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Discretized amplitudemodulated phase-only filter
Oplicr & Laser Technology, Vol. 28, No. 2, pp. 93-100, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0030-3992/96 $15.00 + 0.00
ELSEVIER ADVANCED
0030-3992(95)00066-6
TECHNOLOGY
Discretized modulated K. M. IFTEKHARUDDIN,
amplitudephase-only M. A. KARIM,
filter
P. W. ELOE, A. A. S. AWWAL
The amplitude-modulated phase-only filter (AMPOF) effectively utilizes both amplitude and phase information to obtain a narrower and larger autocorrelation peak. To make it readily suitable for real-time target recognition applications, we consider discretizing both its amplitude and phase. Accordingly, both amplitude and phase of the AMPOF are ternerized to yield improved performance statistics of the correlator. KEYWORDS:
amplitude-modulated
phase-only
filter, optical
correlators,
correlation
peaks
sharper than that of the conventional matched filters. However, several recent studies”m’3 have seriously questioned the validity of totally ignoring the amplitude content of an image. A particular complex filter, namely, an amplitude-modulated phase-only filter (AMPOF) 14,15for example, retains information regarding both phase and amplitude and it has been found to offer better correlation discrimination than the POF.
Introduction The more widely used optical correlators utilize filters that encode the impulse response of an input pattern. Such correlators accordingly generate an output light field that corresponds to the correlated signal between the impulse response and the input’. These correlator architectures largely rely on the appropriate spatial light modulator (SLM) technology* for real-time insertion of the input pattern as well as the filter(s) in the correlator. The magneto-optic SLMs3, for example, are capable of simultaneously displaying both amplitude and phase information. On the other hand, an optically controlled ferroelectric liquid crystal SLM may be used to control the amounts of magnitude and phase modulation independently4,5. Alternatively, a pair of liquid crystal televisions (LCTV) may be used to obtain a complex modulation with independently controllable phase and amplitude6. This particular optical architecture using two LCTVs, polarizers and lenses can be used to realize the particular multilevel phase-multilevel amplitude filters considered herein.
In our effort to explore the flexibility of AMPOF, it is important to note that a number of studies have considered binarizing the POF, that is, phase16J7. Subsequent analysis of binary POF (BPOF)‘8>‘9 have shown that the corresponding correlator is better than the POF-based correlator. Further, multiplying the BPOF phase with a binary amplitude pattern has led to a newer class of POF, namely, a ternary phaseamplitude filter (TPAF)*’ which has been shown to have an even better performance. Unlike POF, however, the AMPOF is a frequency domain complex valued function with distinct amplitude and phase. Accordingly, a separate discretization of the phase and amplitude factors of the AMPOF, even with different discrete levels for each of the components, is quite possible 21. In reality, such multi-level discretization of AMPOF readily lends itself to suitable SLM and/or LCTV implementation. In this paper, we consider ternerizing both the amplitude and the phase of the AMPOF function. We show that discretization does indeed result in an improved statistical performance measure of the correlator.
For some time, only the phase of the filter was considered to be important’ in pattern recognition. The significance of the phase preserving filters has been illustrated by linear optimization* and the bleached spatial filter9 respectively. In fact, a phase-only filter (POF)‘O was found to offer correlation peaks much KMI and MAK are at the University of Dayton, Department of Electrical Engineering and Center for Electra-Optics, 300 College Park, Dayton, OH 45469-0245, USA. KMI is currently at BDM Federal, 1990 Founders Drive, Kettering, OH 45420, USA. PWE is at the University of Dayton, Department of Mathematics, 300 College Park, Dayton, OH 45469-2316, USA. AASA is at Wright State University, Department of Computer Science and Computer Engineering, Dayton, OH 45435, USA. Received 27 April 1995.
Correlator
system
As elaborated, an AMPOF-based system takes on a Vander Lugt type of architecture. In such a configuration, the input scene is first Fourier transformed with the AMPOF function introduced at 93
94
Discretized amplitude-modulated
phase-only
filter: K. M. lftekharuddin
et al.
From (3) and (5) we obtain, p(B) = (1/2~) l” X COs(boiy
Inputimage wansparew Fig. 1
Phak modulator
Amplitude modulator
Optical architecture
PlWW
(PR(!.d, V)}]
(6)
dw
Using the Jacobi-Anger formula23, however, the exponential term can be expanded as
for complex filter implementation
exp[jw{(R(& v)I + A}-’ the Fourier plane of the correlator system. One suitable way to implement AMPOF correlation filters having both multilevel amplitude and multilevel phase is to arrange two liquid crystal televisions (LCTVs) or any other spatial light modulator in a multiplicative format6 as shown in Fig. 1. The first spatial light modulator works as an amplitude modulator while the other functions as a phase modulator. In the case of the LCTV, the polarizers are required to obtain the necessary amplitude modulation.
x cos{bOu where
cos{bOu
(PR(?
-
V)}]
C bU.b(wl{A + IR(u,v)ll)
=
- ‘PR(&
(7)
V)}
l,v=O
E, =
(8)
2,v > 0
and J,(.) is the nth order Bessel function. Accordingly, (6) now becomes ~(4
= 2
%[R(u, V)]cos([bou -
‘PR(%
V)])
(9)
v= 0
Analysis The discretization of the filter function is expected to introduce a non-linearity in the filter plane. The analysis of the effect of discretization can be carried out by considering the defining function for AMPOF14 Lnpof(~,
v)l + A)-’
P(w) exp[jw#(u,
v) =
qpqu,
v>I + A)-’
x exp{-jvRk
v)>
(1)
where IR(u, v)l is the amplitude of the Fourier spectrum R(u, v) of the reference function r(x,y), (P~(u, v) is the phase factor of R(u, v), and D and A are either constants or functions of u and v. When D = 1 and as A approaches zero, the resulting filter function approaches the characteristics of an inverse filter and, therefore, it could yield a Dirac-delta correlation output. The reduced continuous valued linear filter function
where
~11= (O~)W’S_R_ p(w)
K[R(u,
x Jn(wlP + IR(u, v,I>) dw
(10)
Now consider the characteristics of the full-wave (odd) mth law device23 shown in Fig. 2. The full wave (odd) function pa(x) can be described as Dx”‘, PO(X) =
0, I -Dx”,
x>o x=0 x < 0
(11)
where m is a non-negative real number. We may define PO(X) = g(x) + 4(x)
(12)
where Fcontinuous(u,
V)
=
{ IR(u, x
V)I
+
A}-’
exp{-jvR(&
VI>
(2)
can be synthesized as an amplitude and phase modulated bandpass function22 B(u, v) = { IR(u, v)I + A}-’ cos{b@ -
(PR(&
v)}
(134 and
(3)
4(x)=
0,
x>o
-Dxm + Im(w) > 0,
x < 0
where b. is the central frequency of the bandpass function. The non-linear characteristic of the filter plane thresholding device is obtained next by taking the Fourier transform of this modulated signal. Such a transform method of analysis23 of the non-linear system yields a generalized non-linear filter expression. The Fourier transform of (3) leads to
9
1
2D j-
s -cc
p(B) exp(-jwB)
dB
:’ ,’ D
/’ I’ i/i __,’
(4)
w P(w) exp(jwB) dw J’-lx
,’
‘--,‘=I,2
m=O
,’ _,I
*
,’
/’
,’
/”
-D
,’
8’
(5)
/m=l
/‘#
1
-3
where p(B) is any time domain signal and B is a dummy variable. The output of the non-linear system is given by its inverse Fourier transform as p(B) = (1/27r)
,
,’ ,’
00 P(w) =
m=2
Fig. 2
---2D
The mth law transfer characteristics
2
3
X
Discretized amplitude-modulated
phase-only
IE
W =
---------I
KZ[R(% v>lcos([bou - (PR(U,41)
p(B) = 2
rl_____,
Accordingly, appropriate expansion of the cosine term22 yields the output of the filter plane non-linearity and is given by + l)I’({v - m}/2)
DE,,(j)n-mI’(m _____
1 -iv
Fig. 3
&I~(‘) = ~2m{A +
T-----’ C- line
D(j)‘-*r(m
m61
‘lAB)
+ l),
[-D/(-jw)m+‘]I’(m
m< 1
(14b)
respectively where I’(.) is the gamma function. Note that the corresponding inverse transformation contour C_ of (13a) must lie below the w = v (real) axis. We choose C, to be the line w = je+ v and C_ to be the line w = -je+ v as shown in Fig. 3. Substituting 14(a) and 14(b) into (11) results in H,[R(u, v)] = (EJ27r) jT(m
= T{A + Iqu,
m)/2})
(19)
+ 1)
+ l)I’([l - m]/2)
v)lp(i + [{l
x exp{jv[-cpR(u,
(Ida)
and
Q(w)=
-
for odd n. Now considering only the first term (i.e. n = 1) among the odd harmonics for the non-linear correlation output (caused by the thresholded filter), we obtain
Taking Fourier transforms of (13a) and (13b), one obtains + l),
pqu,v)I)mr(i + {(v
x ew{_iv[-wt(u,v) + b0ul)
Inversion contours of the mth law device
G(w) = [D/(jw)“‘l]r(m
(18)
v=o odd
C+ line
+iw
---___________
95
Now using (16) and (17), (9) can be written as
I
u + jE plane
et al.
filter: K. M. lftekharuddin
v) +
- m}/2])
b04)
(20)
This is a generalized expression for the non-linear AMPOF. Next, we may obtain different classes of nonlinear AMPOFs for different values of m. For example, for the linear thresholding as shown in Fig. 2 (i.e. m = l), the thresholded AMPOF can be obtained as P amp&,
v) =PII(B)
= D’M +
x exp{j[bOu
-
IR(u,v)ll-’
(PR(U,
v>l)
(21)
when D’ = (D/n). On the other hand, for the case of hard thresholding (i.e. m = 0), we find the thresholded AMPOF as
jE+m
D je_m {Jn (w/IA + I&(% v)I I)
x
&mpof,thresholded
[./
=
/(jw)“+l} dw jcfoa + C-D)
s jE-m
/(-jw)“+l}
{Jn(wl{A
+ mu>
VI))
dw]
(15)
Now letting v = - w in the second integral of (15) we have, H,[R(u, v)] = [DE, j”I’(m + 1)/r]
J
(u, v) = plO(B) A’j
exp{j[bOu
jE_m
Discretization
4
h
X
+ 1))
IR(KVW>] jP+m J +
jp~
Jn [~/I”+‘1
dt
M
(22)
considerations
/(jw)mf'} dw
l{G
v)])
Threshold line angle (TLA) (the line that divides the phase plane in two discrete levels) plays an extremely
{Jn(wlV + IR(KVI>)
= [{DEn(j)n-m-‘r(m
‘PR(k
where A’ = (40/7r). Note that the thresholded AMPOF expression is nothing but the BPOF function. Hence, this derivation confirms our earlier observation that the hard thresholding of both the amplitude and phase part of the AMPOF results in a BPOF.
je+x
X
-
(16)
for odd n when < = w/(,4 + ]R(u, v)]}. Now by integrating the last term of (16) around the contour shown in Fig. 4, one obtains that
Fig. 4
Integration
contours for (15)
96
Discretized amplitude-modulated phase-only f;ilter: K. M. lftekharuddin et al.
0: 60 60
0
”
Fig. 6
(b)
Truckmage
”
Normalized autocorrelation
Grey level images: (a) tank; and (b) truck
important role in determining BPOF performance7. Accordingly, we shall attempt to explore the effect of different phase discretization values on the detection performance of the filter. First, as discussed in the introduction, we binarize both the amplitude and phase terms of the AMPOF separately. We use 0 and 1 values for the amplitude discretization and 0” and 90” for the phase discretization of the AMPOF. This type of binary quantization amounts to introducing a hard-clipping non-linearity in the filter plane. This particular AMPOF will be referred to as the BB-AMPOF. Next, as an alternative exploration, we binarize the AMPOF phase in some angles other than the above two customary angles, namely 0” and 180”.
with AMPOF
only filter (TB-AMPOF). The TB-AMPOF phase may again consist of two different quantization levels (i.e. 0” and 90”, and 0” and 180’). Further, the phase part may be discretized into three different levels (such as O”, 90” and 180”) along with a ternerized amplitude. This latter AMPOF will be referred to as TT-AMPOF. Obviously, the ternerization process is expected also to introduce a moderate non-linearity (i.e., 0
Fig. 5
0
and discussion
As real-life target recognition problems mostly involve grey level images, we consider here respectively images of a tank and a truck as shown in Figs 5(a) and 5(b). The autocorrelation obtained using the tank image for a continuous AMPOF is first computed. This normalized value of the autocorrelation function is shown in Fig. 6. For consistency, we maintain the same scaling factor throughout the rest of the simulation results. The magnitude response of BB-AMPOF is obtained as shown in Fig. 7. The symmetric nature of the magnitude is evident from Fig. 7. The resulting magnitude levels are obvious manifestations of the filter-plane hardclipping non-linearity. As mentioned earlier, both the choices of BB-AMPOF phase quantization are explored next. The resulting autocorrelation for phase quantization of 0” and 90” and that for 0” and 180” are obtained as shown respectively
1
0.6
$ 0.6
One may notice a subtle similarity between TPAF and BB-AMPOF since both filters involve simultaneous binarization of both the phase and the amplitude at the same discrete levels. Because of the distinct difference between the continuous POF and AMPOF functions, however, one may discretize the AMPOF amplitude at more than just two levels. Consequently, in this work, AMPOF amplitude is ternerized (including at an intermediate level between 0 and 1) while the phase part is only binarized as before. This particular discretization results in a ternary amplitude-modulated binary phase-
2 go.4 0.2
60
"
Fig. 7
Magnitude
0
0
response of BB-AMPOF
"
Discretized amplitude-modulated phase-only filter: K. M. lftekharuddin et al. x 10' 10
1(a)
60
2. (b)
Fig. 8 Autocorrelation with BB-AMPOF (a) 0” and 90”; and (b) 0” and 180”
when phase level is
in Figs S(a) and 8(b). Note that the autocorrelation value shows a considerable enhancement when compared with the continuous case of Fig. 6. It also shows an increment of the noise floor. However, comparison between Figs 8(a) and 8(b) reveals that the latter phase quantization (of 0” and 180’) offers increased correlation and a reduced noise level. An alternative strategy for binarizing the amplitude (while still maintaining the phase quantization of 0” and 180’) may be to obtain an increased non-linearity in the filter plane. This is accomplished as shown by the amplitude response of Fig. 9(a) and the corresponding autocorrelation plot of Fig. 9(b). As expected, the increased non-linearity in the filter plane offers an enhanced autocorrelation at the expense of increased noise floor. The noise floor increment in particular is due to the introduction of a larger binarization error involved in Fig. 9(a) than, for example, that in Fig. 7. A TB-AMPOF is realized next and the corresponding magnitude response is shown in Fig. 10. The symmetric nature of the magnitude is again evident. Figure 10 also confirms that the ternerization process translates itself to a moderate non-linear transformation in the filter plane. The TB-AMPOF phase is realized using 0” and 180” discrete levels. The TT-AMPOF, on the other hand, is realized by incorporating an intermediate phase value of 90”. The respective unnormalized autocorrelation functions are obtained as shown plotted in Figs 1l(a) and 1l(b) respectively. The TT-AMPOF autocorrelation values are found to be significantly larger than those corresponding to TB-AMPOF. More importantly, by
Fig. 9 Alternative discretization for BB-AMPOF when phase level is 0” and 180”; (a) magnitude response; and (b) autocorrelation
0.2.
”
Fig. 10 Magnitude 0” and 180”
0
0
response of TB-AMPOF
”
when phase level is
comparing Figs 1l(b) and 9(b), we find that the ternerization process introduces much less error, thus resulting in a decreased noise floor. The input tank image of Fig. 5(a) is next corrupted with a white Gaussian noise of standard deviation 16, for example. The corresponding autocorrelation obtained using a continuous valued AMPOF is shown in Fig. 12(a). Figure 12(b), on the other hand, shows the autocorrelation obtained using the discretized TTAMPOF for the same noise-added image. Evidently, the target detection capability of TT-AMPOF is comparable to that of the AMPOF even in the case of a moderate noisy environment. In order for exclusive
98
Discretized amplitude-modulated
phase-only
et al.
filter: K. M. lftekharuddin
0.12
XIOS
(a)
3 (a)
0.1
P
2.5.
iO.08
s
$0.06
sfJ 0.04 8 ‘8
c 0.02 0 60
x10'
10 1
5.
(b)
W 4.
Fig. 13 Fig. 11 Autocorrelation (b) TT-AMPOF
with (a) TB-AMPOF;
Noisy correlation
with (a) AMPOF;
and (b) TT-AMPOF
and
1.
(a)
0.6. B
80.6. 6
L
+ 0
P4
0
Y
0 ”
0 noisevariance
x104
Fig. 14
14. 12.
Whout discretization Withdiscrelization
Autocorrelation
versus noise variance with TT-AMPOF
(b)
I 10. g 6. 6 'z 6. L
84 2
600 60
"
Fig. 12 AMPOF;
0
0
visualization of the noise floors of Fig. 12, the respective noisy correlation planes are plotted as shown in Figs 13(a) and 13(b) respectively. The noise enhancement caused by the discretization process is obvious in Fig. 13(b). However, the significant optical efficiency improvement in the correlation peak location still enables one to have a comfortable target detection in the case of extremely noisy conditions. Figure 14 shows the improvement of autocorrelation obtained due to discretization in the noise related case.
"
Autocorrelation (when standard deviation and (b) TT-AMPOF
is 16) with (a)
Finally, the crosscorrelation values are obtained using the noiseless tank and truck images of Figs 5(a) and
Discretized amplitude-modulated
phase-only
filter: K. M. lftekharuddin
et al.
99
introduction in the discretization process. The discretization was not found to affect the noise-related performance of the filter as good correlation performance is observed even in extremely noisy cases. The excellent out-of-class target detection feature of AMPOF is also retained by the TT-AMPOF. These reasons alone may actually provide motivation for realtime implementation of AMPOF, thus facilitating faster automatic target recognition. Acknowledgement Two of the authors (KM1 and MAK) were supported for this research, in part, by the Advanced Research Program Agency (administered by Ohio Aerospace Institute) Grant No. MDA972-93-1-0015. x10'
3.1 1 (b’ f
References
I
I
3
2
%25 6 2 2 L 8 1.5
3
1
4
5 "
0
0
"
6
Fig. 15 Crosscorrelation using tank and truck images with (a) AMPOF; (b) TT-AMPOF
5(b). These are shown in Figs 15(a) and 15(b) for AMPOF and TT-AMPOF respectively. Even though Fig. 15(b) shows some ‘peak’, no distinct isolated crosscorrelation value may be recognized. This is because, there are no clear single correlation peak and side-lobe values that are often necessary to evaluate different statistical performance metrics”~12~‘9~20. Interestingly, therefore, the discretization process does not degrade the excellent crosscorrelation feature of AMPOF. This is confirmed in Fig. 15(b) since no peak is identified.
13
Conclusions
14
The advantage of having discretized the continuous valued complex AMPOF function is assessed in terms of both autocorrelation and crosscorrelation statistics. We have shown that discretization of the amplitude part of the filter function results in improved detection performance (when the correlation value of Fig. 6 is compared with those of Figs 8, 11 and 12). Note that the improved correlation performance is directly related to all other statistical detection performance metrics available in the literature.“~‘2~1920 The intuitive reason of the improvement is that the ternerized amplitude results in AMPOF being more like its continuous counterpart. The choice of different phase levels is often dependent on the scene-specific relative spatial frequency phase alignments and the odd-even sensitivity variation of the discretized filter itselfi8. Target recognition for grey level images using TT-AMPOF also results in better light utilization at the expense of an increased noise floor due to error
12
15
16 17 18
19 20
21 22 23
Gaskill, J. D. Linear Systems, Fourier Transforms and Optics, Wiley, New York (1978) Juday, R. D. Correlation with a spatial light modulator having phase and amplitude cross coupling, Appl Opt, 28 (1989) 4865-4869 Ross, W., Psaltis, D., Anderson, R. Two-dimensional magnetooptic spatial light modulator for signal processing, Opt Eng, 24 (I 983) 485-490 Downe, J. D. Optimal binary phase and amplitude correlation filters for polarization-rotating spatial light modulators, Opt Eng, 32 (1993) 1886-1896 Freeman, M. O., Brown, T. A., Walba, D. M. Quantized complex ferroelectric liquid crystal spatial light modulators, Appl Opt, 31 (1992) 3917-3929 Gregory, D. A., Kirsch, J. C., Tam, E. C. Full complex modulation using liquid-crystal televisions, Appl Opt, 31 (1992) 163-165 Oppenheim, A. V., Lim, J. S. The importance of phase in signals. Proc IEEE, 69 (1981) 529-541 Yu, F. T. Linear optimization in synthesis of nonlinear spatial filters, IEEE Tram If Theory, 17 (1977) 524-521 Gara, A. D. Real-time optical correlation of 3-D scenes, Appl Opt, 16 (1977) 149-153 Horner, J. L., Gianino, P. D. Phase-only matched filtering, Appl Opt, 23 (1984) 812-816 Kaura, M. A., Rhodes, W. T. Optical correlators performance using a phase-with-constrained-magnitude complex spatial filter, Appl Opt, 25 (1990) 2587-2593 Yaroslavsky, L. P. Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recoenition. Aonl Out. 31 (1992) 1677-1679 Mu,%. G., Wang, i. M., Wang: Z. Q. Amplitude-compensated matched filtering, Appl Opt, 27 (1988) 3461-3463 Awwal, A. A. S., Karim, M. A., Jahan, S. R. Improved correlation discrimination using an amplitude modulated phaseonly filter, Appl Opt, 29 (1990) 233-236 Zheng, S. H., Karim, M. A., Gao, M. L. Amplitude-modulated phase-only filter performance in presence of noise, Opt Commun, 89 (1992) 296-305 Psaltis, D., Peak, E., Venkatesh, S. Optical image correlation with binary spatial light modulator, Opt Eng, 3 (1986) 698-704 Horner, J. L., Leger, J. R. Pattern recognition with binary phase-only filters, Appl Opt, 24 (1985) 609-611 Flannery, D. L., Loomis, J. S., Milkovich, M. E. Design elements of binary phase-only correlation filters, Appl Opt, 27 (1988) 4231-4235 Horner, J. L., Javidi, B., Wang, J. Analysis of the binary phaseonly filter, Oat Commun. 91 (1992) 189-192 Flabnery, D.,&Loomis, J. S., Milkoiich, M. E. Transform-ratio ternary phase-amplitude filter formulation for improved correlation discrimination, Appl Opt, 27 (1988) 4079-4083 Goodman, J. W., Silvestri, A. M. Some effects of Fourierdomain phase quantization, IBM J Res Dev 3 (1970) 478-481 Javidi, B. Generalization of the linear matched filter concept to nonlinear filters, Appl Opt, 29 (1990) 1215-1224 Davenport, W. B., Root, W. L. An Introduction to the Theory of Random Signals and Noise Chapter 13. pp. 271-31 I, McGrawHill, New York (1958)
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Discretized amplitude-modulated
phase-only filter: K. M. lfiekharuddin et al.
Biographies
Mohammad A. Karim is the director of the Center for Electra-Optics, Chairperson of the Electrical Engineering Department, and a professor of both electro-optics and electrical engineering at the University of Dayton, USA. He received his BS in physics in 1976 from the University of Dacca, Bangladesh, and MS degrees in physics and electrical engineering, and a PhD in electrical engineering from the University of Alabama, USA, respectively in 1978, 1979 and 198 1. Dr Karim’s ongoing research includes the areas of optical information/image processing, pattern/target recognition, optical systems design, infrared systems characterization, displays, optical computing, and interconnects. He is the author of over 100 journal papers, over 100 conference papers, numerous published book reviews, and the graduate text books Electra-Optical Devices & Systems, Optical Computing: An Introduction and Digital Design: A Pragmatic Approach. Dr Karim edited the reference book Electra-Optical Displays. He served as the guest
editor of four Optical Engineering special issues on electro-optical displays (August 1990), infrared imaging systems (November 1991), acquisition, pointing, and tracking (November 1993) and optical remote sensing and image processing (November 1995), and of one Optics & Laser Technology special issue on optical computing (August 1994). He is a Fellow of the Optical Society of America.
Khan M. Iftekharuddin is employed by BDM Federal, Dayton, Ohio, USA. He received a BSc degree in electrical and electronics engineering from Bangladesh Institute of Technology (Bangladesh) in 1989 and an MS degree in electrical engineering from the University of Dayton, USA, in 1991. He received his PhD degree in electrical engineering at the University of Dayton in 1995. His research interests include digital and/or optical image processing and pattern recognition, optical computing and interconnects, information processing and optical system design. He is a member of IEEE and OSA.
ELSEVIER ADVANCED
0030-3992(95)00066-6
TECHNOLOGY
Discretized modulated K. M. IFTEKHARUDDIN,
amplitudephase-only M. A. KARIM,
filter
P. W. ELOE, A. A. S. AWWAL
The amplitude-modulated phase-only filter (AMPOF) effectively utilizes both amplitude and phase information to obtain a narrower and larger autocorrelation peak. To make it readily suitable for real-time target recognition applications, we consider discretizing both its amplitude and phase. Accordingly, both amplitude and phase of the AMPOF are ternerized to yield improved performance statistics of the correlator. KEYWORDS:
amplitude-modulated
phase-only
filter, optical
correlators,
correlation
peaks
sharper than that of the conventional matched filters. However, several recent studies”m’3 have seriously questioned the validity of totally ignoring the amplitude content of an image. A particular complex filter, namely, an amplitude-modulated phase-only filter (AMPOF) 14,15for example, retains information regarding both phase and amplitude and it has been found to offer better correlation discrimination than the POF.
Introduction The more widely used optical correlators utilize filters that encode the impulse response of an input pattern. Such correlators accordingly generate an output light field that corresponds to the correlated signal between the impulse response and the input’. These correlator architectures largely rely on the appropriate spatial light modulator (SLM) technology* for real-time insertion of the input pattern as well as the filter(s) in the correlator. The magneto-optic SLMs3, for example, are capable of simultaneously displaying both amplitude and phase information. On the other hand, an optically controlled ferroelectric liquid crystal SLM may be used to control the amounts of magnitude and phase modulation independently4,5. Alternatively, a pair of liquid crystal televisions (LCTV) may be used to obtain a complex modulation with independently controllable phase and amplitude6. This particular optical architecture using two LCTVs, polarizers and lenses can be used to realize the particular multilevel phase-multilevel amplitude filters considered herein.
In our effort to explore the flexibility of AMPOF, it is important to note that a number of studies have considered binarizing the POF, that is, phase16J7. Subsequent analysis of binary POF (BPOF)‘8>‘9 have shown that the corresponding correlator is better than the POF-based correlator. Further, multiplying the BPOF phase with a binary amplitude pattern has led to a newer class of POF, namely, a ternary phaseamplitude filter (TPAF)*’ which has been shown to have an even better performance. Unlike POF, however, the AMPOF is a frequency domain complex valued function with distinct amplitude and phase. Accordingly, a separate discretization of the phase and amplitude factors of the AMPOF, even with different discrete levels for each of the components, is quite possible 21. In reality, such multi-level discretization of AMPOF readily lends itself to suitable SLM and/or LCTV implementation. In this paper, we consider ternerizing both the amplitude and the phase of the AMPOF function. We show that discretization does indeed result in an improved statistical performance measure of the correlator.
For some time, only the phase of the filter was considered to be important’ in pattern recognition. The significance of the phase preserving filters has been illustrated by linear optimization* and the bleached spatial filter9 respectively. In fact, a phase-only filter (POF)‘O was found to offer correlation peaks much KMI and MAK are at the University of Dayton, Department of Electrical Engineering and Center for Electra-Optics, 300 College Park, Dayton, OH 45469-0245, USA. KMI is currently at BDM Federal, 1990 Founders Drive, Kettering, OH 45420, USA. PWE is at the University of Dayton, Department of Mathematics, 300 College Park, Dayton, OH 45469-2316, USA. AASA is at Wright State University, Department of Computer Science and Computer Engineering, Dayton, OH 45435, USA. Received 27 April 1995.
Correlator
system
As elaborated, an AMPOF-based system takes on a Vander Lugt type of architecture. In such a configuration, the input scene is first Fourier transformed with the AMPOF function introduced at 93
94
Discretized amplitude-modulated
phase-only
filter: K. M. lftekharuddin
et al.
From (3) and (5) we obtain, p(B) = (1/2~) l” X COs(boiy
Inputimage wansparew Fig. 1
Phak modulator
Amplitude modulator
Optical architecture
PlWW
(PR(!.d, V)}]
(6)
dw
Using the Jacobi-Anger formula23, however, the exponential term can be expanded as
for complex filter implementation
exp[jw{(R(& v)I + A}-’ the Fourier plane of the correlator system. One suitable way to implement AMPOF correlation filters having both multilevel amplitude and multilevel phase is to arrange two liquid crystal televisions (LCTVs) or any other spatial light modulator in a multiplicative format6 as shown in Fig. 1. The first spatial light modulator works as an amplitude modulator while the other functions as a phase modulator. In the case of the LCTV, the polarizers are required to obtain the necessary amplitude modulation.
x cos{bOu where
cos{bOu
(PR(?
-
V)}]
C bU.b(wl{A + IR(u,v)ll)
=
- ‘PR(&
(7)
V)}
l,v=O
E, =
(8)
2,v > 0
and J,(.) is the nth order Bessel function. Accordingly, (6) now becomes ~(4
= 2
%[R(u, V)]cos([bou -
‘PR(%
V)])
(9)
v= 0
Analysis The discretization of the filter function is expected to introduce a non-linearity in the filter plane. The analysis of the effect of discretization can be carried out by considering the defining function for AMPOF14 Lnpof(~,
v)l + A)-’
P(w) exp[jw#(u,
v) =
qpqu,
v>I + A)-’
x exp{-jvRk
v)>
(1)
where IR(u, v)l is the amplitude of the Fourier spectrum R(u, v) of the reference function r(x,y), (P~(u, v) is the phase factor of R(u, v), and D and A are either constants or functions of u and v. When D = 1 and as A approaches zero, the resulting filter function approaches the characteristics of an inverse filter and, therefore, it could yield a Dirac-delta correlation output. The reduced continuous valued linear filter function
where
~11= (O~)W’S_R_ p(w)
K[R(u,
x Jn(wlP + IR(u, v,I>) dw
(10)
Now consider the characteristics of the full-wave (odd) mth law device23 shown in Fig. 2. The full wave (odd) function pa(x) can be described as Dx”‘, PO(X) =
0, I -Dx”,
x>o x=0 x < 0
(11)
where m is a non-negative real number. We may define PO(X) = g(x) + 4(x)
(12)
where Fcontinuous(u,
V)
=
{ IR(u, x
V)I
+
A}-’
exp{-jvR(&
VI>
(2)
can be synthesized as an amplitude and phase modulated bandpass function22 B(u, v) = { IR(u, v)I + A}-’ cos{b@ -
(PR(&
v)}
(134 and
(3)
4(x)=
0,
x>o
-Dxm + Im(w) > 0,
x < 0
where b. is the central frequency of the bandpass function. The non-linear characteristic of the filter plane thresholding device is obtained next by taking the Fourier transform of this modulated signal. Such a transform method of analysis23 of the non-linear system yields a generalized non-linear filter expression. The Fourier transform of (3) leads to
9
1
2D j-
s -cc
p(B) exp(-jwB)
dB
:’ ,’ D
/’ I’ i/i __,’
(4)
w P(w) exp(jwB) dw J’-lx
,’
‘--,‘=I,2
m=O
,’ _,I
*
,’
/’
,’
/”
-D
,’
8’
(5)
/m=l
/‘#
1
-3
where p(B) is any time domain signal and B is a dummy variable. The output of the non-linear system is given by its inverse Fourier transform as p(B) = (1/27r)
,
,’ ,’
00 P(w) =
m=2
Fig. 2
---2D
The mth law transfer characteristics
2
3
X
Discretized amplitude-modulated
phase-only
IE
W =
---------I
KZ[R(% v>lcos([bou - (PR(U,41)
p(B) = 2
rl_____,
Accordingly, appropriate expansion of the cosine term22 yields the output of the filter plane non-linearity and is given by + l)I’({v - m}/2)
DE,,(j)n-mI’(m _____
1 -iv
Fig. 3
&I~(‘) = ~2m{A +
T-----’ C- line
D(j)‘-*r(m
m61
‘lAB)
+ l),
[-D/(-jw)m+‘]I’(m
m< 1
(14b)
respectively where I’(.) is the gamma function. Note that the corresponding inverse transformation contour C_ of (13a) must lie below the w = v (real) axis. We choose C, to be the line w = je+ v and C_ to be the line w = -je+ v as shown in Fig. 3. Substituting 14(a) and 14(b) into (11) results in H,[R(u, v)] = (EJ27r) jT(m
= T{A + Iqu,
m)/2})
(19)
+ 1)
+ l)I’([l - m]/2)
v)lp(i + [{l
x exp{jv[-cpR(u,
(Ida)
and
Q(w)=
-
for odd n. Now considering only the first term (i.e. n = 1) among the odd harmonics for the non-linear correlation output (caused by the thresholded filter), we obtain
Taking Fourier transforms of (13a) and (13b), one obtains + l),
pqu,v)I)mr(i + {(v
x ew{_iv[-wt(u,v) + b0ul)
Inversion contours of the mth law device
G(w) = [D/(jw)“‘l]r(m
(18)
v=o odd
C+ line
+iw
---___________
95
Now using (16) and (17), (9) can be written as
I
u + jE plane
et al.
filter: K. M. lftekharuddin
v) +
- m}/2])
b04)
(20)
This is a generalized expression for the non-linear AMPOF. Next, we may obtain different classes of nonlinear AMPOFs for different values of m. For example, for the linear thresholding as shown in Fig. 2 (i.e. m = l), the thresholded AMPOF can be obtained as P amp&,
v) =PII(B)
= D’M +
x exp{j[bOu
-
IR(u,v)ll-’
(PR(U,
v>l)
(21)
when D’ = (D/n). On the other hand, for the case of hard thresholding (i.e. m = 0), we find the thresholded AMPOF as
jE+m
D je_m {Jn (w/IA + I&(% v)I I)
x
&mpof,thresholded
[./
=
/(jw)“+l} dw jcfoa + C-D)
s jE-m
/(-jw)“+l}
{Jn(wl{A
+ mu>
VI))
dw]
(15)
Now letting v = - w in the second integral of (15) we have, H,[R(u, v)] = [DE, j”I’(m + 1)/r]
J
(u, v) = plO(B) A’j
exp{j[bOu
jE_m
Discretization
4
h
X
+ 1))
IR(KVW>] jP+m J +
jp~
Jn [~/I”+‘1
dt
M
(22)
considerations
/(jw)mf'} dw
l{G
v)])
Threshold line angle (TLA) (the line that divides the phase plane in two discrete levels) plays an extremely
{Jn(wlV + IR(KVI>)
= [{DEn(j)n-m-‘r(m
‘PR(k
where A’ = (40/7r). Note that the thresholded AMPOF expression is nothing but the BPOF function. Hence, this derivation confirms our earlier observation that the hard thresholding of both the amplitude and phase part of the AMPOF results in a BPOF.
je+x
X
-
(16)
for odd n when < = w/(,4 + ]R(u, v)]}. Now by integrating the last term of (16) around the contour shown in Fig. 4, one obtains that
Fig. 4
Integration
contours for (15)
96
Discretized amplitude-modulated phase-only f;ilter: K. M. lftekharuddin et al.
0: 60 60
0
”
Fig. 6
(b)
Truckmage
”
Normalized autocorrelation
Grey level images: (a) tank; and (b) truck
important role in determining BPOF performance7. Accordingly, we shall attempt to explore the effect of different phase discretization values on the detection performance of the filter. First, as discussed in the introduction, we binarize both the amplitude and phase terms of the AMPOF separately. We use 0 and 1 values for the amplitude discretization and 0” and 90” for the phase discretization of the AMPOF. This type of binary quantization amounts to introducing a hard-clipping non-linearity in the filter plane. This particular AMPOF will be referred to as the BB-AMPOF. Next, as an alternative exploration, we binarize the AMPOF phase in some angles other than the above two customary angles, namely 0” and 180”.
with AMPOF
only filter (TB-AMPOF). The TB-AMPOF phase may again consist of two different quantization levels (i.e. 0” and 90”, and 0” and 180’). Further, the phase part may be discretized into three different levels (such as O”, 90” and 180”) along with a ternerized amplitude. This latter AMPOF will be referred to as TT-AMPOF. Obviously, the ternerization process is expected also to introduce a moderate non-linearity (i.e., 0
Fig. 5
0
and discussion
As real-life target recognition problems mostly involve grey level images, we consider here respectively images of a tank and a truck as shown in Figs 5(a) and 5(b). The autocorrelation obtained using the tank image for a continuous AMPOF is first computed. This normalized value of the autocorrelation function is shown in Fig. 6. For consistency, we maintain the same scaling factor throughout the rest of the simulation results. The magnitude response of BB-AMPOF is obtained as shown in Fig. 7. The symmetric nature of the magnitude is evident from Fig. 7. The resulting magnitude levels are obvious manifestations of the filter-plane hardclipping non-linearity. As mentioned earlier, both the choices of BB-AMPOF phase quantization are explored next. The resulting autocorrelation for phase quantization of 0” and 90” and that for 0” and 180” are obtained as shown respectively
1
0.6
$ 0.6
One may notice a subtle similarity between TPAF and BB-AMPOF since both filters involve simultaneous binarization of both the phase and the amplitude at the same discrete levels. Because of the distinct difference between the continuous POF and AMPOF functions, however, one may discretize the AMPOF amplitude at more than just two levels. Consequently, in this work, AMPOF amplitude is ternerized (including at an intermediate level between 0 and 1) while the phase part is only binarized as before. This particular discretization results in a ternary amplitude-modulated binary phase-
2 go.4 0.2
60
"
Fig. 7
Magnitude
0
0
response of BB-AMPOF
"
Discretized amplitude-modulated phase-only filter: K. M. lftekharuddin et al. x 10' 10
1(a)
60
2. (b)
Fig. 8 Autocorrelation with BB-AMPOF (a) 0” and 90”; and (b) 0” and 180”
when phase level is
in Figs S(a) and 8(b). Note that the autocorrelation value shows a considerable enhancement when compared with the continuous case of Fig. 6. It also shows an increment of the noise floor. However, comparison between Figs 8(a) and 8(b) reveals that the latter phase quantization (of 0” and 180’) offers increased correlation and a reduced noise level. An alternative strategy for binarizing the amplitude (while still maintaining the phase quantization of 0” and 180’) may be to obtain an increased non-linearity in the filter plane. This is accomplished as shown by the amplitude response of Fig. 9(a) and the corresponding autocorrelation plot of Fig. 9(b). As expected, the increased non-linearity in the filter plane offers an enhanced autocorrelation at the expense of increased noise floor. The noise floor increment in particular is due to the introduction of a larger binarization error involved in Fig. 9(a) than, for example, that in Fig. 7. A TB-AMPOF is realized next and the corresponding magnitude response is shown in Fig. 10. The symmetric nature of the magnitude is again evident. Figure 10 also confirms that the ternerization process translates itself to a moderate non-linear transformation in the filter plane. The TB-AMPOF phase is realized using 0” and 180” discrete levels. The TT-AMPOF, on the other hand, is realized by incorporating an intermediate phase value of 90”. The respective unnormalized autocorrelation functions are obtained as shown plotted in Figs 1l(a) and 1l(b) respectively. The TT-AMPOF autocorrelation values are found to be significantly larger than those corresponding to TB-AMPOF. More importantly, by
Fig. 9 Alternative discretization for BB-AMPOF when phase level is 0” and 180”; (a) magnitude response; and (b) autocorrelation
0.2.
”
Fig. 10 Magnitude 0” and 180”
0
0
response of TB-AMPOF
”
when phase level is
comparing Figs 1l(b) and 9(b), we find that the ternerization process introduces much less error, thus resulting in a decreased noise floor. The input tank image of Fig. 5(a) is next corrupted with a white Gaussian noise of standard deviation 16, for example. The corresponding autocorrelation obtained using a continuous valued AMPOF is shown in Fig. 12(a). Figure 12(b), on the other hand, shows the autocorrelation obtained using the discretized TTAMPOF for the same noise-added image. Evidently, the target detection capability of TT-AMPOF is comparable to that of the AMPOF even in the case of a moderate noisy environment. In order for exclusive
98
Discretized amplitude-modulated
phase-only
et al.
filter: K. M. lftekharuddin
0.12
XIOS
(a)
3 (a)
0.1
P
2.5.
iO.08
s
$0.06
sfJ 0.04 8 ‘8
c 0.02 0 60
x10'
10 1
5.
(b)
W 4.
Fig. 13 Fig. 11 Autocorrelation (b) TT-AMPOF
with (a) TB-AMPOF;
Noisy correlation
with (a) AMPOF;
and (b) TT-AMPOF
and
1.
(a)
0.6. B
80.6. 6
L
+ 0
P4
0
Y
0 ”
0 noisevariance
x104
Fig. 14
14. 12.
Whout discretization Withdiscrelization
Autocorrelation
versus noise variance with TT-AMPOF
(b)
I 10. g 6. 6 'z 6. L
84 2
600 60
"
Fig. 12 AMPOF;
0
0
visualization of the noise floors of Fig. 12, the respective noisy correlation planes are plotted as shown in Figs 13(a) and 13(b) respectively. The noise enhancement caused by the discretization process is obvious in Fig. 13(b). However, the significant optical efficiency improvement in the correlation peak location still enables one to have a comfortable target detection in the case of extremely noisy conditions. Figure 14 shows the improvement of autocorrelation obtained due to discretization in the noise related case.
"
Autocorrelation (when standard deviation and (b) TT-AMPOF
is 16) with (a)
Finally, the crosscorrelation values are obtained using the noiseless tank and truck images of Figs 5(a) and
Discretized amplitude-modulated
phase-only
filter: K. M. lftekharuddin
et al.
99
introduction in the discretization process. The discretization was not found to affect the noise-related performance of the filter as good correlation performance is observed even in extremely noisy cases. The excellent out-of-class target detection feature of AMPOF is also retained by the TT-AMPOF. These reasons alone may actually provide motivation for realtime implementation of AMPOF, thus facilitating faster automatic target recognition. Acknowledgement Two of the authors (KM1 and MAK) were supported for this research, in part, by the Advanced Research Program Agency (administered by Ohio Aerospace Institute) Grant No. MDA972-93-1-0015. x10'
3.1 1 (b’ f
References
I
I
3
2
%25 6 2 2 L 8 1.5
3
1
4
5 "
0
0
"
6
Fig. 15 Crosscorrelation using tank and truck images with (a) AMPOF; (b) TT-AMPOF
5(b). These are shown in Figs 15(a) and 15(b) for AMPOF and TT-AMPOF respectively. Even though Fig. 15(b) shows some ‘peak’, no distinct isolated crosscorrelation value may be recognized. This is because, there are no clear single correlation peak and side-lobe values that are often necessary to evaluate different statistical performance metrics”~12~‘9~20. Interestingly, therefore, the discretization process does not degrade the excellent crosscorrelation feature of AMPOF. This is confirmed in Fig. 15(b) since no peak is identified.
13
Conclusions
14
The advantage of having discretized the continuous valued complex AMPOF function is assessed in terms of both autocorrelation and crosscorrelation statistics. We have shown that discretization of the amplitude part of the filter function results in improved detection performance (when the correlation value of Fig. 6 is compared with those of Figs 8, 11 and 12). Note that the improved correlation performance is directly related to all other statistical detection performance metrics available in the literature.“~‘2~1920 The intuitive reason of the improvement is that the ternerized amplitude results in AMPOF being more like its continuous counterpart. The choice of different phase levels is often dependent on the scene-specific relative spatial frequency phase alignments and the odd-even sensitivity variation of the discretized filter itselfi8. Target recognition for grey level images using TT-AMPOF also results in better light utilization at the expense of an increased noise floor due to error
12
15
16 17 18
19 20
21 22 23
Gaskill, J. D. Linear Systems, Fourier Transforms and Optics, Wiley, New York (1978) Juday, R. D. Correlation with a spatial light modulator having phase and amplitude cross coupling, Appl Opt, 28 (1989) 4865-4869 Ross, W., Psaltis, D., Anderson, R. Two-dimensional magnetooptic spatial light modulator for signal processing, Opt Eng, 24 (I 983) 485-490 Downe, J. D. Optimal binary phase and amplitude correlation filters for polarization-rotating spatial light modulators, Opt Eng, 32 (1993) 1886-1896 Freeman, M. O., Brown, T. A., Walba, D. M. Quantized complex ferroelectric liquid crystal spatial light modulators, Appl Opt, 31 (1992) 3917-3929 Gregory, D. A., Kirsch, J. C., Tam, E. C. Full complex modulation using liquid-crystal televisions, Appl Opt, 31 (1992) 163-165 Oppenheim, A. V., Lim, J. S. The importance of phase in signals. Proc IEEE, 69 (1981) 529-541 Yu, F. T. Linear optimization in synthesis of nonlinear spatial filters, IEEE Tram If Theory, 17 (1977) 524-521 Gara, A. D. Real-time optical correlation of 3-D scenes, Appl Opt, 16 (1977) 149-153 Horner, J. L., Gianino, P. D. Phase-only matched filtering, Appl Opt, 23 (1984) 812-816 Kaura, M. A., Rhodes, W. T. Optical correlators performance using a phase-with-constrained-magnitude complex spatial filter, Appl Opt, 25 (1990) 2587-2593 Yaroslavsky, L. P. Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recoenition. Aonl Out. 31 (1992) 1677-1679 Mu,%. G., Wang, i. M., Wang: Z. Q. Amplitude-compensated matched filtering, Appl Opt, 27 (1988) 3461-3463 Awwal, A. A. S., Karim, M. A., Jahan, S. R. Improved correlation discrimination using an amplitude modulated phaseonly filter, Appl Opt, 29 (1990) 233-236 Zheng, S. H., Karim, M. A., Gao, M. L. Amplitude-modulated phase-only filter performance in presence of noise, Opt Commun, 89 (1992) 296-305 Psaltis, D., Peak, E., Venkatesh, S. Optical image correlation with binary spatial light modulator, Opt Eng, 3 (1986) 698-704 Horner, J. L., Leger, J. R. Pattern recognition with binary phase-only filters, Appl Opt, 24 (1985) 609-611 Flannery, D. L., Loomis, J. S., Milkovich, M. E. Design elements of binary phase-only correlation filters, Appl Opt, 27 (1988) 4231-4235 Horner, J. L., Javidi, B., Wang, J. Analysis of the binary phaseonly filter, Oat Commun. 91 (1992) 189-192 Flabnery, D.,&Loomis, J. S., Milkoiich, M. E. Transform-ratio ternary phase-amplitude filter formulation for improved correlation discrimination, Appl Opt, 27 (1988) 4079-4083 Goodman, J. W., Silvestri, A. M. Some effects of Fourierdomain phase quantization, IBM J Res Dev 3 (1970) 478-481 Javidi, B. Generalization of the linear matched filter concept to nonlinear filters, Appl Opt, 29 (1990) 1215-1224 Davenport, W. B., Root, W. L. An Introduction to the Theory of Random Signals and Noise Chapter 13. pp. 271-31 I, McGrawHill, New York (1958)
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phase-only filter: K. M. lfiekharuddin et al.
Biographies
Mohammad A. Karim is the director of the Center for Electra-Optics, Chairperson of the Electrical Engineering Department, and a professor of both electro-optics and electrical engineering at the University of Dayton, USA. He received his BS in physics in 1976 from the University of Dacca, Bangladesh, and MS degrees in physics and electrical engineering, and a PhD in electrical engineering from the University of Alabama, USA, respectively in 1978, 1979 and 198 1. Dr Karim’s ongoing research includes the areas of optical information/image processing, pattern/target recognition, optical systems design, infrared systems characterization, displays, optical computing, and interconnects. He is the author of over 100 journal papers, over 100 conference papers, numerous published book reviews, and the graduate text books Electra-Optical Devices & Systems, Optical Computing: An Introduction and Digital Design: A Pragmatic Approach. Dr Karim edited the reference book Electra-Optical Displays. He served as the guest
editor of four Optical Engineering special issues on electro-optical displays (August 1990), infrared imaging systems (November 1991), acquisition, pointing, and tracking (November 1993) and optical remote sensing and image processing (November 1995), and of one Optics & Laser Technology special issue on optical computing (August 1994). He is a Fellow of the Optical Society of America.
Khan M. Iftekharuddin is employed by BDM Federal, Dayton, Ohio, USA. He received a BSc degree in electrical and electronics engineering from Bangladesh Institute of Technology (Bangladesh) in 1989 and an MS degree in electrical engineering from the University of Dayton, USA, in 1991. He received his PhD degree in electrical engineering at the University of Dayton in 1995. His research interests include digital and/or optical image processing and pattern recognition, optical computing and interconnects, information processing and optical system design. He is a member of IEEE and OSA.