MTF evaluation with optical matched filters

Alex Grumet

A modulation transfer function (MTF) measuring technique is described that employs the optical matchedfilter (OMF). It is shown that this technique is mathematically equivalent to the multiple-sine-slit micro-densitometer (MSSM) MTF measurement approach. Similarities and differences in the implementationof the two methods are discussed. Both theoretical and experimental matched filter results are investi-gated. A method for extending the OMF to high spatial frequencies is discussed.

1. Introduction

The modulation transfer function (MTF) (a measureof the depth of modulation vs spatial frequency) of animage storage device or medium has been measured inmany ways. Some of these are described in Refs. 1-7.Reference 7 compares some of the competing MTFmeasuring techniques and comes to the conclusion thatfrom several points of view the multiple-sine-slit mi-crodensitometer (MSSM) has many advantages overthe other MTF measurement approaches. A spatialfrequency version of the MSSM method is essentiallyoptical matched filtering. The equivalence of the twomethods will be mathematically demonstrated, andsome MTF theoretical and experimental results will beconsidered.

II. MSSM Method

In the MSSM method each of a series of sinusoidalgratings stored in the film to be evaluated yields a singlepoint on the MTF curve. In one form of the MSSMmethod each stored grating is scanned with a sinusoidalirradiance pattern of the same spatial frequency gen-erated by the interference of two collimated coherentbeams. The transmitted light flux is collected, inte-grated, and detected by a photodiode. The process isessentially that of autocorrelation if the scanning andstored gratings are identical,

1 r+Y/2 p+X/2oi) = EY/2 r-X19 i(x,y)i(x + u,y)dxdy, (1)XY J- Y/2 S -/2

where i(x,y) is the 1-D amplitude grating with thegrating lines parallel to the y axis, and the scanning uis in the x direction. X and Y define the pupil of the

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

Received 14 November 1975.

grating, and 0ii(u) is the 1-D autocorrelation function.It is pertinent at this point to review some of the prop-erties of the autocorrelation function:

(1) i(0) 2Pi i(U);(2) i(o) = 2;(3) if i(x,y) is periodic in the x direction then ii(u)

is an even function of the same periodicity in thesame direction.

The scanning light pattern i(x + uy) modulation isadjusted to unity by adjusting both interfering beamsto be of equal amplitude. However, the stored gratingcan more accurately be written i(x,y) = io [1 + m(fx)cosr27rfx], where m is the modulation to be measuredand is a function of spatial frequency fi.

Property 3 is of interest because distortion of thestored sinusoidal grating can be determined by scanningall the harmonics in separate scans. The method hasconsiderable resolution realized by scanning u slowly,and high spatial frequencies can be measured accu-rately. The integration together with the large signalrealized by scanning the entire aperture results in a largeSNR at the output.

III. Storage Medium Nonlinearity

Nonlinearity introduced by the D-logE curve ofphotographic film is discussed in Refs. 4 and 7-9. Ingeneral, the distortion increases with both the depth ofmodulation (contrast) and the gamma (slope of the DlogE curve). Adjacency effects are more pronouncedand introduce an additional dropoff of the MTF athigher spatial frequencies. Reference 7 outlines ameans of estimating the nonlinearity introduced by theD-logE curve. If the nonlinearity is known, a singlegrating of known spectral shape could yield the MTF.One possibility is a square wave grating where the nthspatial harmonic is 1/n of the fundamental amplitude.Therefore, correcting for the nonlinear distortion of thenth harmonic nfx and multiplying by n should yield theMTF at harmonically related spatial frequencies withthe use of only one stored square grating.

154 APPLIED OPTICS / Vol. 16, No. 1 / January 1977

IV. Optical Matched Filter (OMF)

The amplitude spectrum of the 1-D grating i(x) isobtained from its Fourier transform10

X(1)Ia(fnin) =ve

and inversely

i(x) exp(-j27rfxx)dx (2)

(3)i(x) a a I(fx) exp(2rfxx)dfx,

abbreviated as a Fourier pair

i(x) I- NJ (4)

The spectral power density of i(x) is definedP 1(f,) = II(fx)12 = I(fx) I*(fx), (5)

where the asterisk denotes complex conjugate. In ad-dition, the autocorelation function and the spectralpower density are a Fourier pair

(6)

as specified by the Wiener-Khintchine theorem. 11

From Eqs. (5) and (6), autocorrelation is seen to beequivalent to passing the spectrum of the stored gratingthrough its conjugate filter. This process is known asmatched filtering and has been described by VanderLugt.12 The right side of Eq. (5) represents the outputimmediately behind the OMF where I(fx) is thematched input-grating amplitude spectrum and I* (fx)is the amplitude transmittance conjugate order of theOMF.

The OMF is an optimum filter in the sense that ityields the peak SNR when the signal is matched and thenoise is either Gaussian or non-Gaussian. When theinput image is matched to that of the filter, the outputfield amplitude (upon inverse Fourier transformation)is the autocorrelation function of the input. When theinput image is mismatched to the filter, the output fieldamplitude is the cross correlation function of the storedimage amplitude (filter) and input image amplitude.The arrangement for fabricating the matched filter isshown in Fig. 1, and the playback arrangement is shownin Fig. 2.

V. Comparison of MSSM and OMF

As Sec. IV indicated, the OMF is mathematicallyequivalent to the MSSM method of MTF measurement.However, the physical configuration of both methodsis quite different. Let us first consider the similarityof both methods: (1) both require a different inputgrating for each point on the MTF; (2) both process theentire aperture of the input grating, therefore, yieldingexcellent signal-to-noise; (3) it is expected that both areaffected by phase distortions in the grating in the samemanner; and (4) both require accurate scale and rota-tional alignment of the grating (with the scanning beamfor MSSM or with the matched filter for OMF). Thedifferences are more dramatic: (1) OMF requires co-herent light, whereas either incoherent1 or coherentlight7 can be used for MSSM; (2) OMF may be scanned

Fig. 1. Schematic of matched filter fabrication arrangement.

INPUTGRATING

TRANSFORMLEN S

MATCHEDFILTER

INVERSETRANSFORMLEN SBEAM

CORRELATIONPLANE

_x

Fig. 2. Schematic of matched filter playback.

at the output, whereas MSSM must be scanned to findsoii (o); (3) OMF requires the preparation of a matchedfilter for each grating, whereas, MSSM requires twointerfering beams; and (4) OMF requires no integratingsphere, whereas MSSM does.

VI. OMF Investigation

The input for all the cases considered were holo-graphically generated square gratings (generated byexciting the film below the threshold and into satura-tion) in a circular aperture on 649F spectroscopic film.The sinusoidal grating may be realized by masking allbut the first harmonic in the OMF. The MTF mea-surements were made on an electrooptical image storagedevice (consisting of a cadmium sulfide and a nematicliquid crystal thin film in intimate contact) rather thanon photographic film. However, the technique is ap-plicable to any image storage (temporary or permanent)medium. A closed circuit TV camera focused on thecorrelation plane served as the detector and permittedthe display of the OMF output. A TV camera is a poordetector for the higher spatial frequencies because ofthe limited video bandwidth and was marginally ef-fective at 20 lp/mm. Three cases are considered: (1) allthe square wave spectrum will be used; (2) only the zeroorder (dc) and first order (fundamental spatial fre-quency) terms will be used; and (3) only the first orderterms are considered.

January 1977 / Vol. 16, No. 1 / APPLIED OPTICS 155

SPATIAL

MATCHEDFI LTERPLANE

INPUTTEST GRATINGIMAGE PLANE

,Pi (U) Pi (fx)

f

A. Square Grating

A 1-D analysis with no loss in generality will be con-sidered. The one dimension is transverse to the gratingat the center of, and in the plane of, the aperture. Thesquare grating of period a may then be representedby

+N 2rect- (x-na)

n=-N a(7)

with (2N + 1) cycles at the center of the circular aper-ture of diameter (2N + 1)a. The rectangular functionis conventionally defined as

rect(x) = I I I /,10 otherwise.

The spectrum of the grating of Eq. (7) is seen to be

I(f) a sinir(2N + 1)af sin(7raf/2)2 siniraf 7raf/2

(8)

.2

= .15......C

CD,

0 AAA 4~AAAA-20 -15 -10

(9)-5 0 5 10 15 20

DISTANCE

and is the matched filter of the grating because i(x) isa real even function with i(x) = i*(x) and i(x) =i(-x).

From Eqs. (5) and (6) qpj(u) I(f)I*(f), wherei*(-x) I*(f). From the convolution theorem i(x) ii*(-x) I(f)I*(f), where @d denotes convolution.Therefore, when an input grating is matched to the fil-ter, the output amplitude is

ii(u) = i(x) i(x),

Fig. 3. Normalized correlation intensity of a square grating (N =8).

(10)

the autocorrelation function of i(x), and is seen to be

= (2N + 1)a22N /fl\F2 1Oii (x)=(2 "l A ) A ) (x-na)] (11)

2 = 2 N (2N + Ja11for the square grating. The triangular function A(x)is conventionally defined as

AW) = I1- xI for lxI 1,t0 otherwise.

(12)

We will normalize the autocorrelation function of 0 ii (x)to (2N + 1) to make it independent of the (2N + 1) cy-cles in the aperture. The normalized function

ii (X)/(2N + 1) will then be squared to compare it withmeasured detector outputs. Figure 3 is a plot of Eq.(11) normalized and squared. Figures 4(a) and 4(b) arephotographs of a TV monitor and oscilloscope, respec-tively, for the same conditions as in Fig. 3. The TVsystem nonlinearity reduced the peaks of the videosignal in Fig. 3 so that it appears as shown in Fig. 4.

B. Sinusoidal Grating

If all the harmonics of the square grating matchedfilter above the first are removed, both a square waveor sinusoidal input grating will play through thematched filter as a sinusoidal grating. To consider thiscase analytically we will convolve a sinusoidal gratingwith a square grating. The sinusoidal grating is rep-resented by

S(X) = E cos2r na) rect (X Inan=-N a a

and the correlation function will be

,pi,(u) = i(x) ED s(x).

(13)

(a)

(b)

Fig. 4. Autocorrelation with a 4-lines/mm grating in laser beam: (a)(14) TV monitor; (b) single TV scan line through center of (a).

156 APPLIED OPTICS / Vol. 16, No. 1 / January 1977

....... ... . ..... ........... . -...... .

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

.2 .04-

o~~~~~~~~~~~~~= .15 .............. .......

Co ::; l ~~~~~~~~~~~~~~~~~~~~~~~~~~...0. .. .. .... *. .... ... .....1 ~~~~~~~~~~~~~~~~~~~~~~~~~.02-

o 0

.05 jt~~~~~~~~~~~jt~~~~~ -~~~.01

0Z2 15 _10 -5 0 5 10 15 20 0

DISTANCE -28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28DISTANCE

Fig. 5. Normalized correlation intensity of a square grating through Fig. 6. Normalia sinusoidal grating matched filter (N = 8). ig. 6 omlzed correlation intensity of a square grating througha sinusoidal grating matched filter with dc suppressed (N = 8).

Solution of the convolution integral is straightforwardbut results in too many terms to list here. The solutionof Eq. (14) is shown in Fig. 5 normalize and squared forN = 8.

C. Sinusoidal Grating with Zero Order Suppression

For transparent electrooptic image storage devices(in the unexcited mode), the full laser beam passesthrough the zero order term of the matched filter andappears at the correlation plane. To avoid this largeaperture correlation superimposed on top of the gratingcorrelations of Figs. 3 and 4, the zero order term issuppressed. This is accomplished with a blankingfunction (that is, the biased spectrum of the aperture)[Eq. (15)] in the matched filter spatial frequencyplane:

lbf W= 1 - sinc(2N + 1)af (15)

The inverse transform of the blanking function is seento be

1 x (a)ib(X) = O(X) + rect . (16)

(2N +)a (2N +)a

Correlation of a square grating with a sinusoidalgrating with blanked zero order is given by

'Pib(U) = i(X) @ S(x) @ ib(X) (17)

and is plotted in Fig. 6 for N = 8, normalized andsquared. The measured curve is shown in Fig. 7. Thediscrepancies between Figs. 6 and 7(b) are probablycontributed by both the TV nonlinearity and the pres-ence of a slight amount of third harmonic in thematched filter. For greater precision the higher spatialharmonics could be physically masked.

The frequency doubling evidenced in Figs. 6 and 7(b)is contributed by negative terms in the convolution in- (b)tegral that result in negative swings in the correlation.Squaring of the negative swings yields the observed Fig. 7. Autocorrelation of 4-lines/mm grating in laser beam, blockeddouble frequency. zero order matched filter: (a) TV monitor; (h singlep TV scan linp.

January 1977 / Vol. 16, No. 1 / APPLIED OPTICS 157

qua

D. Detector Nonlinearity

If the 1-D input grating signal amplitude is

i(x) = io(1 + m cos2irfox) rect (-), (18)X)

where io is the unmodulated laser amplitude and fo isthe grating spatial frequency, and if the grating used tofabricate the matched filter has an amplitude trans-mission

t(x) = (/2 + I/2 cos27rfox) rect (.X), (19)

the matched filter peak output amplitude can be shownto be

;,(0) 1 2 ( + 2) (20)

If the transparency of Eq. (19) is then placed directlyin the laser beam instead of the film under test [whoseoutput amplitude was given by Eq. (18)], the laser beamwill be 100% modulated by the transparency (m = 1),and the correlation peak of Eq. (20) will be larger. Ifthe laser beam amplitude is then reduced to i1 so thatthe new correlation peak qpo(0) is the same as that of Eq.(20) we have

m = 3 (Ix/I()l/2 - 2, (21)

where I, and IO are the unmodulated laser beam in-tensities, respectively, of amplitudes il and io and I, <Io. For m = 0 the ratio I11/0 is seen to be 4/9.

This method of determining the modulation is in-dependent of the detector and its nonlinearities. Thedetector simply serves as a reference indicator. Theratio measurement of I, and Io can be simplified byinserting neutral density filters (such as Kodak'sWrattan filters) to reduce Io to the correct II.

E. Matched Filter Fabrication

Figure 1 is a typical arrangement for fabricating amatched filter. This same arrangement is also used tosuppress the higher grating spatial frequencies as wellas the zero order. To suppress all harmonics above thefirst, the reference beam is adjusted to be exactly equalto the peak of the first harmonic. This may be deter-mined in several ways; a straightforward method issimply to observe the first harmonic interference pat-tern with a microscope at the matched filter plane andadjust for maximum contrast. The contrast of thephase interference pattern on the higher harmonics isseen to be reduced, with the greater reduction on thehigher harmonics. If the exposure for the matchedfilter plate is adjusted so that the third harmonic (thenext harmonic after the fundamental for a square wave)falls below the knee of the D logE curve of the film, allharmonics above the first will not be stored in thematched filter. In practice the D logE curve does nothave a sharp knee, and some third harmonic will be re-corded and stored in the matched filter, probably ac-counting for the discrepancy between Figs. 6 and 7(b).

A physical mask could be placed over all harmonicsabove the fundamental to remove them completely.This was not the case for the data shown in Figs. 4 and7.

To suppress the zero order, a second exposure is madewith the reference beam blocked and the grating re-moved from the input aperture. The resulting matchedfilter has the on-axis zero order saturated so that itsphase information is obliterated.

F. Output Resolution

In both the MSSM and the OMF method, MTF ismeasured by reading 0ii (0). The accuracy with whichu can be set equal to zero, and the resolution Au, willeffect the accuracy of the MTF measurement. In theMSSM method the scan can be stopped at the maxi-mum corresponding to u = 0 and Au = 0. However,since the OMF method has the entire correlationfunction displayed with no scanning, there are two al-ternatives that can be employed. A stationary slit withsome finite Au can be employed or the correlation pat-tern can be scanned and displayed. The electroopticimage storage device under test had a measured MTF6-dB rolloff at 20 lp/mm and presented no substantialproblem for the closed circuit TV. By observing the TVscan line that passed through Xii (0) on an oscilloscope[Figs. 4(b) and 7(b)] the MTF was accurately measured.However, at very high spatial frequencies the TV videobandwidth is not adequate. For example at 2000 lp/mm, one line pair is 0.5 gm wide (of the order of thewavelength of light) and a 1.27-cm (1/2-in.) TV cameratube would require approximately a 360-MHz band-width, far beyond the 4 MHz or 5 MHz of present TV.However, it is difficult but not impossible to view a2000-lp/mm grating with a suitable microscope. At 250power the grating will appear as 8 lp/mm, and this maybe scanned with a 10-,um wide slit to read the peakcorrelation of the matched filter output with a resolu-tion of less than one tenth cycle.

VII. Conclusions

It has been mathematically demonstrated that themultiple-sine-slit microdensitometer (MSSM) and theoptical matched filter (OMF) both measure correlation.No effort has been made here to show exhaustively thecomplete practical equivalence of the two MTF mea-surement methods because the motivation for this studywas MTF evaluation of an image storage device in anexisting matched filter hardware setup. For low spatialfrequencies, implementation of the method isstraightforward. Although there are some practicalproblems for high spatial frequency OMF MTF mea-surements, they do not appear to be insurmountable.

The helpful discussions with R. E. Kopp and M. R.Wohlers are gratefully acknowledged.

The MTF measurements technique described herewas partially supported by the Naval Air. SystemsCommand under contract N00019-73-C-0303.

158 APPLIED OPTICS / Vol. 16, No. 1 / January 1977

References1. J. C. Vienot, Proc. Phys. Soc. 72, 661 (1958).2. R. L. Lamberts, J. Opt. Soc. Am. 49, 425 (1959).3. D. H. Kelly, J. Opt. Soc. Am. 51, 319 (1961).4. R. L. Lamberts, J. Opt. Soc. Am. 51, 982 (1961).5. A. Vander Lugt and R. H. Mitchel, J. Opt. Soc. Am. 57, 372

(1967).6. G. R. Kumar and K. Sayznagi, J. Opt. Soc. Am. 58, 1369

(1968).

7. S. Johansson and K. Biederman, Appl. Opt. 13, 2280 (1974).8. R. L. Lamberts, J. Opt. Soc. Am. 60, 1389 (1970).9. J. S. Wilczynski, Proc. Phys. Soc. 77, 17 (1961).

10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1968).

11. K. S. Miller, Engineering Mathematics (Rinehart, New York,1957) p. 174.

12. A. Vander Lugt, IEEE Trans. Inf. Theory IT-0, 139 (1964). Amore complete version appears as AD 411 473, July 1963.

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continued on page 174

January 1977 / Vol. 16, No. 1 / APPLIED OPTICS 159