RL-TR-97-200 Final Technical Report October 1997
OPTICAL WAVELET TRANSFORM Florida Institute of Technology Samuel P. Kozaitis
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Final
October 1997
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4. TITLE AND SUBTITLE
C - F30602-96-2-0023 PE - 62702F PR - 4600 TA - P4 WU - PY
OPTICAL WAVELET TRANSFORM 6. AUTHOR(S)
Samuel P. Kozaitis 8. PERFORMING ORGANIZATION REPORT NUMBER
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Florida Institute of Technology 150 W. University Blvd Melbourne FL 32901-6988
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RL-TR-97-200
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Rome Laboratory Project Engineer: Mark A. Getbehead/OCPA/(315) 330-4146 12b. DISTRIBUTION CODE
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Approved for Public Release; Distribution Unlimited
13. ABSTRACT (Maximum 200 words)
We described a joint-transform correlator (JTC) that used multiple input images encoded in the spatial domain. This allowed various combinations of cross-correlations between input images to be performed. We extended the theory of a JTC to multiple inputs, and provided experimental results for four inputs with an optically-addressed spatial light modulator in the Fourier plane. We proposed a system to perform multiwavelet feature extraction. For m wavelet scales the output consisted of 4m-l correlation results, one of which is the desired result. Using conventional optics, the space-bandwidth product for wavelet feature extraction can be made the same as for a two-input joint-transform correlator.
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Pattern Recognition, Correlation, Optical Wavelet Transform 17. SECURITY CLASSIFICATION OF REPORT
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UL Standard Form 298 (Rev. 2-89) (EG) Prescribed by ANSI Std.23B.18 Designed using Perform Pro. WHSIDI0R. act 94
1.0 Introduction The main difference between a joint-transform correlator (JTC) and a conventional Vanderlugt (4f) correlator is that the images in the JTC are encoded in the spatial domain. In terms of their output, both types of correlators perform the same operation, the cross-correlation between two images. In a 4f correlator, the Fourier transform of an input image is imaged onto the Fourier transform of a reference image. In a JTC, both input and reference images are input simultaneously in the spatial domain, then the Fourier transform is performed. The JTC has certain advantages when compared to the 4f correlator such as ease of alignment and avoiding spatial filter synthesis. • Although most configurations of the JTC use two input images, the use of multiple input images may allow additional functions to be performed. For example, an image and two different wavelets were used in a JTC for multispectral (multiwavelet) wavelet feature extraction of the image.3 In this approach, different versions of an input image corresponding to different wavelet scales appeared at different locations in the output plane. Another JTC configuration used three inputs, but required an additional electronic optical processing step to implement the wavelet transform.4 In addition, another approach used two SLMs with a holographic mask to produce a wavelet -based JTC.5 Although the wavelet transform has provided an application for a JTC with multiple inputs6, a generalized JTC with multiple inputs may find more applications including those in other areas if its output function was properly described. We described a JTC that used multiple input images encoded in the spatial domain. In the next section we briefly described the general theory of a conventional JTC, and then extended the discussion to multiple inputs. Next, we provided some experimental results to verify the results of the theory. We also described how a multiple-input JTC can be used for multiwavelet analysis and proposed an architecture that has the same space-bandwidth product as a two-input JTC.
2.0 Conventional joint-transform correlator To perform the correlation operation with a JTC, functions are encoded in the input plane.
A schematic diagram of a conventional JTC is shown in Fig. 1. To perform the cross-correlation between the images b(x,y) and d(x,y), they are centered at x = ±oc. A lens produces the Fourier transform when the input plane is illuminated with coherent light. In the Fourier plane, the complex light field is
U = B(p,q)Jap + D(p,q)e-*v
(1)
where B(p,q) is the Fourier transform of b(x,y), and similarly for d(x,y). A square-law device such as a liquid crystal light valve is placed in the Fourier plane before an another lens performs the Fourier transform. Alternatively, a camera can be used to record the power spectrum and display it on a spatial light modulator (SLM) before the Fourier transform is performed. The output intensity distribution from a square-law detector can be written as
\U\2 =Ulf = [B(p,q)Jap + D(p,q)e-jap] [B(p,q)^ap + D (p, q) e~jap]*.
(2)
Multiplying terms, taking the Fourier transform, and grouping terms results in
/ = b(x,y) ®b(x,y) +d(x,y) ®d(x,y) +b(x,y) ®d(x + 2a,y) +d(x,y) ®b(x-2a,y),
(3)
in the output plane, where ® indicates the correlation operation. The first two terms are the auto-
correlations of the input functions and appear on the optical axis. The third and fourth terms are the cross-correlation between the two input functions and appear at* = ±2cc shown in Fig. 1.
3.0 Multiple-input joint transform correlator We considered a JTC with an arbitrary number of input images arranged along a line. We used the JTC as in Fig. 1 but considered n inputs separated by a in the input plane arranged along the z-axis as shown in Fig. 2. The images were labeled aj(x,y) to an(x,y), with the center image labeled as a(n+1)l2(x,y). Using this configuration, the complex light field in the Fourier plane was
n-1
U = Ax (p, q) e
2
■ ,n— 1.
n—3
^Afn ^0 + A2(p,q)e
2
...An + 1(p,q)+...An(p,q)e .
(4)
The output intensity distribution from a square-law detector was written as l£/l2= Uxlf. Multiplying nxn terms, taking the Fourier transform, and grouping terms resulted in In -1 locations in the output plane where a correlation response would occur. The output plane was described as
£ai(x,y) ®ai(x>y)
/ =
Lj=l
n-2 +
a
x
[£"= I i( >y) ®ai + l(x + a,y) + £ at(x,y) ®ai+2(x + 2a,y) n-(n-l)
+...
£
ai(x,y)®an_1(x+(n-l)a,y) n-2
+
a
a
a
[£"= l i+1 (*' y> ® i >0 an+i( >y )
2
2
FIGURE 2. Input plane of multiple-input JTC 11
an(x,y)
up
(n-l)a
(n-l)cc 2a
2a -*
a
a
>
x
ai®an
ai®a3 +a2®a4
+ ... +an.2®an
a
l®a2 +a2®a3 +... +an-i®an
ai®ai a2®SLi fa2®a2 +a3®a2 + ... +... +an®an +an®an.!
a3®a! +a4®a2 +... +an®an_2
t y
FIGURE 3. Output plane of multiple-input JTC
12
an®ai
up
ft
camera
Fourier Transform lens
polarizer SLM
collimator
71
He-Ne laser
BS
iris
Fourier Transform lens input plane polarizer collimator
iris /
Argon laser
/
laser filter
mirror
FIGURE 4. Experimental set-up of multiple input JTC 13
up
FIGURE 5. Input image used in experiment 14
up
a
a
2a
2a
4 a
a
►^—►
a
a x
2(f®f) (f®f) 2( ®f) (f®f) 2(f®f)
4(f®f)
(f®f)
2( ®f) (f®f
FIGURE 6. Location and relative value of correlation response of multiple-input JTC when Fig. 5 was used as the input.
15
up
► x
(a)
Intensity
LL
JuL
Mi
\f\ .fa
Distance along -x axis (b) FIGURE 7. Left hand side of output plane obtained experimentally when Fig. 5 was used as the input plane. 16
up
a
f(x,y)
a
a
f(x,y)
a
Wj(x,y)
w2(x,y)
FIGURE 8. Input plane of multiwavelet example 17
up
a
a
a f®w1
>
< »
f®w2 f®wj +f®w2
< ►
a
2a
2a
a
a w ®f
i>
f®f
(
r®f
f£0^
&f^ *
+w2®wl +wj®W2 +w1®w1 w2®w2
Wj®f w2®f
+w2®f
FIGURE 9. Output plane of multiwavelet example
18
up
FIGURE 10. Image used in simulation experiments. 19
up
FIGURE 11. Output plane of multiwavelet simulation experiment.
20
up
(a)
21
up
(b)
FIGURE 12. Close-up view of left hand side of Fig. 11. Response on left in both cases is for a2=1.2, (a) response of right is for aj + a2, where aj = 0.7 (b) response of right is for aj + a2, where aj = 1.0.
22
up
input image coherent light
SLM
e-e-v \e-fr
wavelet DC image block
fl
detector
E ND filter
FIGURE 13. Schematic diagram of multiple-input JTC used for multiwavelet analysis using two wavelet scales 23
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