Optical quasi-three-dimensional correlation

Transcription

1 Optcal quas-three-dmensonal correlaton Youzh L 1, Joseph Rosen Department of Electrcal and Computer Engneerng, Ben-Guron Unversty of the Negev P. O. Box 653, Beer-Sheva, 84105, Israel ABSTRACT A novel optcal correlator for three-dmensonal (3-D) object recognton s proposed heren. Several projectons of a 3-D scene are recorded under whte lght llumnaton and fused nto a sngle complex two-dmensonal functon. After properly flterng ths functon, t s then coded nto a computer-generated hologram (CGH). When the CGH s coherently llumnated, a correlaton space between the 3-D tested scene and the reference functon s reconstructed, n whch lght peaks ndcate on the exstences and locatons of true targets n the observed 3-D scene. Expermental results are presented. Keywords: Three-dmensonal optcal correlaton, Pattern recognton and feature extracton, Computer-generated hologram. 1. INTRODUCTION Recently, there has been ncreasng nterest n three-dmensonal (3-D) optcal nformaton processng because of ts potental applcatons. A method for performng 3-D optcal correlaton was frst proposed by Bamler and Hofer-Alfes 1, n whch the 3-D observed scene s frst mapped, slce by slce along ts longtudnal axs, onto a large -D plane. Then, conventonal -D optcal correlatons are performed between every slce wth all the others. In ths method the observed scene must be processed wth ntensve dgtal algorthms to reconstruct the 3-D mage nsde the computer memory before any correlaton can be employed. Recently other attempts of 3-D optcal pattern recognton were reported -9. Rosen has proposed to extend the correlaton dmensons from -D to 3-D by ntroducng 3-D optcal Fourer transform (FT),3. By fusng several projectons of the tested scene, a 3-D object functon s frst Fourer transformed, fltered by some 3-D reference flter and fnally nversely Fourer transformed nto the correlaton space. By ths technque a target can be detected and located n ts 3-D envronment. Hs method has been frstly demonstrated wth a 3-D jont transform correlator (JTC) 4. Later, we have proposed a method to mplement 3-D dstorton nvarance for the 3-D pattern recognton by use of a sngle -D synthetc reference functon n the 3-D JTC 5. As an alternatve to the 3-D JTC, we have also proposed a hybrd 3-D correlator wth general complex flters 6. Esteve-Taboada et. al. have suggested to combne the Fourer transform proflometry technque 10 wth a conventonal -D optcal correlator for performng the 3- D object recognton 7. Frauel et. al. have proposed a dfferent 3-D object recognton technque usng dgtal holography 8. Frst, -D varous vewpont projectons of the 3-D observed scene are reconstructed from a dgtal hologram. Then, the system recognzes the true targets ncluded n the tested 3-D scene by use of -D correlaton technques. Matoba et. al. have captured multple perspectves of 3-D objects by a mcrolens array 9. The varous perspectves of the reference and the nput scene are cross-correlated by a -D JTC, whereas the depth nformaton of the 3-D objects s converted to angular nformaton of the perspectves. The man crtcsm on all the varous versons of the 3-D correlators developed at Ben-Guron Unversty -6 (BGU correlators, hereafter) has been drected aganst ther complexty and ther extensve use of dgtal computaton. To overcome these drawbacks, we propose heren a new 3-D pattern recognton system, termed 3-D optcal quas-correlator (OQC). Ths system s a degenerate verson of the hybrd BGU correlator 6. It can stll recognze and locate objects n 3- D space, and thus the 3-D space nvarance property s mantaned. However, the recognton n the 3-D OQC s not done by a complete 3-D correlaton. Therefore, n a few aspects, lke smplcty and operaton speed, the new system s superor over the BGU correlators. However, there s some penalty for ths smplcty by mean of reducton of the 1 l@ee.bgu.ac.l; Tel: ; rosen@ee.bgu.ac.l; Tel: ; Algorthms and Systems for Optcal Informaton Processng VI, Bahram Javd, Demetr Psalts, Edtors, Proceedngs of SPIE Vol (00) 00 SPIE X/0/$

2 system s performances. For nstance, the sgnal to nose rato s reduced compared to ts value n the BGU correlators, as demonstrated n the last secton. The man dfference between the 3-D OQC and the BGU correlators exsts n ther output stage. In the BGU correlators each transversal slce of the 3-D correlaton space s reconstructed separately from a dfferent computer generated hologram (CGH). In the 3-D OQC, on the other hand, only a sngle CGH s used to reconstruct the ntensty dstrbuton of the entre 3-D correlaton space at once. Therefore, the detecton method of the correlaton peaks should be modfed. The correlaton peaks are not necessarly dstrbuted on a sngle transverse plane, whch can be recorded by a conventonal planar detector. Instead, one needs an magng system that s capable of observng the entre 3-D correlaton space. Such system can be, for nstance, the human vson system, whch enables anyone wth a stereoscopc vson to observe the correlaton peaks and estmate ther locatons n the 3-D space. An artfcal magng system, whch mtates ths knd of human ablty, s an addtonal opton that should be consdered n ths context. TESTED OBJECTS o 1 ( x, y, z ) y y L θ z x x o ( x, y, θ ) 3-D to -D COMPRESSION o ( u v ) 3, FILTERING & HOLOGRAPHIC CODING v COMPUTER HOLOGRAM T r ( u, v) CORRELATION SPACE y 0 Plane Wave z 0 f f x 0 Fg. 1 The schematc of the proposed system 8 Proc. of SPIE Vol. 4789

3 . SYSTEM S ANALYSIS The mage capturng processes by the 3-D OQC and by the BGU correlators are dentcal, and such process was already descrbed n Ref. 6. Nevertheless, for the completeness, we repeat the descrpton of ths part n the followng. The complete proposed scheme s shown n Fg. 1. The nput functon contanng several objects s denoted by o ( x, y, z ), where (,, ) 1 xyz are the coordnates of the observed scene. A dgtal camera, located a dstance L from the orgn, captures the objects o 1 ( x, y, z ) from several dfferent ponts of vew. The dgtal camera s shfted n constant angular steps along a horzontal arc centered on the orgn, where the camera s always drected to the orgn. The angle between the optcal axs of the dgtal camera and the z axs s denoted by θ. For each θ, the projected mage o ( x, y, θ ) s recorded nto the computer, where (x, y ) are the coordnates of the mage plane of each camera. Based on smple geometrcal consderatons, the relaton between ( x, y ) and ( x, y, z, θ ) s gven by ( x, y) = M( xcosθ + zsn θ, y) (1) where M s the magnfcaton factor of the dgtal camera. Assumng the dstance L s much longer than the depth of the 3-D nput scene, all the object ponts of the 3-D nput scene are equally maged wth the same magnfcaton factor M. Insde the computer, the processng s dfferent from that of the BGU correlators. In the BGU correlators the computaton process of the angular projectons has yelded 3-D FT of the scene. Then, after approprate flterng and another 3-D FT, we have obtaned 3-D correlaton between a reference and the scene objects. Here the output stage of the OQC s dfferent than that of the BGU correlators, and therefore the computatonal processng s changed. In the OQC output, a Fourer hologram s coherently llumnated, and as a result, the 3-D correlaton space s reconstructed n front of the observer eyes. The desred hologram has only two dmensons, even t represents 3-D scene, and therefore the 3-D data of the angular projectons should be compressed properly nto a two-dmensonal functon. Ths compresson s the goal of the computatonal process as descrbed next. The set of captured mages are compressed n a specal way to a sngle spectral matrx. The dgtal operatons along the horzontal and the vertcal axes are dfferent because the camera collects the angular projectons only along the horzontal axs. As the analyss shows n the followng, the entre spatal spectrum nformaton of every projecton along the x -axs s redundant. For each k-th projecton from the vew angle θ k we just calculate, and store, the spectral content of the u k -th frequency element, where u k and θ k are related by the relaton u k =θ k /a, and a s some chosen parameter. In other words, from the spatal spectrum of each projected mage a dfferent column s pcked up accordng to the abovementoned rule. Of course, there s no need to calculate the complete horzontal spectrum of every projecton, but only the columns from the spectrum, that are used n the new obtaned matrx. Each k-th projecton s multpled by exp( j πxu k λf), and the product s summed up along the rows (.e. along x ) to a sngle column. λ s the wavelength of a plane wave llumnatng the output hologram, and f s the focal length of the sphercal Fourer lens used later n the system s output. As we show later, the parameter a controls the longtudnal magnfcaton of the correlaton space ndependently of the transverse magnfcaton. In ths stage, the entre projected matrces are compressed together to a sngle -D matrx, whch s then Fourer transformed along y -axs, wth a scale factor of 1/( λ f). Thus, n contnuous formalsm, the dmenson of the grabbed data s reduced from three to two n the followng way, o3( u, v) o( x, y, au)exp j π ( ux vy) λ f + dxdy () Note that there s no Fourer transform along x because for each frequency value u the ntegraton s done on a dfferent projecton accordng to the relaton θ = au. It should be noted that although the rule of compressng seems arbtrary, the relaton θ = au leads to the desred Fourer hologram as shown n the followng. The maxmum vew angle θ s chosen to be small (n the current example, t s 16 0 from each sde of z-axs), thus we can employ the small angle approxmaton: cosθ 1 and snθ θ. Substtutng Eq. (1), the small-angle approxmatons and the relaton θ = au nto relaton () yelds Proc. of SPIE Vol

4 3 ( ) o ( u, v) o ( x, y, au)exp j πm ux + vy + au z λf dx dy (3) Let us look on a sngle element from the entre 3-D object functon. Ths nfntesmal element of the sze ( x, y, z ), at pont ( xyz,,, ) and wth brghtness o (,, ) 1 x y z, appears as a sngle element on every projecton plane, but at a dfferent locaton from projecton to projecton accordng to Eq. (1). Therefore, n ths case the spectrum presented n relaton (3) s gven by o ( u, v) o ( x, y, z)exp j πm ux+ vy+ au z λf x y z (4) 3 1 ( ) Next we examne the nfluence of all ponts of the tested scene (,, ) 1 o3 u, v. The tested scene s three-dmensonal, and therefore, the overall dstrbuton of o3 ( uv, ) s obtaned as a 3-D ntegral of all the ponts from the nput scene, as follows, o3( u, v) o1( x, y, z)exp j πm ( ux vy au z) λf + + dxdydz (5) Thus we obtan a -D functon, whch contans 3-D nformaton of the tested scene n a smlar sense that a -D optcal hologram contans the 3-D nformaton of the recorded 3-D scene 11. o3 ( uv, ) s frst multpled by a proper flter, and then the obtaned product s encoded nto a CGH. The CGH, when llumnated by a plane wave, yelds a holographc reconstructon of the correlaton space, as shown n the lower part of Fg. 1. Thus, we can detect the targets, and also locate them n ther 3-D envronment. o x y z on the dstrbuton of ( ) To mplement the 3-D object recognton we next consder the flter functon defned by Fuv (, ) f ( x, y, z)exp j πm( ux vy auz) λf + + dxdydz (6) where f(x,y,z) s the spatal reference functon located at the orgn of the coordnate system (x,y,z), and the astersk denotes complex conjugate. Multplyng o (, ) 3 u v gven by relaton (5) wth the flter functon gven by relaton (6) yelds Tuv (, ) = o( uvfuv, ) (, ) o( xyz,, )exp[ j πmux ( + vy+ auz)/ λfdxdydz ] where 3 1 * f ( ξ, η, ζ)exp[ j πm( uξ + vη+ au ζ)/ λ f ] dξdηdζ = * o1 ( x, y, z) f ( ξ, η, ζ) exp{ [ ( ) ( ) ( ) / } * o1 ( x, y, z) f ( x xˆ, y yˆ, z zˆ) dxdydz j π M u x + ξ + v y + η + au z + ζ λ f dxdydz dξdηdζ = exp[ j πm ( uxˆ+ vyˆ + au zˆ) / λf ] dxdydz ˆ ˆ ˆ = g( xˆ, yˆ, zˆ)exp[ j πm ( uxˆ + vyˆ + au zˆ)/ λf ] dxdydz ˆ ˆ ˆ (7) gxyz ( ˆ, ˆ, ˆ) = o( xyz,, ) f ( x xy ˆ, yz ˆ, zdxdydz ˆ) (8) 1 xˆ = x+ ξ and yˆ = y+ η zˆ = z + ζ The functon gxyz ( ˆ, ˆ, ˆ) s the 3-D correlaton between the tested object functon and the reference functon. The -D functon T(u, v) s generally complex-valued and n ths stage t s stored nsde the computer memory. Lookng over the last lne of relaton (7) we note that there s not symmetry between the horzontal coordnate u and the vertcal coordnate v. In the argument of the exponent one fnds the term au z wthout the term av z. Ths s because the recordng process s not symmetrc snce all the recorded projectons are horzontal-only vewponts. If we could collect all the projectons along the horzontal as well as the vertcal vewponts, relaton (7) would become symmetrcal. In such hypothetcal case the functon T(u, v) could become T( u, v) = g( xˆ, yˆ, zˆ)exp{ j πm[ uxˆ+ vyˆ + a( u + v ) zˆ]/ λ f } dxdydz ˆ ˆ ˆ (9) 30 Proc. of SPIE Vol. 4789

5 In order to understand the way that the correlaton functon gxyz ( ˆ, ˆ, ˆ) can be obtaned from Tuv (, ), we have to look for equvalent optcal system that yelds the same dstrbuton of T( uv, ). Such system s well known n the lterature for the parameter value a=-1/f. When gxyz ( ˆ, ˆ, ˆ) s located n the vcnty of the front focal pont of a sphercal lens, and llumnated by a plane wave, the -D complex ampltude on the back focal plane of ths lens s equal to Tuv (, ) gven by Eq. (9) 1 for a=-1/f. Therefore, based on the prncple of optcal recprocty, f T( uv, ) located at the front focal plane, s encoded as a hologram and ths hologram s llumnated by a plane wave, the 3-D dstrbuton gxyz ( ˆ, ˆ, ˆ) s reconstructed n the vcnty of the back focal pont of the lens. gxyz, ( ˆ, ˆ, ˆ) as mentoned above, s the 3-D spatal dstrbuton of the cross-correlaton between the object functon o 1 ( xˆ, yˆ, z ˆ) and the reference functon f( xˆ, yˆ, z ˆ). Note that choosng a dfferent value for the parameter a changes only the proportons between the transverse and the longtudnal dmensons of the reconstructed space, as already dscussed n Ref. 11. Unfortunately because of the scannng drecton along the horzontal axs only, the dstrbuton we can encode nto a hologram s Tu,v ( ) but not Tuv (, ). As a result the system nherently suffers from astgmatsm, where the correlaton peaks are focused along the x axs n dfferent locatons than they are focused along the y axs. Because of the term of au z the correlaton peaks are focused along the horzontal axs n the locatons of the true objects at the nput space, whle because of mssng of the term of av z the same correlaton peaks are focused along the vertcal axs on the back focal plane of the reconstructng lens. Ths means that f a true object s located somewhere out of the orgn, ts correlaton peak appears out of the back focal plane as a small lne along the vertcal axs nstead of a pont. Identfyng the locaton of an object along the longtudnal axs should be done accordng to the measure of focusng along the horzontal axs only. The astgmatsm descrbed above s one reason for the lower SNR n comparson to the result of the BGU correlators. As mentoned above the astgmatsm s avodable f the scannng of the nput scene s performed along all the drectons of a transversal plane. Therefore the astgmatsm can be consdered as a preventable problem that can be removed wth an mproved technology. However there s unavodable source of nose, whch s nherent to the holographc reconstructon. When there are several true objects, dstrbuted n the scene n dfferent transverse planes, the beam, whch creates a correlaton peak for each true object passes through all the rest of transverse planes and nduces nose on all of them. Ths phenomenon dstngushes the obtaned QOC holographc reconstructon from the real 3-D cross-correlaton dstrbuton obtaned by the BGU correlators, and ths s the reason that ths system s consdered as a quas-correlator only. As mentoned above, the hologram values are memorzed nsde the computer n a form of the complex functon T(u, v). To reconstruct the correlaton space from the complex functon, the computer should modulate some transparency medum wth T(u,v) values. In case the transparency cannot be modulated drectly wth complex values, one of many well-known codng methods of CGH's 13 mght be used. The spatal lght modulator (SLM) we use n ths study can modulate an ncdent wave wth postve gray-tones. Therefore, the complex functon T(u, v) s coded nto a postve real transparency as follows π Tr( u, v) = Re T( u, v) exp ( dxu+ dyv), ( 10 ) λ f where (d x, d y ) s the new orgn pont of the reconstructon space, and T(u,v) s normalzed between 0 and 1. The holographc reconstructon setup s shown n the lower part of Fg. 1. The SLM modulated by T r (u,v) s llumnated by a plane wave, whch propagates through the Fourer lens toward the observer. T r (u,v) gven n Eq. (10) contans, among others, the term T * (u, v). From a Fourer transform of ths term one can get the output correlaton peaks wth the same orentaton as the orgnal objects n the vcnty of the pont ( x0, y0, z0) = ( dx, dy,0), where z 0 = 0 desgnates the back focal plane of the Fourer lens. Proc. of SPIE Vol

6 3. EXPERIMENTAL RESULTS To test our proposed scheme, an experment was carred out, as descrbed next. In ths prelmnary experment, the tested 3-D scene ncluded three objects wth labels on ther front faces, shown n the upper part of the Fg. 1. Two of them wth the letters BGU on ther faces were the true targets that should be detected by the proposed system. The lower front object was located at the pont (x,y,z)=(0,0,5.6)cm, whereas the hgher back object was put at the pont (-3,1.5,-5.4)cm (both wth the letters BGU ). The object wth the letters LR located at the pont (3,8.5,-5.4)cm, was used as the false target. Ths object should be gnored by the system. One dgtal camera observed the scene from a dstance L=10cm from the orgn of the tested scene. The camera was shfted along an arc centered at the orgn. At each poston, the camera was always drected toward the orgn, and captured each projected mage nto a computer. The total vew angle nterval was 3 0, 16 0 for each sde of the z-axs. The camera was shfted by from pont to pont. Thus, sxty-fve projectons of the 3-D tested scene were recorded nto the computer, sxteen of them are depcted n Fg. wth a gap of 0 between every two consecutve projectons. The spectral matrx o 3 (u,v) generated from these data s shown n Fg. 3, where Fg. 3(a) and 3(b) are ts ampltude and phase angle, respectvely. (a) (b) Fg. 16, out of 65 projectons of the tested scene, maged from 16 0 to The reference object was located at the orgn of the scene, and maged nto the computer n the same method as descrbed above,.e. the observed reference object was recorded 65 tmes from dfferent vewponts by the CCD nto the computer. From the recorded projectons of the reference object we calculated a phaseonly flter (POF), frst by calculatng the spectral matrx accordng to Eq. (), and then by takng only the phase dstrbuton of ths matrx. The phase dstrbuton of the POF functon F(u,v) s shown n Fg. 3(c). The multplcaton result T( u, v, ) of o 3 ( uv, ) wth Fuv, (, ) was then encoded as a CGH Tr ( u, v ) accordng to Eq. (10). The central part of the obtaned hologram s shown n Fg. 4. To reconstruct the correlaton space, both smulaton and optcal experment were carred out. Fg. 5 shows the correlaton results of the smulaton, whereas the peaks n Fgs. 5(a) and 5(b) ndcate (c) Fg. 3 The ampltude (a) and the phase functon (b) of the compressed spatal spectrum of the scene. (c) The phase functon of the 3-D POF Fg. 4 The central part of the CGH generated from T(u,v) by use of holographc codng 3 Proc. of SPIE Vol. 4789

7 the exstences and locatons of the front and the back true targets. Fgs. 5(c) and 5(d) are the 3-D plots of the ntensty dstrbutons on the two planes shown n Fgs. 5(a) and 5(b), respectvely. The false target never generated a meanngful peak. In the optcal experment, the obtaned CGH was dsplayed on the SLM and llumnated by a plane wave as shown n the lower part of Fg. 1, the correlaton results were reconstructed from the CGH, and depcted n Fg. 6. Fg. 6(a) recorded at z 0 =.6cm shows the correlaton peak of the front true target, whereas Fg. 6(b) recorded at z 0 =-.5cm depcts the correlaton peak of the back true target. The false target stll never generated a meanngful correlaton peak. The 3-D plots of Fg. 6(a) and 6(b) are depcted n 6(c) and 6(d), respectvely. The expermental results are consstent wth the smulaton results, although the optcal reconstructon system nduces nose, whch s not predcted by the smulaton. From the obtaned correlaton results one can easly detect the two true targets, and locate them as well, n ther 3-D space. The correlaton strength along z 0 -axs s also nvestgated n ths experment. Correlaton peaks ntenstes along z 0 -axs, at ther maxmal transverse ponts, are shown n Fg. 7. (a) (b) (a) (b) (c) (d) Fg. 5 The smulated correlaton results recorded at (a) z 0 = 1.5 pxels (b) z 0 = 1 pxels, (c) and (d) are ther 3-D plots, respectvely. (c) (d) Fg. 6 The expermental correlaton results recorded at (a).6cm and (b)-.5cm along the z 0 -axs. Ther 3-D plots are shown n (c) and (d), respectvely. The proposed system was compared wth the results of the hybrd BGU correlator 6 wth two knds of flters and wth two measures. One measure of the performance s sgnal-to-nose rato (SNR) ndcatng the dscrmnaton capablty and defned by Maxmum correlaton peak ntensty of the true target SNR= (11) Maxmum nose ntensty Another crtera s peak-to-correlaton energy (PCE) 14 ndcatng the correlaton peak sharpness and defned by, Maxmum correlaton peak ntensty of the true target PCE = ( 1) Average correlaton plane energy where the average correlaton plane energy s defned as the sum of the entre correlaton plane ntensty dvded by the total pxels on a correlaton plane. The two crterons are defned on some specfc transverse planes, where the peaks of the true objects are maxmal. Dfferent flters such as the conventonal matched flter and the POF were employed n both 3-D OQC and the hybrd BGU correlator n order to compare ther performances. The results of SNR and PCE are calculated accordngly, and summarzed n table 1. The SNR and PCE for 3-D OQC are summarzed n table 1 column 3 and those for BGU correlators are n column 4. From table 1, one can conclude that although the performances of 3-D OQC are acceptable for dfferent flters, the performances of BGU correlators are superor. Ths s a reasonable penalty Proc. of SPIE Vol

8 we should bear due to the use of less nformaton n 3-D OQC than that n the BGU correlators. However, as mentoned above, when the magng process s carred out along the entre transverse drectons, the generated hologram wll be Tuv (, ) rather than Tuv. (, ) Thus the above-mentoned astgmatsm wll be elmnated, and the performances of 3-D OQC can be mproved. Correlaton Intensty (A.U.) Table 1 Summary of the SNR and PCE between the hybrd BGU correlator and the 3-D OQC for two knds of flters. MF- Matched flter; POF- phase-only flter Fg. 7 The smulated correlaton plot along z 0 -axs. The dash pont and lne denotes the case of the back true object; the astersk denotes the false object, and the sold lne denotes the front true object. z 0 Performance Crtera SNR PCE Flters 3-D OQC BGU Correlators MF POF MF POF CONCLUSION In concluson, we have proposed a novel method named 3-D OQC to mplement 3-D object recognton. By fusng several projectons of the tested scene and wth proper flterng, a -D hologram of the correlaton between the tested scene and the reference functon s dgtally generated. The correlaton results are reconstructed when the sngle hologram s coherently llumnated. The smplcty and speed of the OQC are superor over those of the BGU correlators. However, because of less nformaton used n the 3-D OQC than n the BGU correlators, the method suffers from some penalty of lower SNR and lower PCE n comparson to the BGU correlators. The OQC performances can be mproved when symmetrcal scannng process s carred out. The method of a sngle hologram s general and applcable even to crcumstances where the objects are occluded from some vewponts or ther shapes are changed as the camera moves. Ths generalty statement s based on our knowledge about holograms of 3-D objects whch can mage partally occluded objects from certan ponts of vew. Another unque motve n ths study s the nature of the 3-D correlaton space. For the best of our knowledge, t s the frst tme that someone suggests and demonstrates a reconstructon of 3-D correlaton space from a sngle hologram. Ths nnovaton should be accompaned by a new concept of detecton of the correlaton results. One possble opton s to let a human observer to look over the correlaton space and to estmate the locatons of all the true targets n the scene. 34 Proc. of SPIE Vol. 4789

9 Reference 1. R. Bamler, J. Hofer-Alfes, "Three- and four-dmensonal flter operatons by coherent optcs," Opt. Acta 9, (198).. J. Rosen, "Three-dmensonal optcal Fourer transform and correlaton," Opt. Lett., (1997). 3. J. Rosen, "Three-dmensonal electro-optcal correlator," J. Opt. Soc. Am. A 15, (1998). 4. J. Rosen, "Three-dmensonal jont transform correlator," Appl. Opt. 37, (1998). 5. Y. L, J. Rosen, "Three-dmensonal pattern recognton wth a sngle two-dmensonal synthetc reference functon," Appl. Opt. 39, (000). 6. Y. L, J. Rosen, "Three-dmensonal correlator wth general complex flters," Appl. Opt. 39, (000). 7. J. J. Esteve-Taboada, D. Mas and J. Garca, "Three-dmensonal object recognton by Fourer transform proflometry," Appl. Opt. 38, (1999). 8. Y. Frauel, E. Tajahuerce, M.-A. Castro, and B. Javd, "Dstorton-tolerant three-dmensonal object recognton wth dgtal holography," Appl. Opt. 40, (001). 9. O. Matoba, E. Tajahuerce, and B. Javd, "Real-tme three-dmensonal object recognton wth multple perspectves magng," Appl. Opt. 40, (001). 10. M. Takeda and K. Mutoh, "Fourer transform proflometry for the automatc measurement of 3-D object shapes," Appl. Opt., (1983). 11. Y. L, D. Abookass, and J. Rosen, "Computer-generated holograms of three-dmensonal realstc objects recorded wthout wave nterference," Appl. Opt. 40, (001). 1. J. Rosen, "Computer-generated holograms of mages reconstructed on curved surfaces," Appl. Opt. 38, (1999). 13. O. Bryngdahl and F. Wyrowsk, "Dgtal holography - computer-generated Holograms," Prog. In Optcs, E. Wolf, ed. (North Holland, Amsterdam, 1990) Vol. 8, pp V. K. Vjaya-Kumar and L. Hassebrook, "Performance measures for correlaton flters," Appl. Opt. 9, (1990). Proc. of SPIE Vol