Nonlinearly recorded matched filter: a technique to reducethe false alarm rate

Stephen S. Hsiao

The effect of film nonlinearity in recording a spatial matched filter for optical signal detection is to record adistorted signal rather than the original target signal. This distorted signal could cause a large false alarmrate if it is severely distorted. We propose a method that requires an additional mask immediately beforethe holographic matched filter to convert the original signal to the distorted signal before processing the sig-nal through the nonlinear matched filter. This process will, in theory, eliminate all the false alarm signalcaused by film nonlinearity. The transmittance function of the mask is calculated for a given target signaland given matched filter recording parameters. For a particular choice of recording parameter, the maskcan be fabricated by directly exposing the Fourier spectrum of the target signal. A computer simulationusing a square function as target signal proves the validity of this technique.

IntroductionThe effect of film nonlinearity in recording a general

hologram has been examined by a number of authors." 2

When the hologram is used as an optical spatial filterin optical signal detection,3 the effect of film nonlin-earity causes two difficult problems. In one case, thereis a dynamic range problem since the Fourier spectrumof a real object scene has a dynamic range several ordersof magnitude higher than the standard high resolutionfilm can provide. In the second case, the recordedspectrum is highly distorted so that it may no longerresemble the original signal. To overcome these twodifficulties, various methods have been proposed.Goodman and Strubin4 proposed processing the signalthrough a set of matched filters where each filter cor-responds to a different frequency channel of the inputsignal. Raso5 and Bins etc.6 try to utilize the filmnonlinearity effect to increase the discrimination inoptical filtering. Vander Lugt and Rotz7 proposed atransposed processing method where the input signalplus noise is recorded in the filter rather than the targetsignal itself. In this paper, we propose a method thatdirectly compensates for the kind of distortion that agiven target signal will undergo when it is recorded ina nonlinear matched filter. The compensation isachieved by processing the input signal through anamplitude mask that is placed immediately before theholographic matched filter. The transmittance func-tion of the mask can be calculated using a given non-

The author is with Grumman Aerospace Corporation, ResearchDepartment, Bethpage, New York 11714.

Received 7 April 1976.

linear film model and the recording parameters used tofabricate the matched filter. For a particular recordingparameter, we found that the mask can be fabricatedby directly exposing the Fourier spectrum of the targetsignal. A computer. simulation using a square functionas a target signal proves the validity of this tech-nique.

Direct Compensation MethodIn a linear matched filter system, the normalized

matched filter output C is defined by

Ci f(pq) SF*dpdq.

1, (1)

f £(p,q) IS 2dpdq f , IFI2 dpdq

where S is the input signal spectrum, F* is the conjugateof the target spectrum, p and q are the spatial fre-quency, and B (p,q) is the spatial frequency band overwhich the matched filter is operating. The output Chas to be normalized against the total input signal en-ergy

S X Upq IS I2dpdq

so that the variation of laser power as well as any largeinput signal will not trigger a false alarm detection. Thenormalization against the target signal

f Bx~q IF I2dpdqS B(p,q) iis essentially a calibration factor for each individualmatched filter. Its inclusion is necessary to take careof the diffraction efficiency of the hologram as well asany other system attenuation for a given matched filter.

1166 APPLIED OPTICS / Vol. 16, No. 5 / May 1977

-

.20

.15

U-

,1

.05

0I _ - 2 4 6I vl0 2 4 86 ,0 1

IF[Fig. 1. Transfer curves of amplitude ratio Ta = IGI/IFI vs targetspectrum amplitude IF I for a nonlinearly recorded matched filter.

The value of C as defined by Eq. (1) is less than or equalto one by the Schwarz inequality. It is equal to onewhen the input signal S equals the target signal F.

When the target signal F is recorded by a film witha nonlinear Ta - E characteristic curve, the recordedsignal is no longer equal to the target signal, but ratheris a distorted one G. It was shown in Refs. 1 and 7 thatonly the amplitude of the target signal is distorted. Thephase remains the same, i.e., if F* = IFI exp(-jO), thenG* = IGI exp(-jO), where IF} # IGI andOis the phase.For this reason, the normalized nonlinear matched filteroutput Cn should be

Ko =S fp IFI 2dpdq

S SB(p,q) FG-dpdq

is the calibration factor for the matched filter and K =1 for S = F. All the quantities in Eq. (3) are nowphysically measurable. The value of K may be largerthan one for those signals that are close to G.

In some applications, where G is severely distortedfrom F, the above approach is no longer appropriate.For this case, a direct compensation mask may be usedwhich transforms the incoming signal F into the dis-torted signal G. This distorted signal is the correctinput for the distorted matched filter, achieving thedesired correlation. The compensation mask can be asimple complitude filter, since only the amplitudes ofG and F are different.

The appropriate intensity transmittance of thecompensation mask is given by T1 = T 2 = I G 1 2/I Fl 2.

With this mask placed in front of the nonlinear matchedfilter, the spectrum transmitted by it is given by

St ST. = (IGI/IFI) S.

When S = F, St = G, which is the exact signal that wasrecorded in a nonlinear matched filter. All other sig-nals, after passing through the mask, will be differentfrom G, and they will be eliminated by the subsequentmatched filtering process. The output C,, is now nor-malized in a form of

Cn1SS (pq) StG*dpdq2

.1, (4)

CnS S fB(p,q) 2

5 pq IS I 2dpdq f jf I G 2dpdq

• 1. (2)

Since the recorded target signal is G* rather than F*,the value of C,, will no longer be one when the inputsignal S = F. Let C, (F) be the value of C,, when S =F, and we have C,, (F) < 1 in general. Moreover, thereare an infinite number of signals that could have a Cnlarger than the C, (F). This may cause a large falsealarm rate. In some applications this false alarm con-sideration is usually ignored. There are two reasons forthis. First, the value of C,, (F) is still -larger for thetarget signal than for most of the other signals. Thisis because the phase of the target signal matches thephase of the recorded signal G. Second, those signalsthat may differ substantially from the original signal arenot considered as false alarm signals in most applica-tions. For this reason, the output Cn (F) is usually usedas a normalization factor for the output of other signals.In this case, we define

Cn, I (pq) SGdpdqK = C-= Ko , (3)

C,(F) S SB(p,q) ISI2 dpdq

where

(pq) I St 2dpdq S(pq) G I 2dpdq

where a measurement of total energyS S | St |

2dpdq,

immediately after the mask, is required. This impliesthat an additional processing channel is required in thistechnique.

The fabrication of the mask usually requires a com-puter digitizing of the spectrum of the target signal,calculation of the required intensity transmittancefunction, and printing of the mask function on a filmthat is then developed with unit gamma. The calcu-lation of the required transmittance function Ti = T- = G 1 2 / IF 1 2 for a given matched filter can be carriedout with the help of the so-called transfer curve7' 8 whichspecifies the relationship between I G and IFI (or afunctional of them) for a given film model. For exam-ple, Fig. 1 is a plot of transfer curves of Ta = I G I / IF I vsIFI for a number of reference amplitudes A, accordingto the film model used in Ref. 7. With these transfercurves, the value of Ti = T 0

2 for a given input targetsignal F can be calculated point by point to yield therequired mask function.

An alternative method that may require no computercalculation is to find a film whose Ta - E characteristiccurve resembles the transfer curve of Ta = I G Il / I F I vsIF1 for a given matched filter. To fabricate the mask,the filn is exposed directly by the target spectrum

May 1977 / Vol. 16, No. 5 / APPLIED OPTICS 1167

'A =2

A=3

- 1

-A 4

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

o A: -

LU

o0 ~ ~ ~ 3 40 5

E T. Ez .06

I.-

- A

0

EXPOSURE E OR INTENSITY FOR 12

Fig. 2. Transfer curves of T, = I GI/IFI vs target spectrum intensityIFl2 and Ta - E characteristic curve for the 649-F plate.

function F. After development, the film will have therequired amplitude transmittance function T =I G I / IFI for the compensation mask. This is becausethe transformation carried out by the film Ta - Echaracteristic curve is identical to the required trans-formation carried out by the transfer curve of I G IIIFIvs IF 2. As an example, the dashed line in Fig. 2 is theT- E characteristic curve for Eastman Kodak 649-Ftype 120 plate developed in D-19 for 5 min. In the samefigure, three transfer curves of I G I / I Fl vs I Fl 2 are alsoplotted using the film model adopted by Vander LUgt.7It can be seen that for A = 3, the Ta - E curve resem-bles the GI/l Fl vs F 12 curve. Hence, for the matchedfilter recorded with reference amplitude A = 3, the maskfunction can be fabricated by directly exposing the649-F plate with the Fourier spectrum of the targetsignal F. In the next section, a computer simulation isconducted to reveal the effect of film nonlinearity andto prove the validity of this compensation technique.

Computer SimulationA fast Fourier transform algorithm9 is used to simu-

late the effect of film nonlinearity in matched filterfabrication and the technique of compensation to re-duce the false alarm rate. The simulation is first con-ducted in 1-D using a rectangular function Rect(x/D)as target signal. The simulation is also conducted in2-D using a square target. In the 1-D simulation, thematched filter is first made with a nonlinear film whoseT- E characteristic curve is shown in Fig. 2 by thedashed line. The bias level is set at A = 3. Two ref-erence beams are then chosen to simulate a high and lowpass matched filter. Because the low frequency com-ponents of a real target scene usually have magnitudeseveral orders higher than its high frequency compo-nents, a choice of reference beam in the nonlinearmatched filter fabrication can dictate two types ofmatched filters. When the reference beam is chosen atthe same level as the high frequency components of thetarget spectrum, the recorded target signal exhibits

200

160

I-(nz

(A) z

C.)

0

120

80

40

20

0

3500

3000

I-z

I-

0

2500

2000

1500

1000

500

0

2.5

C,'z

z

(C) O0

1=-

00

2.0

1.5

1.0

.05

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1.0

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z

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C.,

:r0JA:

.08

.06

.04

.02

A

0 40 80 120 160 200

0 40 80 120 160 200

0 40 80 120 160 200

0 40 80 120 160 200

Fig. 3. Output signal for a nonlinearly recorded matched filter: (A)reconstructed signal for the low pass matched filter; (B) reconstructedsignal for the high pass matched filter; (C) output correlation functionfor the low pass matched filter; (D) output correlation function for

the high pass matched filter.

1168 APPLIED OPTICS / Vol. 16, No. 5 / May 1977

u

certain high frequency enhancement. This type ofmatched filter is referred to as a high pass matched fil-ter. On the other hand, when the reference beam is setat a level close to the dc or low frequency componentsof the target spectrum, this type of matched filter isreferred to as a low pass matched filter. For the rec-tangular target function Rect(x/D), we choose a refer-ence beam equal to the third lobe of the Sinc functionSinc(pD) for the high pass matched filter simulation.For the low pass matched filter, the reference beam ischosen to be equal to the dc term in the zero-order lobeof the Sinc function. The nonlinearly recorded targetsignals can be reconstructed for both the high and lowpass matched filters. They are shown in Fig. 3 as (A)low pass and (B) high pass. Note that the recordedtarget signal G is highly distorted in the high passmatched filter as compared to that in the low pass filter,although the diffraction efficiency of the high pass filter(-1.78%) is much larger than that of the low pass filter("0.26%). Note also that by multiplying both filterswith the target signal spectrum and inverse Fouriertransforming the results, we can display the outputautocorrelation functions of both filters in the correla-tion plane. These are shown in Fig. 3 as (C) low passand (D) high pass. The peak correlation for the highpass filter is essentially lower than the low pass filter.

One fundamental property of the matched filter is itsability to detect the stored target signal while rejectingother input signals. To evaluate this property for boththe high and low pass matched filters, a Rect(x/D)function with D = 29 is chosen as a target signal andrecorded for both filters. Once the filters are made, weinput a set of other signals to both filters and plot theoutput correlation function in the correlation plane.For the convenience of simulation, the set of other sig-nals is chosen as the same Rect function, Rect(x/D), butwith D different from D = 29. Hence, the output cor-relation function also represents a scale sensitivity forthe matched filter. Figure 4 is the plot of output cor-

u,2.5llJz-2.0z0

< 1.5-J

LU

cccc1.ol

;0 80 100 120 140

CORRELATION PLANE COORDINATE X

Fig. 4. Output correlation functions for a set of input signals for thehigh pass matched filter: solid line, for D = 29; dashed line, for D less

than 29; dotted line, for D larger than 29.

CA

'_ 2.1uJz0Z 1.!

-J

LUcc 1.1cc0C.

.0

40 60 80 100 120 140CORRELATION PLANE COORDINATE X

160

Fig. 5. Same as Fig. 4 for low pass matched filter.

relation functions for the high pass matched filter. Theoutput correlation functions are normalized accordingto the definition of Eq. (2). The solid line representsthe output for the target signal (D = 29), while thedashed lines (D < 29) and dotted lines (D > 29) repre-sent other input signals whose size is either smaller orlarger than the target. It is interesting to see that largefalse alarms occur, since a set of input signals with D '7 has peak correlation larger than the target peak cor-relation. Notice also that the false alarm peaks did nothappen at the location where the autocorrelation peakis supposed to be. This phenomenon can be understoodif we notice that the distorted target signal shown in Fig.3(B) has two peaks, and these correlate with the in-coming signal. The same correlation function plot forthe low pass matched filter is shown in Fig. 5. Here nofalse alarm signals are present since the recorded targetsignal is not severely distorted. The high pass filter hasbetter discrimination capability than the low pass filter.This can be seen in Fig. 6 where the peak correlation atthe center of the correlation plane is plotted as a func-tion of target size D for both filters (curves 1 and 2 forlow and high filters, respectively). Note that the highpass filter is much more sensitive than the low pass filterfor detecting size changes. In other words, the high passfilter discriminates more than the low pass filter. Thisproperty is well known, aside from the fact that a largefalse alarm signal may occur for this filter. We nowsimulate our compensation technique to see if itworks.

A compensation mask function is calculated bychoosing the same recording parameters and film modelfor which the high pass matched filter is made. For A= 3, the mask function is calculated by using the 649-Fplate Ta - E characteristic curve rather than by thetransfer curve shown in Fig. 1. This simulates thesecond method of fabricating the mask by directly ex-posing the film with the target signal spectrum. Oncethe compensation mask is made, we process the same

May 1977 / Vol. 16, No. 5 / APPLIED OPTICS 1169

D -29

"75

i. 3 -

12

r

O 2.0 URVE

,, 1.6cccc0

1.2CURVE 2

(..08

0 .04-

03 11 19 27 35

OBJECT SIZE D

Fig. 6. On-axis peak correlation as a functior= 3 to D = 61: curve 1, for low pass filter;

filter.

HC"

LLz-

wZ

cc

cc0C.

;0 80 100 12(

CORRELATION PLANE COORE

Fig. 7. Output correlation functions for the hi]using a direct compensation technique for the sai

as in Fig. 4.

I

set of input signals used in the simulation of Fig. 4through both the mask function and the high passmatched filter. The output correlation functions arethen normalized against the total signal energy imme-diately behind the mask according to the definition ofEq. (4). The output correlation functions are plotted inFig. 7. It can be seen that all false alarm peaks thatwere present in Fig. 4 are not shown in Fig. 7. The ratioof the peak target correlation to peak false target cor-relation is about 4. We also simulated the case for A =1.5. The mask function is calculated using the transfercurves. The output correlation functions are similarto those shown in Fig. 7 where no false alarm occurs.The peak target correlation to peak false target corre-lation is about 3. The normalization procedures as

43 51 59 defined by Eqs. (2) and (4) play a crucial role in thesuccess of the compensation technique. For example,if no normalization is used in the simulation, no false

i of input signal sizeD alarm peak exists. The ratio of peak target correlationcurve 2, for high pass to false target peak correlation for both Figs. 4 and 7 is

about 2.2. This means that an intensity increase by afactor of 2.2 for the false target signal can trigger a falsealarm for the system. To eliminate such false alarmpossibilities, a normalization against the total inputsignal energy is used in the definition of Eq. (2). But,this normalization itself introduces a false alarm for anonlinearly recorded target signal. The introductionof mask compensation and normalization against thesignal energy behind the mask eliminates both falsealarm possibilities. Simulations are also conducted in2-D where a square target with a dimension of 29 X 29was recorded for a high pass nonlinear matched filter.The corresponding mask function is also generated.The impulse response of the distorted square target isreconstructed by illuminating the matched filter with

21 41 51 61 a collimated beam, and this is shown in Fig. 8 in 2-D.Two signals, one of the square target itself and the other

0 :40 ; a square letter 0 with the o.d. equal to 29 X 29 (i.d. is 23D E140 1 0 X 23), were processed through the matched filter, first)INATE X with and then without the mask. With no mask com-

thd filtr pensation, the false target (square letter 0) has a cor-me st f tht iter relation peak higher than that of the real target (square)

by a factor of 2.578. With mask compensation, the realtarget has a correlation peak higher than the false target(square letter 0) by a factor of 1.318. In the case of asmall square with a dimension of 9 X 9, the peak targetto false target ratio is 18.4752. The configuration of theoutput correlation functions for the above two falsetargets is also quite different. Figure 9 is the correlationfunction for the square letter 0, and Fig. 10 is for thesmall square (9 X 9).

Fig. 8. Impulse response of a nonlinearly recorded square targetfunction (29 X 29).

SummaryA technique is proposed to reduce the false alarm rate

of a nonlinearly recorded matched filter. The methodis to put an additional mask immediately before thematched filter and normalize the correlation outputagainst the total signal energy that emerges behind themask. The transmittance function of the mask is cal-culated for a given target signal and for the samematched filter recording scheme. For a particular re-cording parameter A = 3, the mask functions can be

1170 APPLIED OPTICS / Vol. 16, No. 5 / May 1977

LFig. 9. Cross-correlation function of a square letter 0 with target

I square function (29 X 29).

Fig. 10. Cross-correlation function of a small square function (9 X9) with target square function (29 X 29).

fabricated by directly exposing a film with the targetsignal spectrum. The technique is very useful for ap-plication to a high pass matched filter which usually hashigh discrimination capability but may have large falsealarms. The requirement of normalization in thistechnique may cause difficulty in some applications. Inthe single character recognition application, where eachcharacter is an isolated signal, the normalization can beapplied easily. In the application of reconnaissancetarget detection, the normalization procedure requiresmany more complicated implementations.

The author thanks Ron Wohlers and Ken Leib fortheir interesting discussion of the work and for theirrecommendations in writing the manuscript. I alsothank Jay Mendelsohn, Mike Rossi, and Marc Hoff fortheir constant help in computer simulation. This workwas supported in part by the Night Vision Laboratory,Fort Belvoir, Virginia, under contract DAAG 53-75-C-0199.

References1. A. Kozma, Opt. Acta 15, 527 (1968).2. A. A. Friesem and J. Zelenka, Appl. Opt. 6, 1755 (1967).3. A. Vander Lugt, IRE Trans. Inf. Theory IT-10, 139 (1964).4. J. W. Goodman and H. B. Strubin, J. Opt. Soc. Am. 63, 50

(1973).5. D. J. Raso, J. Opt. Soc. Am. 58, 432 (1968).6. R. A. Binns, A. Dickinson, and B. M. Watrasiewicz, Appl. Opt. 7,

1047 (1968).7. A. Vander Lugt and F. B. Rotz, Appl. Opt. 9, 215 (1970).8. R. Gonsalves, R. Dumais, and P. Considine, in SPIE Seminar

Proceedings (1974), Vol. 45.9. J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297 (1965).

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May 1977 / Vol. 16, No. 5 / APPLIED OPTICS 1171