2900-394-T Report of Project MICHIGAN SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING A. B. VANDERLUGT July 1963 Radar Laboratory'vtAa&te j Sciece aeW "7ec~oH THE UNIV E RSITY OF MICHIGAN Ann Arbor, Michigan

Institute of Science and Technology The University of Michigan NOTICES Sponsorship. The work reported herein was conducted by the Institute of Science and Technology for the U. S. Army Electronics Command under Project MICHIGAN. Contract DA-36-039 SC-78801, and for the U. S. Air Force under Contract AF 33(616)-8433. Contracts and grants to The University of Michigan for the support of sponsored research by the Institute of Science and Technology are administered through the Office of the Vice-President for Research. Note. The views expressed herein are those of Project MICHIGAN and have not been approved by the Department of the Army. Distribution. Initial distribution is indicated at the end of this document. Distribution control of Project MICHIGAN documents has been delegated by the U. S. Army Electronics Command to the office named below. Please address correspondence concerning distribution of reports to: Commanding Officer U. S. Army Liaison Group Project MICHIGAN The University of Michigan P. 0. Box 618 Ann Arbor, Michigan ASTIA Availability. Qualified requesters may obtain copies of this document from: Armed Services Technical Information Agency Arlington Hall Station Arlington 12, Virginia Final Disposition. After this document has served its purpose, it may be destroyed. Please do not return it to the Institute of Science and Technology. ii

Institute of Science and Technology The University of Michigan PREFACE Project MICHIGAN is a continuing, long-range research and development program for advancing the ArmyVs combat-surveillance and target-acquisition capabilities. The program is carried out by a full-time Institute of Science and Technology staff of specialists in the fields of physics, engineering, mathematics, and psychology, by members of the teaching faculty, by graduate students, and by other research groups and laboratories of The University of Michigan. The emphasis of the Project is upon research in imaging radar, MTI radar, infrared, radio location, image processing, and special investigations. Particular attention is given to all-weather, long-range, high-resolution sensory and location techniques. Project MICHIGAN was established by the U. S. Army Signal Corps at The University of Michigan in 1953 and has received continuing support from the U. S. Army. The Project constitutes a major portion of the diversified program of research conducted by the Institute of Science and Technology in order to make available to government and industry the resources of The University of Michigan and to broaden the educational opportunities for students in the scientific and engineering disciplines. Progress and results described in reports are continually reassessed by Project MICHIGAN. Comments and suggestions from readers are invited. Robert L. Hess Director Project MICHIGAN iii

Institute of Science and Technology The University of Michigan CONTENTS Notices................. Preface................ List of Figures............. Glossary................. Abstract.............. 1. Introduction............. 2. Mathematical Nature of the Problem 3. Optimum Filtering......... 4. Optical Processing Systems.... 4.1. Coherent Optical Systems 4.2. Noncoherent Optical Systems 5. Realization of the Optimum Filter..................... ii..................... vi.................... vii ~ ~ ~ ~ ~ ~ ~ ~ ~ ~..................... 1..................... ~ ~ ~ ~ 2 ~~~~~~.~~~~~~~~~~~~.~~ ~ 3 ~~~~~~~~~~~~~~~~~~~~~ 5.1 @ @ * @ @ @ @ @.2 5.1. Realization of Nonnegative Filters 5.2. Realization of a Real Filter Function 5.3. Realization of the Complex Filter 6. Some Notes on the Performance of Matched Filters.. 6.1. Change in the Scale of the Signal 6.2. Change in the Orientation of the Signal 7. Experimental Results......... 7.1. Detection of Simple Geometrical Shapes 7.2. Detection of Alphanumerics 7.3. Detection of an Isolated Signal in Random Noise 5 8...............10 10 12 13.............17 17 18........... 2o.20 20 21 22 Appendix A. Optimization in Presence of Mutually Exclusive Noise. Appendix B. The Fourier Transforming Property of Lenses..... References............................... Distribution List................................. 24..... 27..... 39.... 40 V

Institute of Science and Technology The University of Michigan FIGURES 1. Mathematical Model for Image Processing................. 2 2. Processing System............................. 3 3. An Optical Fourier Analyzer........................ 6 4. A Coherent Processing System....................... 7 5. A Noncoherent Processing System...................... 9 6. Typical Curve of Density vs. Log Exposure................ 11 7. Sector of Circle Used to Generate Low-Frequency Rejection Filters... 12 8. Liquid Cell.................................. 13 9. Modified Mach- Zehnder Interferometer.................. 14 10. Alternative Optical System for Realizing Complex Filters......... 17 11. Set of Geometrical Shapes.......................... 20 12. Detection of Rectangles........................... 20 13. Reconstruction from a Complex Filter................... 21 14. Cross Correlation of L with the Set of Shapes Shown in Figure 11.....22 15. Convolution of L with the Set of Shapes Shown in Figure 11......... 22 16. Set of Alphanumerics............................ 22 17. Detection of Letter g............................ 22 18. Detection of Isolated Signal in Noise Background.............. 23 19. Processing System When Mutually Exclusive Noise Is Present...... 24 20. Cornu Spiral................................. 31 21. Three-Lens System.............................35 22. Two-Lens System.............................. 36 23. One-Lens System.............................. 36 vi

Institute of Science and Technology The University of Michigan GLOSSARY A k, k', a, b, c d D E f g G h I k m n P. q r R s T x, y,', X7, u, v X 4, y amplitude of illumination; system half-aperture constants distance density exposure; expected value function representing the available data; focal length of lens function representing the scene filter function impulse response of filter indicator function 271/ scaling factor noise; refractive index spatial radian frequencies output of processing system; distance reference wave; correlation function signal transformation; transmission; thickness space coordinates range on signal for space-invariant operation error term lens aberrations phase portion of complex functions; angles wavelength of light gamma of film vii

SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING ABSTRACT This report contains integrated descriptions of the problem of signal detection, the optimum linear filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. Experimental results show that the theory is valid. The appendixes give a treatment of the Fourier transforming property of lenses which is general enough that complete optical systems can be evaluated on the basis of frequency response and region of space-invariant operation. The experimental results obtained to date indicate that this technique provides an excellent two-dimensional filtering capability that will play a key role in problems such as shape recognition and signal detection. 1 INTRODUCTION The fundamental theory of optical spatial filtering has been formulated by several writers [1-3]. The close analogy of spatial filtering to optimum linear filtering theory promised to open up new techniques for realizing those frequency domain filters which could not be synthesized in the time domain because of the realizability contraint. But the advances in spatial filtering since the formulation of the theory have not been what one might have expected. Perhaps the major reason is that, whereas the theoretical formulation tacitly assumed that complex filters could be realized, the actual realization has presented a difficult problem. It was generally assumed that photographic transparencies would play a large role in the realization of spatial filters [2, 3], but any hope for making complex filters on film has seemed remote. Some work was aimed in this direction [4, 5], but it is not directly applicable to the problem treated in this report. Since spatial filters are passive, they can take on all values on or within the unit circle in the complex plane [6]. Early experiments in spatial filtering used occluding filters as bandpass filters to demonstrate the theory. Later, continuous amplitude control, such as Gaussian weighting, was used to show how equalization could be accomplished by optical systems. The next step was to add binary phase control to extend the range of filter values to the entire real line. Binary control was gained by using ruled phase plates, evaporation techniques, or film relief ing techniques. Not only are these techniques awkward to apply, but, more seriously, the impulse response of a real filter is symmetrical. Clearly, if more general filtering schemes are to be performed, a fairly easy method of constructing the general complex filter must be found. This report describes, as one of its major results, a practical technique for realizing a general complex filter even though the filter function is recorded on photographic film. 1

Institute of Science and Technology The University of Michigan In most other approaches to the problem of the realization of the desired complex filter, a complex distribution of light must be analyzed. This report showshow, at the same time, the analysis can be avoided and the complex filter can be realized. This feature is important since the required filter can be found analytically for only a few simple two-dimensional functions. This report contains an integrated description of the problem of signal detection, the optimum filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. The appendixes give some important insight into the Fourier transforming properties of a lens system operating under coherent illumination. The theory, which is more general than usual, is used to discuss and evaluate several optical systems. 2 MATHEMATICAL NATURE OF THE PROBLEM The mathematical model shown in Figure 1 describes the problem of sensing, recording, and processing imagery. The scene will be denoted by g(x, y), an intensity function of two space coordinates. The sensor makes a two-dimensional transformation of g(x, y), and the recording process transforms it further. Because the recording medium is usually photographic film, film grain becomes a secondary source of noise. The processing of the imagery is equivalent to a third transformation on g(x, y), and the output is an estimation of the amount of signal present in g(x, y). FIGURE 1. MATHEMATICAL MODEL FOR IMAGE PROCESSING 2

Institute of Science and Technology The University of Michig It has become common practice to refer to f(x, y) = T2{T1 [g(x, y)]} as the "scene." Although this is not strictly correct, transformation T2 is usually a nonlinear 2 process, and a poorly controlled one at that; therefore, it becomes necessary to operate on the available data f(x, y) rather than to attempt to recover g(x, y) before the processing operation. 3 OPTIMUM FILTERING Suppose n(x, y) is a homogeneous isotropic random process with spectral density N(p, q), and s(x, y) is a known Fourier transformable function of space coordinates. We wish to operate on f(x, y) = s(x, y) + n(x, y) in such a way that we maximize the ratio of peak signal energy to mean square noise energy in the outputo Suppose h(x, y) is the impulse response of a linear, space invariant filter with frequency response H(p, q). It is desirable to determine an optimum filter under the condition stated. See Figure 2. s(x, y) f(x, y) r(x, y) /Tn y)(x, y) n(x, y) Filter FIGURE 2. PROCESSING SYSTEM The signal part of the output is 00 r (x, y) = S( q) H(, q) e (px+q) dp dq -00 and oo MSN (mean square noise)= N(p, q) H(p, q) 2dpdq - 00 r (0, 0)12 We want to find maximum over all {H(p, q)} for the ratio MSN, where the peak signal is taken at (0, 0) for convenience. We have 3

Institute of Science and Technology The University of Michigan oo0 2 2 f S(p, q) H(p, q) dp dq Irs(0, 0)12 4?-r 2(o MSN oo0 1 ffN(p, q) |H(p, q)l2dpdq -0o For nontrivial cases, N(p, q) > for all (p, q), so that it can be treated as the weighting function in the Schwarz inequality. Rewrite Equation 1 in the form iTr ( ) 2 F 2 (p, q)] H(p, q) N(p, q)dpdq IrS(o1 0)12 42 JJ N(pq -s__-cO_______________ -(2) MSN oo { fH(p, q)12N(p, q)dp dq -oo Apply the Schwarz inequality to the numerator of Equation 2 to get.r S0,1 q) N( N(p, q) dpdq I H (p, q)[12N(p, q)dpd {H} MSN C f |H(p,q)2 N(p, q)dpdq -co -o0 We get a maximum on H when H(p, q)= k(p, q) (3) N(p, q) (where the bar indicates a complex conjugate) and the signal-to-noise ratio is S/N = 1__.IS(p, q)12 S/N =N(p, q dp dq (4) -oo From Equation 3 we see that under the given constraints the optimum filter is proportional to the complex conjugate of the signal spectrum divided by the noise spectral density. The output of the system is given by 4

Institute of Science and Technology The University of Michigan r(x, y) = 1 i F(p, q) H(p, q) ej(px+qy) dpdq (5) 4v2 -o0 In case N(p, q) is uniform for all frequencies we have H(p, q) = k' S(p, q) and r(x, y) = if F(p, q) S(p, q) e(P+ dpdq -oo Employing the convolution theorem, we have co r(x, y) = k' jf f(xo, yo) s(-x + x, -y + yO) dxodyO -00 oo = k'J f(x + u, y + v) s(u, v) dudv (6) -o0 which is a cross-correlation process. Optical systems which perform the operations indicated by Equations 5 and 6 will be described in Section 4. An interesting situation arises when the signal and noise are not linearly additive, but are mutually exclusive processes. Then the signal is no longer a known function, but becomes a random process. This case arises when the signal is partially occluded by noise, and also when film grain noise is present. The mutually exclusive noise situation is analyzed in Appendix A. 4 OPTICAL PROCESSING SYSTEMS 4.1. COHERENT OPTICAL SYSTEMS Optimum filter theory makes widespread use of Fourier transform theory to arrive at results easily. Unfortunately, since the synthesis of an electronic filter must be made in the time domain, there are restrictions on the realizability of the filter, as well as some limitations on its performance. The use of a coherent optical system overcomes some of these restrictions because it can display the Fourier analysis of signals as a distribution of light, and one has the option of constructing the filter in either the frequency or the space domain. 5

Institute of Science and Technology The University of Michigan The optical system shown in Figure 3 acts as a two-dimensional Fourier analyzer, as is shown in Appendix B. If f(x, y) denotes the specular amplitude transmission of the transparency in plane P1 and F(p, q) denotes the complex amplitude distribution of light in plane P2, then Collimating - Lens L c " —- 0i...... Point Source of Monochromatic I Li Light Input Spherical Frequency Function Lens Plane f(x, y) F(p, q) FIGURE 3. AN OPTICAL FOURIER ANALYZER on F(p, q) = fff(x, y) ej(px+qy) dxdy (7) -00 when d = f and the illumination is a monochromatic plane wave. In Equation 7 p and q represent spatial frequency variables having the dimensions of radians/unit distance. But the variables in plane P2 are in units of distance which are Afp 217 Xfq 27r where t = the direction parallel to x r7 = the direction parallel to y X = the wavelength of the illumination f = the focal length of the spherical lens In general, the variables (p, q) will be used for emphasis when a distribution of light is a function of frequency, as well as to simplify the notation associated with Fourier transform theory. A transform relationship can exist under a wide variety of conditions. If d # f, then F(~, 7) is modified by a quadratic phase factor which does not affect the intensity of the distribution. Also, convergent or divergent illumination will only relocate plane P2, which changes the scale of F(Q, r7) as well as modifying F(5, 77) by a spherical phase factor. These phase terms serve only to determine the position of the image plane for the input transparency, or else the position 6

Institute of Science and Technology The University of Michigan of the frequency plane (plane P2). These conditions are discussed in Appendix B in connection with the frequency response of optical systems. The fact that a spherical lens can take the Fourier transform of a complex distribution of light allows one to construct an optical system by arranging a sequence of lenses which forms a succession of Fourier transform planes. An image of the input plane can be effected by placing a lens behind plane P2 which takes the transform of F(p, q) (see Figure 4). Since a positive Collimating ff f fLens L. Point Source "of p Li P2 L P Monochromatic 2 2 gmht ptt s h Objective Imaging hhtnLight Input Ln Frequency Output -rLens _ens Plane Plane Plane f(x, y) F(p, q) r(x, y) FIGURE 4. A COHERENT PROCESSING SYSTEM spherical lens always introduces a positive kernel in the transform relationship, we have in plane P3 the distribution r(X, y) = (p q) ej(p+qy) dp dq 47 2 -co = f(-x, -y) We have assumed that the system has unity magnification and sufficient bandwidth to pass the highest spatial frequency in the input function. Note that the output is an inverted image of the input function, which is what one expects from an imaging system operating under any type of illumination. We have also assumed that the lenses have no aberrations and that the system is space-invariant. Limitations placed on the system to assure space invariance for a given bandwidth signal are discussed in Appendix B. In certain conditions the optical system in Figure 4 can be made to operate as a crosscorrelator. Suppose a transparency is placed in plane P2, whose transmission is given by H(p, q). The modified light distribution in this plane is now R(p, q) = F(p, q) H(p, q) 7

Institute of Science and Technology The University of Michigan Lens L2 takes the Fourier transform of this distribution and displays it in plane P3 as o0 r(x, y) = 2 JF(p, q) H(p, q) e(px+qy) dp dq -GO By use of the convolution theorem, o0 r(x, y) = fff(x - u, y - v) h(u, v) dudv -co If N(p, q) is uniform for all (p, q) of interest, the optimum filter is H(p, q) = kl S(p, q), and r(x, y) = kf(x, y) * s(x, y) which is the cross-correlation of the signal with the input transparency (* denotes convolution). If two transparencies are placed in contact, their complex transmissions are multiplicative.' Consequently, to synthesize the filter described in Equation 3, we need to insert a transparency whose transmission is 1/N(p, q) in plane P2 in addition to the transparency representing S(p, q). When this filter is used, the output of the system essentially represents the probability that a signal has occurred at any point in the input; a bright spot in the output indicates high probability, whereas low light levels indicate low probability. This process simultaneously detects all signals with similar orientations in any location in P1, as can be seen by observing that the spectrum of a translated signal is modified by a linear phase factor. This phase factor contains precisely the information required to image the signal at the proper position in the output. In contrast, the system is sensitive to the orientation of the signal; but a rotation of the filter relative to the input will detect these signals sequentially. Other optical configurations will perform the required operation, but the configuration shown in Figure 4 is most convenient and has optimum frequency response. The process could be carried out with two lenses or even a single lens. These configurations are discussed in Appendix B. 4.2. NONCOHERENT OPTICAL SYSTEMS Noncoherent optical systems have limited usefulness, since they can be used only when N(p, q) = N for all (p, q) of interest. This restriction frequently results in processing such a small amount of data that the system becomes impractical. The noncoherent system is further Unless otherwise noted, the word "transmission" will be used in this report to refer to specular amplitude transmission of the transparency. 8

Institute of Science and Technology The University of Michigan limited by the fact that f(x, y) must be real. Since a noncoherent system does not exhibit a frequency plane similar to that of a coherent system, H(p, q) must be realized in the space domain. The problem is to design a system with impulse response oo kf j (px+qy) h(x, y) 47=T- S(p, q) e p dpdq -00 = k1 s(-x, -y) This result can be accomplished easily by using a reference function optical system. In the optical system shown in Figure 5, plane P1 is an extended source of diffuse illumination (not necessarily monochromatic). The transparency representing f(x, y) is placed in plane P2, and the reference function representing h(x, y) is placed in plane P3. Since the signal is real, h(x, y) = s(-x, -y). A ray of light from a point (x1, Y1) in plane P1 passes through the point (x2, y2) in plane P2 and is attenuated by f(x2, y2). This ray passes through the reference f(x, y) h(x, y) fd- d f P1 L1 P2 P3 L2 4 FIGURE 5. A NONCOHERENT PROCESSING SYSTEM function at a point (x3, y3) and is further attenuated by a factor s(-x3, -y3). From the geometry we see that x4 x2 - x3 f d Y4 Y2 Y3 f d The intensity of light in the image of the point (x1, y1) is the summation of all rays parallel to the ray described; i.e., oo 00 r(x4, Y4) = fff(x2, Y2) (-x3, -Y3)dx3 dY3 -00 ff(x2, y2) s f- x - dX2 y2 -00 9

Institute of Science and Technology The University of Michigan By a change of variables we have oo r(x, y) = k f(x + u, y + v) s(u, v)dudv -o0 which is the desired output. The constraint on N(p, q) limits the usefulness of the noncoherent system. 5 REALIZATION OF THE OPTIMUM FILTER Having determined that the optimum filter which maximizes the ratio of peak signal energy to mean square noise energy is given by Equation 3, we must find some method to realize H(p, q). Except for the fact that H(p, q) is usually a complex quantity, photographic film would be the prime candidate for recording the filter. But only when H(p, q) is nonnegative can it be realized on film. Since N(p, q) > 0 for the nontrivial case, it can always be realized on film. Recall that N(p, q) can be found by averaging over an ensemble of sample functions. A better approximation to N(p, q) can usually be found by dividing sample functions into subclasses with distinctly different noise structures. (A class might be all backgrounds consisting of natural terrain, or the structure associated with populated areas, or the background structure associated with radar returns, etc.) Since a priori knowledge exists as to which class of noise functions is being processed, the appropriate N(p, q) can be selected. 5.1. REALIZATION OF NONNEGATIVE FILTERS The first step in realizing any function on photographic film is a brief review of the film's transfer characteristics. A typical curve of density versus long exposure is shown in Figure 6. This curve is characterized in its linear region by Dn= y (log E -log E) (8) n n n o where y is the slope of the straight line E is the exposure n E is the intercept of the straight line 0 D is the intensity density n means that a negative transparency is used The coherent system operates on the transmission of the film so that e-D/2 (9 T = e (9) 10

Institute of Science and Technology The University of Michigan Institute of Science and Technology The University of Michigan H Q / On E0 LOG EXPOSURE FIGURE 6. TYPICAL CURVE OF DENSITY VS. LOG EXPOSURE or T C (10) n n/2 E n If we make E proportional to N(p, q) and require y = 2, we have realized the denominator of H(p, q). The exposure can be made proportional to N(p, q) by photographing a rotating sector of a circle. Since n(x, y) is assumed to be isotropic, N(p, q) is rotationally symmetric. The sector is constructed so that s(p.) N p) p oc N(p) p =Vp2 + q (11) 2vup where s(p) is the arc length at radius p; see Figure 7. This sector, uniformly illuminated, is rotated many times during the exposure interval, with the result that E oc N(p) and T = C/N(p). This process must be modified in order to realize the numerator of H(p, q), since the relationship between T and E is not linear. However, if the negative is contact copied onto another film, the transmission can be made proportional to the original exposure. The exposure on the second film (referred to as a positive transparency) is proportional to T. Applying Equations 8 and 9, we have (y y/2) T =C E (12) p 1 n 11

Institute of Science and Technology The University of Michigan s(p) FIGURE 7. SECTOR OF CIRCLE USED TO GENERATE LOW-FREQUENCY REJECTION FILTERS where T is the specular amplitude transmission of the positive P y 7y is the specular y-product of the process If we require nyp = 2, and make E proportional to the amplitude of the numerator, we realize the numerator of H(p, q). By placing the two transparencies in contact, we realize the entire filter for nonnegative H(p, q). 5.2. REALIZATION OF A REAL FILTER FUNCTION If H(p, q) is real, its values lie on the real axis in the complex plane, with H(p, q)l < 1. We can realize the negative values of H(p, q) by multiplying its magnitude by a phase function which delays the the light by X/2. At the Institute of Science and Technology this is done by film reliefing. When certain films are exposed and bleached, depressions are left in the emulsion. The transparency is immersed in a cell (Figure 8) filled with a liquid whose refractive index is such that the actual thickness is reduced to an optical thickness of X/2. The equation governing the parameters is t= T(nf - ng) (13) where t is the optical thickness of the depressions T is the actual thickness of the depressions nf is the refractive index of liquid n is the refractive index of the emulsion g The resulting phase function is placed in contact with the transparencies representing the magnitude of H(p, q) and N(p, q). All three transparencies are immersed in the liquid cell to minimize unwanted phase errors in the film base. 12

Institute of Science and Technology The University of Michigan Glass Plate e Decalin, nf = 1.4804 Gelatin n = 1.5150 Glass Sides/ FIGURE 8. LIQUID CELL 5.3. REALIZATION OF THE COMPLEX FILTER Since the denominator of H(p, q) can be realized as described in Section 5.1, we will concern ourselves with the problem of realizing the complex numerator of H(p, q). One problem encountered in the realization of the optimum filter is the determination of both the amplitude and phase of oo S(p, q) = Jjs(x, y) exp j(px + qy)dxdy -oc One cannot merely use a spherical lens to take the Fourier transform of s(x, y), as described in Appendix B, because any physical detector measures only the intensity of S(p, q), not its phase. The second problem is to realize S(p, q) after the analysis. Since the phase function is continuous, the reliefing technique described has little value because it requires a continuous relief ing process, which is almost impossible to construct in the two-dimensional case. We will now describe a method of analysis that leads directly to the realization of the complex conjugate of the signal spectrum. A Mach-Zehnder interferometer can be used to determine the phase in a distribution of light by combining that distribution with a reference wave whose amplitude and phase distributions are known. This interferometer is modified for our purposes as shown in Figure 9. 13

Institute of Science and Technology The University of Michigan Collimating Lens L xBeamsplitter Mirror Point Source ~.~.I ofuco |n Reference of -.. Monochromatic Beam Light.. Beamsplitter Mirror P1 L1 P2 Signal Objective Output Plane (x, y) Lens R(p, q) + S(p, q) FIGURE 9. MODIFIED MACH-ZEHNDER INTERFEROMETER The signal s(x, y) whose Fourier transform is to be found is inserted in one beam of the interferometer with a spherical lens. The lens effects the Fourier transform of s(x, y) at its back focal plane, outside the interferometer. A phase delay is placed in the reference beam to maintain temporal coherence. If we temporarily neglect aberrations in the interferometer, the observed output in the back focal plane of the lens is G(p, q)= IR(p, q) + S(p, q)|2 IR(p, q)2 + IS(p, q)2 + R(p, q) S(p, q) + R(p, q) S(p, q) (14) where R(p, q) = R(p, q)J exp j((p, q) is the light coming from the reference beam, and S(p, q) = IS(p, q)| exp jO(p, q) is the signal spectrum. We can rewrite Equation 14 as G(p, q) = R(p, q)|2 + |S(p, q)|2 + 2Re[R(p, q) S(p, q)] = R(p, q)2 + IS(p, q)2 + 2|R(p, q)l|S(p, q) cos [s(p, q) - 0(p, q)] (15) 14

Institute of Science and Technology The University of Michigan Both the amplitude and phase of R(p, q) could be adjusted to determine O(p, q); but since the phase information of S(p, q) is contained in G(p, q), a nonnegative function, we will show how G(p, q) can be recorded on photographic film to realize the optimum filter. The film is exposed so that its transmission is proportional to G(p, q). If we combine this film with the film on which 1/N(p, q) is realized, the transmission of the combination is G(p, q) R(p, q)12 + IS(p, q)12 R(p, q) S(p, q) R(p, q) S(p, q) + + (16) N(p, q) N(p, q) N(p, q) N(p, q) Suppose we require IR(p, q)l to be a constant and O(p, q) to be linear in (p, q); then G(p, q) G~p, q) - A~p q) + H(p, q) -j(bp+cq) N (p, q) e q)+ H(p, q) ebPc (17) IR(p, q)12 + lS(p, q)12 where A(p, q)= R( q) (p, q)12 N(p, q) H(p, q)= (p), q) N(p, q) b, c are constants Thus, the third term of Equation 17 is the desired filter function multiplied by a linear phase factor. The problem is to separate this term from the other two terms. This can be accomplished by inserting the filter represented by Equation 17 into the optical system (Figure 4) at plane P2. Lens L2 performs the separation of the three terms in the output by taking the transform of the light distribution in P2; i.e., output = i G(p, q) ej(px+qy) output = 1 F F (p, q)) (p qy)dpdq (18) 42 N(p, -cc Substituting Equation 17 in Equation 18, we have o output = 1 fF(p, q) A(p, q) e(+q)dpdq 47r -oo + 2 F(p, q) H(p, q) e [(-b)p+(y-c)q] d d 47f -oo + 2 f- F(p, q) H(p, q) e [( +( ddq (19) -oo 15

Institute of Science and Technology The University of Michigan The first term of Equation 19, which appears on the optical axis, is of no particular interest. Nor, in this discussion, is the second term, which appears at x = b, y = c. The third term is r(x + b, y + c), where r(x, y) is defined by Equation 5 and is exactly the output expected from an optimum filter. This term appears with its center displaced from the optical axis by an amount x = -b, y = -c. At this point it will be convenient to set c = 0, since it is arbitrary; but in order to avoid overlap of the three outputs the value of b must be such that b I A, where A is the length of the signal in the x direction. The fact that this output occurs off axis by a distance x = -b is not important, since the output sensor can be located properly to detect it. In passing, a comment will be made on the significance of the second term of Equation 19. If N(p, q) = N for all (p, q), we saw that the third term could be written as a cross-correlation integral (for c = 0); i.e., 00 r(x + b, y)= fI f(u, v) s(x + b + u, y + v)dudv (20) -oo The second term of Equation 19 can then be written as a convolution integral, i.e., co r(x - b) (, y) f(u, v) s(x - b - u, y - v)dudv (21) -oo Thus, when N(p, q) is uniform, the cross-correlation and convolution of the signal with the input function are both displayed in the output plane. If the mirrors or beam splitters in the interferometer have aberrations, the effects expressed in Equation 19 appear, and may degrade the output somewhat. The aberrations in the signal analysis beam can be neglected, because the signal is usually small compared to the aperture of the interferometer. Denote the aberrations in the reference beam by exp j4(p, q)o This aberration can be carried through the analysis to get (for the term of interest) oo r(x + b), y = F(p, q) H(p, q) e'p q) e[(x+b)p+qy dpdq (22) 47u2 -of By use of the convolution theorem, this term can be written as r(x + b, y) = f(x, y) * h(x, y) * 4(x, y) where 4/(x, y) is the transform of exp jI(p, q)o But 4/(x, y) is exactly the same form as the impulse response of a lens having the aberrations exp j4(p, q). Equation 22 is the output of a 16

Institute of Science and Technology T he University of M i ch i a perfect system as "seen" by a lens with aberrations equal to I(p, q). It is apparent that a good quality interferometer is needed if r(x, y) has high-frequency content. (Aberrations in the various lenses can be treated in the same way as imperfections in the interferometer.) There is an alternative optical system which can be used in place of the interferometer to realize G(p, q). In that system, shown in Figure 10, lens L1 collimates a point source of monochromatic light. The signal is placed in one part of the beam with the necessary phase delay. (If the source has sufficient temporal coherence, the phase delay in these systems can be discarded.) Lens L2, placed in the other part of the beam, focuses the light to a point in P2 at a distance b from the center of the signal. Lens L3 simultaneously takes the Fourier transform of the signal and supplies the reference wave. The reference wave automatically has a linear phase component equal to 0(p, q) = bp, and the light distribution in plane P2 is identical to that given by Equation 14. Phase Delay oPoint ~ s(x, y) Source A" L1 L2 L3 P2 FIGURE 10. ALTERNATIVE OPTICAL SYSTEM FOR REALIZING COMPLEX FILTERS 6 SOME NOTES ON THE PERFORMANCE OF MATCHED FILTERS 6.1. CHANGE IN THE SCALE OF THE SIGNAL We will evaluate the performance of a filter matched to a signal when the input is the signal with a change in scale. We will assume white Gaussian noise statistics. Let s(x, y) denote the signal for which the filter was optimized, and let s(mx, my) be the input signal. Then, from Equation 5, the output of the filter is 0o r(x, y) =- ff S(p, q) H(p, q) e(px+qy) dpdq -oo 17

Institute of Science and Technology The University of Michigan when the input signal is s(x, y). The output when s(mx, my) in the input is denoted by rm(x, y). We take as a measure of the change in performance of the filter o00 2 -1ff S(p, q) H(p, q)dpdq I/ \|~y)2 4 2 2 R= ir(xY) 0 2 (23) lr(x, y)l 2 i r x=y=O 12 J4-2 S(p/m, q/m) H(p, q) dp dq -00 Applying the Schwarz inequality to both numerator and denominator and noting that the equality holds in the numerator by Equation 3, we have o0 jfS(p, q)2 dpdq oo 2 00 2 f 1 S(p/m, q/m) dpdq -oo m where m is the scaling factor. 6.2o CHANGE IN THE ORIENTATION OF THE SIGNAL We wish to find some basis for evaluating the performance of a filter matched to a certain signal when the input is the same signal with a different orientation. The output of the system is conveniently expressed in polar coordinates as oo 2 IT rsh(p, )= i s(r, 0) h(r + p, 0 + y)r dr dO (24) Considered as a function of p, the output is a maximum at p = 0. To investigate the effect of signal orientation on the output, we consider the normalized function r sh() f() = rsh (25) (ly) rE~h(0)(25) For signals which are nearly rotationally symmetric, f(y) z1 for all y. Our first task is to find a measure of the degree of nonrotational symmetry (termed orientativeness) or the signal. We begin by forming the function g(r, 0) = s(r, 0) - k(r) 18

Institute of Science and Technology The University of Michigan where k(r) = s(r, 0)dO Thus, g(r, 0) represents a measure of the difference between the signal and a function which has rotational symmetry. We then evaluate Rgh(y) = Rsh() Rkh() But R2f o2 27r Rkh(y) = J- { fo s(r, p) h(r, 0 + y)r dr dO df Since we integrate over all 0, we can write 0 + y = a and let a - > = j, so that Rkh =27. s(r, ) h(r, + 1)rdrdljdf = Ave[Rsh()] (26) and Rgh(y) = Rsh()- Ave[Rsh(Y) We now look at the normalized function R gh(y) fl(Y (27) Rgh(O) and define the "orientativeness" or the signal to be Or= 1 0 <y < ^Y c C where yc is the angle for which fl() = C, 0 < C < 1. By constructing the fi one can always find a y for which fl(y) = C for any signal. The function fl(y) is now the measure of the performance of a filter matched to a signal when the input is a signal which has undergone a change of orientation. This result is highly dependent on the shape of the signal, whereas the result of Section 6.1 was not. 19

Institute of Science and Technology The University of Michigan 7 EXPERIMENTAL RESULTS A few experimental results will serve to illustrate the theory of spatial filtering and indicate the potential to be expected when complex filters can be realizedo A practical result of realizing complex filters by the technique described in Section 5.3 is that the noise rejection capability is better than that of conventional filters, since the minimum transmission of films is not zero when the y-product is fixed, and some noise passes through the filter. In the method described in Section 5.3, the carrier frequency is recorded only for those values of (p, q) for which S(p, q) # 0. Since the noise passing through the filter where S(p, q) = 0 is not deviated into the output of interest, the effective transmission at those points is in effect equal to zero. 7.1. DETECTION OF SIMPLE GEOMETRICAL SHAPES The first example presented is the detection of one of the elementary geometrical shapes shown in Figure 11. Any of the shapes could serve as the signal; we chose the small rectangle first. We realized the complex filter of this signal by the method described in Section 5.3; the output of interest is shown in Figure 12. Note that all signals with proper shape and orientation were detected simultaneously. FIGURE 11. SET OF GEOMETRICAL FIGURE 12. DETECTION OF RECTANGLES SHAPES The second signal selected was the "L" shape, which has a complex spectrum. The Fourier transform of the complex filter, shown in Figure 13, illustrates the fidelity with which the complex filter was realized. The light distribution in the center of the output is the transform of the first two terms of Equation 14. The L in the upper corner is the transform of the third 20

Institute of Science and Technology The University of Michigan FIGURE 13. RECONSTRUCTION FROM A COMPLEX FILTER term, and the other L is the transform of the last term. Note that these two images are inverted and reversed relative to each other, which graphically demonstrates that the Fourier transform of [S(p, q)] = s(x, y) and the Fourier transform of [S(p, q)] = s(-x, -y). Of course, since s(x, y) is real, s(-x, -y) = s(-x, -y). Figure 14 shows the output, which is the cross-correlation of the "L" with the input function. Note the symmetry in the output correlation, which is a necessary feature of cross-correlation. The other L's in the input, having different orientations, do not give as large an output (cf. Section 6.2), but a rotation of the filter relative to the input would sequentially detect them. The output, which is shown for illustration in Figure 15, is the convolution of the signal with the input function. Note that it is asymmetrical, a consequence of convolution unless the signal is even. 7.2. DETECTION OF ALPHANUMERICS The second example is the detection of alphanumerics. An interesting variation of the first example is to record the alphabet (shown in Figure 16) via its complex spectrum as the filter function. It is a simple matter to select any one of the alphanumerics as the signal to be detected and use it as the input signal. The output, when the input is the letter "g," is shown in Figure 17. Since the filter (which is the normal input function in disguise) does not have to be changed while the search is carried out, this technique suggests a method for scanning a printed page for the presence of any particular alphanumeric. 21

Institute of Science and Technology The University of Michigan FIGURE 14. CROSS CORRELATION OF L WITH THE SET OF SHAPES SHOWN IN FIGURE 11 FIGURE 15. CONVOLUTION OF L WITH THE SET OF SHAPES SHOWN IN FIGURE 11 FIGURE 16. SET OF ALPHANUMERICS FIGURE 17. DETECTION OF LETTER g 7.3. DETECTION OF AN ISOLATED SIGNAL IN RANDOM NOISE In the preceding two examples the noise spectral density was uniform enough to be considered white. This final example shows the detection of a signal which is immersed in a noise background with nonuniform spectral density, and demonstrates the power of using a coherent system for signal detection. 22

Institute of Science arnd Technology'I e University of Michigan Institute of Science and Technology The University of Michigan Figure 18(a) shows a figure in a noise background. Since the noise background is predominantly low-frequency, the denominator of the filter must be realized. Figure 18()) shows that the background noise has been completely suppressed and the signal has been detected. I~:ci4,' ". (a) (b) FIGURE 18. 1)DETECTION OF ISOLATED) SIGNAL IN NOISE BACKGROUND. (a) Signal plus noise. (b) Detection of signatl, 23

Appendix A OPTIMIZATION IN PRESENCE OF MUTUALLY EXCLUSIVE NOISE One consideration in optimum filtering of photorecords is that the noise is frequently mutually exclusive and not additive. We can consider the signal g(x, y) to be multiplied by an indicator function I(x, y), which is a random process on x, y having a known autocorrelation function. Since the desired signal is now random rather than known, the optimum filtering process is found by performing a least squares analysis. A block diagram describing the system is shown in Figure 19. Uncorrupted Available Signal I(x, y) Data g(x, y) f_ ^ y) h(x, y) "| -- H(p, q) 1 [I - I(x, y)] -- -- n(x, y) P(x, y) P(P, q) FIGURE 19. PROCESSING SYSTEM WHEN MUTUALLY EXCLUSIVE NOISE IS PRESENT I(x, y) is an indicator function taking on values (0, 1) and is an approximation to a mutually exclusive noise function. P(p, q) is some operation on G(p, q); in our case P(p, q) = 1 for all (p, q) of interest. The problem is to find a filter function H(p, q) which will minimize the expected value of 1e2. We will continue the analysis for one dimension; the two-dimensional extension is obvious. We write E [E(x + u) (x)] = R(u) which is the autocorrelation function of the error term (E is the expected value). R (u) will be expressed in terms of the usual autocorrelation functions of g(x), f(x), and n(x), and it can readily be seen that E(| 12) = R (0). 24

Institute of Science and Technology The University of Michigan A useful relationship between R (u) and its spectral density is given by RE(u) = k S(p) e pudp -.00 and we note immediately that RE(O) = E[el2] = k SE(p)dp -o0 Since S (p) is the Fourier transform of an autocorrelation function, S (p)' 0 for all p, and we can minimize R (O) by minimizing S (p). To find S (0) we write e(x) = f(x) * h(x) -g(x) * p(x) where * denotes convolution. Then RE(u) =E[E(x+ u) E(x)] = E[f(x + u) * h(x + u) - g(x + u) * p(x + u)] [f(x) * h(x) - g(x) * p(x)J R(u) = Rf(x) * h(x) * h(-x) + Rg(x) * p(x) * p(-x) - Rgf(X) * h(x) * p(-x) - Rfg(-X) * p(x) * h(-x) Taking Fourier transforms of both sides and using the result that [Ry(X)* h(x) * p(-x)] = Sy(p)H(p)P(p) we have SE(P) = Sf(p)|H(p)2 + SglP(P)2 - Sgf(p)H(p)P(p) - Sg(p)P(p)H(p) Complete the square in H(p) to get wh2 erelS gf)P (P p) 2 | o(p)P(p)12 Se(P) = Sf(p)/2H(p) - Sg(p)P(p) 2 S(p Sf(p) Sf(P) where Sf(p) is the spectral density of the available data S (p) is the spectral density of the signal g Sgf(p) is the cross-spectral density of the signal with available data P(p) is the desired operation on the signal H(p) is the filter function which is being optimized 25

Institute of Science and Technology The University of Michigan Institute of Science and Technology The University of Michigan We can choose H(p) to minimize S (p). It is apparent that H(p) has no influence on the last two terms of S (p); therefore we take (p) = Sgf(p)P(P) Sf(p) which minimizes S (p) and therefore minimizes E(|eI2). To recover the signal with minimum error, we let P(p) = 1 for all p, and we have Sgf(p) H(p) = gf Sf(P) To get this into a useful form, we write Sgf(p) and Sf(p) in terms of the statistics of the input functions. To determine Sf(p), we have f(x) = n(x) [1 - I(x)] + g(x)I(x) and Rf(x) = E [f(x + u)f(x)] = (1 - 2a) Rn(x) + Rn(x) RI(x) + Rg(x)RI(x) + 2abc - 2bcRI(x) Take Fourier transform of both sides to get Sf(p) = (1 - 2a)Sn(P) + Sn(p) * S(P) + S(p) * S(p) + 2abc6(p) - 2bcS(p) where a = E [I(x)] b = E[g(x)] c = E[n(x)] 6(p) = C sine kA (A = aperture of optical system) A similar analysis for Sgf(p) gives Sgf(p) = aSg(p) + (1 - a)bc6(p) Thus we can now write the optimum filter in terms of the known statistics of the input data and the indicator function, as follows: aS (p) + (1 - a) bc6(p) H(p) = (1 - 2a)Sn(P) S(P) (p) + S(p) * S(p) + 2abc5(p) - 2bcSi(p) This result shows that, in general, the best filtering operation is to attenuate the spectrum heavily where there is little signal energy, and vice-versa. More precise information on H(p) will be obtainable if the autocorrelation function of I(x) is known. 26

Institute of Science and Technology The University of Michigan Appendix B THE FOURIER TRANSFORMING PROPERTY OF LENSES B. 1. DERIVATION OF THE FOURIER TRANSFORM RELATIONSHIP The theory outlined in Section 2 of this report is based on the assumption that a spherical lens can effect the two-dimensional Fourier transform of a complex distribution of light. It is also asserted that the transform relationship holds, to within a phase factor, for a wide variety of positions of the lens relative to the input transparency, so that we can cascade lenses to take successive transforms. The analysis in this appendix is in one dimension; the extension to two dimensions is obvious, though not simple. We begin by assuming that a transparency with complex transmission f(x) is placed in plane P1 at a distance d from the lens in plane P2 (see Diagram 1). Although the transform relationship is found with far less effort if d = f, the results are not general enough to apply in evaluating the performance of combinations of lenses. We proceed from the application of Kirchoff's formulation of Huygens' principle [7], which indicates that the disturbance at a point in P2 due to a disturbance in P1 is given by I d - - - f 2y r 2 A}/ xo P1 P2 P3 DIAGRAM. 1 Af(x ) ejkr Af(xo) (1 + cos ) x(28) 2 x o(28) where X is the wavelength of light A is the amplitude of illumination at point x r is the distance from x to y o o 0 is the obliquity angle, i.e., the angle from x to yo measured from the optical axis f(x ) is the transmission of transparency at x = xo k = 27r/X 27

Institute of Science and Technology The University of Michigan The incoming illumination is E(x) = A(x) exp jq(x); usually A(x) and 4(x) are constant. Since the exact value of 4(x) is unimportant, we can set it equal to zero and regard it as the reference phase. The total contribution at the point yo is g(y) =/iAA f f(x) exp (jkr)(1 + cos 0) dx (29) The obliquity factor is usually neglected by assuming that 0 remains small throughout the integration. Though this may be true when d - f, it is not true when d - 0, and a better reason for neglecting the obliquity factor must be sought. Consider the effect of the contribution from the points in a small region in P1 on a given point in P2. If 1r2 - rl >> X, the contribution at yo averages to zero if f(x2) - f(x1). Thus we can determine the maximum permissible value of 0 as a function of d and the frequency content of f(x). In terms of x, y, and d, the condition that Ir2 - rl >> X implies that 12 + (x - y)2 d + (x + Ax - y) >> X, or that 12xAx + (Ax) - 2yAxl >> 2Xd. 2 Neglecting (Ax) with respect to the other terms, we have Xd Ax >> d (30) x-y Suppose we agree that the obliquity factor can be neglected if 1 + cos 0 99 2_ 0.99 which implies that tan 0 < 0.20. But 0 = arc tan (x - y)/d, so that x y < 0.20 (31) d and from Equation 30 we have that Ax >> 0 Since Ax is the distance over which f(x) must not vary appreciable, the highest allowable frequency in f(x) is ma = 1/Ax << 0.20/h << 375 1/mm max Since the highest frequency encountered in typical input signals is approximately 30 to 50 I/mm, it seems safe to neglect the obliquity factor. Note that this analysis makes no restriction on the relative aperture of the system, the permissible field of view, or the value of d. These restrictions comprise the reasons usually cited for neglecting this factor. If we ignore the obliquity factor, the light distribution in P2 is gy= Azx(32) g(y)= i-A f(x)eJkrdx (32) P1 28

Institute of Science and Technology The University of Michigan Expanding r = Vd + (x - y) by the binomial theorem, we have r=d 1 + xY) 2 ( )4' ] (33) 2 d''' From our previous assumption, (x - y)/d < 0.2, so that we need retain only the first two terms of Equation 33. Since the i/F term in the denominator is relatively insensitive to the small variations in (x - y) /d over the region of integration, we set r = d and take it outside the integral. Also the phase term exp (jkd) is constant and can be dropped from the analysis. The 2 lens at P2 is represented by the phase factor exp [-j(ky /2f)], where f is the focal length of the lens. The P2 plane can be considered the back principal plane of the lens. We will proceed from P2 to P3 in the same way as from P1 to P'. The total contribution at a point 5 in P3 is p1 2 Looking at the exponent only, we have k 2 2 k 2 k 2 2 exponent= j (x2- 2xy+ y) - j- y + j-2-(y -2y + (35) Let f/d = m, and complete the square in Equation 35 to get exponent = 2f [ - (y - ix + jf (yI - /-m yO) - j f( x) (36) 22 2 Substituting Equation 36 in Equation 34, we get A1/ = p dx dy f(x) expjf [/-~ y - (- x + ~)]2 xxp(-Vi exp (-j - x) exp [ —[(1- /) y)} A- - fdx f(x) exp (-j fii x) v(x, ) (37) where v(x, = f dy exp{j 2f [ y - ( x + i )]2 exp {- j [(1 - V-))yj] (38) 2 Evaluating v(x, ~), we let Vm y - (Vm x + 5) = p, and we have v(x, O) = f.J. dp exp [-j- (p (39) m 1.1.~~~~~~~~~~~~~~~~(9 29

Institute of Science and Technology The University of Michigan where u.l. = Vm A - (Vm x + +) 1.1. = -m A - (VmI x + ) 2A = aperture in P2 We can pass the phase factor exp -j4 f (-m) (m x + ) through the integral, since it is not a function of the variable of integration. Thus -jA exp [-j k1 - 2] 2 F(=) 3 ex= J dx f(x) exp [-jfk xj x exp [-j k- ) Vm x] b(x, i) (40) where I, 1.k 2 k I b(x, ) = f exp [j p2] exp [ -jf(1 - J p]dp (41) Note that part of the phase term in the integrand of Equation 4 cancels out, leaving -jA exp [-4j( rViii>/2] -jA ep [ -j f- m dx f(x) exp ( — x b(x, ) (42) F(~): (42) 1 If b(x, t) = k for all (x, t) of interest, and if m = 1, we have an exact Fourier transform relationship between F(Q) and f(x). Further, the integral in Equation 42 is independent of m, which is to be expected since the scale of F(5) is not a function of m. (The manner in which the phase term depends on m will be discussed after b(x, 5) is evaluated.) Since Equation 41 determines the validity of the Fourier transform relationship, we modify b(x, 5) to a Fresnel integral, which is easier to evaluate than the form given. Completing the square in the exponent, we get b(x, e) = exp[-j (1 -2 + m ) ] J exp p p- (1 a ) ~]dp} (43) Change variables by letting p -,a = /f/k v so that b(x, S;i___e_ [-j (1 - 2,i + m) /2] e exp [j(r/2)v2] dv (44) / 1.1. 30

Institute of Science and Technology T he University of M i. hi ga where the new upper and lower limits are u.l. = [A - x - /m] 1.1. = [-A - x - /m] Before evaluating Equation 44, we will discuss the general behavior of the function (45a) (45b) A-u F(u) = exp[j(jr/2)v] dv (A+u) which is a Fresnel integral in its standard form. The evaluation of F(u) cannot be given in a closed form, but the function is tabulated extensively. If the function is written as r.A-u F(u) = iA -(A+u) 2 A-u cos (fr/2)v dv + j A+u) J-(A+u) sin(r/2)v dv I =x + jy (46) one can use a curve known as a Cornu spiral, which plots the first integral against the second (see Figure 20). Values of u are read along the curve, and the corresponding values of x and y are read from the coordinate axis. The magnitude of F(u) is found in the usual way. It can be seen from the curve that F(u) has its greatest change in value near lul = A, so that we have as an approximation u 0.5 0.5 FIGURE 20. CORNU SPIRAL 31

Institute of Science and Technology The University of Michigan F(u) = |u | < A =0 ul >A In Equation 44, u = x + -, and the factor in both limits represents a scaling factor. b(x, ) = V2iff/k exp -j~( - ( m ) 2] (47) which is independent of x as expected, since the given requirement is equivalent to demanding that the system is space invariant, i.e., F(Q) is not a function of the position at which the signal is found in P1. Substituting Equation 47 in Equation 44 and simplifying, we have F() jexA [j k1 - m)2 f(x) ex [ -x] (48) P1 Note that if the input transparency is in the front focal plane of the lens (m = 1) the result is an exact Fourier transform relationship. For m < 1, the factor (1 - m)/m > 0, and the quadratic phase term indicates that the lens is capable of forming a real image of f(x) to the right of the lens at a distance f d = 1 1-m If m > 1, the image is virtual and cannot be imaged to the right of the lens without the aid of a second lens. The remaining task is to evaluate b(x, 5) for the case in which the condition A >> (x + is not satisfied. We want to investigate three different cases, m - oo, m = 1, and m - 0. We can facilitate the analysis by restricting our attention to values of x > 0 and 5 > 0, since similar results will hold for x < 0, 5 < 0. In the first case of interest, m is very large; i.e., the transparency is close to the principal plane of the lens. In this case 5/m is very small and /2m/(kf) (A-x) b(x,) = f(5) { exp(jv2) dv (49) -v2m/(Xf) (A+x) where f/() =rf/k r exp [1 - 21m + m) 2] f(): /f/k exp -j - f Q) W7 Tf m 32

Institute of Science and Technology The University of Michigan The limit on the maximum signal aperture to satisfy space invariance is IxI < A. Hence, in order to obtain maximum system aperture, the input signal should be placed very close to the lens. A second lens is necessary to image f(x), but it can also be selected to maximize system frequency response. This is discussed more fully in Appendix B, Section 3. In the second case to be discussed, m = 1. Then f(t) = Lrf/k and 42/(Xf) (A-x- ) b(x, =)= -{ exp ( v dv (50) -V2/(Xf) (A+x+ ) For small x we see that b(x, 5) reaches its half-amplitude point for 1a1 = A and b(x, ) = 2rf/k I < A =0 |II>A This is the classical result, the so-called aperture limited frequency cutoff. However, this result is valid only for small signal apertures; as the signal aperture increases, the symmetrical band pass decreases and the lens system becomes space variant. It is almost impossible to continue this discussion without placing a limitation on the frequency content of the signal. Suppose the highest frequency of the signal is displayed at o0, which corresponds to a maximum frequency of 27rt Po= Xf rad/mm We then find the largest range on the signal aperture which will retain space invariance. From Equation 44 we see that this condition is satisfied if (x + ~ /m) < A for x > 0 (x - ~o/m) < -A for x < Thus, to have space invariance we must restrict the signal aperture such that |Xi < (A - /m) (51) From Equation 51 it is easy to see the result for m - 0. As m becomes smaller, the input signal is farther from the lens, and the allowable range on IxI rapidly decreases. In those regions where IXI exceeds the limitation imposed by Equation 51, the symmetrical frequency response is reduced, and the result is a space variant operation. For perfect lenses, b(x, 1) adequately describes the space variant characteristics of the lens, but aberrations usually cause the system to be space variant before Equation 51 is violated, except for small values of m. 33

Institute of Science and Technology The University of Michigan This section presents a general discussion of the Fourier transform properties of a lens operating with coherent illumination. The generality allows a determination of the validity of the Fourier transforming property of the lens. Equation 44 relates the frequency response of a lens to the region over which it can be considered space invariant. An application of the results of this section will be used in Section B. 3, in evaluating complete lens systems. B. 2. AN ALTERNATIVE APPROACH TO THE TRANSFORM RELATIONSHIP If Section B.1 masks the fundamental results to be obtained from b(x, t), the following approach may be instructional. It is known that a 6-function is equivalent to a point source in the input plane (Diagram 2). A point source in P1 creates a spherical wave at P2 with radius Point 0 Source X f/m - f - P1 P2 3 DIAGRAM 2 f/m. Each element on the wave can be considered to be a certain frequency, and if the highest 27rt frequency of the signal is P =-, we require that all the energy from that element enter the 0 At lens. Let the element representing P be located a distance (x + y) _ A above the optical axis. Then tan 0 y /m. To get all the light from this element into the lens we Then tan = = or y = /m. To get all the light from this element into the lens, we require that (x + y) < A or x < (A - 5o/m). A similar situation holds for x < 0; so we have Ixl < (A - 5 /m), which is Equation 51. Thus, this approach yields the same major result as the detailed analysis of b(x, 5). The minor oscillation of b(x, 4) is not accounted for, since diffraction effects of the lens aperture were not considered here; otherwise the results would be identical. B.3. EVALUATION OF OPTICAL SYSTEMS In this section, the results from Section B.1 will be used to evaluate three basic systems which perform fundamental spatial filtering operations. We will evaluate the systems on the basis of (1) the maximum frequency that the system will image for a limited region in the input plane, (2) the maximum signal aperture that can be imaged when the input signal is band-limited and the lens system operates as a space invariant system, and (3) minimum total system length. Each system will have unity magnification. 34

Institute of Science and Technology The University of Michigan B.3.1. THREE-LENS SYSTEM. Perhaps the simplest system to analyze is the three-lens system shown in Figure 21. Lens L1 collimates a point source, and the input transparency is placed in plane P1 close to lens L2. From Equation 51 we note that the maximum frequency response of lens L2 is extremely high, but that the maximum frequency that can be imaged is 27T limited by L3 to P x = Af rad/mm. The length of signal which can be imaged with space 3o max =f invariance when the signal is band limited to frequency po is / p Xf\ IXI < A - 2) mm If lens L2 has a focal length of 1/2 f, the total system length is 3f. <z S ~ f 7( f orli 2A L1 P1 L2 P2 3 P3 FIGURE 21. THREE-LENS SYSTEM B.3.2. TWO-LENS SYSTEM. A two-lens system is shown in Figure 22. In this system L1 collimates the point source and the input transparency is placed in plane P1 at a distance 2f from L2. Lens L2 is the transforming lens as well as the imaging lens. The maximum rA frequency imaged by this system is pma = - rad/mm. The maximum signal length to be max Af imaged under space invariance and maximum frequency content po is p Af\ ixi < (A - ~P ) mm The total system length is 5f. B.3.3. ONE-LENS SYSTEM. The one-lens system shown in Figure 23 can also perform the desired operation. In this case the signal is illuminated not be a plane wave, but by a divergent wave of radius r = s - d. The frequency plane is located at a distance t from lens L1, where t = st/(s - f). The image plane is at a distance q = df/(d - f) from lens L1. The maximum frequency passed by the lens for small signal lengths is p = 2rA/(Xd) rad/mm. max The maximum signal length which is imaged when the lens operates under a space invariant condition and the signal is limited to frequencies less than po will now be found. Equation 51 cannot be applied directly, because the illumination is not a plane wave and the frequency plane 35

Institute of Science and Technology The Un'iversity of Michigan Point __t __ I I Source \UHL 2' P P -2f- P 2 P 3 - 2f FIGURE 22. TWO-LENS SYSTEM t L FIGURE 23. ONE-LENS SYSTEM is not located in the back focal plane of the lens. We first find a relationship between the frequency variable p and a distance variable 5 in plane P2. Referring to Diagram 3, we have L1 A1 1 3 DIAGRAM 3 36

Institute of Science and Technology The University of Michigan 2gry P= 2A Xd and _ Y q-t t Therefore r27r t P Xd(q - t) To find the range on the signal aperture we refer to the sketch in Diagram 4. Since 0 is small DIAGRAM 4 (d > f) we can make the approximation that Ay h Ay1. Then, in terms of the maximum frequency in the signal, we have t2fpo = 2r(q - t) and s Y =-dx s -d We require that (y + Ay) < A. Thus 2( t+ < A xs tp 1 _s- d + 27r(q-t)J <A or, for space invariance, Ixl< s d IA-P s -d 2 ft 1t IxI sLA -2vr(q - t)J 37

Institute of Science and Technology T he University of M i ch i a The total system length is s + q, and it is difficult to minimize this distance since the amount of signal aperture is also dependent on it. It is clear, however, that the maximum frequency response of this system is good. If one is willing to work with a small signal aperture, the system length can be kept reasonably short. B.3.4. COMPARISON OF THE THREE SYSTEMS. It will simplify the comparison of these systems if we give numerical values to the system parameters. Suppose that Po/2T = 50 1/mm A = 25 mm X = 5000 A f = 200 mm To fix the parameters for the one-lens system, let d = 2f = q S = 4f t = 4/3 f Substituting these parameters in the one-lens system formula gives Ix <( - The systems are compared in the table. The three-lens system is clearly superior on the basis of all three methods of comparison. The one- and two-lens systems have equal frequency response, but the signal aperture in the one-lens system is only half that of the two-lens system. This waste of system aperture should be avoided by using one of the other two systems; doing so reduces the length of the system also. TABLE: COMPARISON OF THREE OPTICAL SYSTEMS One-Lens System Two-Lens System Three-Lens System P ma2x 125 1/m 125 1/m 250 1/m 27r Range on IXI for 7.5 mm 15 mm 20 mm p or or or a= 50 1/m 30% of aperture 60% of aperture 80% of aperture Total system length 1200 mm 1000 mm 600 mm Total system length 1200 mm 1000 mm 600 mm 38

Institute of Science and Technology The ~University of Michigan REFERENCES 1o T. P. Cheatham, Jr., and A. Kohlenberg, "Optical Filters-Their Equivalence to and Differences from Electrical Networks," IRE Intern Conv Record, 1954, Part 4, Electronic Computers and Information Theory, pp. 6-12. 2. E. O'Neill, "Spatial Filtering in Optics," IRE Trans Inform Theory, Vol. IT-2, No. 2, June 1956, pp. 56-65. 3. A. Marechal, "Filtering of Optical Images," Communication and Information Theory Aspects of Modern Optics, General Electric Company, Electronics Laboratory, Syracuse, New York, August 1962. 4. D. Gabor, "Microscopy by Reconstructed Wave-Fronts," Part I, Proc. Roy. Soc. (London), Ser. A, 194, p. 454, 1949. 5. E. N. Leith, "Reconstructed Wavefronts and Communication Theory," J. Opt. Soc. Am., October 1962, Vol. 52, No. 10, pp. 1123-1130. 6. L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, "Optical Data Processing and Filtering Systems," IRE Trans Inform Theory, Vol. IT-6, No. 3, June 1960, p. 391. 7. M. Born and E. Wolf, Principles of Optics, Pergamon Press, New York, 1959, p. 359. 39

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+ + + AD Div. 6/6 Inst. of Science and Technology, U. of Mich., Ann Arbor SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING, by A. B. VanderLugt. Report of Project MICHIGAN. July 63. 39 p. incl. illus., table, 7 refs. (Report No. 2900-394-T/4594-22-T) (Contracts DA-36-039 SC-78801 and AF 33(616)-8433) Unclassified report This report contains integrated descriptions of the problem of signal detection, the optimum linear filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. Experimental results show that the theory is valid. The appendixes give a treatment of the Fourier transforming property of lenses which is general enough that complete optical systems can be evaluated on the basis of frequency response and region of space-invariant operation. (over) UNCLASSIFIED I. Title: Project MICHIGAN II. VanderLugt, A. B. III. U. S. Army Electronics Command IV. U. S. Air Force V. Contract DA-36-039 SC-78801 VI. Contract AF 33(616)-8433 AD Div. 6/6 Inst. of Science and Technology, U. of Mich., Ann Arbor SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING, by A. B. VanderLugt. Report of Project MICHIGAN. July 63. 39 p. incl. illus., table, 7 refs. (Report No. 2900-394-T/4594-22-T) (Contracts DA-36-039 SC-78801 and AF 33(616)-8433) Unclassified report This report contains integrated descriptions of the problem of signal detection, the optimum linear filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. Experimental results show that the theory is valid. The appendixes give a treatment of the Fourier transforming property of lenses which is general enough that complete optical systems can be evaluated on the basis of frequency response and region of space-invariant operation. (over) UNCLASSIFIED I. Title: Project MICHIGAN II. VanderLugt, A. B. III. U. S. Army Electronics Command IV. U. S. Air Force V. Contract DA-36-039 SC-78801 VI. Contract AF 33(616)-8433 UNCLASSIFIED UNCLASSIFIED + + + AD Div. 6/6 Inst. of Science and Technology, U. of Mich., Ann Arbor SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING, by A. B. VanderLugt. Report of Project MICHIGAN. July 63. 39 p. incl. illus., table, 7 refs. (Report No. 2900-394-T/4594-22-T) (Contracts DA-36-039 SC-78801 and AF 33(616)-8433) Unclassified report This report contains integrated descriptions of the problem of signal detection, the optimum linear filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. Experimental results show that the theory is valid. The appendixes give a treatment of the Fourier transforming property of lenses which is general enough that complete optical systems can be evaluated on the basis of frequency response and region of space-invariant operation. (over) UNCLASSIFIED I. Title: Project MICHIGAN II. VanderLugt, A. B. III. U. S. Army Electronics Command IV. U. S. Air Force V. Contract DA-36-039 SC-78801 VI. Contract AF 33(616)-8433 UNCLASSIFIED AD Div. 6/6 Inst. of Science and Technology, U. of Mich., Ann Arbor SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING, by A. B. VanderLugt. Report of Project MICHIGAN. July 63. 39 p. incl. illus., table, 7 refs. (Report No. 2900-394-T/4594-22-T) (Contracts DA-36-039 SC-78801 and AF 33(616)-8433) Unclassified report This report contains integrated descriptions of the problem of signal detection, the optimum linear filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. Experimental results show that the theory is valid. The appendixes give a treatment of the Fourier transforming property of lenses which is general enough that complete optical systems can be evaluated on the basis of frequency response and region of space-invariant operation. (over) UNCLASSIFIED I. Title: Project MICHIGAN II. VanderLugt, A. B. III. U. S. Army Electronics Command IV. U. S. Air Force V. Contract DA-36-039 SC-78801 VI. Contract AF 33(616)-8433 UNCLASSIFIED + + +

AD The experimental results obtained to data indicate that this technique provides an excellent two-dimensional filtering capability that will play a key role in problems such as shape recognition and signal detection. UNCLASSIFIED AD DESCRIPTORS Optical filters Lenses Interferometers Optical equipment The experimental results obtained to data indicate that this technique provides an excellent two-dimensional filtering capability that will play a key role in problems such as shape recognition and signal detection. UNCLASSIFIED DESCRIPTORS Optical filters Lenses Interferometers Optical equipment UNCLASSIFIED UNCLASSIFIED + AD The experimental results obtained to data indicate that this technique provides an excellent two-dimensional filtering capability that will play a key role in problems such as shape recognition and signal detection. UNCLASSIFIED DESCRIPTORS Optical filters Lenses Interferometers Optical equipment AD The experimental results obtained to data indicate that this technique provides an excellent two-dimensional filtering capability that will play a key role in problems such as shape recognition and signal detection. UNCLASSIFIED DESCRIPTORS Optical filters Lenses Interferometers Optical equipment UNCLASSIFIED UNCLASSIFIED

+ + + AD Div. 6/6 Inst. of Science and Technology, U. of Mich., Ann Arbor SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING, by A. B. VanderLugt. Report of Project MICHIGAN. July 63. 39 p. incl. illus., table, 7 refs. (Report No. 2900-394-T/4594-22-T) (Contracts DA-36-039 SC-78801 and AF 33(616)-8433) Unclassified report This report contains integrated descriptions of the problem of signal detection, the optimum linear filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. Experimental results show that the theory is valid. The appendixes give a treatment of the Fourier transforming property of lenses which is general enough that complete optical systems can be evaluated on the basis of frequency response and region of space-invariant operation. (over) UNCLASSIFIED I. Title: Project MICHIGAN I. VanderLugt, A. B. III. U. S. Army Electronics Command IV. U. S. Air Force V. Contract DA-36-039 SC-78801 VI. Contract AF 33(616)-8433 UNCLASSIFIED AD Div. 6/6 Inst. of Science and Technology, U. of Mich., Ann Arbor SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING, by A. B. VanderLugt. Report of Project MICHIGAN. July 63. 39 p. incl. illus., table, 7 refs. (Report No. 2900-394-T/4594-22-T) (Contracts DA-36-039 SC-78801 and AF 33(616)-8433) Unclassified report This report contains integrated descriptions of the problem of signal detection, the optimum linear filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. Experimental results show that the theory is valid. The appendixes give a treatment of the Fourier transforming property of lenses which is general enough that complete optical systems can be evaluated on the basis of frequency response and region of space-invariant operation. (over) UNCLASSIFIED I. Title: Project MICHIGAN II. VanderLugt, A. B. III. U. S. Army Electronics Command IV. U. S. Air Force V. Contract DA-36-039 SC-78801 VI. Contract AF 33(616)-8433 UNCLASSIFIED + + + AD Div. 6/6 Inst. of Science and Technology, U. of Mich., Ann Arbor SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING, by A. B. VanderLugt. Report of Project MICHIGAN. July 63. 39 p. incl. illus., table, 7 refs. (Report No. 2900-394-T/4594-22-T) (Contracts DA-36-039 SC-78801 and AF 33(616)-8433) Unclassified report This report contains integrated descriptions of the problem of signal detection, the optimum linear filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. Experimental results show that the theory is valid. The appendixes give a treatment of the Fourier transforming property of lenses which is general enough that complete optical systems can be evaluated on the basis of frequency response and region of space-invariant operation. (over) UNCLASSIFIED I. Title: Project MICHIGAN II. VanderLugt, A. B. III. U. S. Army Electronics Command IV. U. S. Air Force V. Contract DA-36-039 SC-78801 VI. Contract AF 33(616)-8433 UNCLASSIFIED AD Div. 6/6 Inst. of Science and Technology, U. of Mich., Ann Arbor SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING, by A. B. VanderLugt. Report of Project MICHIGAN. July 63. 39 p. incl. illus., table, 7 refs. (Report No. 2900-394-T/4594-22-T) (Contracts DA-36-039 SC-78801 and AF 33(616)-8433) Unclassified report This report contains integrated descriptions of the problem of signal detection, the optimum linear filtering process, a coherent optical system which accomplishes this filtering process, and a technique for realizing the required complex filter. Experimental results show that the theory is valid. The appendixes give a treatment of the Fourier transforming property of lenses which is general enough that complete optical systems can be evaluated on the basis of frequency response and region of space-invariant operation. (over) UNCLASSIFIED I. Title: Project MICHIGAN II. VanderLugt, A. B. III. U. S. Army Electronics Command IV. U. S. Air Force V. Contract DA-36-039 SC-78801 VI. Contract AF 33(616)-8433 UNCLASSIFIED + + +

AD The experimental results obtained to data indicate that this technique provides an excellent two-dimensional filtering capability that will play a key role in problems such as shape recognition and signal detection. UNCLASSIFIED AD DESCRIPTORS Optical filters Lenses Interferometers Optical equipment The experimental results obtained to data indicate that this technique provides an excellent two-dimensional filtering capability that will play a key role in problems such as shape recognition and signal detection. UNCLASSIFIED DESCRIPTORS Optical filters Lenses Interferometers Optical equipment UNCLASSIFIED UNCLASSIFIED + 4 -7 —- A-~ O.__ i-; ~.~_~_~..~ 3 ~o ~'~ CA) 0O AD The experimental results obtained to data indicate that this technique provides an excellent two-dimensional filtering capability that will play a key role in problems such as shape recognition and signal detection. UNCLASSIFIED DESCRIPTORS Optical filters Lenses Interferometers Optical equipment AD The experimental results obtained to data indicate that this technique provides an excellent two-dimensional filtering capability that will play a key role in problems such as shape recognition and signal detection. UNCLASSIFIED DESCRIPTORS Optical filters Lenses Interferometers Optical equipment UNCLASSIFIED UNCLASSIFIED

Institute of Science and Technology The University of Michigan July 1963 Report of Project MICHIGAN 2900-394-T SIGNAL DETECTION BY COMPLEX SPATIAL FILTERING A. B. VanderLugt ERRATA Page 6 38 Line Fig. 3 Table Change For f between P1 and L1 read d For m read mm (4 occurrences)