dependence of incoherent system response on object contrast

Dependence of Incoherent System Response onObject Contrast

A. K. Jaiswal and Jagat Bhushan

A relationship between the image contrast and the object contrast for a general periodic or a truncatedperiodic object is derived. The result is illustrated for a periodic object of rectangular wave profile. Ithas been shown that for a truncated sine wave object the image contrast bears a nonlinear relationship tothe object contrast. The contrast dependence of the measured or the apparent transfer function definedas the ratio of the image contrast to object contrast has been investigated.

1. Introduction

It is well known that an incoherent system givesrise to a sinusoidal irradiance distribution in theimage of a continuous sine wave object. The con-trast in the image bears a linear relationship withthe contrast in the object, and the ratio of the two ata certain spatial frequency gives the optical transferfunction (OTF) of the system at that frequency.However, in the usual arrangements for testing ofoptical systems and in the measurement of OTF onlyfinite cycle targets are used. The truncation of thetarget gives rise to an error in the.measurement ofsystem response that has been discussed in past byseveral authors.'- 7 We will show in this paper thattruncation error depends on the object contrast andthe use-of a test target of lower contrast yields betterresults in the measurement of OTF.

In Sec. II, a relationship between the image con-trast and the object contrast for a general periodic ora truncated periodic object is derived. The result isillustrated for a periodic object of rectangular waveprofile. It has been shown that image contrast de-pends nonlinearly on the object contrast if the objectdistribution contains even harmonics. In Sec. III,the incoherent system response for truncated sinewave objects is discussed. The contrast dependenceof the truncation error is presented for some repre-sentative systems.

11. Image Contrast vs Object Contrast

Let us consider a periodic or a truncated periodicobject O(z) of unit contrast where the contrast is de-fined as

The authors are with the Optics Research Group, InstrumentsResearch & Development Establishment, Dehra Dun, India.

Received 2 November 1973.

Co = (Omax - min)/(Omax + omin). (1)

The image contrast is defined in a similar way andwill be denoted by C when C = 1. Consider nowan object of modulation 2b and mean irradiance a,i.e.,

O'(z) = a - b + 2b0(z), (2)

so that C' = b/a. Since incoherent systems are lin-ear in irradiance, the irradiance distribution in theimage of this object is given by

I'(z') = a - b + 2bl(z').

The image contrast is found to be

Ci = b(Imax - Imin)/[a - b + b(Imax + Imin)],

whence the ratio

CiICo = C 2f/[l - C'(1 - 2)],

(3)

(4)

(5)

where I = (max + Imin)/2 denotes the mean irra-diance in the image. It follows from Eq. (5) that theratio C'/C,' depends on C,' except when I = 0.5.

It is easily verified that I = 0.5 for an infinite cyclesine wave object or a square wave object. A moregeneral object with rectangular wave profile is givenby the following Fourier series representation:

0 ( ) +2 Esin(n7rca) OR(Z = a + 2 c i~r),Los(nwaz),

n-1 n(6)

where the object parameter a is such that (1 - a)/arepresents the ratio of the widths of bright and darkspaces. The image irradiance distribution is givenby

IR(Z') = a + 2 T(nw) cos(nwz'),n-i nl

(7)

where T(w) is the normalized OTF of the system.From Eq. (7) we obtain

August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1839

been shown5' 8 that the contrast in the image of a barobject is equal to that of a sinusoidal object near theresolution limit of the system and that the agree-ment between the two is close even in the low fre-quency region. This correspondence between therectangular wave object and the sine wave object isvalid only when the object contrast is unity. For anobject of lower contrast the corresponding value of acould be determined from the equation

sin7ra = r(1 - C + 2aC0')/4. (9)

Hence the use of a general bar target to measure thesine wave response of a system would require thepreparation of a test target with specified contrast,which is not practical.

11. Contrast Error in the Measurement of OTF

Let us consider an N-bar truncated sine waveobject of unit contrast represented by

0(z) = [1 + cos(2r/4L)z] if Izi < 2NL

0.5 10 = otherwise, (10)

C'

Fig. 1. Dependence of image contrast C' on object contrast C'for a rectangular wave object at frequency co = 1.0. (1) a = 0.2,

(2) a = 0.4, (3) a = 0.6, (4) a = 0.8.

I_ Imax - Imin = E 4sin(ngra) T(n)

n odd

2sin(nra) T(nw),

n even

where 4L is the normalized period of the object and zis the normalized distance in the object plane. Theobject spectrum may be obtained by using the Fouri-er transform relationship

LO

(8a)

(8b)

where irradiance maxima and minima have beenevaluated at coz' = 0 and wz' = r, respectively. Itfollows from Eq. (8) that the image modulation isaffected by the odd harmonics, whereas the mean ir-radiance level in the image is affected by the evenharmonics of the fundamental frequency of theobject. For a square wave object with a = 0.5, theeven harmonics are absent and I = 0.5. On theother hand, when a > 0.5, I > 0.5 and the C'/C0' increases with a decrease in C'. The reverseis true when a < 0.5.

Figure 1 represents the dependence of image con-trast on object contrast for rectangular wave objectsof different values of a at frequency w = 1.0. Thenonlinear relationship between C' and C 0' when a# 0.5 is quite evident from the graphs. In Fig. 2 theratio Ci'/C 0' for a = 0.75 has been plotted as a func-tion of the normalized frequency co for differentvalues of C,'. It shows that the system response de-fined as the ratio Ci//C 0' depends on the contrast ofthe target. In both Figs. 1 and 2 the system is as-sumed to be diffraction limited with a circular aper-ture.

For the value of the object parameter a satisfyingthe condition sin7ra = 7ra/2, i.e., for a = 0.60, it has

IC)

C.

0

Fig. 2. Response Ci'/C 0' vs frequency for a rectangular waveobject with a = 0.75. (1) Co = 1.0, (2) C 0' = 0.7, (3) C 0' = 0.4,

(4) Co' = 0.1.

1840 APPLIED OPTICS / Vol. 13, No. 8 / August 1974

1.0

as5

-(

0(w) = 2 fb 0(z) exp(-iwz) dz, (11)

which reduces on using Eq. (10) to

(w), wt) = sin(Nrww/Jt)/{27wo1 - (W/(t)2]}, (12)

where wt = 2r/4L is the normalized frequency of thetarget. The image irradiance distribution is ob-tained by taking the inverse Fourier transform of theproduct of the object spectrum and the transferfunction, i.e.,

I(z') = f O(w)T(w) exp(iwz') d, (13a)

which may be expressed as

I(z') = 2 O(WT(w) cos(coz') d. (13b)

The object spectrum 0(w, wt) a continuous function of co with pAs the value of N increases, thiquency spectrum get narrower an

, O(w, COt) become discrete cospectrum of a continuous sineimage contrast in this case equother hand, when spectrum is coirradiance and hence the image cspatial frequency depend on the ventire frequency range [c.f. Eq. (1for the difference in the image coied object and a continuous one.ference depends among other factransfer function. For truncatedmean irradiance I is found to deling value 0.5, so that image cont.nearly on the object contrast. will investigate the contrast depesured or the apparent transfer ffined as the ratio of the image cortrast for a truncated sine wave objE

We will restrict our attention without loss of generality. Equalan object in a dark background mum for odd N. We will also conwhite background represented by

°1(Z) = [1 - cos(2r/4L)z]

=1which has a minimum at the c(image. To emphasize the role of function we will consider three tydiffraction limited circular apertuerture with obscuration ratio = 7r-phased aperture with = 0.5.ture may be considered to be thegood system. On the other handerture may be considered to reprefor which the transfer function ha,the midfrequency region.9

The images of unit contrast objects were obtainedby numerically evaluating the integral in Eq. (13b)using 64-point Gauss quadrature scheme. The nor-malization of the irradiance was ensured to be suchthat I = 0.5 for the infinite cycle target. The irra-diance maxima and minima were determined direct-ly from the images. It was found that the central re-gion of the image for truncated target gives the bestapproximation to the image of an infinite cycle tar-get. Rence only the central and the first maximumor minimum were used in the calculation of the con-trast. These results in conjunction with Eq. (5)were used to evaluate the image contrast for differ-ent values of C0'. For the object represented by Eq.(14), it is easily verified that

01(Z) = 1 - 0(z); 11(z) = 1 - I(z). (15)

Hence the image contrast in this case could be cal-given by Eq. (12) is culated by using the results of the previous case.eaks at w = 0, +wt. The results for the three types of aperture for co =e bands in the fre- 0.5, 1.0, 1.5; N = 3, 5, 7, 9; and C0 = 1.0, 0.75, 0.5.d in the limit N 0.25 are presented in Tables I-III. TA and TA repre-rresponding to the sent the apparent transfer functions in the two caseswave object. The of dark and white backgrounds, respectively. Theals T(wt). On the percentage error AT/T has also been given againstntinuous the image each value. The last entry in each column for a setontrast at a certain of values of wt and N gives the value of image modu-'alues of T(w) in the lation I, its percentage error and the value of the.3)]. This accounts mean irradiance I.ntrast for a truncat- We will first examine the effect of truncation onEvidently the dif- the image contrast for a unit contrast object in a

tors on the system dark background. It follows from Tables I-III thatobjects the value of for C' = 1, the apparent transfer function TA(wt) is)art from the limit- always more than the system transfer function T(wt).rast depends nonli- The error due to truncation is maximum for N = 3In what follows we in which case the image is a very poor approximationndence of the mea- to the image of an infinite cycle target. As the valueunction TA(wt) de- of N increases, the error T/T steadily decreases,itrast to object con- the improvement being maximum in going from N =ect. 3 to N = 5. The truncation error is found also toto odd values of N depend on the frequency of the target. In most oftion (10) represents the cases considered, it is maximum for t = 1.5 andvith a central maxi- minimum for wt = 0.5. The order of magnitude ofLsider an object in a truncation error is quite different for the three sys-

tems under consideration. For a particular value ofN the error is least in the case of a circular aperture.

if IZi < 2NL It is more for an annular aperture and maximum fora 7r-phased aperture. Barakat and Lerman6 haveotherwise, (14) pointed out that a seven-bar target is an excellent

approximation to the infinite cycle target. It mayentral point of the be said that their conclusion is true for a good sys-the system transfer tem. For a bad system represented by 7r-phased ap-pes of aperture: a erture in our case, even N = 9 is seen to be a poorire, an annular ap- approximation to a continuous sine wave object.0.5, and a centrally The conclusions drawn above are true quite gener-The circular aper- ally. Some exceptions may, however, be noted inrepresentative of a the case of the 7r-phased aperture especially for cot =, the r-phased ap- 1.0, where T(wt) is negative. In this case, for exam-sent a poor system ple, TA(wt) < T(wt) for N = 5 and 9. Moreover, the

a negative lobe in magnitude of the error T/T is found to be less for'N = 7 as compared with N = 9.


Table I. Contrast Dependence of Truncation Error for a Circular Aperture

= 0.5 T(wt) = 0.6850 Wt = 1.0 T(wt) = 0.3910 wt = 1.5 T(Qwt) = 0.1443

N CO' TA error TA error TA error TA error TA error TA error

1.00 0.7018 2.45 0.6713 -2.00 0.4119 5.33 0.3765 -3.70 0.1574 9.08 0.1375 -4.690.75 0.6978 1.87 0.6750 -1.47 0.4071 4.11 0.3806 -2.66 0.1546 7.15 0.1397 -3.16

3 0.50 0.6939 1.30 0.6787 -0.92 0.4024 2.92 0.3848 -1.59 0.1519 5.28 0.1420 -1.580.25 0.6900 0.73 0.6824 -0.38 0.3979 1.75 0.3890 -0.50 0.1493 3.48 0.1444 0.05

0.6862 0.18 0.4889 0.3934 0.61 0.4776 0.1468 1.73 0.46631.0 0.6944 1.37 0.6764 -1.25 0.4020 2.81 0.3815 -2.42 0.1511 4.71 0.1397 -3.150.75 0.6921 1.04 0.6786 -0.93 0.3993 2.13 0.3840 -1.79 0.1496 3.66 0.1411 -2.24

5 0.50 0.6898 0.70 0.6808 -0.61 0.3967 1.45 0.3865 -1.16 0.1481 2.62 0.1424 -1.300.25 0.6876 0.37 0.6831 -0.28 0.3941 0.79 0.3890 -0.52 0.1466 1.61 0.1438 -0.35

0.6853 0.04 0.4935 0.3915 0.13 0.4870 0.1452 0.62 0.48051.0 0.6915 0.95 0.6788 -0.91 0.3986 1.95 0.3841 - 1.78 0.1487 3.07 0.1407 -2.500.75 0.6899 0.72 0.6804 -0.68 0.3967 1.47 0.3858 -1.33 0.1477 2.34 0.1416 - 1.84

7 0.50 0.6883 0.48 0.6819 -0.45 0.3949 0.99 0.3876 -0.87 0.1466 1.62 0.1426 - 1.170.25 0.6867 0.25 0.6835 -0.22 p.39 3 0 0.52 0.3894 -0.41 0.1456 0.91 0.1436 -0.48

0.6851 0.01 0.4954 0.3912 0.05 0.4907 0.1446 0.21 0.48611.0 0.6900 0.73 0.6801 -0.71 0.3967 1.46 0.3854 -1.42 0.1476 2.28 0.1413 -2.050.75 0.6887 0.54 0.6813 -0.54 0.3953 1.09 0.3868 -1.07 0.1468 1.72 0.1421 - 1.53

9 0.50 0.6875 0.36 0.6825 -0.36 0.3938 0.73 0.3882 -0.71 0.1460 1.16 0.1429 -1.000.25 0.6862 0.18 0.6838 -0.18 0.3924 0.36 0.3896 -0.36 0.1452 0.61 0.1436 -0.47

0.6850 0.0 0.4964 0.3910 0.0 0.4928 0.1444 0.07 0.4892

To examine the dependence of truncation error on target of lower contrast is to lower the value of theobject contrast, we expand Eq. (5) as follows: measured or the apparent transfer function. Since

TA is always more than the OTF of the system, a re-Ci'/C0' = TA = I[1 + (2AY)C0' + (2A )2C0

12 + ... ] duction in contrast reduces the error due to trunca-(16) tion. From Eq. (16) it follows that variation of TA

with contrast is more rapid if the value of AI is more.where A = 0.5-I. Clearly the dependence of TA on It may also be noted from the tables that AI is largeC0' is governed by the value of I. From the tables it in those cases where the truncation error is large.is found that 7 is always less than 0.5 if the object is For example, I is maximum for N = 3 for all threein a dark background. Hence the effect of using a systems and decreases as N is increased. It is easily

Table II. Contrast Dependence of Truncation Error for an Annular Aperture

WI = 0.5 T(wt) = 0.3771 WI = 1.0 T(w) = 0.2236 wA = 1.5 T(W) = 0.1924


1.0 0.4027 6.78 0.3679 -2.44 0.2563 14.60 0.2161 -3.33 0.2314 20.29 0.1747 -9.200.75 0.3980 5.53 0.3719 -1.37 0.2504 12.01 0.2205 -1.40 0.2224 15.59 0.1802 -6.33

3 0.50 0.3934 4.31 0.3760 -0.29 0.2449 9.52 0.2250 0.60 0.2141 11.25 0.1861 -3.270.25 0.3889 3.13 0.3802 0.83 0.2396 7.15 0.2296 2.70 0.2063 7.23 0.1924 -0.01

0.3845 1.96 0.4775 0.2345 4.87 0.4576 0.1991 3.48 0.43021.0 0.3898 3.37 0.3699 -1.91 0.2375 6.22 0.2132 -4.65 0.2053 6.71 0.1772 -7.920.75 0.3872 2.68 0.3723 -1.28 0.2342 4.73 0.2160 -3.41 0.2013 4.63 0.1803 -6.31

5 0.50 0.3846 2.00 0.3747 -0.64 0.2309 3.28 0.2188 -2.15 0.1975. 2.63 0.1834 -4.650.25 0.3821 1.33 0.3771 0.01 0.2279 1.86 0.2217 -0.84 0.1938 0.71 0.1868 -2.93

0.3796 0.66 0.4869 0.2247 0.49 0.4731 0.1902 -1.14 0.46321.0 0.3856 2.25 0.3715 -1.49 0.2337 4.52 0.2173 -2.82 0.2003 4.08 0.1782 -7.370.75 0.3838 1.76 0.3732 -1.04 0.2315 3.54 0.2192 -1.96 0.1972 2.50 0.1807 -6.07

7 0.50 0.3820 1.29 0.3749 -0.58 0.2294 2.58 0.2212 -1.08 0.1943 0.96 0.1833 -4.750.25 0.3802 0.81 0.3766 -0.12 0.2273 1.64 0.2232 -0.19 0.1914 -0.53 0.1859 -3.38

0.3784 0.34 0.4907 0.2252 0.72 0.4818 0.1886 -1.98 0.47091.0 0.3833 1.65 0.3724 -1.24 0.2307 3.19 0.2176 -2.66 0.1995 3.69 0.1832 -4.780.75 0.3819 1.28 0.3738 -0.88 0.2290 2.42 0.2192 -1.97 0.1973 2.55 0.1851 -3.80

9 0.50 0.3805 0.91 0.3751 -0.53 0.2273 1.66 0.2208 -1.26 0.1952 1.43 0.1870 -2.800.25 0.3792 0.55 0.3764 -0.17 0.2256 0.92 0.2224 -0.55 0.1931 0.34 0.1890 -1.77

0.3778 0.19 0.4928 0.2240 0.18 0.4854 0.1910 -0.73 0.4787

1842 APPLIED OPTICS / Vol. 13, No. 8 / August 1974

Table Ill. Contrast Dependence of Truncation Error for a 7r-Phased Aperture

cI = 0.5 T(wt) = 0.0760 Wt = 1.0 T(wg) = -0.0556 wt = 1.5 T(wt) = 0.1443


1.0 0.0938 23.39 0.0818 7.68 -0.0423 23.82 -0.0328 40.92 0.2040 41.37 0.1330 -7.850.75 0.0921 21.18 0.0832 9.42 -0.0409 26.48 -0.0338 39.24 0.1912 32.52 0.1390 -3.66

3 0.50 0.0905 19.05 0.0845 11.22 -0.0395 28.96 -0.0348 37.41 0.1799 24.71 0.1456 0.930.25 0.0889 16.99 0.0859 13.08 -0.0382 31.28 -0.0359 35.49 0.1699 17.78 0.1529 5.99

0.0874 15.00 0.4660 -0.0370 33.45 0.4368 0.1610 11.57 0.39461.0 0.0832 9.43 0.0769 1.16 -0.0568 -2.14 -0.0483 13.13 0.1596 10.63 0.1277 -11.500.75 0.0823 8.32 0.0776 2.12 -0.0556 0.0 -0.0492 11.48 0.1548 7.28 0.1310 -9.23

5 0.50 0.0815 7.24 0.0784 3.11 -0.0544 2.16 -0.0502 9.76 0.1502 4.12 0.1344 -6.840.25 0.0807 6.17 0.0791 4.11 -0.0533 4.18 -0.0512 7.97 0.1460 1.15 0.1381 -4.32

0.0799 5.13 0.4804 -0.0522 6.12 0.4596 0.1419 -1.66 0.44451.0 0.0802 5.57 0.0759 -0.14 -0.0551 0.87 -0.0494 11.15 0.1510 4.64 0.1267 -12.180.75 0.0797 4.82 0.0764 0.54 -0.0543 2.29 -0.0500 9.99 0.1475 2.19 0.1293 -10.38

7 0.50 0.0791 4.08 0.0769 1.22 -0.0536 3.66 -0.0507 8.79 0.1441 -0.14 0.1320 -8.500.25 0.0785 3.35 0.0775 1.92 -0.0528 5.00 -0.0514 7.56 0.1409 -2.37 0.1349 -6.55

0.0780 2.63 0.4861 -0.0521 6.29 0.4727 0.1378 -4.50 0.45631.0 0.0789 3.82 0.0756 -0.57 -0.0566 - 1.75 -0.0518 6.77 0.1522 5.49 0.1339 -7.180.75 0.0785 3.25 0.0760 -0.04 -0.0559 -0.60 -0.0524 5.79 0.1497 3.72 0.1360 -5.76

9 0.50 0.0780 2.69 0.0764 0.49 -0.0553 0.52 -0.0529 4.78 0.1472 2.01 0.1381 -4.300.25 0.0776 2.13 0.0768 1.03 -0.0547 1.62 -0.0535 3.75 0.1448 0.36 0.1403 -2.80

0.0772 1.58 0.4892 -0.0541 2.70 0.4782 0.1425 -1.25 0.4681

verified that TA = I for an object of extremely lowcontrast, i.e., in the limit C' - 0. The value of Istabilizes much sooner than the image contrast as Nis increased, and it offers the nearest approximationto the actual transfer function of the system. At St

= 1.5 for annular and r-phased apertures, A T/T be-comes negative as C' - 0. Hence a value of C'may be found for which the truncation error vanishesaltogether. It may be concluded that the use of alow contrast target for the measurement of OTFwould yield better results. A measurement of theimage modulation *_ would invariably give the near-est approximation to the system response. Thiswould, however, require a calibration of the instru-ment so that the irradiance in the image is properlynormalized.

Finally, we shall discuss the nature of the appar-ent transfer function A for a unit contrast object ina white background. It is evident from Eq. (15) thatthe background does not affect the image modula-tion I but alters the mean irradiance level in theimage. The mean irradiance level if the background

is white is given by I1 = 1 - I. Since I is less than0.5, I1 will always be more than 0.5 and therefore TA< TA. This fact is confirmed by the data presentedin Tables I-III. Moreover, since Al is negative inthis case the effect of lowering the object contrastwill be to increase the value of the apparent transferfunction. In the limit C' - 0, TA - I_, which isalso the limiting value of TA.

The authors are grateful to P. K. Katti for valu-able discussions.References1. W. H. Steel, Opt. Acta 3, 65 (1956).2. P. Lacomme, Opt. Acta 7, 331 (1960).3. J. Pastor, Opt. Acta 9, 237 (1962).4. W. N. Charman, Phot. Sci. Eng. 8, 253 (1964).5. F. Kottler and F. H. Perrin, J. Opt. Soc. Am. 56, 377 (1966).6. R. Barakat and S. Lerman, Appl. Opt. 6, 545 (1967).7. J. W. Foreman, G. W. Hunt, and E. K. Lauson, Appl. Opt. 10,

105 (1971).8. P. K. Katti, K. Singh, and K. N. Chopra, Opt. Acta 17, 299

(1970).9. A. K. Jaiswal, Opt. Commun. 9, 161 (1973).


dependence of incoherent system response on object contrast

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