Vol. 11, No. 12/December 1994/J. Opt. Soc. Am. A 3201
Joint transform correlator with incoherent output
David Mendlovic, Zeev Zalevsky, and Naim Konforti
Faculty of Engineering, Tel Aviv University, 69978 Tel Aviv, Israel
Received February 28, 1994; revised manuscript received May 17, 1994; accepted June 16, 1994
A joint transform correlator (JTC) provides a real-time correlation that is useful for pattern recognition.Based on the shearing interferometer configuration, a method for implementing a novel JTC is suggested.The obtained output correlation is spatially incoherent and thus provides better correlation peak quality.The proposed setup is characterized by its simplicity and stability and its ability to operate under real-timeconditions. An experimental demonstration is given.
An optical spatial image processor that is based on thejoint transform correlator (JTC) scheme' does not containany filter. Thus it provides some advantages comparedwith the conventional matched-filter configuration,2 sincethe matched filter must be generated in a complexprocess, has limited resolution, and must have high ac-curacy. The JTC configuration is based on the simul-taneous presentation of two patterns at the input plane,each laterally shifted from the axis center. As Fig. 1shows, a schematic configuration of the JTC contains a4f setup that at its Fourier plane includes a square-lawconvertor device (a device that converts a field distri-bution to an amplitude distribution). The output firstdiffraction orders of such a method are the correlationbetween the two input patterns. Several approachesfor implementing the square-law conversion have beensuggested, such as photofilm,' a spatial light modulator(SLM),3 and liquid-crystal light valves.4 These configu-rations work only with coherent illumination, since onlyin this case do the two lenses perform the optical Fouriertransform operation. The intrinsic advantages of spa-tially incoherent light for optical signal processing havebeen intensively discussed in Ref. 5. It has been shownthat the conventional coherent matched-filter correlatorcan be used with incoherent light.6 For the JTC a white-light configuration has been suggested.7 In what followswe propose another incoherent JTC implementation thatprovides various advantages compared with other opticalprocessors. The suggested implementation is based onthe shearing interferometer. 8
With the shearing interferometer that includes a cornerprism, the Hartley transform can be obtained at the out-put plane,9 following the optical configuration of Fig. 2.This transform is achieved for either spatially incoher-ent or coherent illumination. This setup uses a beamsplitter (B.S.) to divide the light coming from the inputpattern into two paths. One optical path contains the in-formation from the original input. The other path is theoriginal input that was returned from the corner prism;thus it is the original pattern rotated by 180° [g(x, y) be-comes g(-x, -y)]. These two paths are then recombinedtogether by the beam splitter. One can note that every
point in the object plane is split into two diffraction pat-terns according to the two paths. These paths interfereamong themselves in spite of the fact that the preliminaryillumination was spatially incoherent.
Recently, a setup that is based on this concept andperforms a joint transform correlation was suggested.'0
In this setup the joint spectrum was achieved by useof the shearing interferometer. Then, with the use of aCCD camera and a SLM, the joint spectrum was Fouriertransformed with a conventional coherent optical Fouriertransform. Despite the uniqueness of this setup it suf-fers from several disadvantages. First, it cannot performa real-time joint transform correlation, since it is lim-ited by the operating rate of the CCD and SLM devices.Second, the output suffers from the main coherent-lightdisadvantage, i.e., speckles. In our system noises thatare connected with the coherent light and appear atthe Fourier plane affect the output plane. However, atthe output plane we expect the absence of speckle noise(although it appears at the Fourier plane). Since thesuggested setup is to be used in pattern recognition sys-tems, the absence of speckles is a great advantage be-cause a speckle can cause a false alarm in identificationof the object. Thus, in spite of the noises coming fromthe coherence of the light, the absence of the speckles is amajor improvement. Third, the spatial resolution of thejoint spectrum plane is limited by the SLM. The practi-cal meaning of this is that both the spatial resolution ofthe input objects and the lateral distances between themare limited.
We base the following proposed scheme on using co-herent light for finding the joint spectrum, breaking me-chanically the spatial coherence, and then performing thesecond Fourier transform with the incoherent shearing in-terferometer. The final configuration is simple and doesnot need any electrical conversion. Thus it is faster andprovides higher spatial resolution. Furthermore, we areno longer limited by intensity modulation quantizationproblems of the SLM.
2. MATHEMATICAL ANALYSIS
Figure 2 shows the proposed optical setup. Its first partis a conventional coherent 2f system for performing a
3202 J. Opt. Soc. Am. A/Vol. 11, No. 12/December 1994
Input pattern Li
SquarConL.
fl I - fl. a f
L2 Output pattern
f2 -I- -1- -I 1 1I_- -1
Fig. 1. Schematic optical setup for performing the JTC scheme.
A\ Corner prism
Input pattern Li Rotating diffuser Mirror
1 0 | ~NBS
6 f, |1. f, X I MlI- -1 I
Output plane
Fig. 2. Proposed incoherent JTC.
Fourier transform. While presenting the input patternf (x, y) and the reference object g(x, y) side by side asshown in Fig. 3, where yo is the lateral distance betweenthem, we obtain, at the Fourier plane, the joint spectrumof f (x, y) and g(x, y). One can write the input field dis-tribution as
to(x, y) = f (X, y (1)
The joint spectrum that is obtained at the Fourier planeis the field distribution:
ti(at, /3) =F(ar, )exp i f2 )ry,
2 and t are the coordinates of the first path (see Fig. 4),since in Eq. (3) we have assumed that the mirror is notperpendicular but has a small tilted angle of 0 in the 6 - zplane; thus the t and 2 coordinates of the first opticalpath are
= e cos 0 - z sin 0, 2 = z cos 0 + sin 0, (4)
and, for the small-angle approximation, one can write
g= - d,0, 2= di + e0. (5)
The output acquisition device (a CCD camera in ourcase) obtains the output intensity and, with the rotateddiffuser in place (Fig. 2), obtains the time average of theoutput intensity distribution. Thus the intensity trans-formation at the output plane is
I(s, 7) = ((u1 + u2 )(u1 + U2)*)
= (ulu 2*) + (u 2u2*) + (lu 2*) + (u2 *Ul), (6)
Input pattern
fiYo
(a)
+ G (a, 8)exp(i f yo \,Af 2J
where F(a, 13) and G(a, a) are the Fourier transformsof f (x, y) and g(x, y), respectively. (a, /8) is the spatialcoordinate in the Fourier plane, A is the illuminationwavelength, and f is the focal length.
In the same plane a rotated diffuser is placed. Thisdiffuser breaks the spatial coherency by adding a randomphase to the joint spectrum. The temporal coherency(monochromatic light) still exists. Next the beam entersthe shearing lensless Fourier transformation. There aretwo optical paths: the one with the regular mirror (Ml)has an optical length of d,, and the one with the cornerprism has an optical length of d2. With the diffuser theoutput plane is (as a result of the Kirchhoff diffractiontheorem)
forces that B,! = IB21 = IBI) and, by use of the binomialexpansion approximation, obtain
= 2k (xe' + y) + k'O,d(x+y)+'0 (14)
which we arrived at by neglecting the 02 term and bytransferring the intensity into the coordinates x' = x +d0/2 : x and (' = - d0/2. By another transformationof the intensity into the opposite coordinate, I(-6$, -- q),we obtain exactly the cosine transform of the joint spec-trum intensity:
I(e, r7) = 21BI f f dxdylt,(x, y)12
+ J f dxdylt,(x, y)12 COS , (15)
where
k =d (x6+ yn) - kO. (16)
Note that 0 causes a periodical term that does not dependon the input pattern. As we shall see below, this termcontrols the contrast of the output plane and multiplies itby a fringe pattern. Thus one may control the contrast
where Re() is the real part of () and (fx, fy) are theFourier coordinates, and because of the Fourier propertiesfor a real input object,
where
'[F(x, y)G*(x, y)] = f (f, fy) ® g(fx, fy),
F- F(xy)exp(-i2,w A)] =f(fxfY- ), (19)(13)
In order to avoid a spherical phase difference between thepaths, one should choose d = d2 = d (this condition also one obtains the equation
Similarly, we obtain
(IU,12) = 1B 12 f f
(IU212) = B212 J fL
i = k[(d, + eo)2 + (x + d,0 - )2 + (y - 77)2]112
- k[d22 + (X + )2 + ( + 77)2]1/2
Mendlovic et al.
5)
3204 J. Opt. Soc. Am. A/Vol. 11, No. 12/December 1994
where X denotes correlation. In Eq. (20) Ib is the back-ground energy:
Ib= ftL L It,(x, y)12 dxdy. (21)
Fig. 5. Joint spectrum.
Fig. 6. Two white correlation peaks.
Fig. 7. Two black correlation peaks (inverse contrast).
I(6, -q) = 21B112 Ib + cos(k0)[Ad Ad)f(Ad Ad)
'(Ad Ad f'Ad Ad)
,7 o g 26 2,, yo 11+ f 2 2 _ yL '_ 2f2 _ o
\Ad Ad Af I \Ad Ad Afj
+ f i' 2eX 2+ yo ) g d 2 + Y )1(Ad Ad Af Ad Ad Af J
(20)
Along the ± 1 diffraction orders the correlation betweenf and g is obtained. The zero diffraction order containsno interesting information.
3. EXPERIMENTAL RESULTS
The schematic setup of Fig. 2 was constructed experi-mentally. The input pattern consists of two F-18 air-planes separated by 7 mm. The input pattern shown inFig. 3 was illuminated by green (532-nm) coherent col-limated light. A lens Li standing behind the patterncreated its Fourier transform. The focal length of thislens was f, = 140 cm. The Fourier plane was focused onthe diffuser, which was rapidly rotated. The joint spec-trum obtained is shown in Fig. 5, and one can note theinterference fringes that are due to the space betweenthe inputs. We placed a beam splitter 60 mm from thediffuser to splice the light into two paths (one path goesthrough a corner prism, and the other is returned bya regular mirror), as we mentioned in Section 2 above.Both the corner prism and the regular mirror were placed10 mm from the beam splitter. The paths interfere atthe output plane. The output plane was situated 60 mmfrom the beam splitter. Figures 6 and 7 show the cor-relation peaks created by the system. As shown by themathematical development in Section 2, the output con-tains three zones: the zero diffraction order (which is notinteresting) and the ±1 diffraction orders that are thecorrelation of the input objects. According to Eq. (20),on account of the interference fringes, the system canbe aligned to provide a totally black or totally whitecorrelation peak. Note that any state between these twoextremes is possible but not useful. Better correlationresults could be obtained by subtraction of the black peakoutput from the white peak output. Figures 6 and 7 arethe white and black outputs, respectively.
4. CONCLUSIONS
In this paper we have demonstrated the usefulness of op-tics for practical real-time applications that are requiredin optical data processing. The suggested scheme man-aged to obtain high quality correlation peaks in real-timeoperation for spatially incoherent illumination. It shows,to our knowledge for the first time, a JTC square-law con-vertor that is not a bottleneck in the process. Furtherinvestigation toward totally a spatially incoherent JTCcould be conducted by replacement of the first Fouriertransform by another shearing interferometer.
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