Matched filter processing of raster-scanned images:analysis

Stephen S. Hsiao and Martin R. Wohlers

The performance of the matched filter technique on TV raster-scanned images has been analyzed and simu-

lated with two types of input object. The output correlation is a function of bandwidth used for the matched

filter. For a Nyquist bandwidth-matched filter, the output correlation does not decrease substantially (only

about 3-4%) as the object becomes smaller. If the matched filter bandwidth becomes large enough to cover

the total object spectrum, the correlation starts to decrease substantially when the number of scan lines per

resolved image element becomes less than four.

1. IntroductionFundamental theory of the optical spatial-filtering

technique in a coherent optical system has been knownfor years; its application in various areas was recom-mended by several writers.1-3 However, only in the lastdecade was the practical implementation of this tech-nique made feasible by a new method of making com-plex spatial filters3 and by the availability of highlycoherent laser sources. In recent years, the coherentspatial-filtering technique has been successfully usedin signal detection,3 4 character recognition,5 radarsignal processing,6 and image deblurring.7 In appli-cations of the so-called coherent matched-filteringtechnique, various technical problems have been en-countered in system design. Some of these problemshave been analyzed and treated by several authors.4 '810

In a system that we studied for real-time target de-tection, a TV camera is used to project the incomingsignal to an optical transducer before the matched-fil-tering process. The input image often consists of araster of TV scan lines instead of a continuous tone.Since the matched filter is generally made of a well-defined continuous tone target, the TV raster scan linesoften degrade the output correlation peak. To evaluatehow these raster-scanned images can degrade thematched filter performance, we provide in this paperan analysis on the performance of the system for ras-ter-scanned images and a computer simulation of thesystem performance with two types of input object, asquare and a grating. General conclusions can be drawnfrom the results of these simulations, and a great insighton the TV raster effect is revealed.

The authors are with Grumman Aerospace Corporation, ResearchDepartment, Bethpage, New York 11714.

Received 17 January 1976.

11. General Theory of TV Raster Scan Effect

We begin the analysis of the scanning system by as-suming that a specific image, whose field distributionis given by Ui (x,y) [irradiance I(x,y) = I Ui21, isscanned by a sensor whose output is given by theweighting or convolution integral

U.(x,y) = Sw(x - , y - n)Ui(,)dtdn - * U.. (1)

The weighting function co(x,y) is scanned over the inputimage in a specific pattern, which we will assume to bethe standard rectangular TV raster in which there are,say, N discrete values of y at which the sensor is scannedparallel to the x axis. We further assume that the finaldisplay is then formed in the same raster format andconsists of N image lines whose optical fields vary alongthe x axis but are constant across their widths; i.e., weassume that the nth scan line in the output display hasan optical field given by

UO (xy ), for all x and y,, 2< y < y +2t or 2elsewhe+e2

Ud (X,Y) = 0, elsewhere, (2)

where yn is the position of the nth scan line and 1 is thewidth. Note that the regions between the output scanlines have zero optical field strength if we assume thatthe spacing between the lines L is greater than theirwidths. The optical field of an individual scan line maythen be expressed as a multiplication of the sensoroutput U0, with a Dirac delta function located at y = Ynfollowed by a convolution with the rectangular function1(y) = rect(y/l). The total raster-scanned image is thenthe sum of N optical fields that correspond to the Nscan line; it may be written as

N Ud (Y) = (c * UZ) E (y- nL)] * (y),

n=O

(3)

October 1976 / Vol. 15, No. 10 / APPLIED OPTICS 2391

effect of scanning on the output of matched filters.Thus, we assume that the filter is to be matched to theoriginal signal image or that the transfer function of amatched filter system H(wwy) is selected to be equalto Fi*(w,,wy). The performance of this matched filterwith arbitrary inputs can be expressed in terms of thenormalized peak correlation C, which is defined as

CA

f 2 f FFi*dwxdw 2

f~f Fl2 dwxdwyxdwffr JIF 2 d.,d,,, IwJwIFi 2 duxo• 1, (6)

T OVERLAP OVERLAP

(C)

Fig. 1 Fourier transform of TV-scanned output image: (a) Fouriertransform of input image; (b) Fourier transform of sensor output; (c)

Fourier transform of TV-scanned output image.

where we have used Eq. (1) and have assumed that thescan lines start at y = 0.

We may gain some insight into this result by ob-taining the Fourier transform of this expression, whichis

Fd (, y) = sin(w,,/2) [(w* Ui) _ y - nL)] (4)

Or by using the Fourier-series representation for thedelta function

[NEO( n-- a>LeP (i2Lrny)]

we finally haveF() =1 sin(wyl1/2) -; WF" w.I1Y2r (5)

Y L (yt/2) n=- (X L)

where F (wswy) is the Fourier transform of the originalunscanned image U (x,y), and W(wcx,cy) is the Fouriertransform of the weighting function co(x,y). ComparingFd with Fi, we see that the finite thickness I of the out-put scan lines produces a uniform weighting of theoutput transform via the multiplication by the factorsin(wYl/2)/(wy1/2), that the original sensor weightingfunction yields a direct multiplication of the inputimage transform via the function W, and finally that thetotal output involves an infinite sum of displaced rep-licas of the weighted image transform. This spectrumis shown in Fig. 1. The image transform is distorted,in addition to the effects of finite scan line width andsensor weighting, by possible overlap of the shiftedimage replicas. Only in the case where the image hasa finite bandwidth, say B, can the overlap be avoided;this requires that the scanning rate be great enough.Specifically, we get the usual Nyquist requirementthat

(2ir)/L > 2B;

that is, the signal bandwidth in cycles/length should beless than one half (number of scan lines/width of aper-ture). This implies that the maximum resolution of thescanned output image is /2(27r/L).

We now address the basic point of this paper, i.e., the

where F is the input signal spectrum and co2 - wi is theqbandwidth of the matched filter. If F = Fi, the systemis matched to the original unscanned image Fi; and theoutput correlation C has maximum value one. Whenthe input spectrum F equals the TV-scanned imagespectrum Fd, the peak correlation C will be degradedand its value will be less than one. Hence, using thevalue of C, we can simulate the performance of amatched filter system for TV-scanned images. Thissimulation is carried out in the next section.

Ill. Computer Simulation of TV Raster EffectTo simulate the performance of the matched filter

system for TV-scanned images, we first choose a set ofparameters for the TV model. We assume theweighting function is a square window with dimensiona X a. We further assume 1 = L = a. Two objects, anine-cycle grating and a square, are chosen as inputtargets. Their Fourier transforms can be written as

___. __n sin[A wyI A w~\Fi(w.,wy)= 4 2sine(Awy sin[Aw)/9 X sinc (-) (7)

and

Fi(wx,coy) = A2 sinc (A2 ) sinc (Ay),

respectively.A is the size of the target, which we intend to vary

when we substitute Eq. (7) into Eqs. (5) and (6) to cal-culate the value of C. The other system parameters, a,1, L, are kept constant and are set equal to 0.02. A ischosen from 0.001-100 in six steps. This correspondsto numbers of scan lines across the target from N = 0.05to N = 5000. For a Nyquist bandwidth-matched filter[bandwidth = 2(27r/L)], the correlation C did not de-crease much when the object size A became smaller thanthe scanning window size a. This is shown in Fig. 2(curve 1) and Fig. 3 (curve 1) for both targets (gratingtarget in Fig. 2 and square target in Fig. 3). The reasonis that when the object size A becomes smaller, thecorresponding spectrum becomes larger. Because thematched filter has a fixed bandwidth, only the lowerportion of the total spectrum was used in the calculationof the output. Hence, these curves cannot reveal thereal nature of the spectrum overlapping effect when theobject size becomes smaller. But it does indicate thatfor objects whose bandwidth is larger than the Nyquistbandwidth 1/2(27r/L), the degradation of a TV rastereffect is not substantial (only about 3-4%). To see theeffect of spectrum overlapping caused by TV raster

2392 APPLIED OPTICS / Vol. 15, No. 10 / October 1976

F ()

F (0)(A)

WF ()

(B)

S IN P2)

( X12)

WF (0)2L

WF(0) SIN (L)7,

of scan lines decreases, but it peaks up agaip at N = 1.CURVE I for curve 2. This phenomenon is due to the periodic

nature of the object spectrum and the concentration of.8 CR 2i / t -i Al l i at -- l Ispectrum energy in a few spectrum lobes. When the

high energy lobes fall within the negative cycle of the.6 \ /sinc(lwy/2) weighting function in Eq. (5), they provide

.6 ,4.~ -`- ' -'- ''4' -' % '. ' ''44R a nea negative contribution in the over-all correlation cal-C . y . ^ .URVE . ... culation. This causes the dips in curves 2 and 3 (Fig.

.8 HA 2). In any case, the correlation C decreases to about. ,, . rat, f .. tW , , ,. . ,. .90% of its peak value when there are about forty scan

lines across the grating target. Since there are aboutnine grating cycles across the target, this amounts to

A. .~ CC CC. CCCC~C. .C+CC ... CC,+C . about four lines per each resolved image element. Inthe case of the square target, the correlation also drops

| i / ............. to about 90% of its peak value when there are approxi-a 05 0'5 5 50 00 5- mately four scan lines across the square target.

NUMBER OF SCAN LINES

Fig. 2. Correlation output C vs the number of scan lines across thenine-cycle grating for three different bandwidth matched-filters:curve 1: for Nyquist bandwidth ½(2-r/L); curve 2: for a bandwidthten times the Nyquist bandwidth; curve 3: for a variable bandwidth

ten times the bandwidth of (2 x- X 9/A) for the nine-cycle grating.

.8

-

.

_; .

CURVE I

.. . ........ .

CURVE 2 CURVE 3

.6 . .. A ,.... ~~~~~~~~~~~~~~~~~~~~~~~~~....

.~~~~~~~~~~~~~~~~~~~~~~~~~~. ......... ..... ..............

2 --

0.05 0.5 5 50 500NUMBER OF SCAN LINES

Fig. 3 Correlation output C vs the number of scan lines across thesquare object with aperture A for three different bandwidth matchedfilters: curve 1: for Nyquist bandwidth 'A(27r/L); curve 2: for abandwidth ten times the Nyquist bandwidth; curve 3: for a variablebandwidth taken to be twenty times the bandwidth of 27r/A for the

square object.

scanning, a much wider bandwidth is adopted for thecalculation. In one case, we take a fixed bandwidth often times the Nyquist bandwidth /2(27r/L) for thematched filter. The correlation C vs the number ofscan lines is shown in Fig. 2 (curve 2) and Fig. 3 (curve2) for both targets. In another case, we use a variablebandwidth for the matched filter. This variablebandwidth is chosen large enough so that it covers theoriginal object spectrum and varies in accordance withthe object size change. The correlations C for thisbandwidth are also plotted in Fig. 2 (curve 3) and Fig.3 (curve 3). Note that for the grating target (Fig. 2), thecorrelation decreases substantially when the number

In summary, the TV raster effect on the autocorre-lation output of a matched filter system is a function ofbandwidth used for the matched filter. For a Nyquistbandwidth 2(27r/L) (bandwidth based on spacing Lbetween TV scan lines), the output correlation does notdrop substantially (only about 3-4%) as the object be-comes smaller. If the matched filter bandwidth be-comes large enough to cover the total object spectrum,the correlation starts to decrease substantially when thenumber of scan lines per resolved image element be-comes less than four. This number is comparable withthe number that has been identified as needed to rec-ognize military targets in Army application. 1

This work was sponsored, in part, by the Night VisionLaboratory, Fort Belvoir, Va. under Contract DAAK02-74-C-0275.

References1. E. O'Neill, IRE Trans. Inf. Theory IT-2, 36 (1956).2. L. J. Cutrona et al., IRE Trans. Inf. Theory IT-6, 386 (1960).3. A. Vander Lugt, IRE Trans. Inf. Theory IT-10, 139 (1964).4. F. B. Rotz, in International Optical Computing Conference,

Digest of Papers, (IEEE, New York, 1975), p. 162.5. A. Vander Lugt, F. B. Rotz, and A. Klooster, Jr., Optical Elec-

tro-Optical Information Processing, J. T. Tippett et al., Eds.(MIT Press, Cambridge, 1965).

6. L. J. Cutrona et al., Proc. IEEE 54, 1026 (1966).7. G. W. Stroke and R. G. Zech, Phys. Lett. A 25, 89 (1967).8. A. Vander Lugt, Appl. Opt. 5, 1960 (1966).9. A. Vander Lugt, Appl. Opt. 6, 1221 (1967).

10. R. A. Binns, A. Dickinson, and B. M. Watrasiewicz, Appl. Opt.7, 1047 (1968).

11. J. Johnson, Proceedings of Image Intensifier Symposium, Ft.Belvoir, Virginia, October 1958, p. 249.

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