Optimization of parameters in matched spatial filter synthesis David Casasent and Alan Furman Criteria are provided for the selection of various matched spatial filter (MSF) synthesis parameters: the bias exposure (EB); beam balance ratio (K); and spatial frequency band (f*) in which K is set. The signal- to-noise ratio (SNR) and peak intensity (I,) of the output correlation are used as measures of optimum cor- relation and a diffraction pattern sampling unit is used to insure reliable and reproducible data. 1. Introduction Nearly all optical pattern recognition systems use the well-known Vander Lugt matched spatial filter (MSF) and frequency plane correlator.' The three major parameters that must be selected in the synthesis of an MSF are the bias exposure (EB) for the MSF and the spatial frequency band (f*) in which the beam bal- ance ratio (K) (ratio of the intensity of the reference and signal beams) is set. These parameters should be properly chosen to optimize the output correlation for the largest peak intensity (Ip) and SNR with adequate discrimination for unwanted objects (i.e., reduced cross correlations). Although various real-time spatial light modulators 2 are seeing increased usage, film and spectroscopic plates are still the most widely used input and MSF media. Considerable literatures 4 is available on various film properties pertinent to coherent optical processors. Most of these studies have been concerned with the effects of the film's nonlinearity 5 on hologram recon- structions of diffuse objects, 67 the extraneous images present in the reconstructions, 89 etc. The optimum exposures and transmittances of film used in the input' 0 and in hologram recording 7 1' have also been studied. However in all instances above, the goal has been linear recording, and the output has been an image (usually a holographic reconstruction). In this paper, we are concerned with MSF synthesis and in particular the output correlation plane pattern. Limited publi- cations exist in this area. The selectivity of hologram filters as a function of passband characteristics1 2 and the effect of the spatial coherence of the source on the correlation peak intensity1 3 have been several aspects of matched filter synthesis that have been considered. Hard-clipped MSFs have been considered' 4 and small SNR losses encountered. Negligible effects from ex- The authors are with Carnegie-Mellon University, Department of Electrical Engineering, Pittsburgh, Pennsylvania 15213. Received 24 November 1976. traneous frequencies were observed with high spatial carrier frequencies. Transposed processing 15 (in which the roles of the input and MSF are reversed) was also suggested as a system in which the nonlinearity of the MSF film material could be used to suppress common features in inputs and thus enhance character recog- nition. Whereas the use of large K ratios was found to result in linear recording, 4 small K ratios are found to enhance MSF performance and improve filter dis- crimination. 16 Thus the optimum choice of these pa- rameters is expected to be quite different in MSF syn- thesis than in linear recording and holographic recon- struction applications. In this paper we consider the effects of EB, f*, and K in MSF synthesis in achieving the largest Ip and SNR of an output correlation. Following a description of the optical system used (Sec. II), a brief summary of the MSF synthesis and correlation process is included (Sec. III) with emphasis on the importance of EB, f *, and K. Extensive and reliable data from which to assess the effects of these MSF synthesis parameters were ob- tained by use of a diffraction pattern sampling unit (DPSU) in the Fourier transform (FT) plane. 17 This procedure is described in Sec. IV and followed by the highlights of an extensive set of experimental data. Although all data were obtained for aerial imagery, the results are directly applicable to the other data pro- cessing and pattern recognition cases. 11. Optical System A schematic representation of the conventional fre- quency plane correlator used is shown in Fig. 1. A 60-mW He-Ne laser source (X = 633 nm) served as the coherent light source. The variable attenuator (VA) facilitates system alignment and direct viewing of the Fourier transform plane (FTP) and correlation plane (CP) patterns. It provides continuous attenuation without introduction of a gradient in the intensity profile of the beam and without a change in the direc- tion of propagation of the beam as occurs with a neutral density filter. Exposure time can be accurately con- 1662 APPLIED OPTICS/ Vol. 16, No. 6 / June 1977
Transcript
Optimization of parameters in matched spatial filter synthesis
David Casasent and Alan Furman
Criteria are provided for the selection of various matched spatial filter (MSF) synthesis parameters: thebias exposure (EB); beam balance ratio (K); and spatial frequency band (f*) in which K is set. The signal-
to-noise ratio (SNR) and peak intensity (I,) of the output correlation are used as measures of optimum cor-
relation and a diffraction pattern sampling unit is used to insure reliable and reproducible data.
1. Introduction
Nearly all optical pattern recognition systems usethe well-known Vander Lugt matched spatial filter(MSF) and frequency plane correlator.' The threemajor parameters that must be selected in the synthesisof an MSF are the bias exposure (EB) for the MSF andthe spatial frequency band (f*) in which the beam bal-ance ratio (K) (ratio of the intensity of the reference andsignal beams) is set. These parameters should beproperly chosen to optimize the output correlation forthe largest peak intensity (Ip) and SNR with adequatediscrimination for unwanted objects (i.e., reduced crosscorrelations).
Although various real-time spatial light modulators2
are seeing increased usage, film and spectroscopic platesare still the most widely used input and MSF media.Considerable literatures 4 is available on various filmproperties pertinent to coherent optical processors.Most of these studies have been concerned with theeffects of the film's nonlinearity5 on hologram recon-structions of diffuse objects,6 7 the extraneous imagespresent in the reconstructions,8 9 etc. The optimumexposures and transmittances of film used in the input' 0
and in hologram recording7 1' have also been studied.However in all instances above, the goal has been
linear recording, and the output has been an image(usually a holographic reconstruction). In this paper,we are concerned with MSF synthesis and in particularthe output correlation plane pattern. Limited publi-cations exist in this area. The selectivity of hologramfilters as a function of passband characteristics12 andthe effect of the spatial coherence of the source on thecorrelation peak intensity13 have been several aspectsof matched filter synthesis that have been considered.Hard-clipped MSFs have been considered' 4 and smallSNR losses encountered. Negligible effects from ex-
The authors are with Carnegie-Mellon University, Department of
Electrical Engineering, Pittsburgh, Pennsylvania 15213.Received 24 November 1976.
traneous frequencies were observed with high spatialcarrier frequencies. Transposed processing15 (in whichthe roles of the input and MSF are reversed) was alsosuggested as a system in which the nonlinearity of theMSF film material could be used to suppress commonfeatures in inputs and thus enhance character recog-nition. Whereas the use of large K ratios was found toresult in linear recording,4 small K ratios are found toenhance MSF performance and improve filter dis-crimination.16 Thus the optimum choice of these pa-rameters is expected to be quite different in MSF syn-thesis than in linear recording and holographic recon-struction applications.
In this paper we consider the effects of EB, f*, and Kin MSF synthesis in achieving the largest Ip and SNRof an output correlation. Following a description of theoptical system used (Sec. II), a brief summary of theMSF synthesis and correlation process is included (Sec.III) with emphasis on the importance of EB, f *, and K.Extensive and reliable data from which to assess theeffects of these MSF synthesis parameters were ob-tained by use of a diffraction pattern sampling unit(DPSU) in the Fourier transform (FT) plane.17 Thisprocedure is described in Sec. IV and followed by thehighlights of an extensive set of experimental data.Although all data were obtained for aerial imagery, theresults are directly applicable to the other data pro-cessing and pattern recognition cases.
11. Optical System
A schematic representation of the conventional fre-quency plane correlator used is shown in Fig. 1. A60-mW He-Ne laser source (X = 633 nm) served as thecoherent light source. The variable attenuator (VA)facilitates system alignment and direct viewing of theFourier transform plane (FTP) and correlation plane(CP) patterns. It provides continuous attenuationwithout introduction of a gradient in the intensityprofile of the beam and without a change in the direc-tion of propagation of the beam as occurs with a neutraldensity filter. Exposure time can be accurately con-
1662 APPLIED OPTICS / Vol. 16, No. 6 / June 1977
S ; VA The subsequent transmittance t of the MSF afterdevelopment is a function of the t vs E curve for theMSF material used as well as the choice of EB and a.For large a values, G and K are constant over distancesthat are large compared to 1/a.
Many analyses consider the t-E curve to be linear,whereas it is actually nonlinear. The last term in Eq.(2) thus produces a dc term which we include in the
|CP average transmittance to, a fundamental term, andhigher order terms H. The transmittance of the MSFis then
VOL SF1 Ca i t~p u? 0 Ftit1 FtP
$WONAL
Fig. 1. Schematic representation of frequency plane correlator used.Code: VA, variable attenuator; S, shutter; M, mirror; VBS, variablebeam splitter; OL, objective lens; SF, pinhole spatial filter; C, colli-mating lens; FTL, Fourier transform lens; FTP, Fourier transform
plane; CP, correlation plane.
trolled by the electronic shutter (). The variable beamsplitter (VBS) allows easy adjustment of the beambalance ratio (K). Plane wave reference and signalbeams (with less than 5% intensity taper) are producedby the objective lens (OL), pinhole, spatial filter (SF),and collimating lens (C) systems.
Fourier transform lens 1 (FTL1) with focal length f'forms the 2-D Fourier transform at FTP of the ampli-tude transmittance of the input. When this Fouriertransform is interfered with the plane wave referencebeam (at an angle 0 = 300), an MSF is recorded at FTP.With this MSF of the reference function in place atFTP, the reference beam blocked, and a new functionpresent at the input FTL2 with focal length f2 forms atCP the correlation of the reference and input func-tions.
Ill. Theoretical Analysis
The reference beam from C and signal beam fromFTL1 can be described at FTP by
UR(xl,yl) = A exp(-j2lrax 1), (la)
Us(xlyl) = G(xl,yi), (lb)
where the coordinates of the input, FT, and correlationplanes are denoted by (xo,yo), (xl,yl), and (x2,y2 ), re-spectively. A is the constant amplitude of the planewave, a = (sin0)/X, and G(xl,yl) is the complex FT ofthe input function g(xo,yo). During MSF synthesis, theexposure incident on the FTP is
where T is exposure time, i = 27rax, + arg(G), EB =
A2 T is the bias exposure, E = (A2 + IG1 2)T is theaverage exposure, and K = A2 /l G 1 2 is the beam balanceratio.
t = to + (M/2) cost + H
= to + 2d cos + H,
(3a)
(3b)
where M is the peak-to-peak ac amplitude transmit-tance swing or the modulation of the cosine wave. Theamplitude diffraction efficiency d = aI is used ratherthan the intensity diffraction efficiency a7, since ampli-tudes are considered in coherent light until the signalis recorded or detected.
With g(xo,yo) placed at the input and the transmit-tance of the MSF at FTP in Fig. 1 given by Eq. (3b), thelight distribution leaving the FTP is UGt. One term ofinterest in this light distribution is
IG(xi,y1) d(x ,yi) exp(-j27rax 1), (4)
where the spatial dependence of the amplitude dif-fraction efficiency d has been intentionally included.At x2 = af 2 in the correlation plane, FTL2 (of focallength 2) forms the FT of Eq. (4) or the autocorrelationgig. The peak intensity of the correlation is
Ip = If G(xi,y1)d(xi,y1)dxidyIi 2 . (5)
Let us now analyze these results. Since G(xl,yl) isa function of spatial frequency, so are K, M, and d. Mand d also depend on the exact t (E) transfer curve forthe film, the EB value chosen, and K. Since d and Kvary with spatial frequency, the spatial frequency f* atwhich K is measured must be provided. A brief reviewof the myriad of papers on optical pattern recognitionexperiments will show that such data are rarely in-cluded. Once t(E) is known and EB is chosen, d(K) canbe found. Since K = I A/G (x 1,y1) 1 2 varies with spatialfrequency, d is also a function of spatial frequency andcan be found once G(x,,yI) is known.
Curves of vs V (fringe visibility) for the MSF ma-terial are usually available. Since = d2 and V =(2x/K)/(K + 1), /77 vs K curves at various average ex-posures, and Vq vs E curves for actual MSF materialscan be obtained. The linear portion of a constant EOcurve is generally used. For a linear recording material,/ will increase with K, peak at K = 1, and then de-crease. From Eq. (2), we find that linear recording re-quires E < 2EB (where EB corresponds to t = 0.5).This corresponds to K 2 5.8. This result is consistentwith the use of large K values (much larger than 10) tolinearly record data.7 With K < 0.17 we find the re-corded data to be saturated, while for 0.17 < K < 5.8 therecorded data will be clipped. Although clippingimplies that harmonics will be generated, an MSF isgenerally formed with K = 1 since A/11 is then maxi-
June 1977 / Vol. 16, No. 6 / APPLIED OPTICS 1663
Table I. K In All Nine f* Bands for Nine Plates With K = 2 In Each of Nine f* Bands
CorrespondingBands in ring numberswhich K and (spatial K in indicated band
Num- is set frequenciesber to 2 f* in lp/mm) A B C D E F G H I
8 H 24-26 1.5 x 4.6 x 1.1 x 2.54 x 5.9 x 0.16 4.8 x 1.77 4(18.4- 10- 10- 10-2 10-2 10-2 10126.1)
9 I 27-29 7.2 x 2.56 x 5.4 x 1.02 x 2.18 x 6.08 x 1.74 x 6.0 x 1.9(26.2- 10-4 10- 10- 10-2 10-2 10-2 101 10136.7)
mized. However, in MSF synthesis the question is atwhich f* value to set K = 1. From these brief remarks,it is clear that in MSF synthesis EB, K, and f* are im-portant parameters and that one ignores linear re-cording since the MSF is not operated in this portionof the t vs E curve.
IV. Experimental Procedure
The single most useful device that made possible therapid analysis of the effects of K, f*, and EB was thewedge ring detector or diffraction pattern sampling unit(DPSU). We briefly reported on this use of the DPSUearlier.' 7 The DPSU is a 25-mm diam monolithic sili-con photodiode array and readout unit.18 The detectorarray consists of thirty-two wedge-shaped detector el-ements in one semicircle and thirty-two annular shapedelements in the other semicircle. The outputs from thesixty-four detector elements are multiplexed to a highdynamic range amplifier. The intensity incident oneach detector element can be separately read out (dig-itally) by selecting the proper front panel switch.
For this application the DPSU detector array wasplaced in the FT plane FTP, and only the thirty-tworing elements were used. The spatial frequencies fcovered by each ring varied from fmin = Ri/xfI to fmax
= Ro/Xfl, where Ri and Ro are the inner and outer radiiof each ring, and f, is the focal length of the FT lensFTL1. The first two rings are not used since they recordthe dc portion of the FT. The outputs of each set ofthree adjacent rings were summed to produce a measure
of the average power spectrum in nine different spatialfrequency bands. The spatial frequency range coveredby each of the nine sets (bands A to I) of three rings isgiven in column 3 of Table I for f, = 495 mm. Smalldifferences in the values at the edges of the bands aredue to the presence of thin electrodes and insulatorlayers between the ring elements.
To provide a unified description of the use of theDPSU in MSF parameter optimization experiments,one laboratory session scenario is presented in detail.Nine MSFs of an input image were made on nine Kodak649 F plates with K = 2 in a different f* band (from 0.8to 37 cycles/mm) on each plate and with four MSFs atfour different exposure values made in each of the fourquadrants of each plate. Using the DPSU, the inten-sities of the reference and signal beams in each set ofthree rings were recorded for each of the nine plates.Between plates, K was adjusted to about 2 in the nextband by varying the VBS in Fig. 1. By dividing thereading in each ring set (intensity of light in that f*band) with only the reference beam present by thereading with only the signal beam present, a set of ninedifferent values of K (each in a different f* band) wasobtained for each of the nine plates. These K valuesare given in Table I for each of the nine bands (A to I)and for each of the nine plates (1 to 9).
Each of these thirty-six MSFs was then reinserted inthe FT plane with the original image present at theinput. The SNR and Ip of each of the thirty-six outputautocorrelations were then obtained. From this Ip and
1664 APPLIED OPTICS / Vol. 16, No. 6 / June 1977
SNR data and the values in Table I, one can obtain Ipand SNR vs f* for a fixed K = 2 and Ip and SNR vs Kfor a fixed f* and thus determine the optimum K andf* values as well as the effects of a choice of nonoptimumK and f* values. With K and f* fixed for each plate,one can also plot Ip and SNR vs E and thus determinethe optimum exposure. This use of the DPSU thusprovides the necessary data from which to determinethe optimum MSF parameters in a simplified experi-ment and with controlled and reliable results. Theresults of the above experiment and many analogousones are presented and discussed in subsequent sec-tions.
Once the optimum E value has been experimentallydetermined from a sufficiently large representative dataset, the E value can be automatically set using theDPSU. The VBS is adjusted to produce the proper Kratio in a given f* band. The signal beam is thenblocked and the irradiance of the reference beam mea-sured in one ring of the DPSU. The irradiance-timeproduct is then chosen. We generally use the ring 15reading of the DPSU (in relative units) to select theexposure time T. The exposure times for Kodak 649Fand 131 plates that were experimentally found to beoptimum are given in Table II for various ring 15 read-ings. The outputs of several ring readings can also becombined and used to produce a more averaged measureof optimum E once their areas are known.
The output of the DPSU can also be calibrated toprovide absolute intensity readings. However, onlydata in relative units are included in these results.
From the theoretical considerations of Sec. III andthe laboratory scenario described in Table I, it is ap-parent that an extensive experimental program usingrepresentative imagery of the type pertinent to thespecific application involved is essential. The inputimage used in most of the experiments reported hereinis shown in Fig. 2. It is an aerial photograph ofHuntsville, Alabama with 30-,m resolution. It isespecially useful because it contains three specific types
Fig. 2. Typical input image used, aerial photograph of Huntsville,Alabama including rural (A), urban (B), and structured (C) image
sections.
Table II. Exposures for Kodak 649F and 131 PlatesBased on Ring 15 Readings
of image fonts: a low-resolution rural area A (top right);a high-resolution urban area B (center left); and thestructured landing field area C (lower center). Addi-tional input imagery used consisted of a synthetic ap-erture radar (SAR) image of Los Angeles taken with theAPQ 102 radar exhibiting 50-gm resolution and a higherresolution image from a new SAR system with betterthan 15-Am resolution. All imagery were recorded onKodak high-speed holographic film (SO-253) in a35-mm format.
All Ip and SNR data were obtained with a scanningphotometric microscope system with a 90-dB dynamicrange PMT. Because of large fluctuations in thebackground noise level, a slit probe was used for SNRmeasurements. Since this probe averages the back-ground noise level over a considerable area, the SNRvalues presented should be viewed as relative levels forcomparison purposes rather than as absolute values.
V. Experimental Results
A. Optimum Exposure
Several authors7 ,10," have considered the effects ofproper selection of the average exposure and trans-mittance of input imagery and holographic recordings.Thomas10 considered the average transmittance of filmas an input medium only. His analysis was concernedwith linear recording and showed that an average ex-posure to 70% corresponded to the center of the film'st-E curve and yielded the least intermodulation dis-tortion. Large M values corresponding to a highmodulation index were used to insure that intermodu-lation distortion was the dominant noise source. Thelargest ac transmittance (proportional to a) was foundto occur at to 50% with a broad maximum. Theseresults agree with those of Kozma7 and others" whoanalyzed the case of holographic noise and SNR. Theyalso found exposures corresponding to to values of 70%to yield larger SNR values (40 dB vs 26 dB) than expo-sures corresponding to to values of 50%.
From the data of Table II, the optimum bias exposurewas given as the product of the DPSU ring 15 reading(in relative units, with only the reference beam present,and after K had been set) and exposure time. In rela-tive exposure units, this product was unity. We nowconsider how these data were obtained. Experimentalplots of p and SNR vs EB were obtained for three dif-ferent input images each on four different types of filmand using two different optical correlators with different
June 1977 / Vol. 16, No. 6 / APPLIED OPTICS 1665
100l
I-
z
-j
.
10
1
0.1l ' d10 20 30
SNR(DB)
Fig. 3. Ip vs SNR for various bias exposures.
lens types. In all cases, Ip and SNR were maximizedwithin 10% of a relative EB bias exposure of unity. Arapid decrease in Ip (by a factor of 100) was observed foreven a 50% decrease in EB. Thus proper EB selectionis very crucial. Also of interest is the fact that SNRpeaks at the same EB value at which Ip does and thatSNR is slightly less sensitive to EB variations than isIp.
A plot of Ip vs SNR from these data is shown in Fig.3. These data imply that optimizing Ip also optimizesSNR. From the 2:1 slope of the curve, we see that if Ipis increased by 2 dB, SNR will increase by only half asmuch or 1 dB and that the noise level will increase by1 dB also. The proper EB choice is thus crucial sincenoise increases with departures from the optimum EB.This is in contrast to Ip and SNR variations with rota-tion etc.'9 in which we found Ip and SNR to increase atessentially the same rate implying a constant back-ground noise.
From previous remarks, we expect a bias exposurecorresponding to a to = 50-70% to produce the optimumq and hence Ip, the most linear recording and lower in-termodulation distortion and noise and thus larger SNRvalues. For the prior experimental plates, we measuredto (the square root of the ratio of the transmitted toincident light) in a 5-mm diam region of the MSF plateexposed only to the reference beam. For the inputimage of Fig. 2, we found to = 0.50 to yield the optimumIP value, whereas a larger to = 0.65 value was found tobe optimum for the higher 15-,gm resolution SARimage.
These data lie in the expected range of averagetransmittances predicted by others by linear recordingand hologram reconstruction. However, we noted onlya +1-dB fluctuation in Ip over the to = 0.4-0.7 range.This is expected since an MSF is operated in the linearregion of the film's t vs E curve where the slope is larg-est. Small changes in to thus result in large EB changes;EB is expected to be a more sensitive parameter and inaddition the one that is more realistic to control thanto. We thus considered Ip and SNR variations with EBand converted these optimum EB values to to valuesprimarily for the purpose of comparison to prior re-sults.
B. K and f* Dependence (General Imagery)
Nine MSF plates of the full image in Fig. 2 wereformed with K set equal to 2 in a different f* band oneach plate, and data like that of Table II were compiledusing the DPSU as described in Sec. III. Autocorrela-tions of the input and all nine MSFs were formed andSNR and Ip measured for each. These Ip and SNRvalues for each plate are shown in Fig. 4 as a function ofthe spatial frequency band f* in which K was set equalto 2. Both curves peak at nearly the same f* = 4-cy-cles/mm value. This f* value appears to be consistentwith expectations. It must be larger than the 0.06-cycles/mm value set by the 35-mm aperture and lessthan the 10-cycles/mm limit due to the 30-gm resolutionlimit of the imagery. If the image were equally dividedbetween low and high resolution data, f* would occurat 5 cycles/mm. Imagery is generally predominantlylow frequency data, hence the f* = 4-cycles/mm valueseems consistent.
C. K and f* Dependence (Image Fonts)
While some dependence of SNR and Ip on f* existsin Fig. 4, there is only a 3-5-dB variation over the fullf* = 1.5-10-cycles/mm range. This somewhat modestvariation is due to the general nature of the input im-agery whose information content varies rather uni-formly over a broad range of spatial frequencies so thatvariations in K and f* have a less pronounced effectwith predominantly statistical fluctuations.
3CW
10
-7o
M30
(I,
.21 2 3.5 5 741f(CYCLES/MM)
(a)
1 2 35 ! _!1 1.1 155.' - 2 2 5 5 741 10.8 155
f- (CYCLES/MM)
(b)
Fig. 4. I (a) and SNR (b) vs f* for the correlation of the entire image
of Fig. 2 with an MSF formed from the entire image as a function of
the spatial frequency band f* in which K was set to 2.
1666 APPLIED OPTICS / Vol. 16, No. 6 / June 1977
-
w03
0.
1.5 225331J 4.8 7 10Jf* (CYCLES/MM)
(a)
20+
16-
0zCI)
14
12
B
A
15 225 331 48 7 li
f* (CYCLES/ MM)
(b)
Fig. 5. I, (a) and SNR (b) vs f* for the correlation of the three ap-ertured regions (curves A, B, and C) of Fig. 2 with MSFs made, re-
spectively, from each of these input regions only.
It has previously been demonstrated'8 that urban andrural imagery can be distinguished by inspection of theimage's diffraction pattern. In these experiments, itwas shown that an increased intensity at high spatialfrequencies was indicative of finer input resolution (i.e.,urban). It thus seems reasonable to assume that theoptimum f* at which to set K would shift from higherto lower values for urban vs rural imagery.
To verify these assumptions and obtain a better un-derstanding of the dependence of SNR and p on f*,four MSFs of each of the three sections (A, rural; B,urban; C, structured) of the image in Fig. 2 were formedwith K = 2 in three different f* bands. The appropriateinput area was then apertured, correlated with theMSFs made from that area, and Ip and SNR plotted vsf* for all three cases. The results are shown in Fig. 5.
From Fig. 5, we find that SNR and IP for the ruralportion A of the image peak at about f* = 2 cycles/mm,whereas the peak for the urban portion B of the imageoccurs at a larger f* - 7 cycles/mm, with the peak forthe structured image lying between these at f* - 4.8cycles/mm. These results are consistent with expec-
tations since the information content of B is finer thanthat of A with C between the two. The f* 4-cycles/mm location of the peak in Fig. 4 for the full imagecorrelation is approximately the average of the aboveas expected.
These results are also basically consistent with theresults of Binns' 6 who found small K values improveddiscrimination. Since the spatial frequency content ofan image generally decreases with increasing spatialfrequency, so does K. MSFs with low K values thuscorrespond to the case of emphasis on the high spatialfrequency content of an image. As noted previously,the spatial frequency f* at which K is set should bespecified. From Figs. 4 and 5, we can obtain a measureof how low of a K value to select, the 1p and SNR loss tobe expected for a given choice, and what choice tomake.
D. Input Aperture and Space Bandwidth EffectsThe peak SNR for the correlation of the urban region
(curve B) is larger (19 dB) than the correlation of thestructured (curve C) region (18 dB), with the peak SNRfor the rural region (curve A) the lowest (16.5 dB).These results are expected and indicative of the de-pendence of SNR on the space bandwidth (SBW) of theinput. The areas of all three input regions (A, .B, andC) are the same with the urban image B of highest datacontent and hence highest SBW and the rural imagehaving the lowest SBW. These apertured regions areone-sixteenth (12 dB) of the area of the full input. TheSNR values for these cases (Fig. 5) are 12-13 dB belowthe 30-dB peak SNR of the correlation of the entireimage, thus further verifying the dependence of SNRon input SBW. An exact SNR loss of 12 dB does notresult since the entire image is certainly neither totallyrural nor completely urban. The SNR loss for theurban input (B) is less than 12 dB since the SBVV dropis clearly less than 16. The SNR loss for the rural input(A) is similarly more than 12 dB as expected.
The Ip loss between the entire image case (Fig. 4) andthe apertured input case (Fig. 5) is simply due to thereduced input light present. p is expected to decreaseas the square of the ratio of the areas of the input im-agery. The p decrease from about 360 for the entireimage case to about 1.5 for the apertured input caserepresents an Ip loss by a factor of 240 or nearly 162 =256 as expected. This argument assumes a constantdiffraction efficiency which the results seem to verify.Since SNR is relatively independent of the input lightlevel, a similar argument cannot be used to explain theSNR drop.
The selection of K = 2 rather than K = 1 in variousf* bands is arbitrary. If Ip and SNR were plotted vs f*for K = 1 or 4 rather than K = 2, the f* locations of thepeak Ip and SNR values would shift. As long as thesame K value is used for all data, consistent results andreliable data trends emerge. If Ip and SNR are plottedvs K with f* fixed at 4 cycles/mm, the peak will occurnear K = 2. However, if the same plot is repeated forthe f* = 5-cycles/mm band, a different K value will re-sult as expected. These trends are also apparent fromthe data of Table I.
June 1977 / Vol. 16, No. 6 / APPLIED OPTICS 1667
ual- -- AA- #-%4 S S-s J^j
A final experiment was performed to determine theeffects of the input aperture and input SBW. Highresolution full SAR images of an urban and a rural areawere obtained. Two MSFs were made from each ofthese full inputs, one with K = 2 in f* band B(1.7-2.7cycles/mm) and the other with K = 2 in f* band D(4.2-6
cycles/mm). These MSFs and the original images werecorrelated and Ip and SNR measured as the input ap-erture was reduced. The resultant graphs of Ip andSNR vs the fraction of the input area present are shownin Fig. 6 for all four cases.
The slope of all Ip curves [Fig. 6(a)] is 2 as predictedearlier. Curves 1 and 3 (urban input with f* in bandsD and B, respectively) yielded larger Ip values thancurves 2 and 4 (rural input with f* in bands D and B,respectively). Setting f* in bands D is clearly a moreoptimum choice for the urban image than an f* set inband B (compare curves 1 and 3). Conversely, for therural image, setting K in f* band B yields a higher Ipvalue than the choice of f* in band D (compare curves4 and 2). Even though an f* set in band D emphasizesthe high spatial frequency content of the rural image,more data are clearly concentrated in the region of bandB for this image. The slope of all SNR curves [Fig. 6(b)]is one as noted earlier, with the MSFs synthesized withf* in band D yielding larger than those with f * chosenin band B.
Even though these Ip and SNR differences in Fig. 6were only several dB, and complete tests were not per-formed to determine if f* bands B and D were optimumfor the new rural and urban images used, the slope of thecurves are unmistakably 2 and 1, respectively, over theentire range of areas.
E. MSF Aperture Effects
Another practical aspect of an optical pattern rec-ognition system is the spatial extent of the MSF, sincethis affects the packing density of multiple MSFs. Togain quantitative data on the effects of an aperture inthe MSF plane and further insight into the origin of Ipand SNR and their optimization, the two full inputs andfour MSFs in Fig. 6 were again used. The full urbanand rural input images were placed in the input planeand Ip and SNR plotted as the size of an aperture placedin the MSF plane was reduced.
The differences between the MSFs formed with K =
2 in f* bands D and B and the cases of an urban andrural input were now more pronounced. Both Ip andSNR for cases 1 and 2 (f* band D) decreased as soon asthe MSF aperture was reduced, whereas for cases 3 and4 (f* band B) Ip and SNR are rather constant with a farslower drop with decreasing MSF aperture. Ip andSNR for cases 3 and 4 (f* band B) were found to de-crease with MSF aperture at one slope until the widthof the MSF aperture was about 3 cycles/mm (the upperlimit of f* band B) at which point the curves tended todecrease at a faster rate with decreasing MSF aper-ture.
These changes in the slopes of the Ip and SNR curves3 and 4 with decreasing MSF aperture are not sharp, but
the breakpoints can clearly be controlled by varying thespatial frequency f* at which K is set during synthesis
zw
0.uDcr
10
1C
0.1
10 l V 1 110 wu o.1
RAnlOOF INPUT AREA
(a)
0
cioC,
10
4
1 I
U;]a
RATIO OF INPUT AREA
(b)
Fig, 6. Ip (a) and SNR (b) as the area of the input image was re-
duced, for the correlation of an MSF of a full high resolution urban
(curves 1 and 3) and rural (curves 2 and 4) SAR image with K 2 in
a low f* band (curves 3 and 4) and a high f* band (curves 1 and 2).
of the MSF. The rate at which Ip and SNR decreasewith decreasing MSF aperture is less for the rural imagethan for the urban image. From these results, we con-clude that f* should be chosen at a lower spatial fre-quency band (e.g., band B) determined by the requiredhologram packing density. In a limited SBW situation,lower f * values are preferable to enable the system tomake optimum use of the SBW available and yield theoptimum Ip values possible. From SNR consider-ations, higher f* values are advantageous when the MSFaperture and SBW are large to better utilize the SBWpresent. In lower SBW cases and when small inputapertures are used, f* should be reduced again to betterutilize the available input SBW. The effect on SNR ismore pronounced than for Ip.
The Ip values for both urban correlations (curves 1and 3 in Fig. 6) are larger than for both rural correlations(regardless of f*) because of the increased modulationpresent in the urban case. This is expected since thehigher the f* value chosen, the lower the K ratio at a
1668 APPLIED OPTICS / Vol. 16, No. 6 / June 1977
-1Uji
011
given spatial frequency. A point of optimum n in termsof K exists. One can decrease K from this point, de-creasing 7 but increasing SNR since the noise level de-creases faster.
Once again, by realizing that both K and ff* must beprovided to describe MSF synthesis adequately we candescribe and control many effects, such as the use ofapertures in the MSF plane and input SBW.
F. Beam Uniformity EffectsThe effects of the spatial coherence of the collimated
beam on 4p have been previously analyzed.2 0 In thissection we consider the effect of the amplitude unifor-mity of the collimated beam on 4p and SNR. Theclassical collimator produces an output beam with aGaussian amplitude taper rather than a uniform am-plitude distribution. The effects of this taper on theoutput correlation were vividly demonstrated by as-sembling two collimators, one with a 42% taper at 10 mmfrom the center and the second with only a 5% taper overa 40-mm beam diameter. An MSF of the image in Fig.2 was formed and placed in the FTP and urban regionB reinserted in the input plane. As this small inputregion B was moved vertically and horizontally aboutthe input plane, Ip and SNR of the output correlationpeak were measured. This situation is analogous to thecase of a search for the location of a small input patternin the input field.
Because of the severe amplitude taper in the inputbeam, the input light intensity incident on the inputregion B varied with the location of B in the input andso does the portion of the FT lens used. With a perfectFT lens, one would expect 4p to track the taper of theinput beam and SNR to remain constant (both thesignal and noise track the input intensity leaving SNRconstant). Any departures from these results can beattributed to phase errors and aberrations in the lensused. We thus expect beam uniformity to affect 4p andphase errors to affect SNR.
In the transposed correlator,15 an MSF of the fullinput is made, and only the apertured region B of theinput is placed in the input plane. In this correlator,we expect I to track the input intensity as before.Noise will be now constant because an MSF of the fullinput is used, and thus SNR is expected to track theinput intensity also.
Experiments confirmed the above predictions and insome instances showed that the FT lens used had pooroff-axis performance. When an adequate FT lens wasused and a uniform intensity (with 5% taper) collimatedbeam was used, essentially no SNR or 4p variations(other than a few percent) were observed as the locationof the input region was moved about the input field. Auniformly collimated beam was thus used in all exper-iments. It was also found that the amplitude unifor-mity of the signal beam was more important than thatof the reference beam, since spatial variations in thelatter are equivalent to K variations whose effects areminor by comparison.
VI. Summary
The importance of the MSF parameters: bias ex-posure EB, average transmittance to, beam balance ratioK, and the spatial frequency band f* have been theo-retically and experimentally evaluated. The need tospecify the f* at which K is set was vividly demon-strated. Only this approach enables optimum MSFsynthesis for specific image fonts and allows properselection and optimization of the size of the MSF planeaperture in applications requiring multiple MSFs or asearch for a small region in a larger input region. Imagespace bandwidth was found to determine SNR, whereasthe input area was shown to be the.dominant factor indetermination of Ip. The use of MSFs synthesized withdifferent f* bands was also found to allow optimum useto be made of the available SBW and aperture.
A novel use of the diffraction pattern sampling unitdescribed in depth for a typical laboratory scenario wasfound to expedite evaluation of many MSF synthesisparameters and yield reproducible and reliable data.
The support of the Air Force Office of Scientific Re-search (Contract AFOSR 75-2851) for the work reportedis gratefully acknowledged. The authors also thankMike Saverino and Jim Morris who performed most ofthe correlations.
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