Point-Spread Functions, Line-Spread Functions, andEdge-Response Functions Associated with MTFs ofthe Form exp [ - ( o/w ) ]

C. B. Johnson

Point-spread functions, line-spread functions, and edge-response functions associated with modulationtransfer functions (MTFs) of the form exp[-(W/C0)n] are tabulated for values of the MTF index (ri) in therange from n = 1.0 to n = 2.0, in steps of 0.10. The MTFs are also tabulated in this range. Measure-ments of any one of these four image transfer functions which agree with the tabulated valves for a givenset of MTF parameters (wn) allow the other three functions to be estimated by using these tables.

The modulation transfer function [T(w)] is the ab-solute value of the normalized Fourier transform ofthe line-spread function F(x):

T(w) = f F1(x) exp(-iwx)dx / f Fl(x)dxl, (1)

where x is a Cartesian coordinate in the image plane,and is the spatial frequency. The modulationtransfer function is the ratio of the output and inputsinewave intensity modulations at a given spatialfrequency, respectively. For this discussion it is as-sumed that the total fluxes in the point and line im-ages are unity, i.e.,

2r Fp(r)rdr = 1,

and

F(x)dx = 1.

Note that w = 27rfw X and f have units of (rad/mm)and (cycles/mm), respectively.

If the modulation transfer function (MTF) isknown, the inverse transformation yields the line-spread function:

F(x) = T(w) exp(iwx)dw. (2)

For MTFs having the form'T(c)= exp[-(o/wc)n], (3)

Eq. (2) becomes

FI(x) = 2 exp[-(w/c)n] cos wxd, (4)

The author is with the Bendix Research Laboratories, South-field, Michigan 48076.

Received 12 June 1972.

where is the MTF frequency constant and n is theMTF index.

The edge-response function [Fe(x)] is

Fe(x) = Fj(t)dt = 2 f f exp[-(w/ W)n] cos w dtdw

(5)

and the point-spread function [F,(r)] is given by

F,(r) = (1/2wr) f J(wr)T(oj)wdo.

= (1/27r) f JO(wr) exp[-(Gco,)n ]wdw. (6)

Figure 1 shows these four functions graphically forthe cases in which n = 1, 2, and a; for exponential,Gaussian, and rectangular-shaped MTFs, respective-ly. These four functions are also given in Table I forn = 1, 2, and a. For n = , the MTF is perfect(1.0) until the frequency constant wc is reached, andfor > c the MTF is zero. The point-spread func-tion, the line-spread function, and the edge-responsefunction are seen to be physically unrealizable for anMTF of this form. Therefore the MTF itself is notphysically realizable. The maximum value of theMTF index for which a physically realizable situa-tion exists is n = 2. For MTF indices lower than 2the maximum values of the point-spread functionsand the line-spread functions are found to increasemonotonically as the value of n decreases.

If the MTF of an imaging device can be approxi-mated by Eq. (3), then the MTF parameters c, and ncan be found. (A convenient form of graph paperused for determining wc and n is described else-where.2 ) Tables II-V give the MTFs, normalizedpoint-spread functions, line-spread functions, and

May 1973 / Vol. 12, No. 5 / APPLIED OPTICS 1031

Fp r) FQx)

2F0

C -

-

1 X- n=2

2

DISTANCE. 11Iw)0 2 4

RADIUS, {1wr)

x0

DISTANCE. t1/W.)

Fig. 1. Normalized image transfer functions for n = 1, 2, and -.

-. 1.0I

2 TOf) = EXP [.t(WI%)n]

9 0.8 C-)2

2

2 0.4-0

a 0.202

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

SPATIAL FREQUENCY, (l/w,)

Table I. Summary of Image Functionsa

n T(w/c) F(x) Fe(x) Fp(r)

1 exp(-w/wc) (Wc/7r)/(l + Wc2x2 ) [(1/2) + (tan-1wxx)/7r] (Wc2 /2r)/(l + W,2 r2)3

/2

2 exp[- (W/WC)2] (wc/2NFz-)exp(-Wc2 X 2 /4 ) [(1/2) + (erfcox)/2] (Wc2 /47r)exp(- Wc

2 r2/4)= 1.0, 0 < W <WC (WC/r)silcwcx [(1/2) + (Siwcx)/irI (c 2/2r)Ji(wcr)/(wcr)

l/e, w = c0, WC< (

a The actual MTF employed for the case in which n = is T(w/c) = 1.0 for (0 < w < Wc) and T(w/wc) = 0 for (c < co).

Table II. Values of Modulation Transfer Functions for MTF Indices Between 1.00 and 2.00

W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

n 0 wc/5 2wc/5 3wc/5 4wc/5 WC 6w,/5 7wc/5 8wc/5 9wc/5 2wc

1.00 1.00 0.819 0.670 0.549 0.449 0.368 0.301 0.246 0.202 0.165 0.1351.10 1.00 0.843 0.694 0.565 0.457 0.368 0.295 0.235 0.157 0.148 0.1171.20 1.00 0.865 0.717 0.582 0.465 0.368 0.288 0.224 0.172 0.132 0.1001.30 1.00 0.884 0.738 0.598 0.473 0.368 0.282 0.212 0.158 0.117 0.0851.40 1.00 0.900 0.758 0.613 0.481 0.368 0.275 0.202 0.145 0.102 0.0711.50 1.00 0.914 0.776 0.628 0.489 0.368 0.268 0.191 0.132 0.089 0.059

1.60 1.00 0.927 0.794 0.643 0.497 0.368' 0.262 0.180 0.120 0.077 0.0481.70 1.00 0.937 0.810 0.657 0.504 0.368 0.256 0.170 0.108 0.066 0.0391.80 1.00 0.946 0.825 0.671 0.512 0.368 0.249 0.160 0.097 0.056 0.031

1.90 1.00 0.954 0.839 0.685 0.520 0.368 0.243 0.150 0.087 0.047 0.0242.00 1.00 0.961 0.852 0.698 0.527 0.368 0.237 0.141 0.077 0.039 0.018

Table Ill. Values of Normalized Point-Spread Functions for MTF Indices Between 1.00 and 2.00 (units of WC2/2ir)

r

n 0 1/Wc 2/wc 3/wc 4/wc 5/wc 6/wc 7/c 8/wc 9/Wc 10/WC

1.00 1.00 3.54-1 8.94-2 3.16-2 1.43-2 7.54-3 4.44-3 2.83-3 1.91-3 1.35-3 9.85-4

1.10 8.51-1 3.73-1 9.91-2 3.36-2 1.45-2 7.38-3 4.22-3 2.62-3 1.73-3 1.20-3 8.67-4

1.20 7.52-1 3.86-1 1.09-1 3.54-2 1.45-2 7.09-3 3.92-3 2.37-3 1.54-3 1.05-3 7.43-4

1.30 6.83-1 3.93-1 1.20-1 3.72-2 1.43-2 6.68-3 3.56-3 2.10-3 1.32-3 8.86-4 6.19-4

1.40 6.33-1 3.96-1 1.30-1 3.89-2 1.40-2 6.16-3 3.15-3 1.80-3 1.11-3 7.28-4 5.00-41.50 5.95-1 3.97-1 1.41-1 4.08-2 1.35-2 5.53-3 2.70-3 1.49-3 8.95-4 5.75-4 3.89-4

1.60 5.66-1 3.96-1 1.51-1 4.27-2 1.28-2 4.81-3 2.21-3 1.17-3 6.87-4 4.32-4 2.88-41.70 5.44-1 3.95-1 1.60-1 4.49-2 1.20-2 3.99-3 1.70-3 8.62-4 4.90-4 8.02-4 1.98-4

1.80 5.26-1 3.93-1 1.69-1 4.73-2 1.11-2 3.08-3 1.17-3 5.60-4 3.08-4 1.86-4 1.20-41.90 5.12-1 3.91-1 1.77-1 4.99-2 1.01-2 2.07-3 6.23-4 2.72-4 1.44-4 8.50-5 5.39-52.00 5.00-1 3.89-1 1.84-1 5.27-2 9.16-3 9.65-4 6.17-5 2.39-6 5.63-8 8.03-10 6.94-12

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Table IV. Values of Normalized Line-Spread Functions for MTF Indices Between 1.00 and 2.00 (units of we/27r)

x

n 0 1/co, 2/co, 3 /we 4/we 5/oe 6/we, 7/we 8/we 9/we 10/we

1.00 2.00 1.00 4.00-1 2.00-1 1.18-1 7.69-2 5.40-2- 4.00-2 3.08-2 2.44-2 1.98-21.10 1.93 1.07 4.27-1 2.03-1 1.14-1 7.20-2 4.92-2 3.56-2 2.69-2 2.10-2 1.70-21.20 1.88 1.14 4.52-1 2.03-1 1.09-1 6.60-2 4.38-2 3.09-2 2.29-2 1.76-2 1.40-21.30 1.85 1.19 4.78-1 2.02-1 1.02-1 5.94-2 3.81-2 2.62-2 1.90-2 1.43-2 1.12-21.40 1.82 1.23 5.05-1 2.00-1 9.45-2 5.22-2 3.23-2 2.16-2 1.54-2 1.14-2 8.72-31.50 1.80 1.27 5.31-1 1.98-1 8.59-2 4.47-2 2.65-2 1.73-2 1.20-2 8.72-3 6.58-31.60 1.79 1.30 5.58-1 1.95-1 7.65-2 3.68-2 2.08-2 1.31-2 8.90-3 6.36-3 4.73-31.70 1.78 1.32 5.83-1 1.92-1 6.63-2 2.88-2 1.53-2 9.31-3 6.16-3 4.32-3 3.17-31.80 1.78 1.34 6.08-1 1.90-1 5.55-2 2.05-2 1.00-2 5.83-3 3.76-3 2.59-3 1.87-31.90 1.77 1.36 6.31-1 1.88-1 4.41-2 1.21-2 4.98-3 2.72-3 1.71-3 1.15-3 8.22-42.00 1.77 1.38 6.52-1 1.87-1 3.25-2 3.42-3 2.18-4 8.48-6 1.99-7 2.84-9 2.46-11

Table V. Values of Normalized Edge-Response Functions for MTF Indices Between 1.00 and 2.00

x

n 0 4l/(0C +2/ce ±3/ce +4/wc :15/co, +6/coc ±17/wec ±8/we +9/coe + lO/coe

1.00 0.500 0.750 0.852 0.897 0.922 0.937 0.947 0.955 0.960 0.965 0.9680.250 0.148 0.103 0.078 0.063 0.053 0.045 0.040 0.035 0.032

1.10 0.500 0.752 0.862 0.910 0.934 0.948 0.958 0.964 0.969 0.973 0.9760.248 0.138 0.090 0.066 0.052 0.042 0.036 0.031 0.027 0.024

1.20 0.500 0.753 0.872 0.920 0.944 0.958 0.966 0.972 0.976 0.980 0.9820.247 0.128 0.080 0.056 0.042 0.034 0.028 0.024 0.020 0.018

1.30 0.500 0.754 0.880 0.930 0.953 0.966 0.973 0.978 0.982 0.984 0.9860.246 0.120 0.070 0.047 0.034 0.027 0.022 0.018 0.016 0.014

1.40 0.500 0.755 0.888 0.940 0.962 0.973 0.980 0.984 0.987 0.989 0.9900.245 0.112 0.060 0.038, 0.027 0.020 0.016 0.013 0.011 0.010

1.50 0.500 0.756 0.895 0.948 0.969 0.979 0.985 0.988 0.990 0.992 0.9930.244 0.105 0.052 0.031 0.021 0.015 0.012 0.010 0.008 0.007

1.60 0.500 0.757 0.901 0.956 0.976 0.985 0.989 0.991 0.993 0.995 0.9950.243 0.099 0.044 0.024 0.015 0.011 0.009 0.007 0.005 0.005

1.70 0.500 0.758 0.907 0.963 0.982 0.989 0.992 0.994 0.996 0.996 0.9970.242 0.093 0.037 0.018 0.011 0.008 0.006 0.004 0.004 0.003

1.80 0.500 0.758 0.912 0.970 0.988 0.993 0.995 0.997 0.997 0.998 0.9980.242 0.088 0.030 0.012 0.007 0.005 0.003 0.003 0.002 0.002

1.90 0.500 0.759 0.917 0.977 0.993 0.997 0.998 0.997 0.999 0.999 1.0000.241 0.083 0.023 0.007 0.003 0.002 0.002 0.001 0.001 0.000

2.00 0.500 0.760 0.921 0.982 0.998 0.999 0.999 1.000 1.000 1.000 1.0000.240 0.079 0.018 0.002 0.001 0.001 0.000 0.000 0.000 0.000

edge response functions for values of n between 1.00and 2.00, in steps of 0.10. Alternatively, Tables 11-Vmay be used to determine and n from MTF,point-spread function, line-spread function, or edge-response function data. From Table V it is seenthat a useful approximation is that the distance fromthe 50% response point to the 75% response point (orto the 25% response point) is equal to c'1, i.e., F(1/WC) 3/4 for 1.00• n 2.00.

The results obtained for cases in which n = 1, 2,and are generally well known, but the results forvalues of n between 1.00 and 2.00 are not available.It is found in practice' that the value of n is usuallybetween 1.00 and 2.00. In addition, it is often desir-able to convert easily from one image function to an-other. Astronomers, for example, may be given the

MTF of an imaging device by a manufacturer, butthey may wish instead to know the correspondingpoint-spread function for analysis of stellar (point)images. For these reasons, the normalized imagefunctions have been tabulated.

The author gratefully acknowledges the assistanceof R. A. Daigle and J. A. Lindsay, who wrote thecomputer programs for the image function tables.The author also wishes to thank K. L. Hallam,Goddard Space Flight Center, and W. G. Wolber fortheir interest and support. This work was sponsoredby NASA Goddard Space Flight Center.

References1. C. B. Johnson, Photogr. Sci. Eng. 14, 412 (1970).2. C. B. Johnson, IEEE Trans, Elec. Devices ED-20, 80 (1973).

May 1973 / Vol. 12, No. 5 / APPLIED OPTICS 1033