Optical pattern recognition is commonly performedwith a 4f correlator.1 This setup uses a matchedfilter that provides the highest signal-to-noise ratiofor a Gaussian white noise but is not invariant to anyparameter except lateral shifts of the input object.Other invariant properties could be obtained by useof harmonic decompositions. In this method the ref-erence object is decomposed into a set of orthogonalharmonic functions, and the filter source is chosen asa single expansion order. This approach achievesinvariant pattern recognition, but its discriminationabilities are worse than those of the conventionalmatched-filter approach because here the filter isonly a single term out of the full harmonic decompo-sition.Suggested decompositions for obtaining invariant
properties were circular-harmonics for rotation in-variance,2 radial harmonics ~RH! for scale invari-ance,3 and logarithmic harmonics ~LH! for projectioninvariance.4 Later, the harmonic-decompositionmethod was generalized for other distortion proper-
Z. Zalevsky and D. Mendlovic are with the Faculty of Engineer-ing, Tel-Aviv University, 69978 Tel-Aviv, Israel. J. Garcıa is withthe Departament Interuniversitari d’Optıca, Facultat de Fısica,Universitat de Valencia, Calle Doctor Moliner 50, 46100 Burjassot,Spain.Received 4 March 1996; revised manuscript received 19 August
ties by use of deformation harmonics ~DH!.5 Thesemethods are optimized by the choice of the propercenter for harmonic expansion and the proper har-monic order for the filter.6,7Another method that permitted achieving both in-
variant pattern recognition and high discriminationability is the synthetic-discriminant-function ap-proach.8 The problem with this method was that aninvariant property was achieved only in the correla-tion peak itself, and it was commonly followed byunacceptable levels of sidelobes.In this paper we propose a method that permits the
achievement of full invariant pattern recognitionwhile the filter combines several harmonic orders si-multaneously. The method is demonstrated withcircular harmonics, but any other decomposition set~RH, LH, DH! may be used. In this approach eachharmonic order is transmitted with a different wave-length. At the output correlation plane all correla-tion distributions coming from the differentdecomposition orders are added incoherently. Thisfinally provides improved discrimination ability.In Section 2 we provide the necessary details re-
garding circular-harmonic ~CH! decomposition. InSection 3 we explain and prove the suggested algo-rithm. In Section 4 we give an experimental dem-onstration.
2. Circular Harmonic Decomposition
CH decomposition is the first harmonic expansionthat was used for invariant pattern recognition.2 Itinvolves the set of orthogonal functions $exp~iNu!%,
10 February 1997 y Vol. 36, No. 5 y APPLIED OPTICS 1059
which is multiplied by a radial function:
f ~r, u! 5 (N52`
`
fN~r! exp~iNu!, (1)
fN~r! 512p *
0
2p
f ~r, u! exp~2iNu!du, (2)
whereN is the expansion order and f ~r, u! is the inputobject. A single-harmonic impulse response is thusg~r, u! 5 fM~r! exp~iMu!, and it can provide rotation-invariant pattern recognition by use of a single har-monic out of this expansion. The correlationbetween the target and a single-harmonic componentat the origin of the correlation plane is given by
CN 5 2p *0
`
u fN~r!u2rdr. (3)
Correlation of the object with the same object ro-tated at an angle a, in terms of the correlation withdifferent CH components, results in
Ca 5 (N52`
`
CN exp@iNa#. (4)
Equation ~4! permits the interpretation of the rota-tion variance of a conventional matched filter. Thecorrelation-peak contributions of the different CHcomponents get out of phase except for a 5 0 ~whenN Þ 0!. If only one CH is considered, the phasefactor can be neglected when the output-plane inten-sity is taken. However, the simultaneous use of sev-eral harmonics may destroy the correlation peak as aresult of differing phase factors. Multiple CH com-ponents can be used if the phase shift of each term inEq. ~4! is compensated.9 This method, however, canbe applied only to digital correlation or to the use ofspatial multiplexing.10
3. Optical Implementation
The proposed method can be applied with differentharmonic decompositions. We demonstrate it withthe CH expansion used for rotation-invariant patternrecognition, but it can be extended to RH, LH, andDH. The optical setup for performing multiple-harmonic rotation-invariant pattern recognition byuse of wavelength multiplexing is illustrated in Fig.1. The input pattern should be illuminated by sev-
Fig. 1. Suggested optical setup for rotation-invariant pattern rec-ognition.
1060 APPLIED OPTICS y Vol. 36, No. 5 y 10 February 1997
eral spatially coherent wavelengths @for instance, aHe–Ne laser ~red!, a doubled Nd:Yag laser ~green!,and an argon laser ~blue!#.The first part of the setup performs a Fourier trans-
form of the input pattern. A filter is placed in theFourier plane and contains several rings, with eachring being a filter matched to a different order of theCH decomposition. The size of each ring is scaledwith respect to the ratio between the different wave-lengths used for the input illumination because anachromatic lens performs a Fourier transform that isscaled by l1yli for the different wavelengths li @ i.Each ring, which represents a different order of theCH decomposition, is applied to a different wave-length, i.e., each ring corresponds to a different wave-length. Thus, in the output plane the correlationpeaks generated by the different harmonic orders aredisplayed in different wavelengths but at the samelocation and added in their intensities. Hologramswith different spatial carrier frequencies are plottedinside each ring ~each CH order!. The ratio betweenthe grating periods is as follows:
sin a 5li
Ti5
lj
Tj, (5)
where a is the angle of the first diffraction order fromeach grating, li and lj are two different wavelengths,and Ti and Tj are the periods of the two correspond-ing gratings. Equation ~5! ensures that differentwavelengths li and lj will diffract to the same spatiallocation. Because different wavelengths, comingfrom different rings ~different CH orders!, diffract tothe same spatial position, they are added incoher-ently. A schematic illustration of the filter is givenin Fig. 2.The second part of the system performs another
Fourier transform. Thus, in the output plane, animage of the input pattern is obtained after the pat-tern has passed through the CH filter. Each wave-length passes through a different CH order. If manywavelengths are used the output correlation plane is
Fig. 2. Schematic sketch of the CH filter for three-wavelengthmultiplexing.
both rotation invariant and able to provide high dis-crimination comparable with that of the matched fil-ter.As mentioned above each harmonic order ~each
ring! is encoded by a different spatial frequency ~seeFig. 2!. Two contiguous rings should fulfill Eq. ~5!.This means that the first diffraction order comingfrom the grating that has the period T1 and is illu-minated by l1 and the first diffraction order comingfrom the grating that has the period T2 and is illu-minated by l2 should overlap. However, in additionwe expect that the diffracted distributions comingfrom grating T1 illuminated by l2 or from grating T2illuminated by l1 will not overlap with the desireddistribution. Because the overall setup is an imag-ing setup with a magnification of f2yf1, one can trans-late the last restriction ~nonoverlap! to the followingmathematical condition:
f2l1yT1
Î1 2 ~l12yT1
2!2 f2
l2yT1
Î1 2 ~l22yT1
2!$ 2Lx
f2f1, (6)
where f1 and f2 are the focal lengths of the first andsecond parts of the setup, respectively, ~see Fig. 1!and Lx is the size of the input image @the term 2Lx~ f2yf1! is an approximation for the size of the correlationplane#. Using the approximation of sin b ' tan bpermits Eq. ~6! to be simplified to
T1 #f1~Dl!
2Lx, (7)
where Dl is the smallest difference between twowavelengths. Condition ~7! gives us a restriction onthe maximal value for the period of the gratings.In this setup the ring sizes and CH components
must be chosen to maximize the energy contents ofthe filter. A different approach can be used to utilizeevery CH component fully. We can accomplish thisby multiplexing the filters for the different wave-lengths. This multiplexing, although troublesomein computer-generated holograms ~CGH’s!, is easy foroptically recorded holograms ~provided there is asmall number of holograms!. Thus, each CH filtershould be recorded at the same angle for the refer-ence beam but with the use, in each exposure, of adifferent wavelength. This approach will providethe same periods for the carrier frequencies as theabove-described procedure. The final output inten-sity for a target rotated by an angle a, in the notationintroduced in Section 2, is given by
Ia 5 uCau2 5 (N52`
`
uCNl u2, (8)
where the superscript l gives the wavelength withwhich each correlation is obtained. The addition isobtained in the intensity, which is in contrast to thematched-filter case in which there is coherent addi-tion. Nevertheless, now there is no variation withthe rotation angle of the target, and every componentcontributes to the intensity of the correlation peak.
4. Chromatic Aberrations
The experimental problem raised when several wave-lengths are used with achromatic lenses is chromaticaberration.11,12 These aberrations are expressed asshifts of the focal length as a function of wavelength.This shift is caused by the dependence of the refrac-tion index on the wavelength:
1FAL~l!
5n~l! 2 1
FAL~l0!@n~l0! 2 1#, (9)
where FAL~l0! is the focal length of the achromaticlens for the wavelength of l0 and n is the refractiveindex. The dependence of the refractive index n onthe wavelength may be approximated as
n~l! < n~l0! 2 d~l 2 l0!, (10)
where d is a constant.If a zone plate is attached to the achromatic lens,
the overall focal length becomes
1F~l!
51
FZP~l!1
1FAL~l!
, (11)
where FZP~l! is the focal length of the zone plate,whose wavelength dependence may be expressed as
FZP~l! 5l
l0FZP~l0!. (12)
Thus, by using Eqs. ~9!, ~11!, and ~12! one obtains
1F~l!
5 lH 1l0FZP~l0!
2d
@n~l0! 2 1#FAL~l0!J
1n~l0! 1 dl0 2 1FAL~l0!@n~l0! 2 1#
. (13)
The first term on the right-hand side is responsiblefor chromatic aberrations. Choosing
FZP~l0! 5@n~l0! 2 1#FAL~l0!
l0d(14)
ensures the elimination of those aberrations.Keeping the deviation of the wavelength l from the
wavelength l0 small and choosing achromatic lensesmade out of a material with a small d value ensurenegligible aberrations. Note that, for the suggestedwavelength-multiplexing approach, the difference be-tween the various wavelengths does not have to belarge. Small deviations will also ensure the desiredspatial incoherence.
5. Experimental Results
To demonstrate the abilities of the new approach wehave performed several experiments. The referenceobject is decomposed into the CH decomposition, andtwo orders are chosen out of this decomposition ~ac-cording to energy and peak-sharpness consider-ations7!. We chose orders 2 and 5. Obviously,when decomposition into circular harmonics occurs,
10 February 1997 y Vol. 36, No. 5 y APPLIED OPTICS 1061
the proper center ~once again according to energy andpeak-sharpness considerations6! is also chosen.To improve performance we prepared a phase-only
filter out of each circular harmonic. Because thefilter is designed for two orders of circular harmonic,it should be illuminated by two wavelengths. Wechose to work with the doubled Nd:Yag ~532-nm! andHe–Ne ~632.8-nm! lasers. Each circular harmonicwas modulated by a different carrier frequency ~grat-ing! such that conditions ~6! and ~7! are fulfilled. Toencode the phase of the filter we used a CGH.13However, to do so we must first calculate the numberof periods of the grating that enter into each pixel ofthe matrix ~which contains the CH information! thatwe wish to encode. The minimal number of periodsin each pixel of the matrix must be
n0 5LnyNT1
, (15)
where N is the number of pixels inside the matrix ~inour case, N 5 128! and Ln is the size of the filter.Because the fast-Fourier-transform algorithm wasused for the calculations, the following scaling ratioshould be held between the sizes of the input planeand the spectral plane ~see Ref. 14, for instance!:
LxLn 5 lf1N. (16)
Substituting condition ~7! and Eq. ~16! into Eq. ~15!yields
n0 $2l
l1 2 l2. (17)
Calculation with condition ~17! leads to a value of n0$ 10.6. We chose to use 12 periods per pixel. Next,using Lohmann’s CGH method of vertical encoding,we create the filter. Each pixel of the filter contains12 grating periods, encoded so that the compatiblephase of the pixel of the matrix will be reconstructed.Note that the final filter is composed of two matri-
ces: The first one encodes the second CH order, andthe second one encodes the fifth harmonic. The ratiobetween the gratings inside each CH is set accordingto Eq. ~5!. Therefore, the second harmonic will cor-respond to the green laser and the fifth to the red one.Figure 3 illustrates the input pattern scene. The
upper image is of a fault object, and the lower imageis of the reference object, rotated by 90°. In Fig. 4one can see the obtained experimental results. Thecenter image depicts the obtained color output ~greenand red!; this is the correlation peak for both the fifthand second CH’s. On the righthand side one can seethe correlation plane obtained for the second har-monic ~green! and on the lefthand side the correlationobtained for the fifth harmonic ~in red!. Improveddiscrimination ability and stronger correlation peaksare obtained for the multiplexed correlation ~centralimage!, as compared with the single only-red or only-green channels.Figure 5~a! illustrates a three-dimensional plot of
the region of interest of Fig. 4, whereas Fig. 5~b! is a
1062 APPLIED OPTICS y Vol. 36, No. 5 y 10 February 1997
plot of the peaks’ cross sections. One can see here,as well, the improved correlation peak obtained forthe multiplexed correlation. Note that the experi-mental results obtained in this paper are only sup-posed to show that the method works. More resultsare required before evaluation of the performance ofthe method can be carried out with certainty.
6. Conclusion
From the experimental results one can see that theproposed filter provides improved discriminationability and demonstrates a wide range of rotationinvariance. Sharp correlation peaks were obtained.Note that the suggested technique was demonstratedfor circular harmonics and achieved improvedrotation-invariant pattern recognition. The sameprocedure may be applied to LH or RH. In this case,several orders of LH or RH are wavelength multi-plexed ~encoded in the different rings!. In additionto achieving an improved single invariant property,one may obtain several invariant properties simulta-neously. To do so, we choose one harmonic order outof several harmonic types ~for example, CH and RH!,and each harmonic type is multiplexed by a different
Fig. 3. Input scene used in the experiments.
Fig. 4. Experimental output correlation plane.
wavelength ~i.e., each harmonic type is encoded indifferent ring of the filter!. The result is correlationthat is invariant to either rotation or scale.
Z. Zalevsky and D.Mendlovic acknowledge support
Fig. 5. ~a! Three-dimensional plot of the obtained output corre-lation plane. ~b! Plot of the peaks’ cross section.
provided by the Infrastructure Foundation and theEshcol Foundation of the Israeli Ministry of Scienceand Technology.
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