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Coherent lensless matched filter and an application to feature extraction in character recognition Bage, Marc J. 1975

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COHERENT LENSLESS MATCHED FILTER AND AN APPLICATION TO FEATURE EXTRACTION IN CHARACTER RECOGNITION by Marc J. Bage Ingenieur C i v i l Electricien (Courants Faibles) Universite Catholique de Louvain, Belgium, 1969 M.Sc. (E.E.), Universite Laval, Quel-ec, P.Q., 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the required standard Research Supervisor. Members of the Committee., Members of the Department of E l e c t r i c a l Engineering The University of Bri t i s h Columbia August 1975 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th i s thes is fo r f i nanc ia l gain sha l l not be allowed without my writ ten permission. Department of CLeJjC^cJl u ^ c ^ p . The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT Reduction of space variance in coherent matched f i l t e r i n g i s considered in the context of parallel character recognition. The space variance due to vignetting apertures i s eliminated by synthesizing the effect of the second lens of the Vander Lugt matched f i l t e r (MT) inside the hologram. This leads to two types of lensless matched f i l t e r (LLMF) depending on the curvature of the reference beam and the relative scale of the recorded and readout signals. For a given space-bandwidth product, two layouts of recorded and readout signals are found to minimize the space variance due to the volume of the recording medium. They are well suited for line-by-line character recognition. A study of the holographic aber-rations indicates for which of these layouts the space variance of the root mean square (rms) distortion is smallest. Computer-generated plots are used to investigate the space variance of the rms astigmatism of the MF and LLMF. The condition that makes the spherical aberration and coma n u l l i s also derived. Various aspects of the LLMF are examined. The LLMF acts simul-taneously as a MF and a Fresnel zone plate. Therefore, when the hologram is displaced, the LLMF behavior results from the combined displacements of it s f i l t e r component and lens component. In the output plane, the corre-lation component only i s focused by the lens component. The other diffracted output components are unfocused images. They contribute noise that affects the correlation signal. In practical cases of p a r a l l e l processing, the signal-to-noise ratios of the MF and LLMF at the same reference beam angle are comparable and the resolution of available recording media places no restriction on the f i e l d of view or the signals bandwidth. i i Improvement of the recognition performance i s another concern of the thesis. It is shown experimentally that a f i l t e r matched to charac-ter features (FEMF) discriminates better between the similar looking charac-ters 0 and Q than a high-pass f i l t e r matched to their high frequency compo-nents (HPMF). Yet i t s sensitivity to variants i s not as severe as that of the EPMF when the feature extractor i s a low-pass f i l t e r . Also this low-pass f i l t e r i n g reduces the aberrations and allows for a detector with a coarser resolution. THe increased r e l i a b i l i t y of the FEMF i s achieved at the expense of a reduced diffraction efficiency and of a more c r i t i c a l registration of the readout signal. Experimental considerations emphasize the importance of phase accuracy in the input plane. Finally, suggestions for further research are presented. •m TABLE OF CONTENTS Page Abstract i i Table of Contents iv List of Tables v i i List of Figures v i i i Glossary . . . . x i Acknowledgement x v n I INTRODUCTION 1.1 Motivation 1 1.2 Space Invariance: Two Definitions 4 1.2.1 Fil t e r i n g System 4 1.2.2 Fourier-Transforming System . 4 1.3 Previous Research 4 1. A Outline of the Thesis 8 II MATCHED AND LENSLESS MATCHED FILTERS 2.1 Introduction 10 2.2 Matched F i l t e r s : a Review 11 2.2.1 Recording of the MF-1 11 2.2.2 Correlator 14 2.2.3 Minimum Spatial Carrier-Frequency 16 2.2.4 Signals Layout for Parallel Processing . . . . . 16 2.2.5 Improvement of Space Invariance with the MF-2 . . 17 2.2.^ Space-Bandwidth Product 20 2.3 General Matched F i l t e r 21 2.3.1 Recording of the General Matched F i l t e r 21 2.3.2 General Correlator 23 2.3.3 Particular Cases . . . . . . 30 2.3.3a MF-2: 6 = 1 . . . . . . . . . . . 30 2.3.3b LLMF-1: f 2 = 0 31 2. A Lensless Matched F i l t e r 32 2.A.1 LLMF as Fesnel Zone Plate 32 2.A.2 Typical LLMF Configurations 33 2.A.3 Antecedents of the LLMF 37 2.A.A Space Variance of the Stigmatic Code Translator. 39 2.A.5 Limits of the Spatial Carrier-Frequency AA 2.A.5a Upper Bound: Resolution of the Recording Medium . 4 4 2.A.5b A Lower Bound: Shadow Limit A7 2.A.5c Comments on the Spatial Carrier-Frequency. . -48 2.5 Conclusions . . . 50 iv Page III NON-CORRELATION OUTPUT COMPONENTS 3.1 Introduction 51 3.2 Total Output Field Distribution of the LLMF 52 3.3 The MF as a Limiting Case . . . . 54 3.4 Fresnel Diffraction Patterns 54 3.5 Signal-to-Noise Ratio . 57 3.6 Invariance of the Parameters 60 3.7 Conclusions 62 IV SPECTRAL DISPLACEMENT 4.1 Introduction 63 4.2 Transverse Displacement of the F i l t e r 64 4.2.1 General Matched F i l t e r . . . . . . . . . . . . 64 4.2.2 Rectangular Signal 65 4.2.3 Asymptotic Behavior of the LLMF 68 4.2.4 Physical Interpretation . . 70 4.2.5 Comments on Transverse Displacement 71 4.3 Phase Error on Readout Signal 72 4.3.1 Rectangular Signal . 73 4.3.2 Experimental Verification . . 73 4.4 Longitudinal Displacement of the F i l t e r 75 4.4.1 Displacement of the F i l t e r Component . . . . "." 76 4.4.2 Displacement of the Lens Component 77 4.4.3 Comments on Longitudinal Displacement . . . . . 78 4.5 Conclusions . . . . . . . . . . . . . . 79 V SPACE VARIANCE 5.1 Introduction 81 5.2 Volume Effect of the Recording Medium 81 5.2.1 Introduciton 8 1 5.2.2 Diffraction from a Three-Dimensional Hologram . 83 5.2.3 Volume Effect of the Recording Medium . . . . . 8 5 5.2.4 Layout of Recorded and Readout Signals . . . . 88 5.2.5 Numerical Example . . . . . . . . 90 5.2.6 Comments on the Volume Effect 93 5.3 Holographic Aberrations 94 5.3.1 Introduction 94 5.3.2 Third-Order Aberrations of Points Holograms . . 95 5.3.3 Comments on the Five Seidel Aberrations . . . . 99 5.3.4 Standard Deviation of Aberration (SDA) . . . . 100 5.3.5 Aberration and SDA versus Signals Bandwidth . . 102 5.3.6 Space Variance of SDA (or rms Distortion) . . . 104 v Page 5.3.7 Aberration and SDA at constant SBP 108 5.3.8 Space Variance of rms Astigmatism . . . . . . . . 109 5.3.9 Comments on Aberrations. H I 5.4 Conclusions 112 VI FEATURE EXTRACTION 6.1 Introduction HA 6.2 Matched Fil t e r i n g Techniques: a Review HA 6.2.1 Spectral Modifications of the MF 115 6.2.2 Spatial Modifications of the MF 116 6.2.3 High-Pass MF versus Feature Extraction 117 6.3 Selectivity and Tolerance to Variants: Experimental Results . . . 117 6.3.1 Selectivity 117 6.3.2 Translated Signals 120 6.3.3 Rotated Signals 124 6.3.A Different Typefonts 125 6.3.5 Diffraction Efficiency 131 6. A Conclusions 132 VII EXPERIMENTAL CONSIDERATIONS 7.1 Introduction 13A 7.2 Experimental System.. . . 135 7.2.1 Description of the Experimental Set-up 135-7.2.2 Normalization Procedure 139 7.3 Space Variance Experiments or Phase Error of the Illuminating Beam 1A1 7. A Phase Error of the Incoherent-to-Coherent Image Converter (ICIC) -1*5 7.5 Conclusions 147 VIII CONCLUSIONS 8.1 Summary of Results 1 4 9 8.2 Suggestions foi Further Research 151 REFERENCES 153 APPENDIX: Operational Notation for Optical Systems 160 v i LIST OF TABLES Table Page 5.1 Intensity at the diffraction focus of different f i l t e r s in presence of astigmatism I l l 7.1 L i s t of equipment and identification of the symbols of Fig. 7.1 137 v i i LIST OF FIGURES Figure Page 2.1 Arrangement for recording the MF-1 12 2.2 Fil t e r i n g arrangement of the MF-1 14 2.3 Layouts of (a) the recorded and (b) the readout signals for ^ parallel processing, (c) Distribution of the correlations . 2.4 MF-2. (a) Space-invariant recording system, (b) F i l t e r i n g system 18 2.5 Arrangement for recording the general matched f i l t e r . . . . 22 2.6 Fi l t e r i n g arrangement of the general matched f i l t e r . . . . 24 2.7 Block diagram representation of the light propagation from the hologram to the output plane 25 2.8 Interdependence of the LLMF parameters 34 2.9 Typical LLMF-1 arrangement: (a,6,6) = (1,1,-1). (a) Recording, (b) F i l t e r i n g 35 2.10 Typical LLMF-2 arrangement: (a,8,6) = • (a) Recording, (b) Fi l t e r i n g 35 2.11 Gabor's code translator, (a) Recording, (b) Recognition. . . 39 2.12 Stigmatic code translator (D^D S) . (a) Recording. (b) Recognition 40 2.13 Space variance of the stigmatic code translator 43 2.14 Geometry for the discussion of the. medium resolution limit and the shadow limit, (a) LLMF-1. (b) LLMF-2 46 2.15 Limits on the off-axis coordinate x r of the reference point source 49 3.1- Minimum SNR of the LLMF (y = 1.57 and 7.85) and of the MF (y 5 8 °°) • G is the relative off-axis coordinate of the reference point source . 59 4.1 Output intensity versus the transverse displacment Au of the f i l t e r , a = 1, 6 = ±1, X = 632.8 nm, D t = 500 mm. (a) 2a = 0.1 mm. (b) 2a = 1mm. (c) 2a = 3 mm 67 4.2 Actual and asymptotic characteristics of the MF and LLMF. Transverse displacement (Au) and signal size (2a) that reduce the output intensity to a fraction M of i t s value at Au = 0. (a = 1, 6 = ±1, X =632.8 nm, D t = 500 mm). . . 69 4.3 Model of the image translation due to a transverse displace-ment of the holographic lens 70 4.4 Double pulse signal . 74 4.5 Theoretical and experimental variations of the output intensity versus the transverse displacement of the f i l t e r . ,. . . . . 74 4.6 Autocorrelation of "G" when the transverse displacement Au of the readout spectrum i s (a) 0 ym, (b) 180 um . . . . . . 75 v i i i Figure Page 4.7 Model of the image modifications due to a longitudinal displacement of the holographic lens 76 4.8 Asymptotic characteristics of the MF and LLMF. Longi-tudinal displacement (z) and signal size (2a) that reduce the output intensity to a fraction M of i t s value at z = 0. (a = 1, 6 = ±1, X = 632.8 nm, D t = 500 mm) 79 5.1 Direction cosines of the propagation vector R"^  84 5.2 Angular coordinates of the recorded and readout signals. . 89 5.3 Sensitivity of various signals layouts to the volume effect of the recording medium 91 5.4 Recorded and readout signals that, for a given SBP, are least sensitive to the volume effect, (a) "+" layout. (b) " I " layout 92 5.5 Geometry for the computation of the phase of the spher-i c a l wave issued from P 95 5.6 Dependence of the SDA (in X ) on the radius of the hologram aperture, (a) R h = 0.1 mm. (b) R^  = 10 mm 103 5.7 "+" signals layout: Space variance of the SDA (in X/100) when 9 X S = 0.0 and (a) 6 y s = 0.0 or (b) 6 y s = 0.1 . . . . 1 ° 6 5.8 " I " signals layout: Space variance of the SDA (in X/100) when 0 s = 0.1 and (a) 9 X S = -0.05, (b) 8^ = 0.0 or (c) 9X* = 0.05 107 5.9 Space variance of the SDA' (in X/1000). (a) MF-2: a =1. 6 = 1 . (b) LLMF-1: a~= 1, 6 = -1. (c) LLMF-2: a = 0.33, 6 = 0.33 HO 6.1 Set of characters used for selectivity experiments (after 3X reduction). The arrows point to the recorded signals. The others are the readout signals -^j 6.2 Average responses of the 0- and Q-filters to the readout signals of Fig. 6.1. red = LPMF, green = HPMF, blue = FEMF H9 6.3 Correlation pattern of the set of readout signals ^ j _0 with the recorded signal Q. (a) HPMF (exposure time = 1/30 sec), (b) LP;tF (exposure time = 1/4 sec), (c) LPMF (exposure time = 1 sec). . . . . . . ... . . . 120 6.4 Crosssections of the autocorrelation peaks of 0 [(a)* (b)] and of Q [(c),(d)] at constant intensity (0.8 and 0.5). (a) and (c) LPMF. (b) and (d) HPMF . 121 6.5 Crosssections of the crosscorrelation of the t a i l of Q (a) with Q and (b) withO' . . . . . . . . . . . . . . . 1 2 2 6.6 Sensitivity of the LPMF, HPMF and FEMF to only rotation of the readout signal . . . . . . . 124 ix Figure Page 6.7 Set of characters used for typefonts sensitivity experiments (after 3X reduction). (a) The arrows point to the recorded signals. The others are the readout signals, (b) Typefont identification of (a) . . 125 6.8 Average responses of (a) the Q^-f i l t e r and (b) the 0]_-f i l t e r to signals of different typefonts. red = LPMF, green = HPMF 127 6.9 Maximum and minimum responses of the t a i l - f i l t e r to signals of different typefonts 128 6.10 Comparative responses of the 0^-, Qj- and t a i l - f i l t e r to 129 readout signals (a) Q and (b) 0 of five different typefonts. & red = LPMF, green = HPMF, blue = FEMF 130 7.1 Schematic representation of the LLMF-1 used for feature extraction experiments. Insert: variant of this set-up used for space variance experiments 136 7.2 Overall view of the apparatus 138 7.3 Performance of the normalization procedure 141 7.4 Loci of the readout signal positions that yield the same autocorrelation intensity, (a) MF-2. (b) LLMF-1 143 7.5 Variation of the measured autocorrelation p 2 with the ss ICIC wedge angle or fringe frequency, (a) Recorded signal. (b) to (f) Readout signals. 10X magnification 146 7.6 Set-up used for recording the interference pattern of Fig. 7.5 146 GLOSSARY a half width of a one-dimensional aperture ai(p,q) f i e l d distribution incident on lens L^ ' a2(P»l) f i e l d distribution emerging from lens A complex amplitude of the f i e l d incident on the recorded signal B complex amplitude of the reference beam in P^ c(u,v) readout f i e l d in C g t crosscorrelation of s(x,y) with t(x,y) d e = d A ( d c - f 2 ) / d A + d 0 - f 2 ) D ; - V J C M + d o - f 2 ) dj = D71 (j = i,Jl,o,r,s,t) D. distance between P„ and the focus plane of the illuminating beam — o 1 f i D^  distance between lens and P^ D distance between P and either lens L„ (MF) or P, (LLMF) o o l n D r distance between P R and P^ Ds distance between P S and P^ Dt distance between P T and P^ DE diffraction efficiency E[.] expected value of .... E g energy of the recorded signal E t energy of the readout signal A " ^ i 1 (i « 1,2) F i focal length of lens L^ ^ ( i = 1,2) F^ equivalent focal length of the LLMF FEMF feature extractor and matched f i l t e r x i O l 8 - l GSBP h(x,y) HPMF Km) Ku,v) ICIC j k k. 1 "4 H L. 3 LLMF LPMF (m,n) (mo>no) M M, any one of the two components of the MF output f i e l d that are diffracted along the optical axis ( i = 1,2) component of the MF output f i e l d that is diffracted in the minus first-order direction auxiliary function increase in space-bandwidth product impulse response of free space propagation horizontal (&=x) and vertical (£=y) angular width of the set. of recorded (i=s) and readout (i=t) signals. high-pass matched f i l t e r auxiliary integral intensity of the interference pattern of o(u,v) and r(u,v) incoherent-to-coherent image converter * A-D « 2n/\ / propagation unit vector ( i = o,r,s,t) = dV/(A 2o; 2) ^d Qd t/(A 2a 2) angular width of a pixel lens ( i = 1,2) spatial width of the recorded (j=s) or readout signal (j=t) lensless matched f i l t e r (Can be of type -1 or -2. See Fig. 2.9 and 2.10) low-pass matched f i l t e r coordinates in the output plane P Q coordinates of the centre of the correlation pattern output intensity magnification of the MF and LLMF x i i M^  magnification of the LLMF when the hologram i s shifted longitudinally M magnification of the recorded signal s M^  defined by sine 2 M^  = M MF matched f i l t e r (Can be of type -1 or -2. See Fig. 2.1, 2.2 and 2.4) n g refractive index of the recording medium o(u,v) object f i e l d in 0(m,n) f i e l d distribution in output plane P D 0 o£ any one of the two components of 0(m,n) that are diffracted along the optical axis ( i = 1,2) crosscorrelation component of 0(m,n) 0 ^ component of 0(m,n) diffracted in the minus first-order direction OPD optical path difference (p,q) coordinates in the plane of lens p z point source P total number of recorded signals P^  hologram plane P Q output plane P r plane of the reference point source Pg plane of the recorded signal P t plane of the readout signal P x number of columns i n the array of recorded signals Py number of rows in the array of recorded signals Q total number of readout signals Qx number of columns i n the array of readout signals number of rows i n the array of readout signals position vector in the recording medium x i i i r(u,v) reference f i e l d in P n rms root mean square diameter of the hologram over which a is computed s(x,y) recorded signal s^(x,y) any one of the P elementary signals making up s(x,y) sgn(x) = x/|x| sine x = x -* sxn x S(5,n) Fourier transform of s(x,y) SA spherical aberration SBP space-bandwidth product SBPi SBP of the output f i e l d (i=o) , reference f i e l d (i=r) and readout f i e l d (i=t) SDA standard deviation of aberration SNR signal-to-noise ratio t(x,y) readout signal tj(x,y) any one of the Q elementary signals making up t(x,y) T thickness of the recording medium T(£,n) Fourier transform of t(x,y) (u,v) cartesian coordinates in the hologram plane P^ (u,v,z) unit vectors along the coordinates axes in P^ w auxiliary variable w width of each of the pulse of the double pulse signal s W(p,6) aberration of the diffracted f i e l d Wg centre to centre spacing of the pulses of the double pulse signal (x,y) coordinates in P s (x r,y r) coordinates of the reference point source i n P^ xiv x t coordinate of the readout signal X^,Y^,Z^ direction cosines of the propagation unit vector ic^ ( i = o,r,s,t) z longitudinal displacement of the hologram z 1 longitudinal displacement of the output plane a (= d^/d ) ratio of the scales of the recorded and readout signals B = <L/d o s A ( 1 - 6 ) ) 6 = d r/d s 6(x,y) two-dimensional Dirac delta function Au transverse spatial displacement of the hologram Ag transverse spectral displacement of the hologram Ak. vector difference between k. and k. [(i = o and j = r) or 1 ( i = t and j = s)] 1 3 AX^,AY^,AZ£ direction cosines of Ak\ ( i = o,t) e = mg/2a X, = (1 - 6)/2a 0^ horizontal (£=x) and vertical (£=y) angular coordinate of the reference (i=r), the object (i=s) and the readout (i=t) point source 0^ half angle subtended i n P^ by the hologram aperture X wavelength (£»ri) spatial frequency coordinates in P^ ? c cut-off frequency of the recording medium £ maximum spatial frequency of the recorded signal s (p,0) polar coordinates i n half diameter of the object spectrum xv p 2 . normalized intensity of the central value of the crosscorre-s t lation of s(x,y) with t(x,y) aw(R^) standard deviation of aberration T ( U , V ) amplitude transmittance of the hologram x + 1(u,v) component of T ( U , V ) that contributes to the crosscorrelation x + 1^(u) same as T + but with hologram transversally shifted by Au <f> angular width of an elementary signal s^(x,y) or t_.(x,y) <J>.(u,v) phase^of the reference (i=r) , the object (i=s) and the readout (i=t) spherical wave in cJ>o(u,v) phase of the ideal spherical wave diffracted by the hologram 4>._(u,v) first-order approximation of <}>.(u,v) (j = o,r,s,t) <J>. (u,v) third-order component of the third-order approximation of <j>.(u,v) 3 3 (j = o,r,s,t) J $(p>6) phase of the actual f i e l d diffracted by the hologram $l(p»0) first-order approximation of $(p,g) $3(p>6) third-order component of the third-order approximation of $(p>6) $ half angle subtended at P, by the recorded signal s n <Kx,y;d) = exp(jyd(x 2 + y 2)) fi angular bandwidth of the recorded signal a complex conjugate of a a vector (where a is any symbol but i|;(x,y;d)) ij)(x,y;d) complex conjugate of t|>(x,y;d) |a| modulus of a a.b dot product * convolution operator g crosscorrelation operator a indicates a proportionality relationship •*-*• indicates a Fourier transform relationship xv i ACKNOWLEDGEMENT I would like to thank my supervisor, Dr. M.P. Beddoes, for his encouragement and his constant availability during the course of this thesis. I am indebted to Dr. J.P. Duncan, Head of the Department of Mechanical Engineering of UBC, for making available some optical components and the air-suspended table necessary to obtain reliable experimental results, I am grateful to the staff members of this Department for their assistance at various stages of the project, particularly Mr. H.T. Walters for his handling of the purchase of equipment, Mr. H.H. Black for his execu-tion of some of the photographic work required by the project and Miss S. Lund and Mrs. V. Walker for their proficient typing of the thesis. I also wish to thank my fellow graduate students with whom I shared many experiences, particularly Mr. D. Wasson and Mr. J. Yan for their comments and suggestions during the preparation of the manuscript and Dr. A. Soubigou, a fine friend, for helping me with the drawings. Financial assistance in the form of a NRC scholarship, a NRC research assistantship and a departmental teaching assistartship i s acknow-ledged . x v i i 1 I. INTRODUCTION 1.1 Motivation In the early sixties, Leith's and Upatnieks' contributions [1] and the advent of the laser as a powerful source of coherent light brightened up the future of Gabor's invention: holography [2]. Quite a number of potential applications immediately aroused interest: three-dimensional imaging [3], holographic non-destructive testing (HNDT) and pattern recog-nition [A] to name just a few. It i s not u n t i l recently though that holography ventured out of the laboratory to successfully perform imaging [5] and HNDT [6] tasks under controlled commercial and industrial environ-ments. As for i t s pattern recognition promise, i t i s being field-tested i n the holographic fingerprints sorting equipment presently under evaluation at the New York City's police headquarters [7]. Essential to a coherent pattern recognition is the holographic matched f i l t e r (MF) developed in 196A by Vander Lugt as a technique for recording a complex transfer function as a real-valued function on a spatial carrier-frequency [A]. There are many possible uses for the MF. From the onset, recog-nition of alphanumeric characters has been an area of prime interest [8], with applications ranging from Dickinson's and Watrasiewiez's automatic mail sorting [9] to Hard's and Feuk's reading aid for the blind [10]. With characters as input data, the recognition signal of the MF could be used to trigger a speech or spelled-speech generator [11] or to automatic-a l l y feed printed material into a computer. The experimental detection of words for library search purpose has been reported by Holeman [12] and by Vander Lugt [13] while speech identification has been carried out by Leverington [IA]. An analyzer of cloud motion has been implemented by 2 Vander Lugt [15].. Horvath et al.[16] and Thomassen [17] have investigated the recognition of fingerprints. Cluster analysis has been performed by Fourier and Vienot for paleographic and botanic purposes [18], by Vander Lugt for classification of signatures of nuclear detonations [19] and by Lanz et a l . for leukocytes classification [20]. Although most of the techniques discussed and developed in this thesis are presented i n the context of character recognition, they are of general application. The lack of an adequate real-time input transducer is the major roadblock that has hampered the commercial implementation of coherent holographic pattern recognition equipment. The transducer is an incoherent-to-coherent image converter (ICIC) whose role is to impress on a laser beam the information of the readout pattern. After phase correction, films and micro-films are suitable for non real-time processing. Much research i s devoted to the development of a real-time ICIC. Devices like Titus [21], PROM [22,23] and the Membrane Light Modulator [24] are among many of the contenders for the task. While awaiting further advances in the ICIC area, the holographic MF remains attractive. Regardless of the complexity of the pattern to be recognized, fingerprints for example, the synthesis of the f i l t e r always consists of simply recording a Fourier transform hologram of the pattern. Also the hologram has a large information storage capacity. This allows i t to compare the input data simultaneously with a large number P of recorded signals. Moreover, the input data can be made of a large number Q of elementary readout signals. The MF can compute instantaneously and in p a r a l l e l the P x Q crosscorrelations so that i t s processing speed is potentially very high [25].' 3 The more space-invariant i s the f i l t e r i n g system, the larger the number Q of simultaneous readout signals can be (Space invariance is de-fined in section 1.2.1). It follows that high data rate and space invar-iance go hand in hand. Space variance can be caused by vignetting aper-tures, by the volume of the recording medium and by holographic aberrations. Therefore this thesis proposes two lensless matched f i l t e r s (LLMF) that are space-invariant since their only vignetting aperture is located in the Fourier transform plane. It also minimizes the space variance due to the volume of the hologram by determining the optimum layouts of recorded and readout signals in par a l l e l processing. Finally i t compares the aberrations of the MF and LLMF and those of different signals layouts. Parallel processing and high data rate are significant advantages. However they are desirable only i f at the same time the recognition perform-ance i s also satisfying. The simple correlation or template matching opera-tion of the MF and LLMF does not allow for as wide a range of signal varia-tions as slower optical character recognition equipment can successfully recognize. Also i t does not discriminate easily between similar looking characters. A modification of the MF i s needed to improve i t s recognition performance. Often a high-pass matched f i l t e r (HPMF) is used to increase the selectivity. Its sensitivity to variants i s rather high. The thesis looks at an alternative technique: low-pass f i l t e r i n g and feature extrac-tion. It i s very selective and tolerates more variants than the HPMF tech-nique. Its recognition performance i s promising. In summary, i t seems that holographic character recognition i s appropriate for those applications where processing speed i s at a premium, like i n a computer front-end interface, and where the data generation i s well controlled, that i s in a cooperative system. 4 ' 1.2 Space Invariance: Two Definitions Since many arguments of this thesis hinge around the notion of space invariance, i t i s useful to define this concept here. Actually . i t has two different meanings depending on whether i t i s applied to a f i l t -ering system or to a Fourier-transforming system: . 1.2.1 Filtering System A f i l t e r i n g system i s said to be space-invariant or isoplanatic when a transverse translation of the input signal results only in a propor-tional translation of the output signal. In matched f i l t e r i n g , the correla-tion intensity i s considered to be the output signal since the output detector ignores the phase. 1.2.2 Fourier-Transforming System A Fourier-transforming stage is said to be space-invariant when, apart from a linear phase factor, the f i e l d distribution i n the spec-trum plane i s independent of the input signal transverse coordinate. The f i r s t definition is the one that i s most often referred to in the text. 1.3 Previous Research Optical pattern recognition can be carried out with either incoherent or coherent light. The incoherent processors are of two types depending on whether they are based on geometrical considerations or on diffraction theory. In the f i r s t case, the crosscorrelation of two fixed or moving patterns i s computed on the assumption that the light rays follow the rules of geometrical optics (see for example [26, Section 7-3]). Such processors are accurate only when diffraction effects are negligible. 5 Therefore they are not suitable for processing data with a high space-bandwidth product. Armitage [27], Lohmann [28,29] and Lowenthal [30] des-cribed incoherent processors based on diffraction theory. They synthesized f i l t e r s matched to the intensity of either the signal or i t s spectrum [27]. In the latter case, the transverse position of the readout signal i s ren-dered irrelevant and the pupil can be the signal i t s e l f . In the former case, a Fourier transform hologram of the signal can be used as a pupil. The advantages of these schemes are that they can process incoherent patterns (that i s they do not need the ICIC required by the coherent MF) and that the lateral position of the f i l t e r i s not c r i t i c a l . On the debit side however, the incoherent f i l t e r i s not as general as the coherent one. Indeed i t s impulse response must be non-negative or i t s optical transfer function must belong to the class of the autocorrelation functions. Therefore no contrast enhancement by high-pass f i l t e r i n g i s possible. Also negatively weighted patterns [31] and bipolar "principal components" [32] can not be used to increase the selectivity of the incoherent f i l t e r s since only the intensity of their recorded patterns i s relevant. Furthermore, weighting the optical transfer function to account for coloured noise i s not as simple a matter as with the coherent MF. Coherent processors for the purpose of pattern recognition can be divided up into three categories. Two of them perform code-translation operations while the third one computes the crosscorrelation of two pat-terns via matched f i l t e r i n g . The f i r s t code-translation technique i s based on the resolution-retrieving compensation of source effects [33]: since the image reconstructed by a hologram i s the convolution of the reference signal with the crosscorrelation of the recorded and readout signal, the image is identical to the reference signal when the above correlation i s a delta function. An interesting example of this technique is given in 6 [34] where the picture of one of two persons i s reconstructed upon i l l u m i -nation of a hologram by their respective signatures, and vice versa. Watrasiewicz however demonstrated the limitations of this pattern recogni-tion technique by successfully translating a cross into a triangle but not conversely [35]. In general this technique i s not suitable for character recognition because many characters do not have a delta-like autocorrela-tion. Furthermore the crosscorrelations of some look-alike characters (0 & Q) are so similar to their respective autocorrelations that the reconstruct ted code would not provide a positive identification of the readout signal. For example, 0 would ambiguously reconstruct i t s own code as well as that of Q. The choice of a point source as reference signal reduces the image to the crosscorrelation of the recorded and readout signal. The correlation peak provides a quantitative measure of the similarity between the two patterns. A reference point source i s actually used in the MF case [4]. A unique binary combination of reference point sources can also be used for each of the signals to be recorded [36,37]. The second code-translation technique uses a computer-generated Fourier transform hologram to modulate the illuminating spectrum. The transfer function of the f i l t e r i s the ratio of the spectra of the desired output signal and the readout signal [38]. As usually the latter spectrum has many zeroes, the transfer function can only be approximated. Besides, the ambiguity of look-al:.ke characters i s the same as in the f i r s t code-translation technique. Finally the third technique of coherent pattern recognition, which is often investigated and analyzed, uses the matched f i l t e r [4]. It offers the advantage that the synthesis of the f i l t e r consists simply in recording a Fourier transform hologram of the pattern to be detected, 7 regardless of i t s shape complexity. The MF can also be generated by com-puter [39]. In the case of coloured noise, a f i l t e r with real-valued trans-mittance can simply be juxtaposed to the MF used i n the white noise case. The beginning of .Chapter 6 reviews some variants of the matched f i l t e r i n g technique, particularly low- and high-pass f i l t e r i n g and feature; extraction. The MF can be made rotation and scale insensitive by means of a rotating dove prism and a Fourier-transforming stage of varying scale [40]. The detector output i s then averaged over a revolution of the prism and a scan of the scale range. No evaluation of the selectivity of such a f i l t e r i s provided in [40]. Gallagher derived the expression of the trans-fer function of the "matched f i l t e r " which yields the closest (in the mean square sense) approximation of a crosscorrelation under the constraint that the magnitude of the transfer function i s smaller than unity [41]. To avoid saturating the recording medium, a number of authors used the MF in combination with a multiplexing device. The resulting processor i s a multi-channel MF. The channels can be used i n pa r a l l e l when stationary multiple-xing devices are adopted, for example gratings [42] and point holograms [43]. They rather operate in time-sequential fashion when the f i l t e r plane i s swept by means of devices lik e a rotating wedge [37] or a d i g i t a l light deflector [44]. From their comparison of the paral l e l and time-sequential systems, Kock and Rabe [44] concluded that "the par a l l e l multiplex system appears useful mostly for character recognition problems or similar appli-cations with simple input data where the required resolution within each correlation f i e l d i s not too stringent". The parallel multiplexing device i s also simpler and less expansive than the time-sequential device. Furthermore, being stationary, the parallel system allows for a more accurate registration of the readout beams and recorded spectra [45] than the time-sequential system, 8 1.4 Outline of the Thesis In Chapter 2, the principle of the original matched f i l t e r (MF-1) is reviewed along with i t s minimum spatial carrier-frequency and the layout of i t s recorded and readout signals for parallel processing. Then another version of the matchef f i l t e r (MF-2) whose f i r s t Fourier-transforming stage i s space-invariant i s examined. Since i t s second stage would also be space-invariantfrom the point of view of vignetting apertures with a null distance between i t s hologram and i t s second lens, and since space invariance i s essential for a parallel processor, a general matched f i l t e r i s analyzed so as to derive the condition for synthesis of the lens effect inside the holo-gram. This leads to two types of space-invariant lensless matched f i l t e r s (LLMF). The f i r s t one has a converging reference beam (LLMF-1), The second type has a diverging reference beam and appropriately scaled recorded and readout signals (LLMF-2). The LLMF's act as Fresnel zone plates. They are compared to analogous schemes and particularly to Gabor's stigmatic code-translator. A comparison of the resolution of the recording medium of the MF and LLMF is presented as well as some considerations on the spa-t i a l carrier-frequency of the LLMF. In Chapter 3, the non-correlation output terms of the LLMF that are unfocused images are analyzed because their spatial extent governs the minimum spatial carrier-frequency. These terms contribute noise that affects the correlation signal. The signal-to-noise ratio (SNR) of the LLMF is a function of the reference beam angle or the spatial carrier-frequency. The minimum SNR's of the MF and LLMF are compared. In Chapter 4, the dependence of the correlation intensity on the transverse and longitudinal displacement of the hologram i s investi-gated. It i s shown that a displaced LLMF acts as a displaced MF in combination with a displaced lens. Which of these displacements has the predominant effect when a fixed detector i s used, depends upon the signal size or spectrum scale. Also the effect of a phase error i n the input plane and of the resulting spectral mismatch i s reviewed. In Chapter 5, the space variance induced by the volume of the recording medium and by the holographic aberrations i s discussed. The layouts of recorded and readout signals which, for a given space-bandwidth product, are least sensitive to the volume effect are deter-mined. The aberrations of both MF's and both LLMF's are compared. The condition that makes the spherical aberration and coma of the LLMF-1 n u l l is derived. Computer-generated plots are used to determine the maximum root mean square (rms) distortion of some signals layouts and • the'" maximum*"1*^ f i l t e r s . In Chapter 6, the discrimination and sensitivity to variants of the low-pass matched f i l t e r , high-pass matched f i l t e r and low-pass feature extractor are compared experimentally using the characters 0, Q and t a i l of Q. The output resolution and the diffraction efficiency of the different f i l t e r s are considered. In chapter 7, experiments intended to compare the space var-iance of the MF-2 and LLMF-1 are reported. It i s indicated that phase errors in the input plane were the dominant causes of the apparent space variance. They were also responsible for wild variations of the correlation measurements. A corrective measure i s described. In Chapter 8, the findings of this thesis are summarized and suggestions for further research are presented. 10 II MATCHED AND LENSLESS MATCHED FILTERS 2.1 Introduction This chapter reviews the principle of two Vander Lugt matched f i l t e r s (MF) £4,46] and some of,their properties. It proposes two "lensless matched f i l t e r s " (LLMF) whose performances are compared throughout the thesis with that of the two MF's. It considers their space variance from the point of view of vignetting apertures. Many of the equations derived in the following f i r s t -order analysis are basic to the matters discussed in later chapters. F i r s t this chapter considers the original MF [4], hereafter called MF-1, and some of i t s aspects relevant to this thesis. It examines particularly the minimum spatial carrier-frequency and the layout of recorded and readout signals for par a l l e l processing. Then i t discusses another version of the matched f i l t e r [46] which i n the thesis i s referred to as MF-2. The f i r s t Fourier-transforming stage of the MF-2 i s space-invariant. The size of the input f i e l d over which i t s second stage i s space-invariant increases with a reduction of the. distance between the holo-gram and the second lens. Total space invariance of the f i l t e r i s eventually achieved when such distance i s zero. As space invariance i s essential for a parallel processor, this chap.er seeks to make the foregoing distance identi-cally zero by synchesizing the effect of the lens inside the hologram. Exam-ining a particular case of a general matched f i l t e r reveals how the synthesis can be realized. The recording and f i l t e r i n g configurations of the general matched f i l t e r are similar to those of the MF-2 except that the position of i t s reference point source i s not specified a p r i o r i . Two particular cases are of interest. 1) When the reference point source i s coplanar with the recorded signal, the general matched f i l t e r amounts to the MF-2; 11 2) When the second lens Is taken away, an appropriate choice of the reference point source helps in synthesizing the lens effect inside the hologram which simultaneously acts as a matched f i l t e r and a Fresnel zone plate. The resulting "lensless matched f i l t e r " (LLMF) is one of two types depending on whether i t s reference beam converges (LLMF-1) or diverges (LLMF-2) towards the hologram plane [47]. Since their only vignetting aperture is located in the Fourier plane, the two proposed LLMF's are space-invariant. They are compared to analogous schemes and particularly to Gabor's stigmatic code translator [48] which, i n general, i s space-variant. Finally, a comparison of the MF and LLMF from the point of view of the minimum spatial carrier-frequency and the resolution of the recording medium i s presented. 2.2 Matched F i l t e r s : a Review Fi r s t , the original version of the matched f i l t e r [ 4] i s re-viewed along with some of i t s characteristics that are pertinent to this thesis. Then a well known solution for improving the matched f i l t e r ' s space invariance [46] is discussed. Justification for further improvement is also presented. 2.2.1 Recording of the MF-1 Fig. 2.1 illustrates the interferometric set-up used i n i t i a l l y by Vander Lugt [ 4 ] in recording a holographic MF. Throughout this thesis, such type of f i l t e r i s referred to as MF-1. The two arms of the inter-ferometer constitute the object and the reference beams of the hologram. The object beam is a Fourier-transforming stage. Assume a transparency s(x,y) i s located in the front focal plane of lens L^. Then, i n the back focal plane P^ of the latter, the f i e l d distribution o(u,v) i s pro-portional to the Fourier transform of s(x,y) [49]. With the proportionality 12 constant taken as unity, o(u,v) is given by o(u,v) = SU.n) where (£,n) = (u,v) and s(x,y) «-»• S(£,n). (2.1) (2.2) (2.3) The symbol -«-»- indicates a Fourier transform relationship; i s the focal length of L j , and (£,n) are the spatial frequencies in P^. A plane ref-erence wave whose direction of propagation makes an angle 6 x r with the optical axis gives rise in P^ to a f i e l d distribution r(u,v) = exp(-j2irX - 1 sin 0 u) (2.4.a) which, for small 9 X R » i s approximately equal to r(u,v) = exp(-j2iTX" 1 tan 0 u) . (2.4.b) A - 1 tan is the spatial frequency of the carrier. The interferometer can be either of the modified Mach-Zender type with a collimated reference beam incident at angle 6 x r » o r °f the modified Rayleigh type with a reference point source at coordinates (x r,o) in P g. x r and 0^. are related by x = F, tan 0 . r 1 xr (2.4.c) After the substitution of Eq. (2.2) and (2.4.c) into (2.4.o), r(u,v) becomes r(u,v) = exp(-j2ir?x r) . (2.5) Fig. 2.1. Arrangement for recording the MF-1. Without loss of generality, the x axis has been defined as the line joining the reference point source to the origin of P and is assumed to be horizontal; r(u,v) and o(u,v) add up vectorially in P^ where the i n -tensity of their interference pattern is recorded by a photosensitive material. After processing, i f any, i t s amplitude transmittance T ( U , V ) , is taken to be linearly proportional to such intensity T ( U , V ) = |o(u,v) + r(u,v)| 2 = |o(u,v)| 2 + |r(u,v)| 2 + o*(u,v)r(u,v) + o(u,v)r*(u,v). (2.6) In the above equation, the symbol means complex conjugate and the pro-portionality constant has been dropped out. The substitution of Eq. (2.1) and (2.5) into (2.6) yields T ( U , V ) = | s a,n)| 2 + 1.+ S*(5,n) exp(-j2ir?x r) '+ S(£,n) exp(+j2ir£xr) . (2.7) Eq. (2.7) represents the amplitude transmittance of what i s called a Fourier transform hologram because, apart from the linear phase factors that help to separate the reconstructed images, i t s last two terms are the Fourier transform of s(x,y) and i t s coajugate. Moreover the factor S (C»n) represents the transfer function of the optimum linear f i l t e r matched to s(x,y) when the imput signal is imbedded i n linearly additive and independent white noise [50l- Therefore the hologram (2.7) i s also a f i l t e r matched to s(x,y). When the noise i s coloured, the optimum f i l t e r results from the juxtaposition of the hologram (2.7) and a f i l t e r whose amplitude transmittance is inversely proportional to the Wiener spectrum of the noise. However, physical juxtaposition i s not necessary. It can be realized via a "dual frequency-plane processor" that has two f i l t e r planes conjugate of each other [45]. 1 4 Fig. 2 . 2 . F i l t e r i n g arrangement of the MF - 1 . 2 . 2 . 2 Correlator Assume that a readout signal t(x,y) i n P is to be crosscorrelated with s(x,y). Fig. 2 . 2 shows how the crosscorrelation operation can be carried out. Lens L^ displays in P^ a readout f i e l d c(u,v) proportional to the Fourier transform of t(x,y) c(u,v) = T ( C,n) ( 2 . 8 ) where t(x,y) - M - T ( 5 , n ) . ( 2 . 9 ) Multiplying t(u,v) and c(u,v), given respectively by Eq. ( 2 . 7 ) and ( 2 . 8 ) , yields the expression of the f i e l d diffracted by the hologram ccu,v)x(u,v) = T ( S , n ) ( | s ( £ , n ) | 2 + 1 + S * ( C , n ) exp ( - j 2 T r£x r) + S ( C , n ) exp ( j 2 T r?x r)). ( 2 . 1 0 ) The above light distribution i s inverse Fourier transformed by lens L 2 of focal length F 2 i In the back focal plane P Q of L 2, the output f i e l d distribution 0(m,n) i s 0(m,n) -w c(u,v)x(u,v) ( 2 . 1 1 ) or ( 2 . 1 2 ) Note that in P Q the (m,n) axes are oriented i n the direction opposite to the (x,y) and (u,v) axes so that Eq. ( 2 . 1 2 ) is an exact inverse Fourier transform. Again proportionality constants have been disregarded i n 15 Eq. (2.8) and (2.12). With the help of Eq. (2.10) and after expansion of the Fourier transformation, Eq. (2.12) becomes 0(m,n) = 0Ql(m,n) + 0o2(m,n) + 0+1(m,n) + 0_1(m,n) (2.13) F F F F F F where 0 (m,n) = s*(- ^  m,- ^ n)*s (-^  m,- i n ) * t ( ^ m,- i n) , (2.13.a) 01 * 2 r 2 2 2 2 2 F F 0 0 2(i,n) = t ( ^ ^ n ) , (2.13.b) F F F • F F O.-On.n) = [ s ( ^ m . - ^ n X ^ t ^ m,- i n)]*6(m- ^  x.n) , (2.13.c) F F F F F 0 .(m,n) = [ s ^ m , - ^ n ) * t ( ^ m , - i n)]*5(m+^x ,n). (2.13.d) - j . r 2 2 2 2 1 The symbols ® and * are respectively the correlation and convolution operators, and 6(x,y) i s the two-dimensional Dirac delta function. 0_^(m,n) represents the crosscorrelation of s(x,y) with t(x,y) centered about (— x ,o), the image of the reference point source. F l r F F 0.,(m,n) = // s*(m - i-x-hx, n -1+y)t(x,y)dxdy. (2.14) P c F2 r F2 At that particular point, a photodetector measures the intensity of the central value of the crosscorrelation |0+.(!2 x o)|2 = I//' s*(x,y)t(x,y)dxdy|2 (2.15) F l P s to estimate the similarity between the patterns s(x,y) and t(x,y). Then an electronic postprocessor computes the square of the normalized cross-correlation F2 I 0 +I< F T V O ) I p2 k : (2.16) ^st E E s t where E and E. are respectively the energies of s(x,y) and t(x,y), and 16 are defined by E = // |i(x,y)| 2 dxdy i = s or t. (2.17) P s The signal t(x,y) i s identified to belong to the class of signal s(x,y) whenever p 2 f c is larger than some preset threshold. 2.2.3 Minimum Spatial Carrier-Frequency The minimum spatial carrier-frequency or equivalently the min-imum coordinate x r of the reference point source i s ordinarily found by requiring that the correlation pattern be completely separated from the other components of 0(m,n). Let L and L be respectively the horizontal spatial extent of s(x,y) and t(x,y) so that s(x,y) = 0 |x| > L 12 (2.18.a) and t(x,y) = 0 |x| > Lfc/2. (2.18.b) Then the terms O0i(m,n) and 0Q2(m,n), both centered on the optical axis, F2 F2 have a horizontal width respectively equal to — ( 2 L + L ) and —— L F^ S t F t [see Eq. (2.13.a and b)]. Similarly the two f i r s t order terms that are F2 centered about ± ( p — x r » o ) have a same horizontal width equal to (L g + L t ) . Therefore, 0+^(m,n) does not overlap with the other components i f F ? 2L + L. F L + L. F 9 or x > |- L + L-. . (2.20) r 2 s t 2.2.4 Signals Layout for Parallel Processing The a b i l i t y of the MF to compute in p a r a l l e l , or simultaneously, the crosscorrelations of P recorded signals s^(x,y) with Q readout signals t^(x,y) follows from Eq. (2.14) after s(x,y) and t(x,y) are 17 y n ® s ® s l S2 C2 H h t2 t3 fc4 iy O fc6 C6 fc7 fc8 S3 S4 X h X C2 fc4 fcl H '3 *4 — fc7 h *• s,® C6 fc7 fc8 fc5 fc7 fc8 m ® S5 S6 h fc4 ' l C3 fc4 •=6 '7 '8 '5 '6 fc7 '8 (a) (b) (c) Fig. 2.3. Layouts of (a) the recorded and (b) the readout signals for parallel processing, (c) Distribution of the correlations. decomposed into sums of nonoverlapping elementary signals: ? s(x,y) = Z s.(x,y) (2.21) 1=1 1 and Q t(x,y) = £ t (x,y). (2.22) j = l J For the P x Q elementary crosscorrelations not to overlap, the spacing between the s^(x,y) must be such that each of them is allotted an area at least equal to the area over which t(x,y) is different from zero, as illustrated in Fig. 2.3. Thanks to i t s combined capabilities of p a r a l l e l processing [ 8] and storage of large amount of data [51 ], the MF has a potentially high processing speed [25]. This makes i t attractive for character and word [13] recognition applications. 2.2.5 Improvement of Space Invariance with the MF-2 Theoretical [52] and experimental [53] investigations of the f i r s t Fourier-transforming stage of the MF-1 led to the conclusion that, to achieve space invariance,the signals s(x,y) and t(x,y) should be located i n the plane of L^ or i n the path of the converging beam behind L]_. Vignetting of the signals fields i s then avoided and the f i e l d distributions in P^ are, apart from linear phase factors, independent of the transverse coordinate of the signals in P g. In P^, the fields are the quasi-Fourier transforms of the signals, which means that a quadratic phase factor multiplies their Fourier transforms. An exact Fourier trans-form hologram can s t i l l be recorded though by using a reference beam whose curvature cancels the quadratic phase factor of the object beam. Such a beam issues from a point source coplanar with s(x,y) (see Fig. 2.4.a). The crosscorrelation i s obtained by means of lens which images the input plane P g onto the output plane P Q. There an image of t(x,y) i s formed after modulation of i t s spectrum T(£;,n) by S (£,n) (see Fig. 2.4.b). Throughout the thesis, this type of MF i s referred to as MF-X m f Fig. 2.4. MF-2. (a) Space-invariant recording system, (b) F i l t e r i n g system. 19 and is derived in Section 2.3.3.a as a particular case of a general matched f i l t e r . The distance between and lens i s not important since i t affects only a quadratic phase factor multiplying the correlation but not the intensity of the latter. However a distance D. as short as pos-sible is desirable in order to limit the vignetting effect of I^. U l t i -mately = 0 would yield an absolutely space-invariant MF-2. Indeed, a translation of t(x,y) in P g would only result i n a proportional trans-lation of the correlation pattern in P Q without modification of i t s i n -tensity distribution because the only vignetting aperture of the system i s located in the Fourier plane. Lowenthal [54] has derived two conditions under which a coherent processor is space-invariant. They are: 1) The illuminating point source or i t s image must be in the pupil plane; and 2) The pupil and the source, as seen from the image plane, must be at i n f i n i t y . The f i r s t condition is achieved by illuminating the object with a beam focused on the f i l t e r plane and is verified with a MF-2. The second condition, which assumes no vignetting by lens I^, i s to make sure the image i s an exact replica of the object or that there is no quadratic phase factor superimposed on the image. It i s satisfied by = F^y where F^ is the focal length of I^. As only the correlation intensity is of interest in matched f i l -tering, this second condition does not have to be met,and D = 0 i s a more desirable condition as i t reduces vignetting by I^. As Groh [43] and Kock and Rabe [44] put i t , "lens L£ is the bottleneck of the system" for i t has to transmit the total information. This follows easily from space-bandwidth product (SBP) considerations. 20 2.2.6 Space-Bandwidth Product Each elementary signal s^(x,y) and t^ (x,y) is assumed to have a spatial extent L 0 and an angular bandwidth Q along each of the x and s s y axes. For simplicity, though without loss of generality, the recorded signals s^(x,y) are considered grouped into one ve r t i c a l column, P signals high, while the readout signals t^(x,y) form one horizontal line, Q signals wide. Such a layout guarantees that the P x Q crosscorrelations do not overlap in P Q. The SBP of the readout signal i s given by SBP^ = L 2 Q (2.23) t s s x and SBP is that of the total crosscorrelation pattern o SBP = L 2 Q,2 (P + 1)(Q + 1). (2.24) o s s The matched f i l t e r can be thought of as a SBP multiplier whose gain i s defined by ' S B P » S B ? ; " ( P + T + " • « • « (1 + Q - > • ( 2 . 2 5 ) GOT)T, is at least equal to (P + 1) where, to f u l l y exploit the paral l e l SBP processing capability of the f i l t e r , P can be large. Eq. (2.25) means that the second stage of the correlator must be capable of f a i t h f u l l y transmitting G^ -^  times more information than i t s f i r s t stage. This b B P (P + l ) - f o l d increase in SBP calls for as short a as possible to bring the lens closest to the area of the Fourier spectrum where the f i e l d is spatially concentrated r55 ]. For practical reasons though, i t i s d i f f i c u l t to make D equal to zero. Both the hologram and the lens need a physical support the thickness or the design of which may prevent them to come very close, particularly when a mechanism is used to exchange the f i l t e r s and/or when the f i l t e r i s immersed in a liq u i d gate to compensate for i t s thickness variations. In any case , scratches are l i k e l y to damage both the hologram and the lens should they come in contact. The next section investigates the pos s i b i l i t y of making = 0 by synthesizing the effect of the lens into the hologram. Indeed, with D. = 0, the f i l t e r and the lens modulate consecutively the readout f i e l d . Their amplitude transmittances can be combined inside the same hologram t;o yield a space-invariant lensless matched f i l t e r [47] . 2.3 General Matched F i l t e r The general matched f i l t e r i s similar to the MF-2 except for i t s reference beam which i s issued from an arbitrary point source. Two par-ticular cases are of interest. 1) When the point source i s coplanar with the recorded signal, the general matched f i l t e r reduces to the MF-2; and 2) When lens L i s removed from the system, a proper choice of the reference point source reduces the general matched f i l t e r to a "lens-less matched f i l t e r " (LLMF), that i s a matched f i l t e r whose lens L^ i s synthesized inside the hologram [ 47 ] . This paragraph uses extensively the operational notation for optical systems [52]. The Appendix l i s t s the necessary definitions and some useful properties. 2.3.1 Recording of the General Matched F i l t e r The recording configuration i s depicted i n Fig. 2.5. A lens L^ images a point source p z onto a plane P^. The pl?,ne P g, located be-tween L^ and P^ contains a transparency s(x,y) to which the f i l t e r being recorded i s matched. With i/j(x,y;dg) defined i n Eq. (A.2), the f i e l d distribution of the converging beam incident on P i s Aijj(x,y;d ). A i s 8 S i t s complex amplitude at the origin of P , and d i s the reciprocal of the s s distance D between P and P. s s h 22 d = D " 1 s s (2.26) The f i e l d transmitted by the transparency i s A^(x,y;d )s(x,y) and gives rise in P K to an object f i e l d . d o(u,v) = Aij)(u,v;dg)s(u,v) * y5" 4,(u,v;dg) (2.27) where the symbol * i s the convolution operator. With the aid of Eq. (A.9), expanding the convolution of Eq. (2.27) yields the f i n a l form of o(u,v) d o(u,v) = A — tKu,v;d s )sa,n) (2.28) where s(x,y) -«-*• S(£,n) A and (?,n) = (u,v) . XD (2.28.a) (2.28.b) The symbol -<-> indicates a Fourier transform relationship, and X i s the wavelength of the illuminating beam. The reference beam i s a spherical wave centered about a point of coordinates (x^.y^) in a plane P r at a distance from P^. D i s positive or negative depending on whether P g and P r are on the same side or on opposite sides of P^. In other words, a beam diverging (converging) towards P^ is characterized by Br > 0 (D r < 0). Moreover the orientation of the axes i n P r and i n P g are the same or opposite depending on whether D r i s positive or negative. This i s to ensure that, for a given |Dr|, the mean direction of the ref-Fig. 2.5. Arrangement for recording the general matched f i l t e r . erence beam is the same regardless of the actual curvature of the beam. Fig. 2.5 illustrates the case where D r is negative. Now the reference f i e l d in P^ is given by r(u,v) = BiKu - sgn(D r)x r, v - sgn(D r)y r;d r) (2.29.a) where B is a complex amplitude and . D (2.29.b) sgn(Dr) =jp-| • With the aid of Eq. (A.9) and after absorption of the constant factor 4<(xr,yr;dr) into B, r(u,v) can be written r(u,v) = Bi|)(u,v;dr) exp(-jk|d r| (ux r + vy r)) (2.30) where B i s now the complex amplitude at the origin of P^. A hologram i s formed in P^ by recording on a photosensitive material the intensity I(u,v) of the interference pattern betwen o(u,v) and r(u,v) I(u,v) = |o(u,v) + r(u,v)| 2 = |o(u,v)|2+ |r(u,v)| 2 + o*(u,v)r(u,v) + o(u,v) r*(u,v) (2.31) where the symbol means complex conjugate. The amplitude transmittance T ( U , V ) of the processed hologram i s assumed linearly proportional to Ku,v) T ( U , V ) - I(u,v) (2.32) where the proportionality constant i s taken as unity. After the sub-stitution of Eq. (2.28) to (2.31) into (2.32), x(u,v) becomes d 2 d T ( U , V ) |AS(5,n)| 2 + |B|2 +-jT?"A*B S*(e,.n)*(u,v;ds-dr) exp(-jk|d r| d (ux r+vy r))+ ^ 2 - AB*S(?,n)4'(u,v;ds-dr) exp(jk|d r| (uxr+vyr))-. (2.33) 2.3.2 General Correlator As in the recording set-up of Fig. 2.5, the illuminating beam 24 (mQJn0) (m,n) Fig. 2.6. Filtering arrangement of the general matched f i l t e r . i s again a spherical beam converging towards a focal point i n P^. This beam i s modulated in plane P by a readout signal t(ax,ay) as shown in Fig. 2.6.. By analogy with Eq. (2.28), and with Dfc the distance between P and P, , the readout f i e l d incident on P, is t h h c(u,v) = ^ 2 ^ u , v ; d t ) T ( i ' , ^ ) where d S D - i , t t ' a*,n') = (u,v) WD (2.34) (2.34.a) (2.34.b) and t(x,y) -*-»• T(£' . r i 1 ) . (2.34.c) Without loss of generality, the complex amplitude of the illuminating f i e l d in P is assumed unity. The factor a allows selection of different scales for the recorded and readout signals. However, meaningful f i l t e r i n g can take place only i f the scale of their spectra i s identical, that i s i f Combined with Eq. (2.28.b) and (2.34.b), Eq. (2.35) implies that a = d„ D t s d = D s t (2.35) (2.36) This i s the usual relation governing scale searching i n Fourier optics. 25 Substituting Eq. (2.36) into (2.34) gives d t -c(u,v) =-4 ^(u,v;d ) T ( ? , n ) . (2.37) The hologram modulates the readout f i e l d and diffracts a f i e l d whose ex-pression i s found with the help of Eqns. (2.33) and (2.37) to be d d C C U , V ) T ( U , V ) = ^ ( u , v ; d t ) | A | 2 T ( 5 , n ) | s ( 5 , n ) (2.38.a) + _t iKu,v;d ) | B| 2Ta , n ) (2.38.b) + d t d s f(u,v;d -d -djA*B T ( ? , n ) S * ( C , n ) exp(-jk|d |(ux +vy )) (2.38.c) •• 2 2 s r t + d t d s \p(u,v;d -d +d )A B * T ( £ , n ) S ( £ , n ) exp(jk|d |(ux + vy ) ) . (2.38.d) J2^Z s r t r r r The second stage of the general matched f i l t e r consists of a lens of focal length at a distance behind the hologram. Of interest is the f i e l d distribution 0(m,n) in the output plane P Q at a distance D Q behind I^. Fig. 2.7 is a black box diagram of the computations involved in determining 0(m,n). Eq. (2.39) to (2.45) are intermediate steps to-wards the general expression of 0(m,n) in Eq. (2.46). Convolution of C ( U , V ) T ( U , V ) with dJl ^ (u,v;d9) [see Eq. (A.5)] A ~ gives the f i e l d distribution on the l e f t surfi.ce of the lens a 1(p,q) = ^ tKp.q'.d^ )JJ c(u,v) T(u,v) K u,v;d £) exp(-jkd£(up+vq))dudv, X P h (2.39) while multiplication of a^(p,q) by the lens transmittance ijj(p,q;f2^ c(u,v) x(u,v) an(p,q) a 9 (p ,q ) — i|)(p,q;do) 0(m,n) — -^(p»q;f2) Fig. 2.7. Block diagram representation of the light propagation from the, hologram to the output plane. 26 [see Eq. (A.3)] produces the amplitude distribution on i t s right surface a2(.V,<l) = a 1(p,q)i|j(p,q;f 2) . (2.40) In P , the output f i e l d follows again from Eq. (A.5) d 0(m,n) = -r5- ip(m,n;d ) / / a 9(p,qH(p,q;d ) exp(jkd (pm+qn))dpdq. L2 (2.41) In the above equations, the following definitions have been used d £ = D^ 1 , (2.42.a) f 2 = F ^ 1 , (2.42.b) and d = D _ 1 . (2.42.c) o o The positive sign in the exponent of the last integrand results from the choice of opposite orientations for the axes (m,n) and (x,y) re-quired to make Eq. (2.52) an exact inverse Fourier transform. Sub-stituting Eq. (2.39) and (2.40) into (2.41) yields do dA 0(m,n) = -~2- 4'(m,n;do)// i^(p,q;d -f 2 + D 0 ) J 7 c(u,v)T(u,v)^(u,v;d £) L2 P h exp(-jkd^(up+vq))dudv exp(jkdQ(pm+qn))dpdq. (2.43) Changing the order of integration and regrouping the terms allow the integrals of Eq. (2.43) to be rewritten . ' " // c(u,v)x(u,v)4)(u,v;d )// ^ (p,q;d - f +d )exp(jkd (p(m-V )^ P. L 0 d h 2 o + q(n- d& V )))dpdqdudv. d o The fact that lens \>2 has a f i n i t e aperture i s neglected and the limits of integration over L 2 are taken as i n f i n i t e . Then with the aid of Eq. (A. 13) the integral over L2 reduces to y, V V , W f 2 do ' do ' V d° " f2 after a complex coefficient i s discarded. These last two expressions combine with Eq. (A.7) and (A.9) to transform Eq. (2.43) into V i d o 2 V 0 ( m ' n ) = X(dg+d -f,) ^ ' ^ V d0+d - f >// c(u,v ) T(u,v)*(u,v;d J l- d ) *• o 2 Jt o 2 p, a o 1 n V o exp(jk-^-^—3|-y (mu+nv))dudv. (2.44) % o 2 The following definitions d (d -f ) d k ^ o l (2.45.a) e d £+d o-f 2 and A d d0 d' = . ° % (2.45.b) e W f 2 simplify the expression of 0(m,n) to d^ d o ( d £ - f 2 ) 0(m,n) = -r^ - ^ (m,n; -7-3-5 ) / / C ( U , V) T(.u,v)if)(u,v;d )exp(jkd' (mu+nv))dudv. A W £ 2 P, e e n (2.46) It i s readily apparent from Eq. (2.46) and (A.11) that the choice D ^ = F 2 which cancels the output quadratic factor, corresponds to Lowenthal's second condition-for space invariance [see Saction 2.2.5]. For the purpose at hand, no such condition needs to be satisfied. The phase factor is simply dropped out because i t does not affect the intensity of the output f i e l d . The f i n a l expression of the output distribution 0(m,n) results from the following definition d'd. V = X ^ ( 2 ' 4 7 ) and from the insertion of Eq. (2.38) into (2.46) 0(m,n) = 0Q1(m,n) + 0Q2(m,n) + 0+1(m,n) + O^Ou.n) (2.48) 28 where A k 2 d 2 O01(m,n) = - f z 5 - |A|2// |s(?,n)|2T(C,n)'f'(u,v;dt+de)exp(jkd^(um+vn))dudv, h (2.48.a) 0.,(m,n) =k 2|B|2// T(?,n)^(u,v;d +d )exp(jkd'(um+vn))dudv, (2.48.b). h . k 2d _ |d | ' 0+1(m,n) = - f - 5 - A*B// S*(?,n)T(?,n)*(u,v;d s-d r-d t-d e)exp(jkd;(u(m- x ^ P h d + v(n-'-jf y r)))dudv, (2.48 .C) e A k 2^ |d | 0 (m,n) = -j^ AB*//s(5,n)T(?,n)*(u,v;ds-dr+dt+de)exp(jkd;(u(m+ x ) P h U | + v(n+ -j£ y r)))dudv. (2.48.d) The rest of this chapter examines the condition under which the term 0^(m,n) represents the desired crosscorrelation. The other components of 0(m,n) are considered in Chapter 3. The integral in Eq. (2.48.c) represents a cross-correlation whenever the i|) function i s unity, that i s , according to Eq. (A. 11),when d g - d r - d t - d e = 0. (2.49) Defining 6 by 1 A d r D s and solving Eq. (2.49) for d with the aid of Eq. (2.45.a) give the position of the output plane P q . It i s found to correspond to d d (1 - S - a) d = f 9 + / S . T r- (2.51) o 2 d £ - d g ( l - 6 - a) where a satisfies Eq. (2.36). Under condition (2.49), 0+^(m,n) can be expressed solely in terms of the spatial frequency variables (£,n) [see Eq. (2.28.b)] * d' Id I Id I 0+1(m,n) = k2DsXA*B// S (^n)T(?,n)exp(j2T^(5(m- x r)+n(n- ^ y r ) ) ) d S d n . P H (2.52) Carrying out the Fourier transformation yields d' d' 0+1(m,n) = k2 D AA*B C m - , -f n - |6|y ) (2.53) s s where C stands for the crosscorrelation of s(x,y) with t(x,y) S L C .(m,n) = // s*(x+m,y+n)t(x,y)dxdy. (2.54) S T P s At the point of coordinates (m0,n0) defined by A l d r ' l d r l ( mo» no ) P ( d*- V d 7 " ^ ' - ( 2 ' 5 5 ) e e 0_^(m,n) is proportional to the central value of the crosscorrelation 0 + 1(m o,n o) - C8t(0,.0) (2.56) and i s the ultimate information the general matched f i l t e r is to yiel d . The symbol a indicates a proportionality relationship. In Eq. (2.53), d g the reciprocal of the coefficient of m and n, that i s -^j- , i s the magni-e fication of the f i l t e r referred to P S . However, when referred to the input plane P ^ , the f i l t e r magnification i s given by . d d d M f - r F B F - (2-57) s e e Eq. (2.49) i s the condition the general matched f i l t e r has to satisfy in order to display the crosscorrelation of s(x,y) with t(x,y) i n i t s output plane P q . The position of the latter can be derived from Eq. (2.51) where d 'must.be positive i f the correlation i s to be a real image. Two par-ticular recording and f i l t e r i n g combinations that verify Eq. (2.49) are analyzed in the next section. 30 2.3.3 Particular Cases F i r s t , the MF-2 i s derived as a particular case of the general matched f i l t e r by requiring that the reference point source be coplanar with the recorded signal. Then, removing lens from the correlator leads to the condition for the synthesis of the lens effect inside the hologram. 2.3.3.a MF-2: 6=1 The MF-2 whose reference point source and recorded signal are coplanar, i s characterized by D = D g or, according to Eq. (2.50), by 5=1. (2.58) Then the output plane position follows from Eq. (2.51) after insertion of Eq. (2.58) d d a d „ = f 9 ~ "Ta~j— . (2.59.a) o 2 d„+d a I 8 With the aid of Eq. (2.36) and (2.42), this i s equivalent to D - 1 = F - 1 - (D.+DJ-1 . (2.59.b) O 2 & t Shofner and Webb [56] recently derived the equations of the MF-2. In their version of Eq. (2.59), they failed to account for the distance D although i t i s not obvious either from their text or from their figures that i s null or negligible. Eq. (2.59) i s an imaging condition indicating that P q is the conjugate of P g with respect to lens L£. Consequently, the scaled versions of a signal can be detected with the MF-2 by simply shifting the input and output planes and lens L 2 along the optical axis as a r i g i d assembly. Then the distance D t and the scale of the Fourier spectrum vary while the imaging condition is maintained. However, searching mech-anically for the scale of the readout signal i s a slow process compared with correlating the signal simultaneously with some of i t s scaled ver-31 sions stored in the same hologram. Furthermore, the scale of the corre-lation pattern varies with the readout signal scale because the magnification of the f i l t e r i s constant. Indeed, with the aid of Eqns. (2.45.b) and (2.58), Eq. (2.57) becomes: M f - j T ^ h • (2.60) X> t Is constant when the relative positions of P G , P 0 and are constant. Scale variation of the output f i e l d may increase unduly the complexity of the detector and postprocessor. 2.3.3.b LLMF: f 2 = 0 Getting r i d of lens Lg leaves the space between P ^ and P Q free of vignetting components. Such a situation can be derived from the general matched f i l t e r by requiring that the focal length of L2 be i n -f i n i t e : then the lens has no effect on the incident f i e l d and can there-fore be excluded from the design. It follows that the distance D^ has no particular meaning and can be considered zero. Now D Q designates the distance between the hologram and the output plane. With F 2 = ~ or f 2 = 0 , (2.61.a) and D £ = 0 or d £ = °° , (2.61.b) Eq. (2.45) reduces to d = d' = d . (2.62) e e o ' Defining $ by A d D S O and using Eq. (2.36) and (2.50), the correlation condition (2.49) amounts to ct + 8 + 6 = 1. (2.64) This is the condition the general matched f i l t e r has to satisfy to be space-invariant. Throughout the thesis, a f i l t e r that verifies Eq. (2.64) is referred to as "lensless matched f i l t e r " (LLMF) since the f i l t e r i n g operation requires no lens but the one that illuminates the readout signal [47]. The latter lens could hardly be done away with. Under condition (2.64), the LLMF magnification i s obtained by substituting Eq. (2.62) into (2.57) and by using Eq. (2.36) and (2.63) " f - r - r - f - <2-65> o t The coordinates of the correlation centre of the LLMF result from the sub-stitution of Eq. (2.50), (2.62) and (2.63) into (2.55) (mQ,no) = (3" 1 |6|xr, B"1 |fi|yr) . (2.66.a) Before a detailed investigation of the implications of the LLMF condition, i t i s convenient to define here for further reference k 2 as the value of o k e 2 for the LLMF case. With the help of Eq. (2.47), Eq. (2.62) becomes: d d k 2 = k 2 o e • (2.66.b) LLMF 2.4 Lensless Matched F i l t e r A discussion of some aspects peculiar to the LLMF i s now pre-sented, beginning with a physical interpretation in terms of Fresnel zone plates and a description of two types of LLMF. Then follows a review of some precedents of the LLMF and an analysis of the resolution requirements of i t s recording medium. 2.4.1 LLMF as Fresnel Zone Plate The LLMF condition [Eq. (2.64)] i s hardly surprising since a hologram can be thought of as the superposition of a large number of zone plates [57 ]. Indeed, consider that, in a plane P^, a hologram i s made 33 of the interference of two spherical waves. They are issued from an object and a reference point sources located, for example on the optical axis, res-pectively at distances Dg and D r from P^. The hologram is then similar to a Fresnel lens and i t s focusing property i s governed by an equivalent focal length F^ F h = (1 (2.67) or, with Eq. (2.50), F, - D / ( l - 6) . (2.68) n s This confirms that the MF-1 and -2 have no focal power of their own since 6 = 1 . When the above holographic lens is used to image a reconstructing point source at a distance in front of P^, the imaging condition F h = ( l t + i ) _ 1 ( 2 - 6 9 ) determines the distance D Q between the point image and P^. From the iden-t i f i c a t i o n of Eq. (2.67) and (2.68), i t comes h = h + h + h (2.70) D D„ D D s t o r or, after substitution in the above of Eq. (2.36), (2.50) and (2.63), 1 = a + 3 + 6 . (2.64) The imaging condition of the holographic lens id identical to the LLMF condition and confirms that the effect of lens "L^ ^ s b u i l t into the holo-gram. 2.4.2 Typical LLMF Configurations The LLMF condition, which is illustrated in Fig. 2.8, has three degrees of freedom: any two of the three parameters of Eq. (2.64) plus the normalization distance Dg. • Two types of LLMF can be distinguished ac-cording to the sign of D . •/ -3/4 -1/2 -1/4 LLMF-1 converging 0 1/4 1/2 3/4 LLMF-2 diverging reference beam Fig. 2.8. Interdependence of the LLMF parameters. 1) 6 < 0. In this case, the reference beam converges towards P^. In the rest of the thesis, a f i l t e r of this type i s referred to as LLMF-1. Fig. 2-9 illustrates a LLMF-1 with a = B = -6 =1. This corres-ponds to the situation where P t coincides with P g and there i s no difference between the scales of the recorded and readout signals. Then Eq. (2.64) implies that 8 = -5 > 0 (2.71) which means that the output plane P Q coincides with P r, the plane of the reference point source. 2) 6 > 0. In this case, the reference beam diverges towards P^. Hereafter, LLMF-2 is how this type of f i l t e r i s known. Its peculiarity i s that 0 < a, 8, 6 < 1. (2.72) Indeed, Eq. (2.64) states that the sum of the three parameters i s equal to unity. Then Eq. (2.72) follows from the positiveness of each of them, a i s positive since Pfc has to be in front of P^ [see Eq. (2.36)]. 8 i s positive since otherwise the crosscorrelation would be v i r t u a l and the 35 Fig. 2.10. Typical LLMF-2 arrangement: (a,6,6) = (k,k,h)• (a) Recording, (b) F i l t e r i n g . 36 purpose of the MF not f u l f i l l e d . Finally 6 i s positive in the case con-sidered, a < 1 means that the recorded signal must be demagnified with respect to the readout signal. A LLMF-2 i s sketched in Fig. 2.10 with (a, 3, 6) = ( i , 4 , i ) . With both types of LLMF, scale searching is possible although not as conveniently as with the MF-2. Indeed the distance between their input and output planes is no longer constant. This is not a serious drawback since mechanical scale searching has disadvantages of i t s own as pointed out in Section 2.3.3.a. It is noteworthy that the use of a holo-graphically "built-in lens eliminates the optical coatings, the hardware and the precise positioning devices associated with the external lens L2« The f i l t e r has also fewer surfaces to collect dust particles and scratches. The focal power of the hologram i s obtained at the expense of increased complexity of the reference (LLMF-1) or object (LLMF-2) beams. The need for signals of different scales at recording and f i l t e r i n g time precludes the use of the LLMF-2 for those applications where numerous recording and readout operations alternate [15j. As for the focusing lens of the LLMF-1 reference beam i t i s needed only at recording time. Hence, i n applications l i k e character recognition where new f i l t e r s are seldom written, i t i s used for a short period of time and i s time-sharable. Furthermore, the S B P of this lens ( S B P R ) i s much smaller than that of L 2« Indeed, i f | D R | = D S , i t only needs to focus the reference beam to a spot the size of a pixel, that is a resolution element in P G . The pixel's horizontal and v e r t i c a l dimension i is directly related to the angular bandwidth Q of s(x,y). s s Then S B P R i s given by S B P = l2Q2 . (2.73) r s s As I « L and P , Q » 1, i t results from Eq. (2.24) and (2.73) that s s S B P « S B P . (2.74) r o 37 2.A.3 Antecedents of the LLMF In his experiments to restore defocused images, Ragnarsson [58] recorded holographically the interference between a converging reference beam and the object beam diffracted by a circular aperture. The diameter of the latter was chosen to represent the point spread function of a de-focused camera. The hologram was subsequently used in the f i l t e r plane of a quasi-Fourier transforming stage. For the hologram to act as a Fresnel zone plate, he required that the readout signal and the focus point of the reference beam be symmetrically located with respect to the hologram. In this thesis' notation, this amounts to a = -6. This is only a particular case of the general condition derived earlier [see Eq. (2.64)] for the synthesis of lens into the hologram. Furthermore, as the circular aper-ture of the object beam was closer to the hologram than the readout signal, the parameter a of Ragnarsson'r. system [58] was less than unity and he could have used a diverging reference beam by analogy with the LLMF-2. The only claimed advantage of the b u i l t - i n lens i s the reduction of the number of optical elements and noise sources. No reference to space invariance was made. Maloney described a lensless holographic recognition system for spatially incoherent patterns [59]. He recorded a holographic f i l t e r using a converging reference beam. Then a point source at the focus point of the reference beam back-illuminates the hologram and yields the f i l t e r impulse-response, that i s the object or recorded signal. Therefore the intensity response of the f i l t e r to an incoherent pattern i n the plane of the point source i s the convolution of the intensities of the readout and the recorded patterns. The convolution amounts to the crosscorrelation of the readout pattern with the reverse of the recorded pattern. The conditions for spa-ce invariance of an incoherent system are given by Arsenault [60] . Among 38 the possible sets of conditions, the set that applies to Maloney's scheme is 1) that the f i l t e r pupil be the exit pupil for each point of the source, and 2) that the object pupil be on the source pupil and be smaller than i t . The holographic b u i l t - i n lens satisfies the f i r s t condition. Illumination of the readout pattern must be chosen to verify the second condition. Both in Maloney's scheme and in the LLMF case, a converging reference beam i s used to synthesize the lens effect inside the hologram. However the prin-ciples underlying both systems are different. The LLMF i s based on spectral matching in the Fourier transform plane in which i s located an imaging lens. Maloney's incoherent processor consists of a lens with an appropriate pupil. Its impulse response can not be negative. Therefore the range of recorded signals is more restricted than in the LLMF case. Furthermore, since only the intensity of the recorded pattern i s of interest, Maloney used a diffuser in contact with the recorded signal to spread the object f i e l d over the surface of the photographic plate. The diffuse illumination calls for a medium with a finer resolution than in the LLMF case. Gabor proposed [48] a character recognition scheme based on the code translation principle [33] and using a stigmatic or cciverging refer-ence beam. Upon reconstruction, "the stigmatic beam which may be called a 'marker' beam w i l l be reconstructed" f a i t h f u l l y when the readout signal correlates sharply with the recorded one (see Fig. 2.11). The LLMF-1 resembles the above scheme in that both holograms are recorded with a con-verging reference beam. The object and readout beams of Gabor's scheme are unspecified although, for the LLMF's to be space-invariant, they must be of the quasi-Fourier transforming type. As for the LLMF-2, besides using the hologram as a Fresnel lens, i t relies on the variable scale of 39 f a ; (b) Fig. 2.11. Gabor's code translator, (a) Recording, (b) Recognition. the Fourier-transforming stage. A stigmatic code translator has neither this peculiarity of LLMF-2 nor the space invariance of both LLMF's unless i t s object and readout fields are quasi-Fourier transforms. The next para-graph evaluates quantitatively the space variance due to a departure from this condition. 2.A.4 Space Variance of the Stigmatic Code Translator The output f i e l d distribution of the code translation scheme i s expressed in terms of the curvature of the illuminating beam. Then the transverse displacement of a rectangular signal that reduces the intensity of i t s autocorrelation by a fixed amount i s calculated as a function of D^ . The analysis is carried out in one dimension with the rectangular signal modelling an ideal character stroke. A l l constant and irrelevant phase factors are systematically dropped. The symbols have the same meaning and definition as i n Section 2.3. Fig. 2.12 illustrates the system under investigation. A transparency s(x,y) in a plane Pg i s illuminated by a beam converging to a focal point a distance D. behind P so that the f i e l d emerging from P i s [see Eq. (A.14)] s(xH(x;d 4) 40 m v Dt = Ds Fig. 2.12. Stigmatic code translator (D^D S). (a) Recording, (b) Fil t e r i n g . where d. = DT1 (2.75) In a plane P h > a distance Dg behind P g, the object f i e l d [see Eq. (A.16)] o(u) = *(u;d )/ s(x)Kx;d. - d ) exp(-jkd ux)dx (2.76) fa p I S s s r(u) = iKu " sgn(D r)x r;d r) . interferes with a reference beam similar to the one used in Eq. (2.29.a) (2.77) The component of the hologram amplitude transmittance that diffracts into the plus first-order is given by T + 1 ( u ) = o*(u)r(u) = i|<(u,dr - d s)exp(-jk|d r|ux r)/ s*(x)^(x;d j L - d g) P 8 exp(jkd ux)dx. (2.78) 41 For simplicity, the readout signal t(x - x t) i s assumed located in P g where i t has the same scale as s(x) and is centered about x = xfc. Therefore the readout f i e l d expression i s obtained by replacing s(x) by t(x - x t) inside Eq. (2.76) c(u) = i(i(u;d )/ t(x - xjiRxjd. - do)exp(-jkd„ux)dx. (2.79) s p x t> ° s c(u) i s modulated by T (u) so that the f i e l d diffracted into the plus first-order at a distance D q behind the hologram i s , according to Eq. (A.16), O+jOn) = c(m)T + 1(m)*Km;d ) . (2.80) By the substitution of Eq. (2.78) and (2.79) into the above, a regrouping of the terms and a change in the order of integration, 0 + 2(m) becomes 0 + i (m) = / s*(x)<Hx;d, - d )/ t(y - x t)*(y;d - d ) P P s s / ^(u;d + d )exp(-jkd u ( y - x - mg + |6|x ))dudydx. p ° s r H (2.81) With the aid of Eq. (A.11), the integral over P^ reduces to 6(y - (x + mg - |fi|x r)) and 0 +i(m) to 0+1 (m) = 7 s*(x)t(:c - x t + mg - | 6 | x r)exp(-jk(d j L - d g) (mg - |6|xr)x)dx (2.82) P s whenever d = -d o r or g = -6. (2.83) This last condition is identical to the LLMF.condition when a = 1 [see Eq. (2.71)]. With mQ defined by A mQ = x r, (2.84) 4 2 Eq. (2.82) can now be rewritten 0+1(m) = / s*(x)t(x - x t + m - m 0)exp(-jk(d i - d g) (in - m0)x)dx. (2.85) Clearly, at m = m0 + x t, (2.86) O+i(m) i s equal to 0+1 (m0 + xfc) = / s*(x)t(x)dx for any xfc (2.87. a) P s when d = d . (2.87.b) i s , Eq. (2.87.b) means that the illuminating beam is focused i n the f i l t e r plane as i t i s in the LLMF-1 case. On the other hand, i f d^ differs from d s, O+i(mo) is equal to Ofl(m o) = / s*(x)t(x - x t)dx.' (2.88) It i s only when x t is equal to zero that Eq. (2.88) represents the cross-correlation of s(x) with t(x). This implies that the stigmatic code trans-lator requires exact registration of the readout signal and confirms i t s space variance. To evaluate i t quantitatively, the particular case of the autocorrelation of a real rectangular signal of width 2a i s considered. Then s(x) and t(x) are given by s(x) = t(x) =1 |x| < a (2.89) = 0 elsewhere. At the coordinate m = mQ + x t where the correlation peak of a space-invariant correlator would be displayed [see Eq. (2.87)], 0 + 1(m) takes the form 0 + ! ^ + x t) = / s*(x)t(x)exp(-jk(d i - d s)xx t)dx. (2.90) P s After normalization to a maximum value of unity, the solution of the above integral i n the case corresponding to Eq. (2.89) i s Fig. 2.13. Space variance of the stigmatic code translator. 44 the signal displacement x t that reduces the output intensity to a fraction M of i t s nominal value, that is at x = o, results from the comparison of Eq. (2.91) and (2.92) A D ix ' t' 2 air IL ) X s (2.93)' D -5- - 1 D i D i Fig. 2.13 is a plot of |x| versus the relative curvature — of the i l l u m i -s nating beam for M = 1.0, 0.8 and 0.5, and with D g = 500 mm, A = 632.8 nm and 2a = 1 mm. It shows that i f D . and D d i f f e r by 1%, a displacement i s J t of 8.2 mm reduces the output intensity by 20%. However for a 1% difference the same performance is maintained with xfc as large as 82 mm. This confirms that the stigmatic code translator i s space-variant unless the image of the point source is in the f i l t e r plane. Fortunately i t i s relatively easy to focus the illuminating beam in the hologram plane of the LLMF with the above accuracy. Should the illuminating beam be collimated (d^ = o), a displace-ment xfc larger than 82 yra would reduce the output intensity by more than 20%. This illustrates the c r i t i c a l position of the readout signal in a lensless version of Sayar's Fresnel transform f i l t e r [61]. 2.4.5 Limits of the Spatial Carrier Frequency 2.4.5.a Upper Bound: Resolution of the Recording Medium For the MF-1 and -2, the resolution of the recording medium limits only the f i e l d of view of the recorded signal regardless of i t s frequency spectrum because the hologram is of the lensless Fourier transform type where signal and reference point source are coplanar. As this i s not true for the LLMF, i t is necessary to find out how the signal size L g and i t s angular bandwidth ftg are related to the reference beam angle and the 45 cut-off frequency £ of the recording medium. The mask concept introduced by Lukosz [62] is used to that effect. The one-dimensional analysis assumes that the frequency response of the medium is rectangular, or in other words that i t is unity below ? c and zero above. A l l the angles are considered small so that they are a good approximation of their tangents [63,p.218]. Lg — Lg £"2 g —f2 g Let — and be the spatial limits of s(x) , and ^ — and — — be the angular frequencies beyond which i t s spectrum is considered negli-gible. Its maximum spatial frequency £ is related to ti by s s tig n g A l l object rays diffracted at the angular frequency h i t P^ at the same point of coordinate given by ?h " D s T- = X V s ( 2 ' 9 5 ) regardless of the coordinate of the point of s(x) they are issued from. This i s the essence of the Fourier transformation. Imagine that the plane P r is an opaque mask except for a circular (in two dimensions) aperture of radius x = X|D £ (2.96) c 1 r c centered about the reference point source x r« Fig. 2.14.a and b i l l u s t r a t e the situation respectively for the LLMF-1 and -2. Lukosz showed that any object ray, or i t s extension, that reaches the recording medium through the aperture yields resolvable interference fringes. Conversely any ray, or i t s extension, that is intercepted by the mask does not contribute any i n -formation to the hologram and is l o s t . The maximum coordinate x r is found by applying the mask concept 46 to the object ray that gives rise to the fringes of the highest spatial - L s frequency, that is the ray issued from the outermost object point — r — and which hits P, at +p, . The extension of such ray intersects P at a h h r point higher than (x r - x £) i f i t is less inclined on the optical axis than the line joining to the lower edge of the aperture. This condition can be formulated as 2~ ~ Ph , X r - X c * ph  D s + D r where the upper and lower sign correspond respectively to the LLMF-1 and -2. Fig. 2.14. Geometry for the discussion of the medium resolution limit and shadow limit, (a) LLMF-1. (b) LLMF-2. 47 With the aid of Eq. (2.50), the above inequality takes on the general form L p (1 - 6) *r |6l + V (2-97) 2.4.5.b A Lower Bound: Shadow Limit While satisfying Eq.(2.97), x r should be large enough for the reference beam not to cast any object shadow on i t s spectrum. Therefore, a minimum x results from geometrical optics considerations L s . *r + p h ~ 2 + Ph TD > D r s or x r > |6| +-2TfT- ( 2 < 9 8 ) When applied to the MF (6=1), inequalities (2.97) and (2.98) are indepen-dent of the object bandwidth [see Eq. (2.93)]. It can be shown that for a given f i l t e r i n g configuration and readout signal size, the foregoing limits on the spatial carrier-frequency, or on the mean reference beam angle 9xr» are independent of the recording geometry. Indeed, after norm-alization with respect to | | and with the help of Eq. (2.68) and (2.96), Eq. (2.97) and (2.98) can be rewritten p, 9 < -<j> - -r- + AC (2.99) xr s F, c n and Ph 9 > +qb + ~ . (2.100) xr s F, h In the above, 0 is equal to xr n e x r = 1 3 T ' ( 2 ' 1 0 1 ) and <t>s represents the half angle subtended in P^ by the recorded signal * s = l D - ' (2.102) 48 Since the recorded signal is scaled in proportion to D G in order to y i e l d a spectrum scale identical to that of the readout f i e l d , <f> and p^ are constant. Also Eq. (2.69) relates unique to P t and D q . Consequently the limits on 0 are determined by the f i l t e r i n g geometry regardless of the actual recording parameters and particularly of whether the LLMF is of type -1 or -2. 2.4.5.c Comments on the Spatial Carrier—Frequency In the fi n a l account, the minimum x is the maximum of the ' r shadow limit [see Eq, (2.98)] and of the condition for separation of the crosscorrelation [see Eq.(2.20)] x > ^  L + L . (2.20) r 2 s t Chapter 3 shows under what condition Eq. (2.20) is valid for the LLMF too. The paraxial approximation on which the diffraction theory is based limits further <j> to a maximum value of approximately 0.1. Then the half angle s subtended in by the signal is 5°42', and from Eq. (2.102) i t comes L < 0.2 D . (2.103) s s Eq. (2.19.b), (2.97), (2.98) and (2.103) are depicted in Fig. 2.15 where the shaded polygon i s the area that verifies simultaneously the four inequalities. Also represented in the same figure are Eq. (2.97) and (2.98) in the MF case (6 = 1). It appears that i n practice the LLMF does not suffer more limitations than the MF. A numerical example helps appre-ciate this fact. Assuming D R = - D = 500 mm (i.e. 6 = -1) and X = 632.8 nm, a 1000 £/mm frequency, cut-off £ c yields an aperture radius x = 316 mm = 0.632 D , c s and a maximum signal frequency £ g of 40 2./mm results in p, = 12 mm = 0.04 x . h e . 49 Fig. 2.15. Limits on the off-axis coordinate x r of the reference point source. Fig. 2.15 i s drawn according to this numerical example. The limitations due (1—6) to the term | g | are in the order of 8% of x . Therefore c i t i s only with a low-resolution medium that the LLMF may suf-fer limitations of the f i e l d of view or of the frequency con-tent of the signal. The former sets a limit to the recordable amount of data while the latter means the f i l t e r high frequency cut-off may be lower than £ g. This would not affect i t s space invariance though. Fortunately photographic emulsions currently available (Kodak 649 F, Agfa-Gevaert 10E70) have resolution well i n excess of 2000 l/mm and, for the practical range of signals, the reso-lution limit is no more serious for the LLMF than for the MF. In Fig. 2.15, Eq. (2.20) i s plotted for an arbitrary value of L t. However i t i s l i k e l y that with small L g, the readout signals would have a large extend L t so as to take advantage of the parallel processing capabi-l i t y of the correlator. In that case, the condition for separation of the correlation would outweigh the shadow limi t . This i s already true for large L_ whatever L_. 50 In summary, in most practical situations the spatial carrier-frequencies of the LLMF and MF are subject to the same constraints, and the LLMF can easily be constructed with the recording media available to date. 2.5 Conclusions Two LLMF's have been designed that combine inside the same holo-gram a MF and a Fresnel zone plate. This was achieved through the use of either a converging reference beam (LLMF-1) or a diverging one with an appropriately scaled object (LLMF-2). In most practical cases, available recording media have sufficient resolution to record the data without undue restrictions on either i t s f i e l d of view or i t s frequency content. Their spatial carrier-frequency is essentially limited by the separation of the crosscorrelation as with the MF rather than by the shadow li m i t . As the only vignetting aperture of the LLMF's is the hologram i t s e l f and as i t is located in the Fourier plane, the LLMF's are space-invariant. However aberrations of the illuminating beams can result in a loss of space invariance as is discussed in Chapter 7 that describes the experi-mental apparatus. Also holographic third-order aberrations and the volume effect of the recording medium introduce some degree of space variance. This is considered in Chapter 5. Meanwhile the next chapter concentrates on the terms diffracted by the LLMF's into the zeroth-and minus f i r s t -order directions. It evaluates their spatial extent and determine the minimum spatial carrier-frequency that separates the crosscorrelation term from the other output components. 51 III NON-CORRELATION OUTPUT COMPONENTS 3.1 Introduction The MF reference beam i s chosen to cancel any object beam curvature in the film plane. The resulting hologram is of the Fourier transform type. A l l i t s output components are focused by lens L 2 in the output plane P Q (see Fig. 2.4). Consideration of the spatial extent of the focused images leads to the minimum value of x^, or of the minimum spatial carrier-frequency, that separates the crosscorrelation from the other output terms [see Eq. (2.20)]. The LLMF reference beam is selected to provide the f i l t e r with a focal power that enables i t to focus the crosscorrelation in P o without the external lens L 2 (see Fig. 2.9 and 2.10). The non-correlation output components are the unfocused images or the Fresnel diffraction patterns of the corresponding MF output components. The minimum x^ , that separates the crosscorrelation from the unfocused images is given by Eq. (2.20) as in the MF case when a parameter y> defined later in this chapter, is larger than 10 because then the Fresnel diffraction patterns are con-centrated over the geometrical projections of the MF output components. Y i s a function of the width of the MF output components relative to the distances D g and Dr. In most applications of paral l e l character recogni-tion, the condition y » 10 is satisfied. The performance of the LLMF i s measured experimentally i n terms of signal-to-noise ratio (SNR)- for some values of y. The signal i s the correlation peak intensity and the noise i s the intensity of the unfocused terms at the position of the correlation peak. The SNR increases with y and with the reference beam angle or spatial carrier-frequency. For Y = 10, the SNR of the LLMF is very similar to that of the MF at the same reference beam angle. Finally, i t i s shown that for a given f i l t e r i n g 52 configuration the minimum value of x r is the same whether the LLMF i s of type -1 or -2. 3.2 Total Output Field Distribution of the LLMF To reflect that the one-dimensional impulse response of free space is [see Eq. (A.16)] h(x) = /( f) Kx;d), (3.1) a l l the spatially related coefficients of Eq. (2.48) are replaced by their square roots. Under the LLMF condition, the one-dimensional expression of the output f i e l d distribution results from the substitution of Eq. (2.64) and (2.66.b) into (2.48) 0(m) = O01(m) + 002(m) + 0 + 1(m) + 0_1(m) (3.2) d where 0 0 1(m) = -f-k |A|2 / T(£) | S(g) | 2^(u;d c(l-6))exp(jkd um)du, *h (3.2.a) 0 0 2(m) = k j B | 2 / TaH(u;d s(l-6))exp(jkd 0um)du, (3.2.b) p h 0,.(m) = (-f) k A*B / T(0S*(?)exp(jkd ou(m- l^- lx r))du, (3.2.c) o d_% . |d 0_n(m) = (-f) k AB* '/ T(OS(0^(u;2d s(l-6))exp(jkd ou(m+'—'x r))du. (3.2.d) P h The correlation term of interest, 0+^(m), i s centered about I d r l : m = m = '-r~ x . In the MF case, the minimum value of x that separates o d r r v o 0+^(m) from the other components of 0(m) is given by Eq. (2.20) in terms of the spatial widths of s(x) and t(x). For comparison between the MF and LLMF cases, Eq. (3.2) i s now expressed i n terms of the spatial s i g -nals s(x) and t(x). Eq. (3.3) to (3.8) are algebraic manipulations and definitions that lead to the wanted expression Eq. (3.9). 53 The integrals of Eq. (3.2) are of the form l(m) = / iKu;w)G(£)exp(jkd0um)du. (3.3) P h With the aid of Eq. (2.28.b) and (2.63), I(m) can be rewritten I(m) = XDS / ^(C;X2D2w)G(Oexp(j2TrmB5)d5. (3.4) P h When the Fourier transformation is carried out, Eq. (3.4) becomes I(m) = AD g ( m e)*e j 1 T / A (Aw)"^ ijJ(mB ;d2w"*) . (3.5) s s s The l e f t and right arguments of the correlation operator are the inverse Fourier transforms of G(£) and * K£;A 2D 2w) respectively g ( x ) ^ G ( 0 , (3.6) e J 7 r / A (Aw)"*2 d $(x;d 2w - 1) iKS,A 2D 2w). (3.7) s s s Eq. (3.7) follows readily from Eq. (A.13). The expansion of the convolution in Eq. (3.5) yields ^ d 2 Km) = (V e J 7 r A / g(m0 - x)i|;(x;^) dx. (3.8) P s The substitution of Eq. (3.8) into (3.2) results in d h d Ooi(m) = ( A ( 1! 6 )) k 2|A| 2 / g01(m6 - x ) ^ J ( x ~ ) d x , (3.9.a) P s AD H d O02(m) = <Ji^)) k l M 2 / 8 0 2 0"B - x ) K x ; ^ ) d x , (3.9.b) P s 0+1(m) = (AD )** k A*BC t(mB - |fi|x ), (3.9.c) s o st v • d 0_!(m) = (2(1-6))" 2 kiAB* / g^i(m3 + |6|xr - x ) f ( x ; 2 ( 1 _ { ) ) s (3.9.d) where k j - ' k e ^ 4 (3.10) 1 o 5 4 and g 0 1(x) = t(x)*s(x)*s*(-x) «-* TCO.I S(C) | 2 , ( 3 . 1 1 . a ) g 0 2(x) = t(x) -w T(?), ( 3 . 1 1 . b ) . g_i(x) = t(x)*s(x) T(?)S(?), ( 3 . 1 1 . c ) . C g t(x) = t(x)8s(x) ^ T ( 0 S * ( 5 ) . ( 3 . 1 1 . d ) ' 3 . 3 The MF as a Limiting Case When 6 = 1 , the LLMF condition [see Eq. ( 2 . 6 4 ) ] reduces to D = - D . ( 3 . 1 2 ) o t This means that, in the absence of lens , the output terms of the MF ( 6 = 1 ) are virtual images in the plane P . Since L^ i s simply supposed to generate a real image of 0(m) without modification of the f i e l d d i s t r i -bution, Eq. ( 3 . 9 ) with 6 = 1 can be considered as a description of the MF output components. With the aid of Eq. (A. 1 2 ) they are Ooi(m) « goi(mB), ( 3 . 1 3 . a ) 0 0 2(m) « g02(im3), ( 3 . 1 3 . b ) 0+2(m) - C s t(m6 - X ] . ) , ( 3 . 1 3 . c ) 0_j(m) oc g_I(mB + x r ) . ( 3 . 1 3 . d ) As indicated by Eq. ( 2 . 2 0 ) , the minimum value of x^ . that ensures the spatial separation of 0+i(m) from the other components of 0(m) results from the consideration of the spatial extent of the g functions x r > | L g + L t. ( 2 . 2 0 ) 3 . 4 Fresnel Diffraction Patterns O Q I and 0 Q 2 > as given by Eq. ( 2 . 4 8 . a and b) and ( 2 . 6 2 ) , are exact inverse Fourier transforms whenever the last argument of their functions i s n u l l 55 d + d • = 0. (3.14) o t Specifically they are the focused images of ggj and gQ2 respectively. Let d^ designate the solution of Eq. (3.14) and l e t be i t s reciprocal D ' £ ^ r = - ^ - = - D . (3.15) o d o d t t In the plane P q where D q satisfies the LLMF condition, O Q I and O Q ^ are Fresnel diffraction patterns centered on the optical axis. It can be shown that they seem to be produced by apertures of transmittance gg\ and g02 located a distance ( D Q - D ^ ) from P Q and illuminated by a beam converg-ing to a focal point a distance | D ^ J behind the apertures. The details of the proof which follows from the use of the operational notation of the Appendix are omitted here. The same discussion applies to 0 . It i s the focused image of g when d - d + d + d = 0. (3.16) o r s t With defined as the reciprocal of the solution of Eq. (3.16), the particular case a = -6 = 1 implies D " = - D /3. (3.17) o s In P , 0 is a Fresnel diffraction pattern centered about -m , the re-o -1 o flection of the image of the reference point source with respect to the optical axis. It seems to be produced by an aperture g located a distance ( D - D " ) from P and illuminated by a beam converging to a focal point a O O O J o o distance |D^| from the aperture. The spatial extent of the Fresnel patterns is considered in the particular case of the autocorrelation of a s l i t of width 2a s(x) = t(x) =1 |x| < a (3.18) = 0 elsewhere. 56 Hereafter the term 0Q2 > whose expression i s the well-known Fresnel integral, i s examined in detail. The discussion and the conclusions relative to can be extended to the more complex expressions of O Q J and 0.^  when go i and g respectively are substituted for g o 2 - With the normalization para-meters y and £ defined by A Ttd h T = 2a<nd)) (3-19) and e = mB/2a, (3.20) Eq. (3.9.b) becomes O 0 2(e) - f " 0 ' 5 e x p ( - j Y 2 ( ^ ) 2 ) d ( ^ ) . (3.21) e+0.5 Z a The above integral is proportional to the length of the chord joining the points (e-0.5) and (e+0.5) on the Cornu spiral (e.g. see [ 64]). When the s l i t i s wide compared to /XDg(l-6) , for some value of e, the chord's ends are close to the spiral's eyes and 0 Q 2 ( e ) i s essentially confined within the geometrical projection of the s l i t . This occurs typically when ((e+0.5) - (e-0.5))Y>/?) » 1 IT or Y »* /(f) = 1.25, " (3.22) and the major diffraction effects take place when e i s within a distance ~~^^) from the edge of the geometrical shadow of the s l i t [192, p. 376] |e - 0.5| = ^ C f > . (3.23) For example, with 2a = 10 mm, Dg = 103mm, X = 632.8 nm and 6 = -1, Eq. (3.19) gives Y = 15.76 and Eq. (3.23) implies that the diffraction pattern is 1.59 times larger than the geometrical projection of the s l i t . Accord-ing to Eq. (3.23), the larger Y the smaller the area around the geometrical shadow boundary that is affected by diffraction. 6 = 1 makes the parameter 57 Y of the MF equal to i n f i n i t y [see Eq. (3.19)]. As seen in Section 3.3, the MF output fields are indeed limited by their geometrical projections. Whenever the LLMF output components can be considered confined within the geometrical projections of the corresponding MF output compo-nents, the LLMF can use the same minimum reference beam angle as the MF and Eq. (2.20) is valid for the LLMF as well as for the MF. With the above numerical values for Dg, X and 6 , a 1-mm s l i t i s not wide since Eq.(3.19) gives y = 1.57 which does not satisfy inequality (3.22). Should the LLMF be used to detect such a s l i t , i t would require a larger x r or larger reference beam angle than the MF. Yet most character recognition applica-tions use simultaneously many recorded and/or readout signals to take advantage of the parallel processing capability of the f i l t e r . Even i f individual signals are too small for their y to satisfy the inequality (3.22), in parallel processing the fields g o i , g02 a n d g-l a t e wide and their associated y's may be large. In most practical situationsi the minimum reference beam angle of the MF is adequate also for the LLMF and Eq. (2.20) i s valid i n both cases. 3.5 Signal-to-Noise Ratio To appreciate the importance of the parameter Y» the signal-to-noise ratio (SNR) of the LLMF was measured experimentally for some values of Y« The signal was the peak intensity of the autocorrelation of a s l i t . The noise was the energy contributed at the autocorrelation peak by the non-correlation terms of 0(m). Such noise i s deterministic. The noise of the LLMF differs from that of Gabor's in-line hologram: in the l a t t e r , unfocused images ruin the focused image; in the former an increase of the reference beam angle moves the correlation pattern further away from the optical axis so that, by virtue of the conservation of irradiance, the 58 energy of the unfocused images O Q I , 0Q2 a n ^ 0_j eventually makes i n s i g n i f i -cant contribution to the output detector in P Q. The experimental evaluation of the SNR calls for independent measurements of the signal and of the noise. A 13°-reference beam angle makes xr(=230 mm) large relative to 2a(< 10 mm) and separates completely the correlation pattern from the other terms. This allows for the measure-ment of the signal. The f i e l d diffracted about the optical axis is then measured at a small coordinate e from the optical axis in the direction of the correlation pattern. Such a f i e l d would be the noise in the case 2a where x^ . = [ s e e E9- (3.20) with m replaced by m0 (Eq. 2.66.a)] i f the 0-i term was not displayed far from the optical axis due to the large reference beam angle. Such a term is usually much weaker than the zeroth-order terms. Also i t is often weaker than the correlation term 0 + i since the volume nature of the hologram does not allow simultaneous diffraction of 0-i and O+i with maximum efficiency [65 ]. Moreover, in the MF case, 0- 2. plays no role in determining the lower limit of x^. Therefore the zeroth-order terms are considered a good approximation of the noise. The experimental minimum SNR expressed in dB is shown in Fig. 3.1 for some values of y(1.57, 7.85 and «>) and for some values of E (2 and 4). It i s the ratio of the signal to the maximum noise measured at or beyond a particular e. The experimental values were measured with a Gamma Sc i e n t i f i c 2900 MR scanning autophotometer. They were obtained tfith a LLMF-1 (6 = -1) and a MF-2 (6 = 1) characterized by a = 1, X = 632.8 nm and Dg = 103mm. They were recorded with a ratio of reference to object beam intensities equal to unity on the optical axis. The Y = 1.57 and 7.85 cases correspond to the LLMF-1 of s l i t s respectively 1- and 5-mm wide. At the rightmost edge of Fig. 3.1 are represented the values of the SNR of the MF-2 (Y = 00) . In the latter case, Eq. (2.20) predicts that the autocorrelation of a s l i t 59 4 minimum SNR dB 30 20 A to O X o O 6=2 X £=4 1.57 7.85 CD If Fig. 3.1. Minimum SNR of the LLMF (y = 1.57 and 7.85) and of the MF (y = °°) . e i s the relative off-axis coordinate of the reference point source. 60 i s separated from the other terms when e > 2.5. Beyond this l i m i t , the SNR is theoretically i n f i n i t e . Therefore, the SNR measurements at y = °° indicate the practical order of magnitude of the SNR. At y = 1.57, the SNR at z = 4 is only 16.2 dB, i l l u s t r a t i n g the earlier comment that, for small y, the fields extend further than the geometrical projections of the g functions. At y = 7.85 though, the SNR is larger than 25 dB. at e = 2. It i s of the same order of magnitude as the SNR of the MF at the same value of e. This confirms that the lower limit of x r, given by Eq. (2.20), applies to the LLMF as well as to the MF when y = 10. Throughout the thesis, i t is assumed that the hologram is positive or that i t s amplitude transmittance is simply proportional to the intensity I(u,v) of the interference pattern [see Eq. (2.32)]. This assumption does not affect the validity of the analysis and conclusions of the various chapters even i f in practice the hologram i s negative. When i t comes to measuring the zeroth-order noise i t must be borne in mind that the actual hologram i s negative and has an amplitude transmittance T ( U , V ) = k' - k" I(u,v) (3.24) where k' and k" are constants. Consequently the zeroth-order output term is not (O 0i + O 0 2 ) but rather k'O 0 2 - k"(O 0i + O 0 2) = (k' - k")0 02- - k"O0.i The term k'O 02 i s the f i e l d diffracted by the readout signal and simply attenuated by the unexposed recording medium. Its spatial extent i s the same as that of k"0o2» Although k'0Q2 is often strong, Fig. 3.1 shows that the measured SNR when y = 10 i s similar to that of the MF-2 at the same reference beam angle. 3.6 Invariance of the Parameters The integrals of Eq. (3.9.a and d) could be expressed in terms 61 of y and e as was done for Eq. (3.9.b). Although these parameters are defined in terms of the recording geometry (2a, DG, <5) , i t is shown below that they can also be expressed in terms of the readout geometry. It follows that, for a given readout system, the spatial extent of the compo-nents of 0(m) do not depend on whether the LLMF i s of type -1 or -2. The' conclusions of this chapter are independent of the type of LLMF considered. Because of the variable scale property of the f i r s t Fourier-transforming stage, the recorded signal size is proportional to DG. Then the angle 2$ g subtended at the hologram by the recorded signal is constant 2$ = 2a/D0 = constant. (3.25) s s By the substitution of Eq. (2.63) i n (3.20), e becomes — - — J T ' ( 3 - 2 6 ) O S e 2a D 2c; Since the LLMF condition [Eq. (2.64)] implies that 1 - 6 = a + B, Y can be rewritten with the aid of Eq. (2.36) and (2.63) (3.27) TTD D . h y - 2*s <uF+V> • (3'28) o t' For later use in chapter 4, the parameter C i s defined as C = (1 - 6)/2a. (3.29) Like Y» S can be transformed into , D + D s ° c Since $ g is constant, Eq. (3.28), (3.26) and (3.30) show that Y» E and x, are independent of the LLMF type and are determined by the f i l t e r i n g configuration. 62 3.7 Conclusions The expressions of the non-correlation output terms of the LLMF have been derived. These terms are the Fresnel diffraction patterns of the corresponding MF output terms. The spatial extent of the unfocused images governs the minimum value of x or the reference beam angle. In the p a r t i -cular case of the autocorrelation of a s l i t , the diffracted fields are mostly confined within the geometrical projections of the M F output terms when Y> the relative size of the s l i t , i s in the order of 10 or more. Experimental measurements confirms that for y = 10 the SNR of the LLMF i s of the same order of magnitude as that of the MF at the same e (the relative value of x r ) . In most practical cases of parallel character recognition, the equivalent y can be considered equal to or larger than 10. In that case, the minimum reference beam angle that separates the correlation pattern from the other output components i s the same for the M F and for the LLMF. Phase holograms that have a better diffraction efficiency than their amplitude counterparts should improve the SNR at a given reference beam angle. IV. SPECTRAL DISPLACEMENT 4.1 Introduction The intensity of the correlation peak decreases with the posi-tioning error of various components of the MF such as transverse and longitudinal displacements of the hologram (that i s displacements respec-tively normal or parallel to the optical axis). This problem has been investigated theoretically [45,61,66 ] and experimentally [17]. The LLMF too is affected by a displacement of the hologram. This chapter derives the equations that relate the correlation intensity of the LLMF to a mismatch of the recorded and readout spectra. Via a numerical example, i t compares the sensitivity of the MF and LLMF to such a spectral dis-placement. It was suspected that the MF and LLMF might behave somewhat differently because a LLMF displacement amounts also to a displacement of i t s b u i l t - i n lens. Indeed, this chapter confirms that the LLMF be-havior arises from the combined displacements of i t s f i l t e r component and lens component. Spectral mismatch i s caused either by transverse or longitud-inal registration errors of the hologram or by a phase error due either to the illuminating beam or to the medium supporting the daca (s(x,y) or t(ax,ay)). The registration errors are l i k e l y to occur when the hologram is repositioned after remote processing of the photographic emulsion or when different f i l t e r s are interchanged either by physical replacement or by multiplexing in a sequential multichannel processor [37, 44]. They could also be caused by the dynamic i n s t a b i l i t y of a creeping or vibrating apparatus. This chapter treats only static errors. Without loss of generality, the analysis i s carried out i n one dimension and irrelevant constants and phase factors are simply discarded. 64 4.2 Transverse Displacement of the F i l t e r The expression of the output amplitude of the general matched f i l t e r with transverse misalignment is derived f i r s t . Then the performance of the MF and LLMF are compared for the particular case of a rectangular signal. It is shown that the LLMF behavior arises from the combined displacements of i t s f i l t e r component and i t s lens component. 4.2.1 General Matched F i l t e r The third term of Eq. (2.33) represents the useful component of the general matched f i l t e r transmittance x + 1(u) = Mu;d g(l-6)) S*U) exp(-jk|d r|x ru). (4.1) Should the f i l t e r be transversally shifted by Au, i t s transmittance T _ J ^ ( U ) would be x + 1 A(u) = x + 1(u-Au) = ^(u;d g(l-6)) S * a-A£) exp(-jkd gu( | 6 |x^ (1-6)Au)) where A? = Au/XD . (4.3) s According to Eq. (2.37) and (2.46), the readout f i e l d c(u) c(u) = ^(ujd t) T(5) (4.4) yields an output f i e l d distribution 0+^(m) 0+1(m) = / c(u) T + 1 A(uH(u;d e) exp(jkd^mu)du. (4.5) P h By the substitution of Eq. (4.2) and (4.4) into (4.5), 0+^(m) becomes 0.,(m) = / ty (u;d (1-01-6-^) )T(£) S*(?-A£) exp( jkd * u(m-|^r |x +^ s (1-6) Au) )du. + 1 S d e d' r d' h s e e (4.6) As i n Eq. (2.49), the last argument of the ty function i s made equal to zero l _ a _ d _ e _ 6 = 0. (4.7) d s Then expansion of the Fourier transforms yields ° 4 . i ( m ) = // t(x)s*(x')exp(-j2TTA5x') / exp(j2 u£(x'-x+ de(m-| dr|x + ds (1-6) P s \ dT d1" d' b n s e e Au)))dudx'dx. ( 4 . 8 ) With the assumption that the hologram aperture i s large enough for the integral over P^ to be the Fourier Transform of a Dirac delta function, Eq. ( 4 , 8 ) becomes 0 (m) = / s*(x)t(x-i-de_l(m-|^r| x + ^ (1-6)Au)) exp(-j2TTA?x)dx. ( 4 . 9 ) + 1 P d d' r d' s s e e At the nominal position of the detector [see Eq. (2.55)] m = m = x d / d' (4.10) o r r e , the output amplitude i s 0,(m) = / s*(x)t(x+(l-6)Au) exp ( - j 2 T r A£x)dx. (4.11) +1 o p s 4.2.2 Rectangular Signal For the MF in the white noise case, Vander Lugt showed [45] that the rate o..' decrease of 10,, (m ) | 2 with small Au i s maximum for the ' +1 o ' autocorrelation of a real rectangular signal. Hereafter follows a compar-ison of the performances of the MF and LLMF for that particular signal which, considering the character recognition application, i s a better representation of an ideal character stroke than a gaussian signal [61]. Let 2a be the width of s(x) and t(x) s(x) = t(x) = 1 |x| < a (4.12) = 0 elsewhere so that Eq. (4.11) becomes ° + i (V -a - ( l - 6 ) A u A u 1 // , ^ e x p ( - j 2 7 r A £ x ) d x 0 < g < (4.13) -a 66 a 1 Au exp(+j2TTAt;x)dx - < ~ < 0 -a-(l - 6)Au (4.13.cont'd) = 0 elsewhere . The amplitude of 0.,(m ) is an even function of A£ or Au and, after norm-+1 o alization to unity at Au = 0, can be expressed as irAu2a,, / n r v|Au t sin i ( i - (i-«) Sr) TrAu2a 0 + 1(m o) = V — 0 < - l 2 ^ l K 1-6 X D a t (4.14) = 0 elsewhere . For the MF, 6 = 1 and 0+^(mo) varies as sine x as reported i n the l i t e r -ature. With the definitions of y and r,[see Eq. (3.25) and (3.52)], Eq. (4.14) can be rewritten as follows O ^ O n )'• s i * ( Y 2 C A u ( l - r , J A u l ) ) 0 < \^\ < C l +1 O 2 v A Y^Au ( 4 > 1 5 ) = 0 elsewhere. Under this form, i t i s easy to see that 0 .. (m ) does not depend upon the + 1 o particular type of LLMF since i t was proven i n Section 3.6 that, for a given f i l t e r i n g configuration, y and £ are independent of the recording parameters. Although the product r,y 2 is the same for the MF and LLMF, Eq. (4.15) makes their analytical comparison somewhat d i f f i c u l t as, for the MF, y = 0 0 and t, = 0, Therefore both f i l t e r s are compared via numer-i c a l examples. The square of Eq. (4.14) was simulated on a computer IBM 360/70 with X = 632.8 nm, D = 500 mm and a = 1. For the LLMF, 6 was chosen equal to unity. Fig. 4.1.a, b and c are Calcomp plots representing the intensity at the nominal position of the detector of each f i l t e r ver-sus the f i l t e r displacement Au, with the signal width 2a equal to 0.1, 67 \ \ \ \ \ \ \ MF LLMF 2A- 0.1 MM D T = 500 MM (a) 1 i I i 0 . 0 10.0 20.0 30.0 10.0 FILTER DISPLACEMENT: A l l [MICRONS) 50.0 0 . 0 (b) 0 . 0 MF LLMF i 1 =i " r 100.0 200.0 300.0 400.0 F I L T E R DISPLACEMENT: AUIHI'CRONS) 500.0 (c) 20.0 40.0 60.0 60.0 100.0 F I L T E R DISPLACEMENT: AU(MICRONS) Fig. 4.1. Output intensity versus the transverse displacement Au of the f i l t e r , a = 1, 6 = ±1, X = 632.8 nm, Dt = 500 mm. (a) 2a = 0.1 mm. (b) 2a = 1 mm. (c) 2a = 3 mm. 68 1 and 3 mm respectively. It appears that the LLMF behaves very much l i k e the MF for large signal width but is more sensitive to Au than the MF when the width is small. A look at the scale of the horizontal axes of the three plots of Fig. 4.1 reveals that there exists an optimum size 2a at which the LLMF i s least sensitive to Au. Such an optimum i s determined by returning to Eq. (4.14) and computing the asymptotic behavior of 0 + 1(m Q). 4.2.3 Asymptotic Behavior of the LLMF When 2a is large compared to (1 - 6 )Au, Eq. (4.14) can be approx-imated by 0,, (m ) +1 o large 2a fc i:Au2a s i n c TD^c" (4.16) which is identical to 0 + 1(m Q) when 6 = 1 . This case corresponds to Fig. 4.l.c where the MF and LLMF have similar performance. When 2a and Au are comparable, their product i s small with respect to XD^a/ir. Therefore the sine function can be approximated by i t s argument and Eq. (4.14) tends towards - 1 - (1-6)||H| 0 < , A U | - 1 < l2a' '2a1 1 -6 small 2a (4.17) = 0 elsewhere. This case corresponds to Fig. 4.1.a where the intensity decreases quad-ra t i c a l l y with Au. That Au which asymptotically yields an output inten-sity M can be expressed in terms of 2a with the aid of Eq. (4.16) log Au = - log 2a + log(M AD h) , 8 (4.18) large 2a where M^  i s defined by sinc ML = vfo , (4.19) 69 10001 s 3. 700 10, Au  N / / / NT* / / s ^ \ \ / / / / 1 / /> * V \ ^ * \ V X \ 2a on 1 mm 10 Fig. 4.2. Actual and asymptotic characteristics of the MF and LLMF. Transverse displacement (Au) and signal size (2a) that reduce the output intensity to a fraction M of i t s value at Au = 0. a = 1, 6 = ±1, X = 632.8 nm, D t = 500 mm. and with the aid of Eq. (4.17) log Au = log 2a + log 1 - -M 1-6 (4.20) I - * small 2a In Fig. 4.2, the straight lines with negative slope represent the MF char-acteristic and, in the lower right corner, the asymptotic characteristic of the LLMF [see Eq. (4.18)]. Eq. (4.20) i s depicted by the solid lines with positive slope which approximate the LLMF behavior for M = 0.5 and 0.8. The dashed curves and line are the real behavior of the LLMF as given by Eq. (4.14) for M = 0, 0.5 and 0.8. About the transition between the two characteristics l i e s the region of smallest sensitivity to Au. It corresponds to a signal width 2a^ , found by equating Eq. (4.18) and (4.20) , x j 2 a T = [M L X D s ( 1 - 6 ) I TT(1- *fa) (4.21) The value of y in the transition region results from the substitution of Eq. (4.21) into (3.23) Y T =1 1- Mi (4.22) With M = 0.8 (i.e. ^  = 0.809) and with X = 632.8 mm and D g = 500 mm, Eq. (4.21) and (4.22) give respectively 2a T =1.24 mm and y^ = 2,76. This means that the largest Au that yields 80% of the nominal correlation intensity i s 55 ym. This occurs when the signal size i s 1.24 mm and rep-resents only a 15% reduction from the corresponding Au of the MF. 4.2.4 Physical Interpretation The twofold behavior of the LLMF can be explained as the result of two conflicting mechanisms. On one hand, a broad signal has a narrow spectrum and i t does not take a large Au for a significant spectral mis-match to occur. The same i s true for the MF. On the other hand, as the spectrum of a narrow signal is broad, a positioning error amounts prin-cipally to a shift of the b u i l t - i n lens and of the output distribution. As the correlation pattern i s narrow, the output image could easily miss the detector. With reference to Fig. 4.3 where P and P q are conjugate planes, a lens shift of Au results in a proportional image shift of Am A m = Au | D t * D o | = Au = Au 1-61 . (4.23) The fixed detector measures an output amplitude which decreases linearly Fig. 4.3. Model of the image translation due to a transverse displacement the holographic lens. with |Am| as long as the latter is smaller than half the width of the t r i angular autocorrelation |Am| < l ^ a g - 1 ) ) . (4.24) When |Am| does not satisfy the inequality (4.24), the measured output amplitude is null. With the aid of Eq. (4.23), the above inequality can be expressed as |Au| < 2a/(1-6). (4.25) The variation of output amplitude just described is identical to the asymptotic variation of Eq. (4.17). 4.2.5 Comments on Transverse Displacement From Eq. (4.9), i t is clear that the following choice of detec-tor position d m = m - ~ (1-6)Au (4.26) O Q ' e removes any difference between the MF and LLMF characteristics. This choice corresponds to the situation where the detector has a large num-ber of resolution elements that are logically grouped into cells of large si. e. Then a signal is detected whenever its correlation peak lies within a cell [15]. Thus the positioning accuracy of the readout signal does not have to be better than the size of the cell (when M^  = 1). The case of the fixed detector describes the response of a single resolution element when the position as well as the presence of the signal are im-portant, as for example in geometrical feature extraction [42, 43]. Such a f i l t e r being naturally highly selective, i t is recorded as a low-pass MF (LPMF) and fortunately has a correlation peak two or three times broader than i f selectivity was achieved through a high-pass MF (HPMF) [67, see also Section 6.3.2]. It must also be kept in mind that, while a reduction of the signal size relaxes the tolerance on the f i l t e r (MF) or f i l t e r component (LLMF) registration, i t also places stricter 72 requirements on the position of the input signal and on the resolution of the output detector. It i s expected that the decomposition of the LLMF behavior i n white noise into a lens-effect [Eq. (4.20)] and a f i l t e r - e f f e c t [Eq. (4.18)] is valid also in nonuniform noise. Indeed, as for the MF, dividing the LLMF transmittance by the power spectral density of the noise affects only the factor S (£) of Eq. (4.1). Furthermore, Vander Lugt showed [45] that the MF tolerance to a later a l displacement i s an order of magnitude more stringent in a coloured noise situation than in the white noise case. In Fig. 4.2, this tenfold decrease i n tolerance could be accounted for by shifting the negative sloped lines one decade downwards, leaving the lens-effect lines unchanged. Now, 0.4 mm would be the signal width at which the LLMF is least sensitive to Au, resulting in a 20% output intensity reduction for Au approximately equal to 20 urn. A "dual frequency plane processor" has been proposed [45] as a mean of maintaining relatively high the tolerance of the MF to Au even i n nonuniform noise. This scheme has two f i l t e r planes conjugate to each other with respect to an extra relay lens. Therefore i t has no LLMF counterpart as the addition of the lens would defeat i t s i n t r i n s i c space invariance. 4.3 Phase Error on Readout Signal In this section, the f i l t e r i s supposed perfectly registered and i t is the readout spectrum that i s transversally displaced because of a phase error either in the medium supporting the readout data or on the beam illuminating the same. Both the MF and LLMF behave identically. It i s shown that a transverse shift of the spectrum i s not as severe as reported by Dickinson and Watrasiewicz [66 ]. Experimental ver i f i c a t i o n supports this view. 73 4.3.1 Rectangular Signal A linear phase variation on top of the readout signal t(ax) exp ( - j 2 T r A£x) (4.27) results in a transverse shift Au of the readout spectrum,and the readout f i e l d c( u) = i K u i d t ) T ( C + A £ ) (4.28) does not match the recorded spectrum. Therefore the output amplitude distribution at the nominal detector position i s 0 + 1(m o) = / s*(x)t(x) exp ( - j 2 T T A£x)dx (4.29) P s which i s the same whether the MF or LLMF is used, since the b u i l t - i n lens of the latter i s not translated. Eq. (4.29) i s identical to Eq. (4.11) with 6 = 1 since i t is the relative displacement of the f i l t e r and the readout f i e l d that i s relevant. In the case of a rectangular signal of width 2a, 0+^(mQ) becomes 0,..(m ) = sinc(irAS2a) (4.30) +1 o after normalization. Eq. (4.30) i s identical to Dickinson's and Watrasiewicz's findings [66].. However, their calculated actual displacement i s too stringent by a fac-tor 2ir, as they estimate the f i r s t zero of Eq. (4.30) at Av = 26 ym for 2a = 2mm, D = 500 mm and X = 632.8 nm. Using Eq. (4.3), the f i r s t zero s occurs rather at Au = 158 ym and this i s corroborated by the curve "L/F = 4.10" 3" of [45, Fig. 2]. 4.3.2 Experimental Verification In the course of experimentation i t was observed that a trans-lation of the readout signal in i t s plane P produced substantial v a r i -ations of the correlation peak•intensity and serious distortions of the correlation pattern. Concurrent with a horizontal translation i n P^  74 was a non linear horizontal shift of the readout spectrum caused by a piecewise linear phase error of the illuminating beam. The signals s(x) and t(x)(a=l) used in one such experiment were the printed character "G" so that in a one dimensional analysis s(x) and t(x) are taken as a hori-zontal crossection of "G" or the double pulse function of Fig. 4.4. Evaluation of Eq. (4.29) for this particular signal yields 0,, (m ) = sinc(TTA£ w ) cos(irW A ? ) . (4.31) + 1 c s s w and W are respectively the width of each pulse and their centre to 6 S centre distance. When w << W , the central lobe of 0,..(in ) can be ap-s s' +1 o' r proximated by 0+1(nO = cos(TrWsAt;) |AC| 1 . (4.32) With W = 1.36 mm and w =0.22 mm, the maximum error of approximation s s Fig. 4.5. Theoretical and experimental variations of the output intensity versus the transverse displacement of the f i l t e r . 75 (4.32) is less than 3%. The experimental and theoretical values of |0.,(m )| 2 versus Au are plotted in Fig. 4.5 with A = 632.8 nm and D = +1 o s 779 mm. The two curves are in good agreement, especially in view of the crude approximation of "G" by a one-dimensional double pulse and of the d i f f i c u l t y of determining the position of the central value of the autocorrelation for large Au. Indeed, in this case, the correlation pattern is severely distorted as can be seen in Fig. 4.6 which presents pictures of the autocorrelation of G for Au = 0 (Fig. 4.6.a) and 180 ym (Fig. 4.6.b). taining reproducible correlations measurements and in investigating ex-perimentally the space variance of the MF and LLMF. These matters are discussed further in chapter 7. 4.4 Longitudinal Displacement of the F i l t e r That the longitudinal registration of the MF is much less c r i t i c a l than the tranverse one is well known [ ] and i t is the intent of this section to show that the same applies to the LLMF as well. As the expression of the output intensity versus the longitudinal displace-ment does not lend i t s e l f to easy interpretation, i t is assumed that, as in Section 4.2, the LLMF behavior results from the combined displace-ments of i t s f i l t e r component and i t s lens component. Only the asymp-totic behaviors are derived. (b) Fig. 4.6. Autocorrelation of "G" when the transverse displacement Au of the readout spectrum is (a) 0 ym, (b) 180 ym. Phase error on the readout signal was a major problem in ob-76 4.4.1 Displacement of the F i l t e r Component A f i l t e r translated by z from the focal plane l i e s in the Fres-2a nel diffraction region and i s space-variant. If the signal width — is much smaller than the distance Dfc, the readout f i e l d i s a slowly varying function of z and, for small z, can be approximated by the Fourier trans-form of the readout signal, shifted l a t e r a l l y by [see Fig. 4.7] Clearly, Au is a function of the readout signal coordinate xfc and the degree of spectral mismatch is space-variant. Now that the effect of a z displacement has been related to a spectral transverse registration similar to that due to phase error i n P t, the result of Section 4.3 can readily be used. In white noise, the output amplitude i s found by sub-stituting Eq. (4.3) and (4.33) into (4.30) (4.33) trzx^a 0.,(m ) = sinc(irA£2a) = s i n c ( - r r r — — ) . (4.34) With M equal to |0 + 1(m o)| 2 and defined by Eq. (4.19), i t comes M L A D T 2 A log z = l o g ( — - )- log 2a - log x (4.35) 1 I » 0+1(m)\ Fig. 4.7. Model of the image modifications due to a longitudinal displacement of the holographic lens. 7 7 4 . A . 2 Displacement of the Lens Component Similarly to the physical interpretation of Section A . 2 . 4 , the longitudinal displacement z of the holographic lens of focal length F^ [see Eq. ( 2 . 6 8 ) ] results in a displacement z' of the output plane, a new f i l t e r magnification and a latera l shift of the image to a new position m^ . The imaging condition implies (see Fig. 4 . 7 ) 1 + „ 1 _ , = j k = j r - + J r - . ( A . 3 6 ) D +z ' D -z-z 1 F D^ ' D t o h t o When a term in the second power of the errors z and z' is neglected, Eq. ( A .36) combines with Eq. (2.65) to yield — = M 2 - 1 . ( A . 3 7 ) z f i f the magnification of the f i l t e r when z = 0 . When z 0 , i t s magr nification becomes z M , 1 -D - z - z D K = £ • 7 M f — . ( A . 3 8 ) f D t + Z f • 1 + z/Dt With z = 0 , the image or triangular autocorrelation of the rectangular signal of width 2a, i s centered about m = M , x. ( A . 3 9 ) o r t and has an extent 2(2a3 - 1). The new image is now centered about m' = m M L / M , ( A . A O ) o o f f It has an extent 2(2a3 - 1 ) M ^ / M _ and, by virtue of the conservation of radiance, has an amplitude larger than in the z = 0 case by a factor ( M j / M p . In the particular case = 1 , mQ = xfc and i n f i r s t approx-imation z' = 0 . (For ^ 1 , the following analysis would also be valid as long as z' is small and the correlation i s a slowly varying function of z'.) At mQ, the nominal position of the photodector, the amplitude 78 distribution is now given by 0+i ( , Bo ) - < g f > * ( 1 " l ^ o l ^ T i g " - ( 4 ' 4 1 ) With Mf = 1 and with the help of Eq. (A.37) to (4.40), Eq. (4.41) becomes 1+z/D , 2ooc |z/D | - < H S ; » »- d-UpL' <4-"> and can be approximated by 0 + 1(m o) a 1 - 2a(^)|§-| (4.43) with less than 0.1% of error even when z is as large as 1 mm and Dfc = 500 mm. . Once more with M = 0,, (in ) 2 and a = 1, Eq. (4.43) amounts to ~ r l O log|z| = l o g ( 1 2 ^ Dt) + log 2a - log x t . (4.44) For the sake of completeness, Eq. (4.41) to (4.43) should include a note to the effect that 0+^(mQ) is zero when the triangular autocorrelation i s off the detector. 4.4.3 Comments on Longitudinal Displacement From Eq. (4.35) and (4.44), i t is obvious that the f i l t e r d is-placement is limited by the maximum excursion x^ of the readout signal or by the aperture size in P- . Eq. (4.35) representing the displacement of the f i l t e r component is the actual characteristic of the MF and the asymptotic behavior of the LLMF, while Eq. (4.44) is peculiar to the LLMF and i t s b u i l t - i n lens displacement. Both equations are plotted in Fig. 4.8 as solid lines for M = 0.5 and as dashed ones for M = 0.8, with xfc varying from 10 to 100 mnr. The region of v a l i d i t y and of interest i s approximately z s. 1 mm. Note that, with the chosen parameters, the trans-i t i o n between the two asymptotic behaviors occurs at the same value of 2a^ , or f° r transverse and longitudinal displacement. A comparison of Fig. 4.2 and 4.8 clearly shows that the transverse registration of the 79 Fig. 4.8. Asymptotic characteristics of the MF and LLMF. Longitudinal displacement (z) and signal size (2a) that reduce the output intensity to a fraction M of i t s value at z = 0. LLMF is much more c r i t i c a l than the longitudinal one, since with 2a = 1.24 mm and even with xfc as large as 100 mm, the output intensity i s down to 80% of i t s nominal value for a longitudinal displacement of 200 ym or for a transverse one of 55 ym. Therefore, with the LLMF as well as with the MF, most of the attention and effort must be devoted to the accurate transverse positioning of the holographic f i l t e r . 4.5 Conclusions The transverse and longitudinal positioning accuracy of the hologram of the LLMF has been investigated and compared to the corresponding MF requirements [ 4 5 ] . As with the MF, the transverse accuracy i s much more c r i t i c a l than the longitudinal one. Two cases are to be distinguished depending upon the detection procedure. 1) When the output plane i s divided up into a number of cells within which the position of the correlation pattern i s unimportant, the LLMF behaves like the MF and the f i l t e r displacement reduces the correlation intensity by spectral mismatch. A signal demagnification broadens the spectrum and relaxes the constraints on the f i l t e r registration. Toler-ance on the readout signal registration i s smaller though for the LLMF than for the MF. 2) When the correlation is measured with a fixed detector, as for example in feature extraction, the displacement of the b u i l t - i n lens of the LLMF shifts the correlation pattern with respect to the detector and degrades the performance. This effect i s more pronounced for small than for large signals and a compromise between the spectral mismatch and the lens effect has been indicated. Section 4.3 dealing with the phase error of the input signal has presented theoretical and experimental evidence that Dickinson and Watrasiewicz overestimated the spectral registration accuracy by a factor 2ir [66]. The experimental considerations of Chapter 7 are largely based on the analysis of Section 4.3. 81 V. SPACE VARIANCE 5.1 Introduction The LLMF's described in Chapter 2 are space-variant in the sense that their only vignetting aperture is the hologram i t s e l f (see Section 2.3.5). Neither their f i r s t nor their second stages contain additional vignetting components. As the hologram is located in the Fourier transform plane P^, i t vignets, and more specifically, low-pass f i l t e r s uniformly the readout f i e l d , no matter where the readout signal is located in the input plane P as long as the Fourier transformation is valid [26, p.80]. However, the non-zero thickness of the recording medium and the third-order holographic aberrations destroy such space invariance. Indeed the diffraction efficiency of a medium that cannot be considered thin and the holographic aberrations are functions of the readout signal coordinates. Section 5.2 computes in three dimensions the minimum diffraction intensity of various configurations of recorded and readout signals. It finds those that are least space-variant, or least sensitive to the volume effect of the medium. Section 5.3 determines the aberrations of the MF-i and -LLMF-i ( i = 1,2). Then computer-generated plots are used to compare the space variance of the root mean square (rms) distortion of the best signals confi-gurations found in Section 5.2 and that of the rms astigmatism of MF-2, LLMF-1 and LLMF-2. 5.2 Volume Effect of the Recording Medium  5.2.1 Introduction The hologram of the previous chapters was supposed to be an i n -f i n i t e l y thin two-dimensional grating. However the thinnest photographic 8 2 emulsions available to date are 6-ym deep or about ten-wavelength thick. The volume holograms recorded in such emulsions behave like three-dimen-sional gratings and maximum diffraction occurs only i n the direction pre-dicted by Bragg's condition. Leith and his co-workers [65] have analyzed the angular sensitivity of the volume hologram of two plane waves, that is the variation of the diffracted intensity versus the angular mismatch between the readout wave and one of the recorded waves. Their assumption of low diffraction efficiency i s j u s t i f i e d in the case where an amplitude hologram is recorded i n photographic emulsions. Their analysis is two-dimensional in that they consider one longitudinal and only one transverse coordinate. In other words, they assume the direction of propagation of the readout wave is i n the plane formed by the directions of propagation of the two recorded waves. In this section, the analysis i s extended to three dimensions. Indeed, in most applications and particularly in character recognition, the data of the matched f i l t e r i s tvo-dimensional. Then, for a fixed number of recorded and readout signals, that is at constant SBP, any attempt to mini-mize the volume effect due to a horizontal angular mismatch increases the vertical mismatch, and vice cersa. Furthermore the reference-to-object beam angle, whicn is a function of the horizontal width of the recorded and readout signals, also governs the volume effect. This means that among a l l the recorded and readout signals configurations that satisfy the condi-tion given i n Section 2.2.4 for the non overlapping of P x Q crosscorre-lations, some are least sensitive to the volume nature of the holographic medium. Two such configurations well suited for line-by-line character recognition are found. 83 5.2.2 Diffraction from a Three-Dimensional Hologram Without loss of generality, the following analysis considers that the recorded and readout beams are plane waves. Indeed, any f i e l d can be Fourier analyzed into a sum of plane waves. Then an actual hologram is the superposition of many plane waves holograms and the diffracted f i e l d is the superposition of the f i e l d diffracted by each elementary hologram upon illumination by each of the plane wave components of the readout f i e l d . Let k and k_ be the unit vectors in the direction of propagation S L of two plane waves inside a medium of refractive index n e, and l e t r be the vector position of a point of coordinate (u,v,z) i n the medium r = uu + w + zz (5.1) where u,v and z are respectively the unit vectors along the u,v and z axes. Within the medium, the waves generate the fields o(u,v,z) = exp(jkn e k s.r) (5.2) and r(u,v,z) = exp(jkn e k r . r ) . (5.3) The symbol . i s the dot product operator, and k = 2u/X where X i s the i l l u m i -nating wavelength in air. The amplitude transmittance of the volume holo-gram that contributes to diffraction into the plus first-order is x + 1(u,v,z> = 0*(u,v,z)r(u,v,z) = exp(jkn e(k r - k s ) . r ) . (5.4) A readout plane wave c(u,v,z) = exp(jkn ek t.r) (5.5) whose direction of propagation within the medium is p a r a l l e l to the unit vector k t, is modulated by T + ^ ( U , V , Z ) and generates a first-order diffracted wave. The latter i s computed using the scalar Kirchhoff diffraction integral over the volume of the medium and, at a point corresponding to the vector position r Q , is given by 84 °+l ( ?o ) = ^volume T + 1 C u,v,z)c(u,v,z)exp(jkn e|r Q - r|) du (5.6) where irrelevant factors have been dropped, dco is a volume element and | | stands for modulus. Eq. (5.6) assumes the diffraction efficiency i s weak so that the same f i e l d i s incident on any section of the medium. "Leith and his co-workers [65] assume that the only significant contribution to the integral in Eq. (5.6) arises from plane waves propagating i n a direction k D and adding up vectorially. Then, by the substitution of Eq. (5.4) and (5.5) into (5.6), becomes 0,. (IO = / exp(jkn e(k r - k s + k t - k ).r)du (5.7) "*"-L ° volume after a constant phase factor exp(jkn £ k 0 . r 0 ) has been disregarded. Each vector k^ can be decomposed into k\ = Xj.ii + Y ±v + Z ±z i = o,s,t,r (5.8) where the direction cosines are (see Fig. 5.1) ^ = sin 6 x i cos 6 y i (5 . 9 .a) Y± = sin 6 y i (5.9.b) (5 .9.c) Z. = COS 6^ COS 0y£. V Fig. 5.1. Direction cosines of the propagation vector k.. Now Eq. (5.7) can be written - r+Pu r + P h 0 + 1(k o) = / h exp(jkn e(X r - X g + X t - Xo)u)duJ exp(jkn e(Y r T 2 - Y g + Y t - Y Q)v)dv / T exp(jkn e(Z r - Z s + Z t - Z D)z)dz. 2 (5.10) 2p^ i s the transverse width of the hologram aperture along each of the axes u and v, and T is the thickness of the medium. Normalizing the solution of the integrals of Eq. (5.10) to a maximum value of unity yields °+l( Eo ) = sinc(kn e p h(X r - X g + Xfc - X 0))sine(kn e p h(Y r - Y s + Y t - Y D)) sinc(kn e|(Z r - Z s + Z t - Z Q)). (5.11) From the above equations, i t i s clear that maximum diffraction occurs in a direction k such that o X Q = X r - Xg + X t (5.12.a) Y = Y - Y + Y. (5.12.b) o r s t Z D = Z r - Z s + Z t. (5.12.c) This i s Bragg's condition where Eq. (5.12.a) and (5.12.b) are the conditions for diffraction from a plane grating while Eq. (5.12.c) combined with the two previous equations says that the maximum occurs in a direction such that the readout f i e l d seems to be reflected by the grating planes. 5.2.3 Volume Effect of the Recording Medium To examine how the thickness of the medium affects the diffraction intensity, the plane grating conditions [eq. (5.12.a) and (5.12.b)] are assumed satisfied and the intensity of 0 +^(k Q) is computed as a function of of the angular mismatch between k t and k g. If the readout direction k t differs slightly from k g, that i s i f kfc = k s + Ak t, (5.13) 86 then the output direction k Q departs slightly from k r and i s given by k 0 = k r + A k 0 > ( 5 1 A ) In Eq. (5.13) and (5.14), the moduli of Ak arid Ak are much smaller than t o unity. The vectors Ak^ and Ak Q can be decomposed as follows Ak ± = AX±u + AY±v + AZ ±z . i = o,t (5.15) Then Eq. (5.9) implies that AXi = - Y ±sin 6„,.A6„,. + Z^A9.„. (5.16.a) x i y l ' ~ i - " x i AY± = cos e y iA6 y l (5.16.b) AZ. = -Y.cos e x.A6 y i - X.A6 x l. (5.16.c) When Eq. (5.13) to (5.15) are substituted back into Eq. (5.12.a) and (5.12.b), the plane grating conditions are satisfied by AXt = AX0 (5.17.a) AYt = AY0. (5.17.b) Then the output intensity, seen as a function of k t takes the form |0 + 1(k t) | 2 = sinc 2(M L) (5.18.a) kn„T with ^ = -f- (AZfc - AZ Q). (5.18.b) Since l A k t l a n d l A k o l a r e s m a l l » t n e approximations 6 x t - 6xs5 V S V (5.19.a) e _ e ; e - e (5.i9.b) xo xr yo yr can be substituted into Eq. (5.17) to yield cos 0 A0 yo cos 0 y r yt 2s- A0 (5.20.a) 8 7 A 6xo - ( Y r cos 6 y r " Y s s i n exs> Z r + z7 AG x t. ( 5 . 2 0 . b ) With Eq. (5.16.c), (5.19) and (5.20), (AZfc - AZQ) is equal to X r X cos 6 v r cos 9 V Q  A Z t " A Z o = - ^ ) A6 x t + [Y r c X s r 6 y r Y S (1 + x r X. z tan 6 x r) - Y s cos 8^(1 + ^  tan B^JJAB t (5.21) With the aid of Eq. (5.9) and of the following identities 1 + tan a tan b = (tan a - tan b)/tan(a - b) (5.22.a) 1 + tan 2a = cos" 2a, (5.22.b) Eq. (5.21) reduces to AZ t - AZ 0 = cos 6 X S cos 0y S[A8 x t(tan 8 x r - tan 6 X S) + A0 y t tan 6 y r tan 6 x r - tan 9 X S < cos exs cos exr " t a n eys TanTexr - 9 x s) ) ] -(5.23) If the angles are either small or close to 180°, (AZfc - AZQ) can be approxi-mated by AZ t - AZ Q * A9 x t(tan 6 x r - tan 9 X S) + A9 y t(tan 9 y r - tan 9 y s ) . (5.24) Let 0 X£ and 6y^ be redefined as the angles of the propagation direction i n the a i r . Then, with the aid of Snell's law, Eq. (5.18) eventually becomes l ° + 1 ( k t ) | 2 = sinc 2(M L) TTT with ML = ^ - ( A 8 x t ( t a n 9 x r - tan 9 x g) + A0 y t(tan 9 y r - tan 0 y s ) ) . (5.25) 88 5.2.4 Layout of Recorded and Readout Signals Eq. (5.25) indicates that the volume effect of the medium is a function of four mismatches, namely the horizontal and vert i c a l angular mis-matches between k t and k g on the one hand, and between k r and k g on the other. Consequently i t depends on the distribution of the recorded and read-out signals in the plane P g of the matched f i l t e r . For a fixed number of signals, decreasing some of these mismatches necessarily increases the others and a compromise must be found. For this purpose, the maximum of ' m a X \ = l n 7 ( l A e x t l l e x r " 6 x s l + NyJ'V " 6 y s I > < 5- 2 6> is minimized and the maximum diffraction intensity obtainable i n the worst case i s computed | 0 + 1 ( k t ) | 2 = sinc 2(minimax 1^). (5.27) Various signals layouts that satisfy the condition given i n Section 2.2.4 for the non^-overlapping of P x Q crosscorrelations are compared i n order to determine which are least sensitive to the volume effect of the hologram. Let the P recorded signals and the Q readout signals be regularly distributed over two rectangular arrays, respectively P x and Q x signals wide, and Py and Qy signals high, so that P = P xP y (5.28.a) and Q = QxQy (5.28.b) With each side L g of a square elementary signal subtending an angle ^ at lens Li of the MF-1 4> - L s / F l f (5.29) the readout signals f i l l a' rectangle whose sides subtend the angles H t x and Hty given by H t x = Qx<|> (5.30.a) H t y = Qy<r. (5.30.b) 89 As each recorded signal i s allotted on area at least larger than the rec-tangle of readout signals, the sides of the rectangle of recorded signals subtend respectively the angles H g x and H g y (see Fig. 2.3) H s x - ((P x - 1)Q X + 1H, (5.31.a) H g y = ((P y - l ) Q y + l)cb. (5.31.b) ; The maximum of |A8 x t| occurs when the rightmost readout signal is cross-correlated with the leftmost recorded signal or vice versa, and a similar argument applies to the maximum of |A0 y t| (see Fig. 5.2). It i s obvious that these maxima are minimized when the centres of the two rectangles coincide. Then they are minimax |A6xt| = ( H s x + H t x)/2, (5.32.a) minimax |A6 y t| = ( H s y + H t y)/2. As for the maximum of | 6 'sy '  Aty>l^' (5.32.b) , i t i s minimized when the rectangle of recorded signals i s vertically centered about y r , in which case i t comes minimax |0 y r - 8 y s| = Hsy/2 ys i (5.33) recorded signals max\9x^Q> xs\ sx y — • * < t axrPyr % P l ft. max\A9x{\ readout signals • e. y *> 9„ Fig. 5.2. Angular coordinates of the recorded and readout signals. 90 Dividing both sides of Eq. (2.20) by Fj gives the angular condition for the separation of the crosscorrelation from the zeroth order term 6 x r * 1 ' 5 H s x + H t x ' <5-34> When used with the equal sign, Eq. (5.34) minimizes the maximum of ] 8 x r - 9 X S| and yields minimax |0xr - 6 X S| = 6 x r + Hgx/2 = 2H S X + H t x. (5.35) Finally the substitution of Eq. (5.30) to (5.35) into (5.26) gives minimax ^  = [(P XQ X + 1)(2(P XQ X + 1) - Qx) + (P yQ y + 1) ((P yQ y + 1) - Qy)/2]. (5.36) With P and Q fixed, P x and P y on the one hand, and Q x and Qy on the other are not independent, and any reduction of some of the angular mismatches i n -creases the others. A numerical evaluation of Eq. (5.27) points to possible trade-offs. 5.2.5 Numerical Example Let the number of signals be fixed at P = Q = 32. x;'or constant values of P XQ X, Fig. 5.3 represents the intensity | 0 + 1 ( k t ) | 2 [see Eq. (5.27)] versus Q x when is given by Eq. (5.36). The solid and dashed lines corres-pond respectively to the 6-and 15-ym emulsion thickness of the Kodak 120-02 or Agfa-Gevaert 10E70 plates and of the Kodak 649F plate. n g is set equal to the refractive index of the latter emulsion (i.e. n g = 1.52) and X is taken equal to 632.8 um. Finally, with F. = 500 mm and L c = 2 mm, Eq. (5.28) results in $ = 0.004. Even for T = 6 ym, the intensity diffracted in the direction corres-ponding to the minimax is substantially lower than 1.0 when P XQ X = 8 or 64. This i s due respectively to excessive v e r t i c a l and horizontal angular 91 I O^jfminimax MJ 0.5 J 1 1 1 1 1 1 B » 1 2 4 8 16 32 Qx Fig. 5.3. Sensitivity of various signals layouts to the volume effect of the recording medium. mismatches. The signals layouts that correspond to the maxima of the curves at P XQ X = 16 (i.e. when Q x = 1) and at P XQ X = 32 (i.e. when Q x = 32) are the most interesting ones as they both reduce the diffraction inten-sity only b^ 2.2% when T = 6 ym and 13% when T = 15 ym. The i l l u s t r a t i o n of these layouts in Fig. 5.4 motivates their appellations of "+" and " I " layouts. With the help of Eq. (5.28) and since P = Q = 32, they are char-acterized as follows layout: P XQ X = 32 and Q x = 32; Q y = 1, P x = l , . P y = 32, " I " layout: P XQ X = 16 and Q x = 1; Q y = 32, P x = 16, P y = 2. With an appropriate orientation of the data, these two layouts are well 9 2 H4-H-t I' 1 readout signals I I I I I I I I II I T T T T T T T T T T I I I I I I I l I 111 "TO c to tJ O o QJ 32 33.5 L reference —. _© point source (a) I I I I I I I Irfal I I I I I I I recorded signals i i i i i i i IHHI II I I I I I c 3 O tJ Q3 t '75 = f32 . = / \ 25 L 32 reference o point source px=16 P f 2 Qx»7 Qf32 (b) Fig. 5.4. Recorded and readout signals that, for a given SBP, are least sensitive to the volume effect, (a) "+" layout, (b) " I " layout. 93 suited for character recognition on a line by line basis. With PxQx = 32, values of Q„ different from 32 result i n poorer performance because of A. the increased | 0 x r - ^ x s l * This is also compounded by a concurrent re-duction of I 0 - 0 I which, when P„ = 1, is as low as 0.5d>. This effect i yr ys 1 ' y ' i s more pronounced as T increases and, at T = 15 ym, the diffracted i n -tensity can be as much as 13% lower than in the "+" or "I" configurations. 5.2.6 Comments on the Volume Effect The volume effect of the recording medium has been calculated in three dimensions for a fixed number of recorded and readout signals. The following condition for i t s minimization has been derived: the set of read-out and recorded signals must have a common center and the latter must be symmetrically distributed with respect to the ve r t i c a l coordinate of the reference point source. A numerical example has illustrated the importance of a careful selection of the signals layout. Although the "+" and " I " configurations are least sensitive to the volume effect, departing from them results only in a slight reduction of performance when T = 6 ym and P xQ x =32. In this case, a layout that makes more efficient use of the illuminating beam power could be conveniently used, for example P x = = 8 and Py = Q x = 4. However for T = 15 ym the latter choice reduces the d i f -fracted intensity by 27% although the "+" and " I " configurations reduce i t only by 13%. It must also be bore in mind that immersion in a liquid gate increases approximately eight times the emulsion.thickness [68 ]. Section 5.3 calculates the standard deviation of the matched f i l t e r s ab-errations and reveals the superiority of the " I " layout over the "+" one. Leith et a l . [ 65] demonstrated that transmission and reflection volume holograms exhibit the same angular sensitivity [see Eq. (5.24)] and that the high spectral selectivity of the la t t e r , combined with the shrinkage 94 of a processed emulsion, results in a colour shift towards shorter wave-lengths. Righini et a l . [69] experimented with a reswelling process made necessary by their use of the same wavelength to record and readout their reflection matched f i l t e r . This is a reflection type of MF-1 whereby a unique lens Fourier transforms the readout signal and inverse Fourier trans-forms the f i e l d retro-diffracted. The distance between the hologram and lens I12 of such a f i l t e r cannot be less than the focal length of L^. Vig-netting by lenses and I^, or actually by a same component, would be eliminated with a reflection type of LLMF. For the same overall system length, the registration accuracy of the hologram of a reflection MF or LLMF is half that of their transmission counterparts. Also relevant to the space variance of a matched f i l t e r i s the low angular sensitivity of reflection phase holograms achievable with high diffraction efficiency as shown by Kogelnik [70]. 5.3 Holographic Aberrations  5.3.1 Introduction Meier [71] has shown that when the readout f i e l d i s neither the duplicate nor the conjugate of one of the recorded f i e l d s , none of the plus and minus first-order images are reconstructed free of aberrations. There-fore, in parallel processing of characters using a MF or LLMF, aberrations w i l l affect a l l the crosscorrelations but the one associated with the read-out character that coincides with a recorded signal. Meier's paper, though not the f i r s t one to study aberrations (see [72]) is often considered a landmark of this topic. His analysis is followed to compare the aberrations of MF-i and LLMF-i ( i = 1,2). When the aberrations are small, their effect on the correlation intensity can be deduced from the standard deviation of aberrations (SDA). Also SDA 95 is a common measure of wavefront quality. It i s calculated as a function of the readout signal position. The space variance of the root mean square (rms) distortion i s determined for each of the "+" and " I " layouts, and the rms astigmatisms of different f i l t e r s are compared. Since volume effect considerations (see Section 5.2.4) requires a small reference to object beam angle and since accuracy of the Fourier trans-formation requires paraxial object and readout beams [26,p.80], the non-paraxial case [ 7 3 , 7 4 ] is not considered here. It is neither a practice to use a readout wavelength different from the recording one, nor to alter the hologram trans-verse dimensions as doing so would create more problems than i t could solve. Therefore Meier's parameters m (the hologram scaling) and u (the wavelength ratio) are taken as unity and chromatic aberrations do not exist [ 7 5 ] . 5.3.2 Third-Order Aberrations of Points Holograms , Holographic aberrations are caused by the mismatching of the reference, the object and the readout wavefronts. Thi3 can be seen by considering the case of three spherical waves and by expanding to the second order the distances between the three point sources and the hologram. The aberration of the reconstructed wavefront i s defined as the difference between the third-order expansion of the phase of the wavefront actually diffracted by the hologram and that of an ideal spherical wavefront issued from a point source whose coordinates are predicted by the f i r s t order approximation. Fig. 5.5. Geometry for the computation of the phase of the spherical wave issued from P. 96 In plane P^, the phase of the wavefront issued from a point P of coordinates (x^y^^^) (see Fig. 5.5) is given by <f>±(u,v) = |^ (PQ - PO) = f % ( u -x ±) 2+ (v - y.) 2+ D|)* - (x£ + y 2 + D2)*]. (5.37) As i n Section 2.3.1, is considered positive i f the f i e l d diverges to-wards P^ and negative otherwise. It is also assumed that the phase at the origin of P^ i s zero and that the amplitude distribution i s uniform over P^. With a second-order expansion of the square root terms (1 ± x ) * = 1 ± |x - |x 2 ± (5.38) <{>^ (u,v) can be approximated by ^(u,v) a ^ ( u . v ) + <j>13(u,v) (5.39) where ^ ( u . v ) = -^-( (u-x ±) 2+(v-y ±) 2 - ( x 2 + y 2)) (5.39.a) i = - T ^ r((u-x.) 2+(v-y_.) 2) 2- (x? + y_ 2) 2]. (5.39.b) and *13<ufv) = - ^ [ ( ( u - ^ ' + C v - y i ) 2 ) 2 " <x| + ^ - 1 In the above equations, the subscripts 1 and 3 refer to the power of D^" in the series expansion. By the cartesian to polar coordinates transform-ation . p 2 = u 2 + v 2 and tan 6 = v/u , (5.40) Eq. (5.39.a and b) becomes $±1(p>&) = -jjp (P 2 ~ 2p(x1cos8 + y ±sine)) (5.41.a) i 2 1 and <f>13(p,9) = - ~p [p 4 - ^ ( X j C O s e + y^inG) i + -2^p2(xi2cose + y^ 2sin6 + 2x^y^ sin9 cos0) + | p 2 ( x 2 + y 2) - |p(x 2 + y 2 ) ( x i C o s e + y^ind) ]. (5.41.b) In matched f i l t e r i n g , the diffracted term of interest i s r(u,v)o*(u,v)c(u,v) 97 [see Section 2.3.2]. Its phase is given by $(u,v) = <J>r(u,v) - cj>s(u,v) + <j,t(u,v) (5.42) in the case of three point sources at coordinates (x^,y ^ . , D ^ ) with i res-pectively equal to r, s and t. After each component of Eq. (5.42) is expanded by means of Eq. (5.39) and (5.41), $(u,v) is approximated by *(p,6) = $ i(p , e ) + * 3 ( p , e ) (5.43) where ^ ( p . e ) = (p 2 - 2p (x0cos9 + y Qsin9)), (5.43.a) o * 3(p»6) = * r 3 ( p , 6 ) - • 8 3(p-»9) + * t 3 ( p . e > . (5.43.b) D = D (a + 6 - 1 ) _ 1 , o s or, with the aid of Eq. (2-64), D = D 3 _ 1, (5.43.c) o s ' and x = 3 - 1(x - cot - 6x ) ; y = 3 - 1 ( y - ay^ - 6y ) . (5.43.d) o s t r o s • t r ^^P'^) represents the first-order or quadratic approximation of the phase of a spherical wave centered about the Gaussian image point ( x 0 , y 0 , D 0 ) . Th0. sign convention for D Q is chosen identical to that used in Section 2.3.3.b, that is D q is positive when the wavefront $(p,9) converges to-wards and is negative otherwise. The spherical wave is called a ref-erence sphere and its phase <f>Q(p,9) in Pjj can also be expanded to the third-order by substituting "o'! for " i " in Eq. (5.41). Meier defines the aberration W(p,9) as the difference between the third-order expansions of the phase of the reference sphere and of the diffracted wavefront W(P,9) = <f>o(p,9) - *(p,e) = <f>ol(p,9) + 4>o3(P,9) - ^ ( p . e ) - * 3 ( p , e ) . (5.44) From the definition of (x Q, y Q, D Q ) [see Eq. (5.43 .C and d)], i t comes *6l(P»6) = *l(p,6) (5.45) so that W(p,6) reduces to w(p,e) = • O 3 ( P . Q ) - * 3 ( p . e ) - ( 5 - 4 6 > After the substitution of Eq. (5.43) and (5.41.b) into (5.46), W(p,0) can be expressed in terms of the five Seidel aberrations W(p,0) = 2TT(^ ) t ^ C ^ - ) 3 SA (spherical aberration) - 4-(-Pr-)2(C cos 6 + Cv sin 0 ) (coma) 2. x ' - •|(^-)(A x xcos 2 6 + Ay y s i n 2 0 + 2Axy sin 6 cos 6) (astigmatism) + -|-(^-)FC ( f i e l d curvature) - i(D cos 0 + D y sin 0)] (distortion)(5.47) where SA = (cx3+ 8 3 + 6 3 - 1) = -3(a + 5)(1 - 6) (1 - a), (5.47.a) c £ = 0 £ t ( a 2 - B 2 ) - e A s a - e 2 ) + e £ r(6 2 - e 2 ) ; a = x , y ) , (5.47.b) A . „ = 6 i ^ n ^ a - 0 i 9 0 + e. e„ 6 + B ( e , - e , + 0 . ) ( e 0 < . Tc£ kt It ks £s kr £s kt ks kr It ~ 9£s + 6 J l r ) ; ( k , J l = X ' y ) > (5.47.c) FC = A ^ + A^, (5.47.d) D * = 8 £ t < 8 £ t + 9kt> " 6 £ s ( 9 L + 6ks> + 6 £ r ( e L + 6£r> - < 9 U " 6£s + V ( ( 9 k t ~ 6ks + V 2 + < 8£t - 9£s + V 2 ) (k,£ = x,y but k ^  £). (5.47.e) 0 ^ is defined as the angular coordinate of the source " i " 99 A " i 8 ^ = — with I = x or y and i = r,s,t, (5.47.f) i and i s identical to the angle 0 ^  (£ = x,y) of Fig. 5.1 when i t i s small enough for the tangent to be approximated by i t s argument. 5.3.3 Comments on the Five Seidel Aberrations Spherical aberration (SA) depends only upon the distances along the optical axis and is space-invariant. Coma i s independent of the readout signal coordinates (8 . 8 ) when a = ±6. This condition which b xt yt makes coma space-invariant amounts to D t = ± D q or M^  = ±1 [see Eq. (2.65)]. The + sign corresponds to the LLMF with a = 0.5(1 - <5) and the f i l t e r plane midway between the input and output planes while the - sign corresponds to the MF since the conjugate of the output plane with respect to lens L 2 coincides with the input plane. Distortion i s common to a l l MF-i and LLMF-i ( i = 1,2) with a same reference to object beam angle because i t is only a function of the angular coordinates of the point sources. A l l aberrations are nu l l when the readout source i s either identi-cal to the object one, that i s 6.1 = 8 (£ = x,y) and a = 1, or the conju-gate of the reference source, that i s 8 = - 8 (i = x,y) and a = -6. Although the f i r s t situation can occur in MF-1, MF-2 and LLMF-1, the second can only be realized in LLMF-1 (6 < 0) since a i s positive. Therefore i t i s anticipated that, for some positions of the readout source, the LLMF-1 has less aberration than the other f i l t e r s . Also the distortion term, being independent of a, is zero in the two above situations. It has been shown [46] that f i e l d curvature of the MF-2 is zero when the centre of the ideal reference sphere belongs to a plane normal to the axis of the reference beam. A review of each f i l t e r type i s in order. For the MF-1, D^ = 0 0 ( i = r,s,t) and a l l the wavefronts are plane waves so that aberration con-sists only of the distortion term. The latter i s common to a l l four types of f i l t e r s . As <S = 1 for the MF-2, SA is n u l l and so i s coma when a = 1. 100 With 6 = -I and a = 1, SA and coma of the LLMF-1 are n u l l too. Finally, for the LLMF-2, none of the aberration terms can be made zero, except dis-tortion when the readout source coincides with either the recorded source or the conjugate of the reference one. The aberrations considered i n this section are those due to the holographic f i l t e r i t s e l f . Although the MF as well as the LLMF-1 can be free of SA and coma, the correlation output of the LLMF-1 i s inherently free of such aberrations while both MF's require the extra lens L 2 to be corrected for SA and coma i f i t is not to degrade the f i e l d diffracted by the hologram. Aberrations due to the glass plate supporting the photographic emulsion or to lens L 2 are not considered here. Furthermore i t i s assumed that the reference beam shaping lens of the LLMF-1 is well corrected. As the latter i s needed at recording time only, i t is possible that for those applications where a new f i l t e r i s seldom recorded, as in character rec-ognition, the economics of the problem favors the LLMF-1. 5.3.4 Standard Deviation of Aberration (SDA) In order to compare the aberrations of the MF-i and LLMF-i ( i = 1,2), o w(R^), the standard deviation of aberration (SDA) over the circular aperture of the hologram defined by p - R^ , i s computed a w(R h) = (E[W2(p,9)] - E 2[W(p,e)])* (5.48) where E[.] indicates the spatial average of the argument. The rms value of aberration, or of the optical path difference (OPD) between the actual wavefront and the Gaussian reference sphere, i s indeed used to characterize the quality of an optical system [76]. Preston, in his computer-aided design of a Fourier-transforming lens, relates SBP and rms OPD [77, p.46]. Also when the aberration is small, the intensity at the centre of the reference sphere, normalized to the aberration-free case, i s given by 101 [78, p. 463] I(x . y , D ) = 1 - o 2 . (5 . 4 9 ) Equality (5 . 4 9 ) is valid when the third and higher moments of W(p,9) are negligible compared to i t s f i r s t and second ones. Born and Wolf c a l l the expression (a /k2) the 1 'mean square deformation". Let 6 be defined as half the angle subtended i n P by the holo-gram aperture eh ~ W ( 5 - 5 0 ) Then with the aid of Eq. (5 . 4 7 ) and of the following identities 2TT 2TT 2TT / cos 26 d9 = / sin 20 d6 9 / sin 2 2 e d9 E u, (5.51.a) o o o 2ir 2ir J cos^e d6 = / sin^ e de E 3 T T / 4 , (5.51.b) o o the average aberration and i t s second moment are found to be equal to .. 2ir R h E[w(p ,e)] = ^ -z- J / w( p , e ) p d P d e n o o K e e 2 q A = T <T^ ) <—)((—) 7 T  + A + A + F C > < 5 ' 5 2 > 4 X a a 3 xx yy and 2TT E [ w 2 ( P , e ) ] = / / w 2 ( p , e ) p d P d e * \ O o ,TT \ * V 6 SA2 • < A J U a } 80 + ¥lT> ( C x + C y + f (Axx + Ayy + F C ) ) + l £ > ( f ( A - + Ayy)+ X x + Ayy + F C 2 ) + f ( A ^ .+ FC(A x x + A y y ) ) + C xD x + C yD y ) + 0.25(Dx + D 2) ] (5.53) 102 A Fortran program was written to simulate Eq. (5.48), (5.52) and (5.53) on an IBM 360/70 computer. The simulation results pertain to three categories: the f i r s t one relates SDA to the signal bandwidth, the second one establishes the space variance of SDA, that i s i t s variation with the angular coordin-ate 0 (£ = x,y) of the readout signal, while space variance of rms astig-matism is considered in the last category. 5.3.5 Aberration and SDA versus Signals Bandwidth It i s clear from Eq. (5.47) and (5.50) to (5.53) that W(p,6) and Ow(Rji) increase with the hologram aperture (p or R^) and with the signals bandwidth. This is illustrated i n Fig. 5.6 which, for different values of R^ > represents the SDA cr^R^), expressed i n wavelength, versus 8 . Fig. 5.6.a corresponds to R^ /A = 160 or, with X = 632.8 nm, to R^  = 0.1 mm, while in Fig. 5.6.b, R^  i s equal to 10 mm. ° w^h) seems to increase l i n -early with R^ , meaning that distortion i s the major component of SDA since in Eq. (5.53), only (D2 + D2) is not multiplied by 8 h or Rh/Dfc. It was noted earlier that for anyone of the four f i l t e r s , distortion i s zero whenever the readout signal coincides with the recorded one, i.e. 9 = 9 for £ = x,y, or i s the conjugate of the reference source, i.e. 0 = -9 for H = x,y. Such behavior is indeed observed in Fig. 5.6.a where o" (R^) = 0 at 8 = 0.0 and -0.2 and where the four f i l t e r s ' plots overlap. It i s not u n t i l R^ i s equal to 10 mm that local differences between the four f i l t e r s appear as then the other components of SDA which vary with the second, fourth and sixth power of p or R^  become more important. Because of the two zero's of a , SDA raises much faster when 0 ^ departs from 9 w' xt xs towards the reference signal than i n the opposite direction. The high-pass MF (HPMF) that many authors [79, 80] use to i n -crease the selectivity of the holographic correlator, has a larger effective - 0 . 2 -0 .15 o.oQ^-j-o (a) - 0 . 2 -0 .15 - 0 . i -o.os o.oQ^jO. r 0.1 0.1S -I 0.2 (b) Fig. 5.6. Dependence of the SDA (in X) on the radius of the hologram aperture, (a) = 0.1 mm. (b) = 10 mm. o LO 104 aperture than the low-pass MF (LPMF) since the energy density of the signals i s larger at low than at high frequencies. Consequently i t suffers more from aberrations. Means to increase the selectivity of the LPMF w i l l be considered in Chapter 6. 5.3.6 Space Variance of SDA (or rms Distortion) With SDA representing essentially the rms distortion, i t s v a r i -ation with 8 (l = x,y) is now computed for the "+" and " I " signals lay-outs. Primary distortion results in a transverse displacement of the d i f -fraction focus from the Gaussian image point with no reduction of i t s i n -tensity, and i n this thesis notation, the maximum displacement [78, p.475] i s equal to 8"1 max OPD distortion For example, with = 0.1 mm and Dfc = 500 mm, 8^  = 2 * IO - 4 and a maximum OPD of IX (=632.8 nm) yields a 3.16 mm transverse displacement of the diffraction focus. This i s a rather important displacement which must be accounted for in the output detector design. Winzer and Kachel reached a similar conclusion in their comparison of the transverse coordinates of the readout signal and the correlation in the MF-1 case [81 ]. The com-parison between the signals layouts is based on the rms OPD a/2ir since i t is easier to compute than the maximum OPD. In f i r s t approximation, these two quantities are supposed related by a constant factor. Without loss, of generality, the rest of this chapter considers the signals as point sources grouped in rows or columns of zero width. With <(> = 4 x 10 - 3and Q x = 32, Eq. (5.30.a) yields H = 0.128. tx This figure i s rounded up to 0.2 and the angles subtended by the lines of 105 point sources are increased proportionally. Then the "+" and " I " config-urations are respectively characterized by H = 0.0; H = 0.2 and 9 = 0.0; le I < 0.1 sx sy xs ' ys 1 H = 0.2; H = 0.0 and le J < 0.1; 9 = 0.0 (5.54) tx ty 1 xt 1 yt 9 = | H + H = 0.2; 9 = 0.0, xr 2 sx tx yr ' and H = 0.1; H = 0.2 or le I < 0.05; 9 = ±0.1 sx sy ' xs ys \ x = H t y = °' 2 ° r 6 x t = °-0; ' V ' -< 0 , 1 < 5 ' 5 5 ) 9 = 4 H + H = 0.15; 9 = 0.0 . \ xr 2 sx tx yr The space variance of the SDA of the "+" layout, expressed in 100-th of X, is plotted i n Fig. 5.7.a and b with 9 x g =0.0 and respectively 9 =0.0 and 0.1 as indicated by the *. The cases 9 = -0.1 and +0.1 ys ys are symmetrical with respect to the 9 axis. The interval between the lines of constant SDA i s 0.02X when < 0.4X and 0.1X above this thres-2TT hold. With 9 x t and 6 varying from -0.1 to +0.1, i t follows from sym-metry considerations that 6 _ = 0 minimizes the maximum SDA sc that yt minimax (•?—). = 0.91X Z T T + (5.56) when (6 „,6 J = (0.1, 0.0) and (9 ,9 ) = (0.0, ±0.1). xt yt xs ys For | e x t | < 0.05 and |e | < 0.1,.thr SDA of the " I " layout is plotted in Fig. 5.8.a,b and c with 9 y g = 0.1 and respectively 9 x g = -0.05, 0.0, +0.05. Minimizatioa of the volume effect requires that the sets of readout and recorded signals be centered about the same point, or that 9 = 0.0. Then the maximum of (—) occurs at (9 .9 • ) = (0.0, +0.1) xt 2TT xt yt when (9 ,9 ) = (-0.05, ±0.1). It i s equal to xs ys max(f-) = 0.73X 2T: R (5.57) Fig. 5.7. "+" signals layout: Space variance of the SDA (in X/100) when 0 =0.0 and (a) 0 = 0.0 or (b) 0 = 0.1. xs ys ys e X R = 0.15 SDfl : I C ? *<3fr/2TT Rtf\ = 160 -o.os -0.025 -0.0 e X T 0.025 0.05 (a) 8 X R = 0 . 1 5 SDfl : \&*au/2Tt Rtf\ = 160 9 X R = 0 . 1 5 SDfl : l O 2 * ^ / 2 * Rrf\ = 160 -0.05 0.025 -0.0 E X T 0.02S Fig. 5.8. " I " signals layout: Space variance of the SDA (in A/100) when 9„ g = 0.1 and (a) 9 = -0.05, (b) 0 ^ = 0.0 or. (c) 9^ = 0.05. 0.05 o 108 and i s 20% smaller than in the "+" layout. However, further examination of Fig. 5.8.a reveals that the choice of 0 = -0.05 minimizes the maxi-mum of (a/2tr) so that minimax (|--) = 0.47A- (5.58). This i s only 51% of the SDA of the "+" configuration [Eq. (5.56)]. A l -though this choice of 0 ^  results in half the rms distortion of the "+" xt layout, i t i s not optimum from the volume effect point of view. Indeed, Eq. (5.32.a) must now be replaced by max |A0 xJ = H s x, (5.59) and i n the worst case, the intensity of the diffracted wave i s equal to 96.3 or 78.9% of i t s nominal value depending whether T = 6 or 15 ym. This represents an 8% reduction from the minimum when T = 15 ym but only a 1% reduction when T = 6 ym. Clearly the best compromise between the reduc-tion of the volume effect and of the aberrations must be determined with regard to the particular problem at hand. 5.3.7 Aberration and SDA at Constant SBP How the aberrations depend upon the size of the signals i s ex-amined hereafter. Indeed a scale reduction by a factor Mg broadens pro-portionally the spectra and the hologram aperture and, as explained above, increases the aberrations. On the other hand, for a fixed amount of data, that i s at constant SBP, the angles subtended by the signals and the mini-mum reference to object beam angle are reduced by M and so are the angular mismatches. This i s turn reduces the aberrations. The overall effect of a Mg scale reduction is obtained by re-placing in Eq. (5.47), (5.50), (5.52) and (5.53) p, P^ and 0 £ f c respec-tively by pMg, HnMg and Q^M"1 U = x,y; k = r , s , t ) . The following con-clusions can be drawn 109 1) . SA and coma increase respectively by a factor M1* and M2 , 2) distortion decreases by a factor M2 and 3) astigmatism and f i e l d curvature are unaffected by the scale change. As SA and coma are zero for the MF-2 and LLMF-1 with a = 1, the net effecjt of the signal demagnification i s a reduction by M| of i t s distortion com-1 ponents. It i s therefore possible that, for some value of M ,astigmatism and f i e l d curvature would be the predominant aberrations. Since astigma-tism reduces the intensity of the diffraction focus, i t i s worth examining in i t s own right. 5.3.8 Space Variance of rms Astigmatism Making FC = D = D = 0 in Eq. (5.52) and (5.53), the SDA due x y to astigmatism (and to SA and coma in the LLMF-2 case) has been computed for (8 ,0 ) = (0.0,0.0). It i s denoted by SDA' and i t s space variance xs ys J r is plotted in 1000-th of A in Fig. 5.9.a, b and c, respectively for MF-2 (a = 1, 6=1), LLMF-1 (a = 1, 6 = -1) and LLMF-2 (ct = 0.33, 6 = 0.33). With = 1.0 mm, the hologram aperture i s ten times larger than in Fig. 5.7 and 5.8. When < 0.04A, the interval between the curves of constant SDA' is 0.002A, while i t i s 0.01A above the same li m i t . As noted in Sec-tion 5.3.3, the astigmatism of the LLMF-1 is smaller than that of the MF-2 over most of the area to the l e f t of the recorded signal, that i s i n the direction opposite to the reference source,but i s larger on the right side of 0 . Over the displayed range of readout signal positions (i.e. |0„ I < 0.1 (l = x,y)),the maximum rms OPD occurs at 0 = ±0 ^  = 0.1. 'A t 1 x t yt The intensity of the diffraction focus> corresponding to that readout signal is given by Eq. (5.49) and is l i s t e d for each f i l t e r i n Table 5.1. It shows that the LLMF-2 suffers from more severe astigmatism and i s more 110 SDA': 103xcrw/2n RH/X = 1 6 0 0 9 H = 2 . 0 x l O - 3 8 X R = 0 . 2 0 SDA': 103xOv/2TT = 1600 9 H = 2.0*10- 3 6 X R = 0.20 i 1 r -0.1 -0.015 -0.05 -0.025 L L H F : a = l .00 ; p = l.00 o.o i n a SDA': MpxcijrtTr RH/X = 1600 \ 0.025 0.05 0.075 0.1 6 H = 2 . 0 x 1 0 ~ 3 8 X R = 0 . 2 0 o-I 1 1 1 -0.1 -0.015 -0.05 -0.025 LLMF:a=0.33:0=0.34 0.0 0.025 O.OS 0.075 0.1 (a) (b) Fig. 5.9. Space variance of the SDA' (in A/1000), (a) MF - 2 : a = 1, 6 (b) LLMF-1: a = 1, 6 = -1. (c) LLMF - 2 : a = 0.33, 6 = 0.33. (c) = 1. I l l space-variant than the MF-2 and LLMF-1 whose performances are practically identical. Moreover, a Mg demagnification of the signals would increase i t s non-zero SA and coma respectively by Mg and M2. Therefore, the LLMF-2 appears less desirable than the LLMF-1. F i l t e r type Maximum (i.e. @ rms OPD of 0~ =±e =0. xt yt SDA' 1) Normalized intensity of diffraction focus MF-2 (a = 1, 6 = 1) - 0.02A 0.984 LLMF--1 (a = 1, 6 = 1) ~ 0.03A 0.964 LLMF--2 (a = - j , 6 -¥ ~ 0.06A 0.858 Table 5.1 Intensity at diffraction focus of different f i l t e r s in presence of astigmatism. 5.3.9 Comments on Aberrations. Distortion i s the same for a l l four f i l t e r s and the maximum rms distortion of the " I " signals layout i s 20% smaller than that of the "+" one. Furthermore a variant of the " I " configuration exhibits only half the maximum rms distortion of the "+" one. This i s interesting i n the case where the diffraction efficiency does not depend strongly on the volume effect. It is relevant since reducing the distortion simplifies the detec-tor design by reduction of the later a l displacement of the correlation from i t s nominal position. At constant SBP, astigmatism does not depend on the signal scale. The rms astigmatism of the LLMF-1 is smaller or larger than that of the MF-2 according to the position of the readout signal relative to the recorded signal and the reference source. In the case ill u s t r a t e d , the maximum rms astigmatism of the MF-2 yields a 1.6% reduction of the correlation intensity while the reduction i s 3.6% with the LLMF-1 and 14.2% with the LLMF-2. 112 Therefore,, the MF-2 and the LLMF-1 can be considered to have similar performance. On the other hand, SA and coma of the LLMF-2 are not zero. At constant SBP, they would increase as Mg and Mg with a Mg signal demag-nification* Also, the LLMF-2 rms astimatism is more space-variant than that of the LLMF-1. The tolerable level of aberration i s eventually determined by the acuity ratio of the correlator defined as the ratio of the autocorrelation to the maximum of the cross-correlations. This ratio is reduced by an increase of the detector size relative to the correlation size but at the same time, the measured intensity is less sensitive to the blur or the aberrations of the correlation. As aberrations increase with the signals bandwidth, the next chapter seeks to improve the acuity ratio by using low-pass f i l t e r s matched to geometrical features rather than high-pass matched f i l t e r s . 5.4 Conclusions Two factors that contribute to space variance have been investi-gated. They are the volume effect of the recording medium and the third-order holographic aberrations. It has been shown in Section 5.2 that for a given SBP, two configurations of the recorded and readout signals mini-mize the maximum reduction of diffraction intensity. They have been called the "+" and " I " layouts and are well suited for character recog-nition on a line-by-line basis. In Section 5.3, the aberrations of MF-i and LLMF-i ( i = 1,2) have been calculated and the conditions that make SA and coma of the LLMF-1 zero have been determined. The dependence of aberrations on signal bandwidth and SBP has been pointed to. Via com-puter-generated plots, the space variance of the rms distortion, which i s common to a l l f i l t e r s , has been found smaller for the " I " layout than for 113 the "+" one, and the space variance of the rms astigmatism larger for the LLMF-2 than for the MF-2 or LLMF-1 whose performances are almost identical. 114 VI FEATURE EXTRACTION 6.1 Introduction The simple template matching operation of the MF is not well suited for discrimination between similar looking characters like 0 and Q. A re-view of different matched f i l t e r i n g techniques intended to improve the recognition performance i s presented. Among them, two that are easily implemented, are compared experimentally. One consists i n matching the f i l t e r to the fine structure of the signal by emphasizing somehow i t s high frequency components. The f i l t e r i s referred to as the high-pass MF (HPMF) (as opposed to the conventional low-pass MF (LPMF)). The other extracts geometrical features. This f i l t e r i s known as the feature extractor and matched f i l t e r (FEMF). With the HPMF, increase i n discrimination i s ob-tained at the expense of severe restrictions on the range of variants that can be properly identified. Experimental evidence shows that the FEMF is more selective than the HPMF and tolerates a wider range of variants than the same. An unambiguous discrimination between 0 and Q is achieved using a f i l t e r matched to the t a i l of Q. Recording this f i l t e r as a low-pass f i l t e r helps maintaining the discrimination even though the readout signals belong to four typefonts that dif f e r in shape and pitch from the recorded one. The FEMF appears conditionally less sensitive than the HPMF to a misorientation of the readout signal. Also i t i s shown that the resolution of the FEMF detector can be coarser than that of the HPMF one. The FEMF discrimination and tolerance to variants i s achieved at the expense of a higher registration accuracy of the readout signal and a reduced diffraction efficiency. 6.2 Matched Filtering Techniques: a Review Tailoring the matched f i l t e r i n g operation to particular needs or 115 applications has been accomplished through spectral or spatial modifications of the basic MF. Spectral modifications consist in juxtaposing an additional f i l t e r to the MF or in synthesizing i t into the holographic f i l t e r . Spatial modifications involve either a postprocessing of the correlation signals or the holographic recording not of the signals to be detected but rather of related patterns. 6.2.1 Spectral Modifications of the MF As soon as the holographic MF came into being, researchers inves-tigated ways of improving i t s selectivity by emphasizing the high frequency components of the spectrum. A character i s recognized on the basis of i t s outline or discontinuities rather than i t s total area. The physical im-plementation of this idea takes the form of a high-pass f i l t e r [ 8], an additional derivative f i l t e r [79] or low frequency stop [82], or a low frequency overexposed MF [80,67.,56]. Overexposure i s brought about by a low ratio of reference to object beam intensity on the axis. A detailed analysis of the effect of overexposure on selectivity i s presented i n [83], An adaptive technique of overexposure, known as transpose processing [84], saturates the f i l t e r only at those frequencies common to the set of simul-taneous unknown signals. A pass-band f i l t e r that cancels the crosscorrelation terms i n some particular cases i s described i n [85]. Besides experimenting with the overexposure technique, Binns and his coworkers improved the discrimination between 0 and Q by using an additional s l i t f i l t e r oriented at 90° to the t a i l of Q [67]. Similarly, decomposition of the spectrum i n annular areas of different diameter and wedges of different orientations leads to the recognition of character features [86]. Additional f i l t e r s have also been designed to improve the f i l t e r sensitivity to a scale change [87] or to account for the random position of the readout signal [88]. 116 6.2.2 Spatial Modifications of the MF Three types of postprocessors have been reported on. The f i r s t one combines linearly a l l the output signals of a standard MF so as to make the crosscorrelation terms zero [89]- It was shown recently that the dis-crimination capability of this technique in presence of random noise i s adequate but that in presence of signal-related noise i t s performance i s rather poor [90]. The second postprocessor type combines log i c a l l y the re-constructed images of the many reference point sources of a binary code translator [36,37]. Finally the last type enhances the discrimination by measuring the time derivative of the crosscorrelation generated by a se-quential multichannel MF [37]. The performance of the MF is governed also by the types of recorded patterns. Thanks to the high information capacity of a hologram, insensi-t i v i t y to character variants can be bu i l t into the MF through the recording of a large, number of variants [36]. Bipolar weighted patterns have been synthesized so that the crosscorrelations do not exceed a certain threshold [31]. Their negatively weighted areas are analogous to the penalty areas of Highleyman's pattern recognition scheme [91]. Other complex patterns, which are called principal components and are designed to yield binary correlation peaks that form a code word, have been suggested [32]. Fin-a l l y , the combined geometrical feature extraction and multichannel corre-lation, that Groh proposed [43], has been demonstrated experimentally by Winzer and Douklias [42]. Feature extraction with a single channel co-herent [92,93] or incoherent [59] correlator has also been reported. Groh suggested that the different features be recorded with appropriately trans-lated reference point sources so as to combine optically the responses of a l l the features of a character. Yao and Lee implemented a f i l t e r for the 117 combined operations of subtraction and correlation [94]. Their method i s not directly suited for character recognition though for i t detects fea-tures that are the difference of two simultaneous readout patterns occu-pying specific positions. 6.2.3 High-Pass MF versus Feature Extraction Of a l l the techniques reviewed above, two are easily implemented: they are the high-pass matched f i l t e r i n g , obtained by overexposure, and the geometrical feature extraction. The next section compares experi-mentally the selectivity and tolerance to variants of the HPMF and FEMF. In the following, the normalized intensity of the correlation function is simply referred to as correlation, unless otherwise stated. The normali-zation procedure is explained in Section 6.3.1. It is also useful to define here the acuity ratio as the ratio of the correlation with the correct recorded signal to the maximum of the correlation with the incor-rect signals. 6.3 Selectivity and Tolerance to Variants: Experimental Results  6.3.1 Selectivity Selectivity experiments were performed to compare the discrimination 0 0 0 0 0 0 Fig. 6.1. Set of characters used for selectivity experiments (after 3X reduction). The arrows point to the recorded signals. The others are the readout signals. Q Q Q Q Q Q Q Q Q Q Q Q 118 between similar looking characters (0 and Q) that can be achieved with the LPMF, HPMF and FEMF. Fig. 6.1 presents the set of eighteen (18) samples of 0 and seventeen (17) samples of Q that were used to that effect. They are of the IBM Prestige E l i t e 72 typefont whose pitch i s twelve (12) char-acters per inch. The fourth symbol on the f i r s t row of Q's represents the t a i l of Q. It was obtained by masking the 0-component of the Q that had . been previously typed at that place. The arrows point to those signals that were recorded using a LLMF-1 with a = 1 and 6 = -1. By varying the ratio K of the reference to object beam intensities on the optical axis, the 0- and Q-filters, namely f i l t e r s matched to 0 and Q, were recorded as LPMF (K=5) or HPMF (K=l/7). Similarly the t a i l - f i l t e r , that i s the f i l t e r matched to the t a i l of Q, was recorded as a LPMF with K = 2.8. The re-corded and readout signals actually used in the experiments were three times smaller than those of Fig. 6.1. This 3X reduction was necessary to lessen the effect of the phase error introduced by the photographic plate on which the signals were recorded (see Section 7.4). According to Eq. (2.16), each correlation peak was normalized with respect to the readout signal energy. For the 0- and Q-filters, they were further normalized by taking the autocorrelation of the recorded signal equal to unity. As for the intensity response of the t a i l f i l t e r , i t was taken as unity when the readout signal was the same Q that had been recorded In the HPMF or LPMF. Fig. 6.2 depicts the average responses of the 0-, Q- and t a i l -f i l t e r to a l l the readout characters of Fig. 6.1 that belong to a same class. The averaging does not encompass the response of a f i l t e r to the particular character used in i t s recording. The red, green and blue bars correspond respectively to the responses of the LPMF, HPMF and FEMF. At most, the standard deviation of measurements was 0.05. Fig. 6.2 shows the HPMF improvement in discrimination between 0 and Q i s a paltry 6 or 7% r/.O 0.8 10.6 -OA [0.2 •O-filter Q- filter H \ tail-filter] 0 Q 0 Q 0 Q 0 readout signal Q Fig. 6.2. Average responses of the 0- and Q-f l i t e r to the readout signals of Fig. 6.1. red = LPMF, green = HPMF, blue = FEMF. over the LPMF performance. On the other hand, the minimum response of the t a i l - f i l t e r to Q was 0.9 while 0.08 was i t s maximum response to 0, an 82%* difference. Although with the HPMF the acuity ratio of the 0 readout signal i s 1.31 compared to 1.21 with the LPMF, the ratio of the average correlations of Q and 0 with the t a i l i s 20. It is clear that the FEMF is significantly more selective than the HPMF when the readout signals are reasonably well printed characters Q and 0 of the same typefont as the recorded signals and similarly oriented and positioned. Before ex-amining the performance of the different f i l t e r s when the readout signals depart from such an ideal situation, a note on the experimental procedure is in order. The responses of the LPMF and HPMF are the peak intensities of the 120 respective correlation. The response of the FEMF is the intensity mea-sured by a fixed photodetector when the readout signal occupies a prede-termined position. The four corner Q's of Fig. 6.1 where translated in the input plane so as to place the correlation peak at the detector and the corresponding coordinates of the translation stages were recorded. The coordinates that bring the other Q's and O's in the same position were computed by extrapolation, and the correlations were measured with the readout signals accordingly registered. This procedure reflects the fact that, with the FEMF, the position as well as the presence of the feature i s relevant. With the LPMF and HPMF, the detection i s based on the inten-sity of the correlation peak which is searched for within a certain area. 6.3.2 Translated Signals It i s easy to understand that the HPMF yields a narrower correla-tion peak than the LPMF since i t i s matched to the character outline. The photographs of Fig. 6.3 i l l u s t r a t e this point. They represent the output of a Q- f i l t e r when the readout signals are 0 Fig. 6.3.a corresponds to the HPMF while the LPMF response i s shown in Fig. 6.3.b and c with m (a) (b) (c) Fig. 6.3. Correlation pattern of the set of readout signals 0 with the recorded signal Q. (a) HPMF (exposure time = 1/30 sec), (b) LPMF (exposure time =1/4 sec), (c) LPMF (exposure time = 1 sec). 121 Fig. 6.4. Crosssections of the autocorrelation peaks of 0 [(a),(b)] and of Q [(c),(d)] at constant intensity (0.8 and 0.5). (a) and (c) LPMF. (b) and (d) HPMF. 122 different exposure times. A quantitative appreciation of this effect i s offered in Fig. 6.4. It represents the l o c i of signal translations about i t s recording position that result in a constant output intensity. Plots a and b correspond to the autocorrelation of 0 as produced respectively by the LPMF and HPMF, while the autocorrelation of Q is similarly depicted by plots c and d. They were obtained by translating the input signal while keeping the detector fixed. These plots are equivalent to crosssections of the correlation peak at constant intensity, that i s at 80% and 50% of the peak intensity. The minimum width of the Q-peak at 50% is 135 or 100 ym depending whether the f i l t e r i s a LPMF or HPMF. This means that the HPMF needs a finer resolution in the output plane than the LPMF. With a scanning detector, this could result in slower processing speed. The curves of Fig. 6.5 show the sensitivity of the FEMF to a s i g -nal translation. They are equivalent to constant intensity crosssections of the correlation peak of Q with the t a i l (plot a) and of the correlation ^S. 10fjm 1 0 \ \ * io 0.05/. y i—i Wfjm '0.075 6 ^ 70.5 I fo.8 <*> \ N?" \ to 3 \ 6 \9 (a) -6 7-3 / y 7 / io-/ s (b) ' '• ; o . j j ^ ^  9 0 . 2 ^ ' " ' ' Fig. 6.5. Crosssections of the crosscorrelation of the t a i l of Q (a) with Q and (b) with 0. 123 "sink" of 0 with the same (plot b) . The Q signal was the same as that of Fig. 6.4 while the 0 that had given the worst or highest (7.5%) crosscor-relation with the t a i l in the experiments of Section 6.3.1 was used here. The displacements are measured from the positions computed by extrapolation. A translation of Q in the direction of the positive diagonal, i.e. at 90°, to the t a i l direction, reduces the performance fastest. 115 ym is the maximum range over which the correlation i s larger than 50%. On the other hand, the correlation of 0 with the t a i l increases fastest in the downwards direction of the negative diagonal, that i s in a direction p a r a l l e l to the t a i l . Here a displacement of 130 ym from the nominal position does not yield more than 30% of the nominal correlation of Q with the t a i l . The displacements given in this paragraph must be considered r e l -ative to the signal size. The printed characters are 1.6 mm wide by 2 mm high and their average stroke width i s in the order of 360 ym. The actual signals were reduced three times giving an average stroke width of 120 ym and a stroke correlation extending over 240 ym. This indicates that the output resolution of the LPMF or FEMF must be i n the order of a character stroke width while i t must be about 25% finer in the HPMF case. However according to Binns and his coworkers [67], the resolution of the HPMF detector must be three times finer than that of the LPMF one. The photo-graphs of Fig. 6.3 corroborate their findings. The difference between the results of Fig. 6.4 and those of Binn et a l . i s due to different experimental conditions. Their f i l t e r s are characterized by K=l and 1/70 which makes their HPMF more selective than reported here. They indeed measured crosscorrelations of 0 and Q equal to 80% or 60% of the auto-correlation of Q respectively for the LPMF and HPMF. Moreover, their photodetector had a 6-ym aperture compared to 50-ym in the above experi-ments. As the signals of Fig. 6.1 were further reduced three times, the 1 2 4 relatively large detector aperture engulfs a sizeable portion of the cen-t r a l area of the crosscorrelation pattern and more than the peak value i s detected. Therefore, the f i l t e r selectivity i s reduced and so is the acuity ratio [95, p. 206], It follows that the measured signal i s r e l -atively insensitive to transverse displacements and to other variants. In the above experiments, the use of a large detector aperture has led to the underestimation of the HPMF selectivity. The following sections show that even in this situation, the HPMF is s t i l l too selective to detect variants unambiguously. The coarse resolution is adequate though for the FEMF selectivity and tolerance to variants. 6.3.3 Rotated Signals The dependence of the autocorrelation of 0 and Q on the angular position of the readout signal i s illustrated in Fig. 6.6 where the LPMF appears more tolerant than the HPMF. A 20° rotation reduces the auto-correlation of 0 to 83% and 78% of i t s value at 0°, respectively for the LPMF and HPMF. The performance reduction-is not too severe because 0 is almost rota-tionally symmetrical. As the same i s not true for Q, i t s autocorrelation f a l l s off more abruptly. At 20° i t i s only 61% and 35% of i t s value at 0° depending whether the f i l t e r i s a LPMF or HPMF. Also shown in Fig. 6.6 is the variation of the cor-relation of Q with the t a i l which at 20° is s t i l l 90% of i t s nominal value. FEMF HPMF angular rotation 0 4 '8 '12 '16 '20 24° Fig. 6.6 Sensitivity of the LPMF, HPMF, and FEMF to only rotation of the readout signal 125 Due to the experimental procedure however, the latter curve i s not directly comparable to those relative to 0 and Q. The photodetector was kept fixed at the position of the correlation peak at 0°. After each two degrees rotation, the readout signal was also shifted i n i t s plane so as to bring the correlation peak back on the detector. With the 0- and Q-f i l t e r s , this means the signals were rotated approximately about their centroids, although with the t a i l - f i l t e r , Q was rotated about the centroid of i t s t a i l . The c.urves of Fig. 6.6 are for the rotation of the readout signal about the centroid of their respective recorded signals. In a practical case, the variation w i l l be more severe because the rotation of a character about i t s feature centroid i s accompanied by translation. With this caution, the FEMF is less rotation-sensitive than the HPMF. 6.3.4 Different Typefonts Filtering signals of typefonts different from the recorded one . gives a measure of the signal distortion that the LPMF, HPMF and FEMF can Q 0 Q. 0. Q Q 0 0 \ Q 0 0 0 Q q l °1 °1 °1 °1 q l q2 q2 q2 q4 q4 q4 Q Q Q Q °2 °2 °2 °4 °4 °4 q3 q 3 q 3 q 5 q 5 q 5 0 0 0 0 °3 °3 °3 °5 °5 °5 q l H °1 °2 q2 q l Q Q Q Q (12 pitch) Q 0 1: Prestige E l i t e 72 2: Artisan 12 72 (12 pitch) Q 0 0 0 0 0 3: Letter Gothic (12 pitch) Q 0 4: Courier 72 (10 pitch) Q 0 5: Delegate (10 pitch) Q 0 0 0 o_ Q (a) (b) Fig. 6.7. Set of characters used for typefonts sensitivity experiments (after 3X reduction). (a) The arrows point to the recorded signals. The others are the readout signals, (b) Typefont identification of (a). 126 tolerate. Fig. 6.7.a shows the set of signals used in this experiment after a three times reduction. Again the arrows point to the recorded signals. Fig. 6.7.b identifies the symbols of Fig. 6.7.a: "o,q and t" stand for "0,Q and t a i l of Q" and the subscripts refer to one of the five l i s t e d typefonts. As before, the recorded signals are of the typefont 1 or IBM Prestige E l i t e 72 of 12 pitch. Typefonts 2 (IBM Artisan 12 72) and 3 (IBM Letter Gothic) also have a 12 pitch. The 10 pitch of the typefont 4 (IBM Courier 72) and 5 (IBM Delegate) enables the simulation of a 20% scale variation with respect to typefont 1. The peaks of the crosscorrelations of 0^ and with a l l the characters of Fig. 6.7.a were measured and normalized as i n Section 6.3.1. The bar graphs of Fig. 6.8 a and b i l l u s t r a t e respectively the average response of the 0^- and Q^ - f i l t e r s to the various classes of signals. The red bars correspond to the LPMF while the green ones correspond to the HPMF. The averaging does not encompass the autocorrelation of the recorded signals. Compared to the LPMF, the HPMF increases the discrim-ination somewhat between 0^ and Q^ . It i s also more sensitive than the LPMF to a change of typefont and particularly to a pitch or scale change. The maximum and minimum responses of a LPMF for the t a i l of (K=4.4) to the same signals are shown in Fig. 6.9. Here the correlations were measured with the readout signals registered at the same position found by interpolation between the translation stages coordinates for which the four corner Q^'s yielded maximum correlation (see Section 6.3.1). Clearly, on the basis of the t a i l f i l t e r alone, the discrimination between 0 and Q i s adequate for a l l five typefonts. With the HPMF however, the d i f f e r -ence between the 0 and Q responses decreases when the typefont is altered and their magnitude decreases significantly so that ambiguity with other characters of a larger alphabet would l i k e l y occur. Qj-filter 1.0 0.84 0 5 + 0 . 4 + 0.2+ 'LPMF •HPMF 1.0 0.8 X 0 . 6 " + 0 . 4 + 0 . 2 + Oj-fitter °5 °4 °3 °2 °1 Q1 CL CL Q, 0 - reac/oc/f CL Q OL GL Q 1 2 3 4 5 sjgna/ 5 4 3 2 1 Oj Q 2 Q 3 a 4 Q 5 Fig. 6.8. Average responses of (a) the Q - f i l t e r and (b) the 0 - f i l t e r to signals of different typefonts. red = LPMF, green = HPMF. 1.2 °5 °4 °3 °2 °1 Q/ Q2 Q3 Q4 °5 readout signal F i g . 6.9. Maximum and minimum responses of the t a i l - f i l t e r to si g n a l s of d i f f e r e n t typefonts. QJOJ 0 7 0 ; v 0 1 QJOJ 0 7 0 7 . 0 ; 0 ; 0 7 0 ; 0 7 , 0 7 Q ; 0 7 OjO^ > ty Q 7 O 7 o7o7 , cy* 1.0 1 0.6] 0.6 i 0.4 0.2 0.0. 0, Fig. 6.10.b. Comparative responses of the 0 Q^ - and t a i l - f i l t e r to readout signals 0 of five different typefonts. red - LPMF, green - HPMF, blue - FEMF. o. 0. i •o, J c o .1 to o Ol QJOJ QJOJ S 0j QJOJ QJOJ . 0 J QJOJ QJOJ . 0 J QJOJ QJOJ . 0 J QJOJ QJOJ . 0 J J 131 In Fig. 6.10, the information of Fig. 6.8 and 6.9 is presented in a form that i s directly relevant to the decision process: given an unknown signal, how do the responses of the different f i l t e r s compare? Again, the red stripes are the average responses of the LPMF and the green bars those of the HPMF. The blue slanted stripes indicate the maximum and the minimum responses of the t a i l f i l t e r . It appears that the responses of the 0-^ - and Q^ -HPMF become more similar and smaller when the typefont variation increases. With the LPMF, discrimination i s not conserved either but the magnitude of the responses does not decrease as fast as with the HPMF. As for the t a i l f i l t e r , i t s maximum response to a 0 is 14% and i t s minimum response to a Q i s 83%, a minimum response ratio of 5.9. This means that the presence or absence of the t a i l fea-ture i s clearly indicated for a l l typefonts. The t a i l feature is sufficient to discriminate between 0 and Q once the choice among a large alphabet has been narrowed down to this alternative. To arrive at this point, the feature 0 (LPMF) would be used. In the worst case above (Q,. ® 0^-LPMF), i t s response i s 33.5%. whether this i s good enough to infer that the readout signal belongs to the 0 or Q class must be determined with regard to the other patterns. The fea-tures should be designed with this in mind. In the above experiments, the feature 0 was not optimized in any way. 6.3.5 Diffraction Efficiency The improved selectivity and tolerance to variants of the FEMF is obtained at the expense of a much reduced diffraction efficiency (DE). The latter i s defined as the ratio of the correlation peak power to the readout signal power. There are two reasons for this. The f i r s t one i s that the DE of the HPMF i s about twice as large as that of the LPMF [67]. The second reason has to do with the feature correlation i t s e l f . 132 If the amplitude transmittance of the characters is binary, the signal energy is proportional to i t s area of non-zero transmittance. Since the energy of 0^ was measured to be 90% of that of Q^ , i t can be concluded that the area of the t a i l of Q^  i s 10% of the area. It follows that the amplitude of the crosscorrelation of with i t s t a i l i s 10% of i t s autocorrelation amplitude. Therefore the ratio of their intensities i s 1%. Indeed, with as readout signal, the following observations were made DE of the HPMF = 2.2 x DE of the LPMF and DE of the FEMF = 0.014 x DE of the LPMF. This means that the DE of the FEMF is 1/158 times smaller than that of the HPMF. Fig. 6.3 illustrates this point. It shows the.output f i e l d d i s t r i -bution of different Q-filters when illuminated by the set of readout s i g -Fig. 6.3.a corresponds to the HPMF and was exposed for 1/30 nals Q . 0 sec. Fig. 6.3.b and c correspond to the LPMF and were exposed respectively for 1/4 and 1 sec. The energy in the peak of the correlation of 0 with the t a i l i s significantly smaller than that in the autocorrelation peak of Q. The DE of the FEMF i s feature dependent and i t i s l i k e l y that the example of the t a i l of Q i s a worst case as probably no smaller feature than that w i l l ever be used. 6.4 Conclusions The experiments just described reveal that an appropriate choice of geometrical features can lead to a significant improvement of the s e l -e ctivity between similar looking characters over the HPMF performance. Furthermore, contrarily to the latter, the FEMF does not f o r f e i t the abi l i t y to recognize variants of the recorded signals. This i s so because the FEMF is a low-pass f i l t e r . Indeed, Winzer's experiments with the HPMF 133 of a s l i t , that i s a character feature, demonstrate the intolerance to size and rotation of a high-pass feature extractor and i t s sensitivity to nonlinearities of the recording medium [96]. The FEMF has been shown to tolerate more typefonts variations and to be conditionally less sensitive to the readout signal orientation than the HPMF. Also, the FEMF does not yield as narrow correlation peaks as the HPMF and can therefore accomodate an output detector with a coarser resolution. A further benefit of re-cording the FEMF as a low-pass f i l t e r has been noted in Section 5.3.5. Indeed, low-pass f i l t e r s have a smaller aperture than their high-pass counterparts and usually low frequencies carry more energy than the high ones. For these reasons, the low-pass FEMF is less prone to aberrations than the HPMF. On the other hand, the FEMF has two drawbacks of i t s own. F i r s t , i t s readout signal must be positioned with an accuracy in the order of a character stroke width since the location as well as the presence of geometrical features is relevant. The corresponding accuracy of the HPMF is In the order of a character width. The second shortcoming of the FEMF is i t s low diffraction efficiency. These two disadvantages seem to be the price of a much improved r e l i a b i l i t y . 134 VII EXPERIMENTAL CONSIDERATIONS 7.1 Introduction A description of the apparatus used in the experiments of this thesis is presented in this chapter and a normalization procedure is des-cribed. The latter was made necessary by the d i f f i c u l t y of obtaining reproducible results when more than one f i l t e r and more than one readout signal were used at a time. The normalization procedure provided a mean of comparing the performance of different f i l t e r s and readout signals. The parallel processing capability of the MF and LLMF could not be exploited because of the phase error of the illuminating beam. As seen in Section 4.3, this error results in a transverse shift of the readout spectrum that reduces the correlation intensity. This phase error was evidenced during the experiments intended to compare the space variance of the MF-2 and LLMF-1, and made such comparison impossible with the available optical components. The phase error of the illuminating beam was responsible for the observed space variance and i s proposed as an explanation of the abnormal vertical space variance and deformation of the correlation pattern observed by Douklias and Shamir [97]. Another phase error, due this time to the thickness variation of the input transparency, had to be circumvented before reproducible cor-relation measurements could be obtained. This phase error i s evidenced by the variable frequency of the fringes of the interference between two beams reflected respectively off the f i r s t and second surface of the transparency. Demagnification of the signals was a quick and inexpensive method of reducing the effect of this phase error and of obtaining repro-ducible results. 135 7.2 Experimental System 7.2.1 Description of the Experimental Set-up Fig. 7.1 i s a schematic representation of the LLMF-1 (a = 1,6 = -1) used for feature extraction experiments. .Shown in the insert i s a variant of this set-up used for the space variance experiments described i n Section 7.3 The longitudinal distances along the beam axes are at 1/8 scale while for the sake of cla r i t y transverse dimensions are greatly exaggerated. The components and..symbols of Fig. 7.1 are identified in Table 7.1. The reference beam consists of the spatial filter-beam expander SF2 and the lens L r which focuses the reference beam in the output plane P D. The pinhole of SF£ is situated at the same optical distance from the f i l t e r plane P^ as the input plane P g so that removal of lens L^ leaves the reference beam ready for the MF-2 recording. The object beam, which is also the readout beam, is the Fourier-transforming stage made up of the spatial filter-beam expander SFn and the lens L . The input plane P i s located at an optical distance D g from P^ where D g ~ 500 mm for feature extraction experiments and 779 mm for space variance tests. After a 3X reduction (see Section 7.4), the recorded and readout signals are recorded as a negative transparency on a Kodak 649F plate with a minimum contrast of 100. The plate position and orientation i n P can be adjusted with a X-Y-o) micropositioning device MPD. X and Y are the horizontal and ve r t i c a l transverse directions and to is the angular rotation about the optical axis. During the f i l t e r i n g operation, the reference beam i s blocked off and a crosscorrelation pattern i s displayed i n the output plane P Q. The la t t e r and the input plane P are symmetrical of each other with respect to P^(Do = D t). The correlation intensity i s measured with the 50-ym fiber optic of a Gamma Scientific 2900 MR scanning microphotometer. Its eye-piece L., images the correlation pattern and the fiber optics on the p = P0 TVM DVM- DVM-DEC PDP-12 7 7 / autophoto-meter 2900 MR DVM, optical path electrical path longit udinal scale: 1/8 Fig. 7.1. Schematic representation of the LLMF-1 used for feature extraction experiments. Insert: variant of this set-up used for space variance experiments. 1 3 7 Symbol Identification A1' A2 Amplifiers acting as level converter between respectively PM or PD and ADC ADC Analog to d i g i t a l converter of the PDP-12 CBS Cubic beam split t e r (side = 25.4 mm) DVM Digital voltmeter FO Fiber optic of scanning eyepiece; aperture = 50 ym HPH Holographic plate holder; Jodon Engineering Associates Inc., Model # MPH-45 Laser Spectra-Physics, Model #125, He-Ne, 50 mW L s Collimator; Spectra-Physics, Model # 333; diameter = 24.5 mm; E.F.L. = 85 mm L r Reference beam shaping lens; Oriel Corporation, Model # A-18-141-60; diameter = 10 mm; F.L. = 100 mm L n Imaging lens of normalization channel; diameter = 25.4 mm; F.L. = 200 mm L i Lens of scanning eyepiece. See PM M Front surface mirror MPD X-Y-u) micropositioning device consisting of 2 Ardel-Kinamatic translation modules, Model # TT-103; micrometer graduation = 10 ym 1 Ardel-Kinamatic polar rotation module, Model # RT-200; . di a l graduation = 1 minute PD Photodetector; Alphametrics, Model # P 1101S PDP-12 Digital Equipment Corporation Computer PM Microphotometer; Gamma Scien t i f i c , Model 2900 MR, includes scanning eyepiece with FO, PMT and DVM^  PMT Photomultiplier tube; S ^ l l photocathode S F 1 Spatial filter-beam expander; Data Optics Inc., Model # 5002, 20X power, aperture = 10 ym SF 2 Spatial filter-beam expander; Gaertner S c i e n t i f i c Corp., Model # R250F, 10X power, aperture = 25 ym TTY Teletype TVC Television camera TVC Television monitor VBS Variable beam s p l i t t e r ; Jodon Engineering Associates Inc., Model # VBS-200 Table 7.1 List of equipment and identification of the symbols of Fig. 7.1. Fig. 7.2. Overall view of the apparatus. 139 sensitive area of a TV camera TVC for visual observation on a TV monitor TVM. The 5X microscope objective of the microphotometer and the glass dust cover of i t s eyepiece were found to cause intolerable distortion and inter-ference fringes. So they were removed at the expense of the resolution. Fig. 7 . 2 i s an overall view of the experimental apparatus. It shows the air-suspended granite table that isolates the set-up from vibra-tions, and the wooden partition that reduces somewhat the' air turbulences induced by the air-cooled laser. 7 . 2 . 2 Normalization Procedure With the experimental set-up of Fig. 7.1, only one f i l t e r and one readout signal at a time could be used because of the phase errors of the illuminating beam (see Section 7.3) and of the signals transparency (see Section 7.4). Therefore the comparison of different f i l t e r s and readout signals required a normalization channel for implementation of Eq. ( 2.16). In an actual system though, this would not be necessary as the relative responses of a bank of f i l t e r s to the same readout signal are independent of i t s energy E f c. As for the normalization with respect to the energy of the recorded signal E , i t can be done electronically after detection of the f i l t e r ' s outputs. It follows that the cubic beam s p l i t t e r CBS inserted between P g and P^ to provide tae normalization channel would not be present in an actual system and that i t s f i r s t Fourier-transforming stage would be space-invariant. The normalization channel measures E f c i n one of two modes. For feature extraction experiments, a lens L r images the input plane P s onto a f i e l d stop that prevents the light of a l l the readout signals but the one located at the recording position, from reaching the photodetector PD (see Fig. 7.1). For space variance experiments, a unique readout signal 140 scans the input f i e l d and i t s energy i s measured with PD located i n the Fourier plane of the normalization beam (see insert of Fig. 7.1). Ampli-fiers A^ and bring the outputs of the Gamma Scientifi c photometer and PD respectively, up to the operating range (-1 to +1 Volt) of the analog to di g i t a l converter ADC. The d i g i t a l computer PDP-12 directs the ADC to sample the correlation and normalization channels within an interval of 22.5 usee. It averages typically sixteen (16) samples of each channel over a period of 2.6 msec. Then i t calculates the ratio of the average correlation intensity over the average E^. A signal proportional to such ratio i s returned by the computer and displayed on the d i g i t a l voltmeter DVM-j. Then the computer begins a new cycle of sampling, computation and display. Observation of DVM^  helps to position the fiber optic at the peak of the correlation pattern. The PDP-12 which i s located in the room above the holographic laboratory is remotely controlled v i a a teletype TTY and operates in one of the following modes: 1) display mode: continuous sampling, ratio computation and display, 2) storage mode: storage of last computed ratio, 3) text mode: storage of typed-in comments and information relative to the stored ratios, 4) offset mode: sampling of both channels i n absence of light s i g -nals to determine their computed zero's, 5) printout mode: l i s t i n g of comments and stored ratios after fur-ther normalization with respect to one of them. The overall performance of the normalization procedure i s illu s t r a t e d i n Fig. 7.3 which is a plot of the computed ratio p 2 of a correlation inten-sity to i t s corresponding readout signal energy E t when E t i s changed by 141 ti 1.0 0.75 0.5 0.25-0 o 0:25 0:5 0:75 1:0' Fig. 7.3. Performance of the normalization procedure. means of the variable beam splitter VBS. As expected is fairly in-sensitive to E and is constant within 2.5% over the used range of high signal energy. Variation of p2 over a two hours time lapse does not St exceed 2.5%. 7.3 Space Variance Experiments or Phase Er r o r of the Illuminating Beam > To investigate experimentally the space variance of the MF-1, Douklias and Shamir recorded a high-pass filter matched to a printed char-acter and bleached the hologram [97]. Then they measured the intensity of the autocorrelation peak as the readout signal scanned the input field. The wild intensity variations led them to conclude that the MF space var-iance "is much more severe than would be predicted from an idealized theory on volume holograms" and that "the variation of the autocorrelation func-tion is not monotonic as would roughly be the effect of conventional aberrations and seems to be unpredictable." Furthermore they observed "a marked change in the form of the autocorrelation function, as the a rbitrary unit 142 object position i s changed", as well as larger variations when the signal displacement is normal to the plane of incidence than when i t i s p a r a l l e l to the same. These conclusions which refute the p a r a l l e l processing capability of an actual MF w i l l be reviewed in the light of the results of experiments undertaken in an attempt to compare the space variance of the MF and LLMF. The system described in Section 7.2 was used to measure the space variance of the MF and LLMF. To limit the effect of aberrations and of emulsion thickness, a small reference to object beam angle (8.8°) and a long distance P g - ^ (779 mm) were chosen, and holograms were recorded in the 6 ym thick-emulsion of Kodak 120-02 plates. HPMF's of a character "G" were recorded as absorption holograms using a MF-2 and a LLMF-1 (a = - 6=1). Fig. 7.4.a and b represent the l o c i of the readout signal posi-tions that yielded a same autocorrelation intensity with respectively the MF-2 and LLMF-1. The position of the recorded signal is the origin of the axes that are graduated in mm. The reference point source is located to the l e f t and outside of the picture at (x r,y r) = (-120 mm,o). Both figures'show the same general trend except for small local differences imputed to a slight transverse adjustment of the pinhole of SF^ in between the two experiments. Furthermore these figures are analogous to Douklias'. In Douklias' case, the signal scanned an input f i e l d of 50 mm i n diameter compared to 20 mm for Fig. 7.4, and the Fourier-transformation scales were governed respectively by a 500 mm focal length and a 779 mm distance between P g and P^ . The difference between Douklias' experimental parameters and those of this experiment, and the analogy between Douklias' results and Fig. 7.4 suggest that the curves reflect the circular symmetry of the illuminating beam in Pc. Fig. 7.4. Loci of the readout signal positions that yield the autocorrelation intensity, (a) MF-2. (b) LLMF-1. same 144 Two similarities between Douklias' results and Fig. 7.4 are note-worthy. They are 1) the strong influence of a Y-displacement which is not predicted by the volume theory when y = y [see Eq. (5.25)], and TC S 2) the deformation of the correlation pattern at low correlation value which is not predicted either by the volume theory. After the experiments of Fig. 7.4, i t was noted that shifting the readout signal in i t s plane resulted in a nonlinear nonmonotonic shift of i t s spectrum which degrades the matched f i l t e r i n g operation. Positions of low correlation were positively associated with shifted spectra. Section 4.3 has investigated the spectral mismatch and associated reduction of correlation intensity due to a phase error on the input signal or equiva-lently on the illuminating beam. The experimental verification and the pictures of deformed correlation patterns presented there are relative to the experiments reported here. Consequently the curves of Fig. 7 . 4 .a and b il l u s t r a t e not the space variance due to volume effect, aberrations or vignetting, but rather the intensity variation due to a phase error on the illuminating beam. I n i t i a l experiments had shown the LLMF-2 to be less space-variant than the MF-2 which in turn was less space-variant than the LLMF-1 [47]. It is now believed the same phase error was res-ponsible for the reported apparent space variance. In view of the unexplained correlation deformations and Y-space variance reported by Douklias and Shamir, and in view of the fact that both abnormalities were also observed and explained by the author in relation to an imperfect illuminating beam, i t is questionable whether Douklias' and Shamir's results can be interpreted in terms of volume effect. It i s f e l t that space variance i s probably not as severe as 145 they reported [97]. Fig. 7.4 emphasizes the need for optical components, and particularly a spatial filter-beam expander SF^ and a lens L^, of high quality i f true space variance is to be measured. Only then could the MF and LLMF be compared experimentally. 7.4 Phase Error of the Incoherent-to-Coherent Image Converter (ICIC) It was reported in Chapter 6 that the feature extraction experi-ments were performed with signals reduced three times from their type-written size. This demagnification proved necessary for obtaining repro-ducible measurements. Indeed, without size reduction, the eighteen (18) readout signals 0 resulted in "autocorrelation" measurements ranging from 10 to 100% of that of the recorded signal. Such a wild variation was traced to a transverse displacement of the readout spectra with respect to the recorded one. High and low correlation measurements were seen to correspond respectively to small and large spectrum s h i f t . This matter was examined in Section 4.3 dealing with phase error in the input plane. Again the one-dimensional double pulse of Fig. 4.4 can be used to relate loss of performance and spectrum s h i f t . One of the 0's yielded an auto-correlation intensity equal to 55% of that of the recorded signal and i t s spectrum was shifted v e r t i c a l l y by 35 pm. Eq. (4.32) with Wg = 1.8 mm, A = 632.8 mm and r> g = 500 mm predicts the same performance (55%) for a 41 urn displacement. In view of the crude approximation of 0 by a one-dimensional pulse, the agreement between the two displacements i s s a t i s -factory. This time, the phase error on the input signal was introduced not by the illuminating beam as in Section 7.3 since the readout signals were in turn positioned at the location of the recorded signal, but rather by the ICIC,that is the Kodak 649 F plate on which the signals were recorded. 146 (a) p 2 = 1.00 ss (b) p 2 = 0.92 ss (c) p 2 = 0.65 ss — E X*%m^ (d) p 2 = 0.55 ss (e) p z = 0.37 ss (f) p 2 = 0.18 ss Fig. 7.5 Variation of the measured autocorrelation p 2 with the ICIC s s wedge angle or fringe frequency, (a) Recorded signal, (b) to (f) Readout signals. 10X Magnification. emulsion glass pi a te 1 1 147 Fig. 7.5 shows photographs of the interference fringes produced when a collimated beam is incident at 45° on the non-emulsion side of the ICIC and is reflected off the emulsion and off the glass plate (see F i g . 7.6). Locally the ICIC can be considered as wedge-shaped with the fringes fre-quency proportional to the wedge angle. It i s the difference i n wedge angle between the recorded signal shown in Fig. 7.5.a and the other read-out signals (Fig. 7.5.b to f) that governs the amount of relative spectrum shift and eventually the intensity of the normalized autocorrelation p 2 g (indicated below each frame). For example, the 0 signal of F i g . 7 . 5 . f which shows practically no fringes i s recorded over an almost f l a t area of the ICIC and performs poorly (18%) when correlated with a signal re-corded on a steeper wedge (Fig. 7.5.a). Immersing the photographic plate in a liquid gate would have alleviated this problem but a 3 X signal re-duction prior to recording on Kodak 649F plates was a quick and inexpensive solution that offered acceptable results. Indeed, scaling down the s i g -nals broadens their spectra and lessens the relative importance of a spec-trum s h i f t . The standard deviation of the correlation of the reduced signals was no more than 5%. 7.5 Conclusions This chapter has pointed out the importance of phase accuracy in the input plane of the MF and LLMF. Indeed a phase error there shifts the readout spectrum with respect to the recorded one and reduces the correlation by spectral mismatch. The phase error of the illuminating beam prevented the comparison of the space variance of the MF - 2 and LLMF-1. Optical components of higher quality than those available must be used i f such comparison is to be meaningful. The phase error caused by the thickness variation of the ICIC, or input transparency, was also responsible for non-reproducible results. After i t s effect had been lessened by the of signals reduced three times, the reproducibility of the results w quite satisfactory. 149 VIII CONCLUSIONS 8.1 Summary of Results This thesis has investigated the space invariance and the detection r e l i a b i l i t y of the holographic matched f i l t e r in the context of character recognition. It has presented solutions towards their improvements. Parallel processing, which i s the most attractive advan-tage of the holographic matched f i l t e r , depends upon i t s space invariance. Two lensless matched f i l t e r s (LLMF) have been designed. They are space-invariant as far as vignetting apertures are concerned. The analysis of the space variance induced by the volume of the recording medium and by the holographic aberrations has led to the selection of the optimum lay-out of recorded and readout signals. Improved detection r e l i a b i l i t y has been achieved experimentally by means of feature extraction using a low-pass f i l t e r . The following conclusions can be drawn from the study. i) Both LLMF's act as a combination of a matched f i l t e r and a Fresnel zone plate. They are synthesized either by means of a converging reference beam (LLMF-1) or by appropriate scaling of the recorded and readout signals (LLMF-2) . T.iey are space-invariant because their only vignetting aperture is located in the Fourier transform plane. In the f i l t e r i n g mode they use one less lens than the Vander Lugt matched f i l t e r (MF-2) and therefore have fewer positioning devices and optical coatings. Having fewer surfaces to collect dust particles and scratches helps in reducing spurious diffraction and simplifies the maintenance. A l l these factors contribute to a cost reduction of the holographic matched f i l t e r . 1 5 0 2 ) The resolution of currently available photographic recording media is sufficient not to restric t the f i e l d of view or the frequency content of the signals. 3 ) The non-correlation output terms are unfocused images. They contribute noise at the position of the signal or correlation peak. When Y , the relative width of the signals, i s in the order of 1 0 or more, the measured SNR of the LLMF is similar to the SNR of the MF-2 at the same reference beam angle. In practical cases of paral l e l processing, the condition Y > 1 0 would be sat i s f i e d and the lower limit on the reference beam angle would be the same for the MF and LLMF. 4 ) In presence of small displacements of the hologram, the LLMF behavior results from the combined displacements of i t s f i l t e r component and of i t s lens component. In the case of a fixed detector, there exists an optimum signal size for which the correlation signal i s least sensi-tive to the f i l t e r displacement. 5) Parallel processing at constant space-bandwidth product has been found to be least sensitive to the volume effect of the recording medium;, or least space-variant, for two layouts of recorded and readout signals. They have been called the "+" and " I " layouts and are well suited for line-by-line character recognition. 6) The space variance of the rms distortion of the " I " layout i s smaller than that of the "+" layout. 7 ) At unit magnification, the correlation produced by the LLMF-1 is free of spherical aberration and coma. The same holds true for the MF-2 only i f i t s second lens is corrected for the same aberrations. 8 ) The maximum rms astigmatism of the LLMF-1 is only sl i g h t l y larger than that of the MF-2 while the LLMF-2 suffers from more severe rms astig-1 5 1 matism than the other f i l t e r s . Except for aberrations, both LLMF's behave identically in each of the aspects considered in the thesis. 9) Experiments with the characters 0, Q and the t a i l of Q reveal that the acuity ratio of the high-pass matched f i l t e r (HPMF) is only 1.31 although the ratio of the responses of Q and 0 to the t a i l - f i l t e r (or feature extractor and matched f i l t e r FEMF) i s equal to 20. 10) Contrary to the HPMF, the improved discrimination of the FEMF does not compromise i t s a b i l i t y to recognize some variants l i k e charac-ters of different type fonts. This is because the FEMF is recorded as a low-pass f i l t e r . For the same reason, the FEMF uses a coarser output plane resolution and has smaller aberrations than the HPMF. 11) The performance of the FEMF was obtained at the expense of a reduced diffraction efficiency and increased accuracy of the readout signal registration. 12) Phase accuracy in the input plane is of the utmost importance as spectral mismatches have been shown experimentally to have drastic effects on the space variance of the matched f i l t e r s and on the repro-ducibility of the correlations. 8.2 Suggestions for Further Research The following aspects of the LLMF and of the FEMF are worth examining further. 1) For a same overall length, the f i l t e r component of a reflection type of LLMF does not have to be positioned with as great an accuracy as when a transmission type of LLMF i s used. Furthermore such LLMF would bring the "second lens" into the f i l t e r plane, which i s not possible with a reflection type of MF. 152 2) Having a greater diffraction efficiency than their amplitude counterparts, phase holograms could be used to increase the SNR of the LLMF. Reflection phase holograms could also decrease the space variance due to the volume of the recording medium. 3) The effect of the emulsion shrinkage must be considered in the LLMF with particular attention to the reflection type of LLMF. 4) Experimental comparison of the space variance of the MF and LLMF would establish the relative importance of the effects of the aberrations, of the volume of the recording medium and of the vignetting apertures. 5) Experimental verification of the LLMF behavior in presence of transverse and longitudinal displacements of the hologram i s needed. 6) Investigation of the space variance of the rms astigmatism of the "+" and " I " layouts would complement the aberration study of chapter 5. 7) Experiments with more features and more characters are needed to assess the performance of th:>. feature extraction - and low-pass f i l t e r i n g technique, and to evaluate the probability of error that can be achieved with a holographic character recognition equipment. 8) Increasing the detector size relative to the signal size reduces the acuity ratio or discrimination. 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(A.3.b) Similarly, free space propagation over a distance D is characterized by an impulse response h(x,y) = e j ( k D " ^ d iKx,y;d) (A.4.a) where d A D - 1 . (A.4.b) Often the phase factor which is independent of the transverse coordinates x and y can be disregarded and h(x,y) becomes h(x,y) = y *(x,y;d) . (A.5) Hereafter are lis t e d some properties of the 4* function that follow readily from the definition ( A . 2 ) : ?(x,y;d) = i|)(x,y;-d) , (A.6) iKax,ay;d) = tf'(x,y;a2d) , (A.7) i K x . y ^ ) ^(x,y;d 2) = iKx,y,d1+d2) , (A.8) i|»(x-u,y-v;d) = ijj(x,y;d)i|;(u,v;d) exp(-jkd(ux+vy)) , (A.9) 161 iKx;d)*(y;d) = *(x,y;d) . (A.10) The limits of 4i(x,y;d) are lim iKx,y;d) = 1 and (A.ll) d^ o lim d_ ,Kx,y;d) = 5(x,y) (A.12) where 6(x,y) i s the two dimensional Dirac delta function. The Fourier transform of the \p function often occurs under the form X - d2 ff^ \l>(x,y;d1) exp(-jkd2(ux+vy)) dxdy = j iKu,v;-j^ -) . (A.13) The operational notation can be applied to one dimensional systems (i.e. only one transverse coordinate i s considered) with Eq. (A.3) and (A.4) properly modified [ 98, p. 316]: x(x) = *(x;f) (A.14) and : h(x) = e J ( k D " 4} /l-j-) <Kx;d) . (A. 15) Accordingly, Eq. (A.5) reduces to h(x) = /(-^) ^(x;d) . (A.16) 

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