportional to the hth power of the Fourier transform magnitude of the reference signal. The correct phase information of the filter function is obtained for the first order harmonic term (z‘= I ) which yields
f(k+ I) tR(a>P)I' “‘“(~)=2~+‘r(l-(l-li)/2)r(l+(l+h)/2) xexp(i[S”(Y-_~(~.‘,)]i,
(5)
where the subscript 1 denotes the first order harmonic term and we have retained only the exponential component of the nonlinear filter that produces the nonlinear correlation. Various types of nonlinear filters may be produced by varying the severity of the nonlinearity. A hard clipping nonlinear filter (I, = 0) will produce a continuous phase-only filter [7-91 for the first order harmonic term (L’= I )_ It is noted that the nonlinear filter produced by the hard clipping operation is a binary function. However, the first order intermodulation term for L’= I produces a continuous phaseonly filter. For example. a binary SLM may be used to read out the hard clipped filter which will produce a continuous phase-only filter for the first order harmonic term. For a given input signal, the light pattern 7‘( cy, p) leaving the filter plane is the product of the Fourier transform of the input signal S( (Y,p) and the nonlinear filter function gk (E). The light pattern pattern leaving the filter plane corresponding to the first order harmonic term (P= 1 ) that produces the nonlinear correlation signals is given by
xexp/i[.\i,cr+~,(cr.B)-O,(cr,P)]).
I”
I February
(6)
where the subscripts 1k are as defined earlier. The inverse Fourier transform of the above equation will produce the nonlinear correlation signal at the output plane. In section 3. we shall investigate the nonlinear correlation signals for different degrees of the nonlinearity.
Volume 75, number
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OPTICS
For the joint power spectrum based nonlinear processors, the nonlinearity will affect both the input signal and the reference signal. Nonlinear joint transform processors use thresholding on the joint power spectrum of the input signal s(x-x0, _)J)and the reference signal v(x+xO, y). The implementation of the nonlinear JTC using an electrically addressed SLM is shown in fig. 1 (b). Plane P, is the input plane that contains the reference signal and the input signal displayed on SLM,. The images are then Fourier transformed by lens FTL, and the interference between the Fourier transforms is produced at plane PZ. The intensity of the Fourier transform interference is obtained by a CCD array located at plane P2 and is thresholded using a nonlinear network. A SLM located at plane P3 is used to read out the thresholded joint power spectrum provided by the nonlinear network. The correlation function can be produced at plane P4 by taking the inverse Fourier transform of the thresholded interference intensity distribution at plane P3. The light distribution intensity formed on the CCD array at the back focal plane of the transform lens FTL, is E(a.p)=S2(cu,8)+R2(~,/3)+2R(a,P)
=
z
U(k+
“=, 2&r( I-
1) [R(atP)S(wP) lk (c-k)/2)I-(
Wk+ 1) [R(a. P)S(a, P) lk g1”(E)=I-(l-(l-X-)/2)I-(1+(l+h-)/2)
xcos[2xo~+~S(~,8)--~(~,P)l
(9)
If the input signal and the reference signal are the same, then eq. (9) will produce the thresholded joint power spectrum for autocorrelation signals, i.e., 2T(k+l)[R(a,/?)]*’
(7)
In the classical case, the inverse Fourier transform of eq. (7 ) can produce the cross-correlation signals at the output plane. In nonlinear JTC, the Fourier transform interference intensity provided by the CCD array is thresholded before the inverse Fourier transform operation is applied. The CCD array at the Fourier plane is connected to a SLM through a thresholding network so that the thresholded interference intensity distribution can be read out by coherent light. In a different implementation, a variable contrast optically addressed SLM can be used to threshold the joint power spectrum [ lo]. Given a full wave (odd) kth law device, the output of the nonlinearity (threshold joint power spectrum) for the cross-product terms is [ 1 ] gk:k(F)
1990
For v= 1, the nonlinear system has preserved the phase of the cross-product term [&.(a, /II) -eR(cx, /I) and only the amplitude is affected. It is evident from eq. (8) that in nonlinear joint power spectrum based processors, the nonlinearity affects the amplitude spectra of both the input signal and the reference signal. This is different from nonlinear matched filter based processors where the nonlinearity affects the amplitude spectra of the reference signal only [see eq. (6) 1. For the kth law nonlinear JTC, the envelope of each harmonic term is proportional to the kth power of the product of the Fourier transform magnitudes of the input signal and the reference signal. The correct phase information of the joint power spectrum is obtained for the first order harmonic term (v= 1 ) which yields
S(cu,B)
xcos[2x,a+~,(~,p)-~,(~,p)l.
I February
COMMUNICATIONS
1-t (v+k)/2)
L’ odd
xcos[2vsoa+v~,(a,p)-z;~,(cu,P)l. (8)
XCOS(2&(Y).
(10)
It can be seen from eq. (9) that for k=+, the thresholded interference intensity will produce an autocorrelation signal that is identical to the autocorrelation signal obtain by a phase-only matched filter without synthesizing a phase-only matched filter.
3. Computer simulation results A numerical analysis of the nonlinear JTC and the nonlinearly transformed matched filter based optical correlator is provided to study the correlation signals. The nonlinear correlators are computed using a kth law device. For the nonlinear filters [see eq. ( 5 ) ] this corresponds to mapping the matched filter function [ eq. (2) ] using a nonlinearity at the Fourier plane that has an input/output characteristics of a kth law device. For the nonlinear JTC, the joint 11
Volume 75, number
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power spectrum is thresholded using a lath law device [see eq. (S)]. To study the performance of the proposed system, we have used a 128 x 128 point 2D fast Fourier transform (FFT) and the results are plotted using a 3D plotting subroutine. The autocorrelation tests are performed for the character F as shown in fig. 2 (a). Fig. 2(b) and (c) illustrate the correlation signals produced with k=0.5 using the nonlinear JTC and the nonlinearly transformed filter based correlator respectively. Table 1 illustrates the results of the autocorrelation tests for the nonlinear JTC and the nonlinear filter based correlator. It can be seen from this table that both types of nonlinear correlations will increase the autocorrelation peak intensity compared to the linear case. The nonlinearity will also reduce the correlation sidelobe intensity. The nonlinearity will result in a reduction in the autocorrelation width and will produce well-defined autocorrelation signals for small k. For a given degree of the nonlinearity (give li), the nonlinear JTC produces higher peak intensity, lower sidelobes and narrower width compared with the nonlinear filter based correlator. This is evident from eq. (9) which describes the thresholding effects on the joint power spectrum of the JTC. The nonlinearity affects the Fourier transform magnitudes of both the input signal and the reference signal. For compression types of the nonlinearity (kc 1 ), the magnitudes of the high spatial frequencies in the input signals Fourier magnitudes are increased which result in higher peak intensities. lower sidelobes and narrower width. For nonlinear filter based correlation. the nonlin-
I February
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1990
F b
C
Fig. 2. Nonlinear autocorrelation stgnals for character F produced with k=0.5. (a) Image used in the autocorrelation tests. (b) Nonlinear joint transform correlator. (c) Nonlinear filter based correlator.
Table I Autocorrelation results. Ri is the first order autocorrelation peak intensity; .SL’ IS the largest sidelobe intensity; k is the degree of nonlinearity given by eq. (3); fwhm is the full correlation width at the half maximum; JTC is joint transform correlation Correlator
linear nonlinear nonlinear nonlinear nonlinear
12
k
1 0.8 0.5 0.3 0
Nonlinear
filter based correlation
Nonlinear
JTC R.$/SL’
G
R;/SL’
fwhm (x,-y’)
Ri
1 1.8 5 11.5 52
3.4 4.2 6.2 9 17
(4-4) (4-4) (3-3) (2-2) (1-I)
2.6 22 219 3x104
I
2.7 4.2 19 21 65
fwhm (x’ -j” 4-4 ) 3-3) 2-1) 2-l ) 2-1)
)
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earity affects the Fourier magnitude of the reference signal only [see eq. (6) 1. The Fourier magnitude of the input signal in the light field leaning the Fourier plane remains unaffected by the nonlinearity. This explains the better correlation performance of the nonlinear JTC compared with the nonlinear filter based correlator for a given degree of the nonlinearity.
4. Conclusion We have compared the performance of the nonlinear JTC with that of the nonlinearly transformed filter based correlator. The results for both types of nonlinear correlator are listed in table 1. It was found that for both types of nonlinear correlators, as the severity of the nonlinearity increases, the autocorrelation peak intensity increases, the sidelobe intensity decreases and the correlation width decreases. In nonlinear JTC, the nonlinearity affects the Fourier magnitudes of both the input scene and the reference function. The effect of a decompression type of the nonlinearity on the joint power spectrum is to emphasize the Fourier magnitudes at high spatial frequencies to produce a better correlation. For the nonlinearly transformed filter based correlator, the nonlinearity affects the Fourier magnitude of the reference function only. As a result, for a given nonlinearity, the nonlinear JTC produce a better correla-
1 February
1990
tion performance compared with the nonlinear filter based correlator. Both types of nonlinear correlators produce higher peak intensity, lower sidelobes and narrower width compared with the linear correlator.
Acknowledgement I wish to thank Jack and Carol Ruiz for their help with the computer simulations.
References [ 1] B. Javidi, Appl. Optics 28 (1989) 2358. [2] B. Javidi, Opt. Eng. 28 (1989) 267. [ 31 B. Javidi, Generalization of the linear matched filter concept to nonlinear matched filters, Appl. Optics to be published. [4] B. Javidi, Nonlinear filter based correlation, in: Proc. SPIE, Vol. 115 1 (August, I989 ); see also Appl. Optics 28 ( 1989) no. 22. [ 51 C.S. Weaver and J.W. Goodman, Appl. Optics 5 ( 1966) 445. [6] E.G. Paek,Opt. Eng. (1989) 519. [ 71 J.L. Homer and P.D. Gianino. Appl. Optics 24 ( 1985) 2889. [8] H. John Caulfield, Appl. Opt. 21 (1982) 4391. [ 91 KC. Macukow, Pattern recognition using phase-only information, in: Technical Digest of the OSA Annual Meeting in Santa Clara (November, 1988) p. 132. [lo] C. Warde, Variable gamma spatial light modulator, in: Technical Digest of OSA, Vol. 11 (March, 1987) pp. 1 l14.
13
I
OPTICS
COMMUNICATIONS
I February
1990
COMPARISON OF NONLINEAR JOINT TRANSFORM CORRELATOR AND NONLINEAR MATCHED FILTER BASED CORRELATOR Bahram JAVIDI Department qfElectr:cal Engineering, Unrversity qf Connecticut, Storrs. CT 06269. USA Received
19 July 1989
We compare the performance of the nonlinear joint transform correlator, the nonlinearly transformed filter based correlator and the conventional linear correlator. The pattern recognition systems will be compared in the areas of correlation peak, peak to sidelobe ratio and the correlation width. We show that compared to the linear correlator, both types of nonlinear correlators produce better correlation performance
1. Introduction
Recently, we introduced two types of nonlinear image correlators with performance substantially superior to that of the conventional linear optical correlators [ l-41. The first type [ 1,2] is joint Fourier transform based [ 51 which uses nonlinearity at the Fourier plane to threshold the joint power spectrum. The performance of the nonlinear joint transform optical correlator (JTC) has been compared with the classic joint transform image correlator. We have shown that the nonlinear correlator can provide a much higher autocorrelation peak intensity, smaller correlation sidelobes, narrower correlation width and better discrimination sensitivity. The nonlinear JTC concept has been extended to nonlinearly transformed matched filter based optical correlator [ 3,4]. The nonlinearly transformed matched filter is computed electronically by expressing the linear matched filter as an amplitude and phase modulated bandpass waveform using the amplitllde and phase of the Fourier transform of the refcrcnce function. The modulated bandpass real function is then transformed using a general type of nonlinearity. The nonlinear filter is displayed on an electrically addressed SLM located at the Fourier plane of an optical correlator. The nonlinear matched filter can provide high peak intensity, small sidelobes and well-defined correlation spots. The non8
linear filter can improve the performance of the filter based associative processors [ 61. In this paper, we compare the performance of the nonlinear JTC and the nonlinearly transformed matched filter based correlators. The nonlinear correlators will be compared in the areas of correlation peak, correlation peak to sidelobe ratio, correlation width and discrimination sensitivity. The nonlinear correlators will be investigated for kth law device nonlinearity. The effects of the various degrees of the nonlinearity on the performance of the nonlinear correlators will be determined. We show that in nonlinear JTC, the nonlinearity affects the Fourier magnitude of both the input signal and the reference sigthe nal. In nonlinear filter based correlator, nonlinearity affects the Fourier magnitude of the reference signal only.
2. Analysis In this section. we provide mathematical expressions for the two types of nonlinear correlators. The correlation functions will be determined in terms of the transfer characteristics used for the nonlinear transformation. The nonlinear transformed filter based correlator is shown in fig. 1. Plane P, is the input plane that contains the input signal displayed on SLM,. Plane Pz is the Fourier plane that contains the nonlinear
0030-4018/90/$03.50
0 Elsevier Science Publishers
B.V. (North-Holland)
Volume 75, number
OPTICS
1
I February
COMMUNICATIONS
1990
k&Law Nonllnearltv
a
Nonlinear Flltergtz) Correlation Input
Plane
"
-1
Electrically Addressed
Input SLM
SLM
Filter plane
<
f
f
f
b
f
*
FTL, Linearly
--+
Polarized Coherent Light Plane Klh
Law
Nonlinearity
Linearly Polarlied Coherent Llght
SLM
Fig.
1. (a) Nonlinear
matched
filter based optical correlator.
filter displayed on SLM2. The output functions produced at plane P,. The input signal and the erence signal are denoted by s(x, y) and r(x, y), spectively. The Fourier transforms of the signals represented by
are refreare
(la)
(b) Nonlinear
joint transform
FT[r(x,y)l=R(a,P)
correlator.
exp[ih(wP)l
7
(lb)
where (a, /3) are the angular spatial frequency coordinates and S( (Y,8) exp [ i&( cr, /I) ] and R(cu, P) exp[ iq3R(Ly, p) ] correspond to the Fourier transforms of the input signals s(x, y) and T(X, y ), respectively. Here, R(a, p) and S(a, /3) are the am9
Volume 75, number
I
OPTICS
COMMUNICATIONS
plitude spectrum and QR( (Y,,!?) and &( a!, /J’) are the phases of the Fourier transforms. We express the linear matched filter as an amplitude and phase modulated bandpass function E(CX./J)=R(a,B)
.
cos[S(jcX-d,(cu,p)]
(2)
where _Y,,is the central frequency of the bandpass function. It can be seen from the above equation that E( cy. p) is a real function and can be displayed on an electrically addressed spatial light modulator located at the Fourier plane of an optical correlator. In the linear case. the inverse Fourier transform of the product of the input signal Fourier transform and the above equation can produce the linear correlation signals at the output plane. The nonlinearly transformed matched filter is obtained by thresholding the modulated bandpass function with a nonlinear mapping and displaying the thresholded result at the Fourier plane on an electrically addressed SLM. The thresholded function can be considered as the output of a nonlinear system. The nonlinear characteristics of the thrcsholding network is denoted by g(E) where E is the modulated bandpass function given by eq. (3). An expression for the thresholded filter can be obtained by a similar approach as the analysis of nonlinear systems using the transform method. The amplitude modulation of the nonlinear matched filter depends on the type of the nonlinearity that is used. To obtain a specific expression for the nonlinear matched filter. we consider full wave (odd ) lith law device nonlinearity g(E) =f?” . =-
E>O.
/fYIA,
Ii‘tO.
(3)
where g(E) is the transfer characteristic of the nonlinearity, E is the linear matched filter input of the nonlinearity and k is a nonnegative real number. The nonlinear matched filter produced at the output of the h-th law device is given by [4,5] gk(E)=
5 ,,=I
Wk+
1) [R(w/j)
2Af( I - (zb--!i)/z)r(
I + (z,+li)/2)
, odd
xcos[z’~~~s,cu-L’~~(~.p)]
where r is the gamma function stands for the lith law device. IO
(4)
.
and the subscript
I\
1990
It can be seen from this equation that each harmonic term is phase modulated by E’times the phase of the matched filter function and the higher order filters are diffracted to 11x,,.The envelope of each harmonic term is proportional to the hth power of the Fourier transform magnitude of the reference signal. The correct phase information of the filter function is obtained for the first order harmonic term (z‘= I ) which yields
f(k+ I) tR(a>P)I' “‘“(~)=2~+‘r(l-(l-li)/2)r(l+(l+h)/2) xexp(i[S”(Y-_~(~.‘,)]i,
(5)
where the subscript 1 denotes the first order harmonic term and we have retained only the exponential component of the nonlinear filter that produces the nonlinear correlation. Various types of nonlinear filters may be produced by varying the severity of the nonlinearity. A hard clipping nonlinear filter (I, = 0) will produce a continuous phase-only filter [7-91 for the first order harmonic term (L’= I )_ It is noted that the nonlinear filter produced by the hard clipping operation is a binary function. However, the first order intermodulation term for L’= I produces a continuous phaseonly filter. For example. a binary SLM may be used to read out the hard clipped filter which will produce a continuous phase-only filter for the first order harmonic term. For a given input signal, the light pattern 7‘( cy, p) leaving the filter plane is the product of the Fourier transform of the input signal S( (Y,p) and the nonlinear filter function gk (E). The light pattern pattern leaving the filter plane corresponding to the first order harmonic term (P= 1 ) that produces the nonlinear correlation signals is given by
xexp/i[.\i,cr+~,(cr.B)-O,(cr,P)]).
I”
I February
(6)
where the subscripts 1k are as defined earlier. The inverse Fourier transform of the above equation will produce the nonlinear correlation signal at the output plane. In section 3. we shall investigate the nonlinear correlation signals for different degrees of the nonlinearity.
Volume 75, number
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OPTICS
For the joint power spectrum based nonlinear processors, the nonlinearity will affect both the input signal and the reference signal. Nonlinear joint transform processors use thresholding on the joint power spectrum of the input signal s(x-x0, _)J)and the reference signal v(x+xO, y). The implementation of the nonlinear JTC using an electrically addressed SLM is shown in fig. 1 (b). Plane P, is the input plane that contains the reference signal and the input signal displayed on SLM,. The images are then Fourier transformed by lens FTL, and the interference between the Fourier transforms is produced at plane PZ. The intensity of the Fourier transform interference is obtained by a CCD array located at plane P2 and is thresholded using a nonlinear network. A SLM located at plane P3 is used to read out the thresholded joint power spectrum provided by the nonlinear network. The correlation function can be produced at plane P4 by taking the inverse Fourier transform of the thresholded interference intensity distribution at plane P3. The light distribution intensity formed on the CCD array at the back focal plane of the transform lens FTL, is E(a.p)=S2(cu,8)+R2(~,/3)+2R(a,P)
=
z
U(k+
“=, 2&r( I-
1) [R(atP)S(wP) lk (c-k)/2)I-(
Wk+ 1) [R(a. P)S(a, P) lk g1”(E)=I-(l-(l-X-)/2)I-(1+(l+h-)/2)
xcos[2xo~+~S(~,8)--~(~,P)l
(9)
If the input signal and the reference signal are the same, then eq. (9) will produce the thresholded joint power spectrum for autocorrelation signals, i.e., 2T(k+l)[R(a,/?)]*’
(7)
In the classical case, the inverse Fourier transform of eq. (7 ) can produce the cross-correlation signals at the output plane. In nonlinear JTC, the Fourier transform interference intensity provided by the CCD array is thresholded before the inverse Fourier transform operation is applied. The CCD array at the Fourier plane is connected to a SLM through a thresholding network so that the thresholded interference intensity distribution can be read out by coherent light. In a different implementation, a variable contrast optically addressed SLM can be used to threshold the joint power spectrum [ lo]. Given a full wave (odd) kth law device, the output of the nonlinearity (threshold joint power spectrum) for the cross-product terms is [ 1 ] gk:k(F)
1990
For v= 1, the nonlinear system has preserved the phase of the cross-product term [&.(a, /II) -eR(cx, /I) and only the amplitude is affected. It is evident from eq. (8) that in nonlinear joint power spectrum based processors, the nonlinearity affects the amplitude spectra of both the input signal and the reference signal. This is different from nonlinear matched filter based processors where the nonlinearity affects the amplitude spectra of the reference signal only [see eq. (6) 1. For the kth law nonlinear JTC, the envelope of each harmonic term is proportional to the kth power of the product of the Fourier transform magnitudes of the input signal and the reference signal. The correct phase information of the joint power spectrum is obtained for the first order harmonic term (v= 1 ) which yields
S(cu,B)
xcos[2x,a+~,(~,p)-~,(~,p)l.
I February
COMMUNICATIONS
1-t (v+k)/2)
L’ odd
xcos[2vsoa+v~,(a,p)-z;~,(cu,P)l. (8)
XCOS(2&(Y).
(10)
It can be seen from eq. (9) that for k=+, the thresholded interference intensity will produce an autocorrelation signal that is identical to the autocorrelation signal obtain by a phase-only matched filter without synthesizing a phase-only matched filter.
3. Computer simulation results A numerical analysis of the nonlinear JTC and the nonlinearly transformed matched filter based optical correlator is provided to study the correlation signals. The nonlinear correlators are computed using a kth law device. For the nonlinear filters [see eq. ( 5 ) ] this corresponds to mapping the matched filter function [ eq. (2) ] using a nonlinearity at the Fourier plane that has an input/output characteristics of a kth law device. For the nonlinear JTC, the joint 11
Volume 75, number
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COMMUNICATIONS
power spectrum is thresholded using a lath law device [see eq. (S)]. To study the performance of the proposed system, we have used a 128 x 128 point 2D fast Fourier transform (FFT) and the results are plotted using a 3D plotting subroutine. The autocorrelation tests are performed for the character F as shown in fig. 2 (a). Fig. 2(b) and (c) illustrate the correlation signals produced with k=0.5 using the nonlinear JTC and the nonlinearly transformed filter based correlator respectively. Table 1 illustrates the results of the autocorrelation tests for the nonlinear JTC and the nonlinear filter based correlator. It can be seen from this table that both types of nonlinear correlations will increase the autocorrelation peak intensity compared to the linear case. The nonlinearity will also reduce the correlation sidelobe intensity. The nonlinearity will result in a reduction in the autocorrelation width and will produce well-defined autocorrelation signals for small k. For a given degree of the nonlinearity (give li), the nonlinear JTC produces higher peak intensity, lower sidelobes and narrower width compared with the nonlinear filter based correlator. This is evident from eq. (9) which describes the thresholding effects on the joint power spectrum of the JTC. The nonlinearity affects the Fourier transform magnitudes of both the input signal and the reference signal. For compression types of the nonlinearity (kc 1 ), the magnitudes of the high spatial frequencies in the input signals Fourier magnitudes are increased which result in higher peak intensities. lower sidelobes and narrower width. For nonlinear filter based correlation. the nonlin-
I February
a
1990
F b
C
Fig. 2. Nonlinear autocorrelation stgnals for character F produced with k=0.5. (a) Image used in the autocorrelation tests. (b) Nonlinear joint transform correlator. (c) Nonlinear filter based correlator.
Table I Autocorrelation results. Ri is the first order autocorrelation peak intensity; .SL’ IS the largest sidelobe intensity; k is the degree of nonlinearity given by eq. (3); fwhm is the full correlation width at the half maximum; JTC is joint transform correlation Correlator
linear nonlinear nonlinear nonlinear nonlinear
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k
1 0.8 0.5 0.3 0
Nonlinear
filter based correlation
Nonlinear
JTC R.$/SL’
G
R;/SL’
fwhm (x,-y’)
Ri
1 1.8 5 11.5 52
3.4 4.2 6.2 9 17
(4-4) (4-4) (3-3) (2-2) (1-I)
2.6 22 219 3x104
I
2.7 4.2 19 21 65
fwhm (x’ -j” 4-4 ) 3-3) 2-1) 2-l ) 2-1)
)
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earity affects the Fourier magnitude of the reference signal only [see eq. (6) 1. The Fourier magnitude of the input signal in the light field leaning the Fourier plane remains unaffected by the nonlinearity. This explains the better correlation performance of the nonlinear JTC compared with the nonlinear filter based correlator for a given degree of the nonlinearity.
4. Conclusion We have compared the performance of the nonlinear JTC with that of the nonlinearly transformed filter based correlator. The results for both types of nonlinear correlator are listed in table 1. It was found that for both types of nonlinear correlators, as the severity of the nonlinearity increases, the autocorrelation peak intensity increases, the sidelobe intensity decreases and the correlation width decreases. In nonlinear JTC, the nonlinearity affects the Fourier magnitudes of both the input scene and the reference function. The effect of a decompression type of the nonlinearity on the joint power spectrum is to emphasize the Fourier magnitudes at high spatial frequencies to produce a better correlation. For the nonlinearly transformed filter based correlator, the nonlinearity affects the Fourier magnitude of the reference function only. As a result, for a given nonlinearity, the nonlinear JTC produce a better correla-
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tion performance compared with the nonlinear filter based correlator. Both types of nonlinear correlators produce higher peak intensity, lower sidelobes and narrower width compared with the linear correlator.
Acknowledgement I wish to thank Jack and Carol Ruiz for their help with the computer simulations.
References [ 1] B. Javidi, Appl. Optics 28 (1989) 2358. [2] B. Javidi, Opt. Eng. 28 (1989) 267. [ 31 B. Javidi, Generalization of the linear matched filter concept to nonlinear matched filters, Appl. Optics to be published. [4] B. Javidi, Nonlinear filter based correlation, in: Proc. SPIE, Vol. 115 1 (August, I989 ); see also Appl. Optics 28 ( 1989) no. 22. [ 51 C.S. Weaver and J.W. Goodman, Appl. Optics 5 ( 1966) 445. [6] E.G. Paek,Opt. Eng. (1989) 519. [ 71 J.L. Homer and P.D. Gianino. Appl. Optics 24 ( 1985) 2889. [8] H. John Caulfield, Appl. Opt. 21 (1982) 4391. [ 91 KC. Macukow, Pattern recognition using phase-only information, in: Technical Digest of the OSA Annual Meeting in Santa Clara (November, 1988) p. 132. [lo] C. Warde, Variable gamma spatial light modulator, in: Technical Digest of OSA, Vol. 11 (March, 1987) pp. 1 l14.
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