Reference

NBS

PUBLICATIONS

IMBSIR 84-2893

A 1 1 1 b E b D b 1 b

The Modulation Transfer Function

for Two-Point and Periodic

Objects Using Gaussian and

Lorentzian Resolution Functions

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, Maryland 20899

June 1 984

U.S. DEPARTMENT OF COMMERCE

-QC

100

. U56

84-2893

1984

''EAU OF STANDARDS

V

national bureau

OF STANDARDS

LIBRARY NBSIR 84-2893

THE MODULATION TRANSFER FUNCTION

FOR TWO-POINT AND PERIODIC

OBJECTS USING GAUSSIAN AND

LORENTZIAN RESOLUTION FUNCTIONS

. u

no,

R. A. Scrack

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, Maryland 20899

June 1 984

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director
II
The Modulation Transfer Function for Two-Point and Periodic Objects

Using Gaussian and Lorentzian Resolution Functions

by

Roald A . Schrack

The National Bureau of Standards

Washington, DC 20234

This paper presents an analytical study of the effect of Gaussian- and

Lorentzi an-shaped line spread functions in non-coherent noise-free imaging

systems. A mathematic development is given for the calculation of the

Modulation Transfer Function (MTF). This technique is used to calculate the

MTF for two-point and periodic objects using Gaussian and Lorentzian resolution functions. Figures and graphs are used to illustrate the comparison of

the results. Relationships between the results obtained are developed that

are useful in the interpretation of experiments used to determine the resolu

tion of experimental systems. The development covers only noise-free, incoherent, one-dimensional systems.
IIIIIII
Introduction

In the development of optical or electronic imaging systems some technique is usually employed to determine the resolution of the system--that is,

its ability to present detail in the image. The resolving power of optical

instruments is frequently given in terms of the ability to resolve two point

sources. Rayleigh's criterion is probably the most widely known example of

this approach. A technique that gives a much greater amount of information is

the determination of the modulation transfer function of a system. The

modulation transfer function shows how well an imaging system will handle

sinusoidal images as a function of the frequency of the sinusoidal image.

The work presented in this paper was done in connection with the development of an imaging system used in neutron radiography employing a microchannel

plate electron multiplier. The resolution of the system was found to be

dependent on a number of variables in the optical and electronic components of

7 the system. In this paper the mathematical development of the modulation

transfer functions are given with comparisons to results for two-point image

resolutions for mathematical models that closely approximate the experimental

systems. The material developed in this paper was found to be useful in the

development of the neutron radiography system and should be of value in other

similar systems that are linear and stationary, having response functions that

are real and symmetric.

The resolution of many systems can be given as a response function of the

system. The image produced by the system is then obtained by a convolution of

the object and response functions. The Lorentz distribution and the Gaussian
distribution are analytic distributions that are similar in shape to response

functions that are encountered in many systems. Many of the response functions one encounters have characteristics that lie between the Lorentzian and

Gaussian distributions so that a study of the characteristics of these two

functions is frequently helpful in predicting the characteristics of these

systems.

To simplify the discussion and mathematics this paper will consider only

one dimensional non-coherent noise-free imaging systems.

Tests of two-dimensional imaging systems are frequently done with patterns of lines that effectively reduce the system to one dimension. The resolution of a system is frequently described in terms of the system response to

a single line. This single line —or impulse response is called a Line Spread

Function. The two-dimensional analog is termed a Point Spread Function. If

the impulse response is Gaussian in shape, the Line Spread Function and Point

Spread Function are both Gaussian with the same width. This similarity is not

true for resolution functions that are Lorentzian in shape.

The systems are assumed to be noncoherent in order to avoid the complexities of phase and interference effects. These effects can be ignored if the

width of the line spread function is much greater than wavelength of the radiation that carries the image. Noise has also been ignored in these calculations but must, of course, be considered in cases where the signal to noise

ratio is poor.

Two types of periodic objects and images are considered: square and sine

waves. The square wave is used because it most nearly represents the sharp

- 2-
change in contrast of black and white objects frequently used as test patterns. Sine waves are considered because from them any other type of image

contrast change can be generated by Fourier analysis.

In this paper the properties of systems having Gaussian- and Lorentzianshaped Line Spread Functions when imaging objects with periodic sinusoidal and

square wave intensity distributions are given. These results are compared

with the imaging results obtained for objects having only two cycles.

Experimentalists making resolution measurements should find many of the relationships developed useful in the interpretation of results.

Description

The Modulation Transfer Function (MTF) describes the response of an optical system in producing images of different spatial frequency. It is analogous to the frequency band pass curve used to describe the response of an

electrical circuit to electrical signals. In analogy to the gain of an electrical circuit the MTF shows the "visibility" of an image as a function of

spatial frequency. Visibility (V) is defined in terms of the relative intensity in the image:

V = (H+

- HJ / (H+

+ H_)

where H +is the maximum intensity and H_ is the minimum intensity of the

image. The MTF is the response of system to a periodic object of sinusoidal

intensity variation. Because of the experimental interest the response to

periodic objects having square wave intensity variation will also be developed. The spatial frequency is usually given in cycles per mm. The term "line

pairs per mm" is equivalent to cycles per mm.

- 3

-
The image produced by a system for a delta function object is termed

the Line Spread Function (LSF) of the system. The symbol G will be used to

denote the LSF distribution. The LSF is defined to have unit area:

G- 60 4 x - 1- • -CO

The image H, of any other object is the convolution of the line spread

function, G and the object intensity distribution, I:

He*) « 6-cm ® Iw = / (x-^)cUj

-CP

Since it is usually difficult to get a delta function object of sufficient

intensity to produce an experimentally useful image the LSF of an optical system

is frequently determined by using a step function object (."knife edge"). The

image of the "knife edge" object will be the cumulative integral of the LSF, Q

) = J-(Co

The derivative of Q is then the LSF, G. - G-M

The MTF and the LSF are complementary ways of describing the response of a

system. Both may be determined by experimental methods but it is easiest to

measure the LSF. The MTF may then be obtained from the LSF by calculation.

Calculation of MTF for Sinusoidal Objects

For an object having a sinusoidal intensity variation of one frequency

the intensity may be written: X(x) = 0.5* (

I + Su^ X )

The image, H produced by an optical system will then have an intensity:

H (X) = 0.5 (( f- V\ h X

)

- 4

-
and the "visibility", V of H is given by:

v =

- H (3TT/Z) _ |V\

note that:

where

and

H (TT/2.) V M (3TT/2) |^\‘

+oo H(*) = f g^j ch

f -CO H[ir/z)- U(W/z) =0.5j^ G Oj) )-

^

"3 )]

5

^ (^.TT/i-y

) — 5

(^v ^TT/x (Las — <2oS wTr/z. 5

^^

Using the above relationships it can be shown that the visibility, V

is the Fourier transform of the LSF:

tc^

vc-fO - y

^ = syyoy

The most commonly encountered LSF forms are the Gaussian and the Lorentzian.

The Fourier transform of the Gaussian is a Gaussian. The Fourier transform of a

Lorentzian is an exponential. In Appendix A a number of properties of these

functions and their transforms are given.

let the Fourier transform of I(x), the object distribution be

and the Fourier transform of H(x), the image distribution be

Then the MTF is defined to be:

= <1 (7)

MTF (p) = &Lf)/ J(f)

but 1<i [?>) m] = & ooj = tT{r wj = J-tyMif)

thus mff Cf) =

and for objects with sinusoidal distribution the image visibility is the MTF.

V(p) = MTF(p)

- 5 -
Calculation of Visibility for Square Wave Objects

Strictly speaking the visibility is equivalent to the MTF only for sinusoidal objects. The frequency response of a system to a square wave input is

frequently desired however, because most test object distributions are square

wave in nature, i.e., alternate bands of black and white.

The square wave distribution can be expanded in terms of a Fourier series

of sinusoidal components. Consider the square wave given by the following

conditions:

I(x) = 1.; - ir/2^x < n/2

I ( x ) = 0 ; ir/2 < x 1 3 it/ 2

I(x+2-mn) = I(x), m = 1, 2, 3

The cosine expansion for this square wave is given by:

where

note that for I(o) = 1. = 0.5 [1 + J(1

- 1/3 + 1/5 - 1/7 + 1/9 + )]

yields the known series: it = 4(1 - 1/3 + 1/5 - 1/7 + 1/9 + ).

Note that vn= i

The image H(x) is the convolution of the LSF and I(x)

H(x) = G(x) 8 I(x).

Expand H(x) into its sinusoidal components: = 2 ~yr\ - 1 , 3 ,^ 7

let H q

= 0.5, Hm

= 0.5 bm

(G(x) 8 cos mx)

but G(x) 8 cos mx = G(mx) 8 cos x

- 6 -
thus +°°

Hm

(x)= J

G(mZ) cos (x-Z) dZ

— oo and cos(x-Z) = cos x cos Z + sin x sin Z

.

max: (o) = Jjr (mi) CoS £ di_ = LTM?)} = XJ(-y„-

thus V (p) =M(0hHM = 2 b-Jftm?)

H(0)+H(ir‘) m=l

m

m odd

Square Wave Visibility for Lorentzian LSF

The Lorentzian LSF is given in Appendix A as G (x) = air/ (a 2*- x

2

) . The

Fourier transform is:

J

= X(y) - *f.p (_- XTTcLy)

The visibility of a square wave object is then

CO Q&

Vt1 }) «£ - Z bm [Wf (-2F<vmj)] w =

VW=I

J • m*,

or V(u) = Z bvw Qrj=(-zrr^M )J

The values of V(p) are calculated in Appendix B. The results are

plotted in Fig. 1 and shown in Table 1. The abscissa p is:

p

= ya = y -r

2 where y is the number of cycles per unit length (usually given as line pairs

per mm) and r is the full -width half-maximum of the LSF: r = 2a.

- 7-
Thus to find the visibility for the value of line pairs per mm that is the

1 r reciprocal of the full-width half maximum of the LSF we let p = ya = y • ^

= 0.5

and get a value of V(0.5) = 0.055.

This corresponds to a density ratio of

D

= - log io(I f /l-j ) = " log 10 (0 .055) = 1.26.

For comparison purposes Fig. 1 also shows the visibility of a sinusoidal

object.

Square Wave Visibility for Gaussian LSF

The Gaussian LSF is given in Appendix A as

G

g (x) = B exp(-u 2

)

.

The Fourier transform is = -iKi0

) = ex-jo and

the visibility of a square wave object is then:

o? P=> ^ (yrv-jo) = E k. L- (^ip) 1 }

: va = 3_,

J

The values of V(p) are calculated in Appendix C, plotted in Fig. 2 and shown

in Table I.

The abscissa p is:

P = Ty/c

where y is the number of cycles per unit length, r is the full width halfmaximum of the Gaussian LSF and the constant c = 2/ | 1 n( .5) | = 1.67.

To find the visibility for the value of y that is the reciprocal of r we let

and get a value of V( 0.6) = 0.036.

This corresponds to a density ratio of 1.44.

- 8-
Comparison of Sinusoidal and Square Wave Visibility

Because of the different width parameters used in the normal representation

of the Lorentzian and Gaussian LSF it is not easy to compare their relative

visibilities as shown in Figs. 1 and 2. A common basis of comparison is the

full-width half-maximum of the LSF distributions, r . Figure 3. shows the

visibility for both distributions plotted as a function of p where

P

= ry

and y is the number of cycles per unit length. Note that the visibility for the

Gaussian LSF is greater than that for the Lorentzian distribution for values of

p less than 0.9. For values above that the visibility for the Gaussian

distribution drops rapidly in comparison to the Lorentzian.

The ratio of visibility of the sine wave image. to the visibil ity. of the

square wave image is shown for the two LSF in Fig. 4. Note that for p = ry

greater than about 0.6 the ratio for both approaches tt/4 because by then only

the first term in the square wave expansion is contributing significantly.

Visibility of Two-Cycle Objects

Up to this point the image visibility has been considered for periodic

objects having a large number of cycles. In many cases it is interesting to

know what the resolution of the system is for discerning the existence of two

objects.

The formalism used up to this point does not easily allow for the calculation of the two cycle visibility. Numerical calculations of the convolution of

different LSF and object distributions were performed. The cases of Gaussian

and Lorentzian LSF and sine and square wave object distributions of several

cycles were calculated and were found to agree with calculations based on the

- 9 -
analytic expressions derived in the previous sections. The computer program

was then modified to do the case of the two cycle object distributions. The

image amplitude distributions were obtained for different values of the width

of the LSF. Figure 5 shows the image amplitude distributions obtained for a

convolution of the Lorentzian LSF and a two cycle square wave object. The

visibility, V, is defined in the same manner as before:

with H_ being measured at the midpoint of the distribution. When the curvature at the midpoint becomes zero, V is taken to be zero. The visibility is

not defined for cases of negative curvature at the midpoint. The visibility

as determined from Fig. 5 is shown as a function of cycles per unit length' y

as a dashed line labeled L 2 in Fig. 3. Note that for values of p = yr less

than 1 there is little difference between the results for a multi-cycle object

and a two cycle object. The difference becomes quite large however as p

approaches 1.5, where V = 0 for the two cycle case.

Figure 6 shows the image amplitude distributions obtained for a convolution of the Gaussian LSF and a two cycle square wave. The visibility as

determined from Fig. 6 is shown as a dashed line labeled G 2 in Fig. 3. In

this case the visibility goes to zero for p

= 1.03.

- 10 -
Most test objects have a pattern with a square wave intensity distribution. If the intensity distribution has a rectangular wave shape—i „e . , other

than a 50% duty cycle, the visibility of the image will be greater and the

value of y for which the visibility goes to zero will be slightly altered.

Figure 7 shows the image amplitude distribution obtained for a convolution of the Gaussian LSF and a two cycle delta function. The visibility is

about 40% greater than for the comparable square wave and the visibility goes

to zero at p = 1.1.

The square wave and the delta function are the most extreme cases of the rectangular wave shape so that the results for other values of the duty cycle will

lie between these cases.

- 11 -
FIGURES

1. Visibility of Image formed by Lorentzian LSF.

2. Visibility of Image formed by Gaussian LSF.

3. Comparison of Lorentzian and Gaussian visibilities.

4. Comparison of Ratios of visibility for sinusoidal and square wave objects.

5. Image distribution for convolution of Lorentzian and two cycle square wave

object.

6. Image distribution for convolution of Gaussian and two cycle square wave

object.

7. Image distribution for convolution of Gaussian and two cycle delta function

object.
p

Figure 1. Visibility of image formed by convolution with Lorentzian line spread

faction. Results are shown for sinusoidal and square wave objects.

The abscissa is p

= ry/2.
p

Figure 2. Visibility of image formed by convolution with Gaussian line spread

function. Results are shown - for sinuso idal and square wave objects

The abscissa is p = Fy/ 2 f

) 1 n ( 0 . 5 )

|

'

.
p

Figure 3c Comparison of visibility of image of square wave object for Gaussian

and Lorentzian line spread functions. The curves labeled Goo and l*

show the image visibility for a square wave object with an infinite

number of cycles for Gaussian and Lorentzian line spread functions.

The dashed curves labeled G 2 and L

2 show the image visibility for

objects having only two cycles. The abscissa p

= Ty

.
sine /

v

square

p

= r

y

Figure 4. Ratio of visibility of sinusoidal and square wave objects for

Gaussian and Lorentzian Line Spread Functions. The ordinate is the

ratio of visibility of the sine wave object to the visibility of the

square wave object. The curves labeled G and L refer respectively to the ratio obtained for Gaussian and Lorentzian line spread

functions. The abscissa p = yr where r is the full-width halfmaximum of the line spread function and y is the line pairs per

unit length. It is interesting to note that the shape of the ratio curves replicates the shape of the line spread functions.
X

Figure 5* Convolution of Lorentzian line spread function and two cycles of a square wave object. The wavelength of the square wave is c = 1/y.

Amplitude of the convolution is shown as a function of position for

five different values of p = Ty. Note that the curvature of the

convolution at the midpoint of the two square waves goes to zero for

p = 1.5. The visibility as a function of p obtained from these curves is shown as the dashed line in Fig. 1.
RELATIVE

AMPLITUDE

X

Figure 6. Convolution of Gaussian line spread function and two cycles of a square wave object. The wavelength of the square wave is c = 1/y.

Amplitude of the convolution is shown as a function of position for

five different values of p

= ry. Note that the curvature of the

convolution at the midpoint of the two square waves is closest to zero for p = 1. Analytic extrapolation indicates the curvature is zero at p = 1.03. The visibility as a function of p obtained

from these curves is shown as the dashed line in Fig. 2.
Figure 7„ Convolution of Gaussian line spread function and two cycle delta

function object . The amplitude of the convolution is shown as a function of position for four different values of p

= Ty = r/c

where c is the spacing between the delta functions. Note that the

visibility goes to zero at p = 1.1.

I

!
TABLE I. Visibility of Images Derived From Convolution of LSF With

Sinusoidal and Square Wave Objects*

LSF Gaussian Lorentzian

P V(sine) V(square) V(sine) V(square)

0 1.00 1.00 1.00 1.00

0.05 0.98 0.99 0.73 0.81

0.10 0.91 0.99 0.53 0.62

0.15 0.80 0.96 0.39 0.47

0.20 0.67 0.85 0.28 0.35

0.25 0.54 0.69 0.21 0.26

0.30 0.41 0.52 0.15 0.19

0.35 0.30 0.38 0.11 0.14

0.40 0.21 0.26 0.081- 0.10

0.45 0.14 0.17 0.059 0.075

0.50 0.085 0.11 0.043 0.054

0.55 0.051 0.064 0.032 0.040

0.60 0.029 0.036 0.023 0.029

0.65 0.015 0.020 0.017 0.021

0.70 0.008 0.010 0.012 0.016

0.75 0.004 0.005 0.009 0.011

p = ry/2 for the Lorentzian LSF and

p = ry/1.67 for the Gaussian LSF.
APPENDIX A

INTRODUCTION

The line shape functions most commonly encountered are the Gaussian and

Lorentzian. The Gaussian shape is the most frequently encountered and is

closely associated with counting type experimental data and other situations

where Poisson statistics obtain. It is also the result when many different

distributions are convoluted together as might occur in an experiment where

the total response function of the system is a result of a sequence of

response functions affecting the observed result. The second function considered is the Lorentzian distribution, which is frequently the response function associated with experiments in which scattering plays a role in producing

the observed result.

The Gaussian Distribution " -

Figure A shows the shape of the Gaussian distribution G(x) and its integral Q(x), where x

The form of the response function of a system may be determined by

obtaining the image of a "knife edge" object. The "knife edge" is mathematically represented by 0 for x < 0 and 1. for x >_ 0. The image is the distribution Q(x). The derivative of this distribution is then the line shape function G(x). If the functional form of the line spread function is known to be

Gaussian, the full-width half-maximum r characterizing the Gaussian may be

obtained directly from the observed image Q without differentiation. Figure

A shows that r is the distance between the 0.12 and 0.88 amplitude points on

the image Q. It should be noted that the distance between the 0.25 and 0.75

amplitude points is r/1.74 for the Gaussian distribution.

c>Qw = G- l%)

b Z
The following table shows the relationship between a number of different

mathematical forms of G(x) that are commonly encountered and their fourier

transforms^} (p) . G(x) is normalized to unit area. The full-width halfmaximum of G(x) is r o

G(x) = B exp(-u Z

) 9 J<J (p) = exp(-p 2

).

the Fourier transform

where

of G(x) is

£

G(x)|

J*

1xp G(x) dx.

= ^(P)

let C = 2 / |1 n ( . 5 ) | = 1.67

width

parameter

full-width

at halfmaximumstandard

devi at ion B u p

a /T Co a 1/ ( / 2 it a) x/(/T a) / 2 way

r r r/(/T C) r) Cx/r irTy/C

X c/n 1// 2x / X/ IT f~\ X wy//"!

The Fourier transform of the convolution of two functions is the product of

the transforms:

G a (x) 0 G 2 (x) J^Gi(x)j tfi' |^G 2 (x)j =/i/x(p) J 2 (P)
using this relationship it is easy to show that the convolution of two

Gaussians characteri zed by Tj and r

2 produces a Gaussian distribution

characterized by width r

3 where

The Lorentzian Distribution

Figure B shows the shape of the Lorentzian distribution L(x) and its

integral Q(x). Note that for the Lorentzian line shape function r is the

distance between the 0.25 and 0.75 amplitude points on the "knife edge" image

distribution Q.

The most frequently encountered mathematical form for the Lorentzian

distribution normalized to unit area is

The convolution of two Lorentzians of width r

x and r2 produces a Lorentzian of

L(x) * < a/lT >

where the full -width half-maximum r = 2a.

The Fourier transform of L(x) is ^(y):

ly I)

width r

3 where
APPENDIX B. Visibility of a Square Wave Folded with a Lorentzian

The visibility V of the image of a square wave object formed by a system with a Lorentzian line spread function is given by

oetA. •yn - • V17 2* t~0 )~] /-yn.

where p = Ty/2.

r is the full width at half maximum of the Lorentzian line shape

function

and y is the inverse wavelength of the object usually given as line

pairs per mm.

The program listed for the Texas Instruments calculator model TI-59 will

calculate five terms of the expansion for V(p). To operate the code load the

starting value for p into storage register 01, press RST and R/S. When done

the calculator will print four numbers:

1. P

2. V(p)

3. exp(-2irp)

4. R(L) = exp(-2Trp)/V(p)

The value of exp(-2Trp) is the visibility for a sine wave object. The

ratio R(L) is then the ratio of the visibilities of sine and square wave

objects

After printing the four numbers the program will cause p to be incremented by 0.05 and restarted.

The program will continue until stopped externally.

Table I gives the numerical output produced by the code.
APPENDIX C. Visibility of a Square Wave Folded with a Gaussian,,

The visibility V of the image of a square wave object formed by a system with a Gaussian line spread function is given by

where p

= Fy/ ( 2 / | In (.5) | ) = Ty/1 „67 ~ J~2 ay

r is the full width at half maximum of the Gaussian line shape

function

The program listed for the Texas Instruments calculator model TI-59 will

calculate seven terms of the expansion for V(p). To operate the code load the

starting value of p into storage register 01, press RST and R/S. As with

the program described in Appendix B; when done the calculator will print the

four numbers:

and y is the inverse wavelength of the object usually given as line

pairs per mm.

1. P

2. V ( p

)

3. exp(-( irp) 2

) 4. R(G) = exp(-( irp) 2 )/V(p)

Table I gives the numerical output produced by the code.
Relative

Area

Q(x)

Relative

Fig. A. The Gaussian Line Shape Function G(x) and its integral 0(x).

The abscissa scale is in units of half the full-width halfmaximum, r. Dashed lines indicate that r is the distance

between the 0.12 and 0.88 relative amplitude points on the

0 distribution.
Relative

Area

Q(x)

Relative

Amplitude

L(x)

Fig. B. The Lorentzian Line Shape Function L(x) and its integral Q(x).

The abscissa scale is in units of half the full -width halfmaximum, r. Dashed lines indicate that r is the distance

between the 0.25 and0.75 relative amplitude points on the Q

distribution.
Program listing for TX-59 to calculate the visibility of

images produced by square and sine wave objects and a Gaussian

line spread function.

000 43 RCL

000 1 0

002 65 x 003 U d iLL 004 65 x 005 8 9 vi 006 95 = 007 42 STD

nos 02 02

009 01 1 0 1

Q

42 STO

Oi 1 0 4 0 4

0 i 2 71 SBR

0 i 3 15 E

0 1

42 STO

015 06 06

016 42 STO

017 03' 03

n i h 03 3

0 i 9 42 STO

020 04 04

021 71 SBR

022 15 E

cnzo 94 - 024 44 SUM

025 03 03

026 05 5

027 42 STO

0 2 3 0 4 0 4

0 2 9 71 SBR

030 15 E

031 44 SUM

0 3 2 0 3 0 3

U o 034U f 42 STD

035 0 4 0 4

036 71 SBR

037 15 E

038 94 +/-

0 3 9 44 SUM

n - n 03 03

041 09 9

0 4 2 42 STD

0 - 3 04 04

044 7l SBR

0-5 15 E

0 4 6 44 SUM

ijr? 0 3 O 3

0 4 3 01 1 0 4 9 01 1 050 42 STO

051 04 04

052 71 SBR

053 1 5 E

0 5 4 94 +x - 055 44 SUM

056 Uo U 3

057 01 1 058 03 j

059 42 STO

0 6 O 04 04

061 71 SBR

062 15 E

063 sr

-J J -r 064 02 hmm 065 95 = 066 44 SUM

067 0 3 03

063 43 RCL

069 0 3 0 3

070 65 x 071 04 4

072 55 - 073 39 n 074 95

075 42 STO

076 05 05

077 93 RDV

hj 7 8 43 RCL

079 01 01

080 65 x 081 02

082 95 =: 083 99 PRT

084 43 RCL

085 05 05

086 99 PRT

087 43 RCL

088 06 0 6

089 99 PRT

090 IT =T

• J — 09! 43 RCL

092 05 05

093 95 =

0 4 9 8 PRT

0 9 5 43 RCL

0 9 6 01 01

097 O trJ

4- 0 9 8 83 P 099 0 0 0

1 00 02 XL 101 05 zr 102 95 = 103 42 STO

104 01 01

105 6 1 GTO

106 00 0 0

107 0 0 00

108 !' 6 LBL

108 15 E

110 •=* -

• J •.1*

1 1

«

112 43 RCL

113 02 02

114 65 X

115 43 RCL

116 04 04

117 54 **:

1 1

-J Vi

119 94 +x - 120 £ £> IHv

121 LN2

122 54 ’) 123 55

124 43 RCL

125 04 04

126 95 = 127 92 RTN

1 23 91 RxS

-129 00 0

130 00 0

131 00 o

132 00 0

133 00 • o
*

Program listing for TX-59 to calculate the visibility of images

produced by square and sine wave objects and a Lorentzian

line spread function

.

019 03 i 020 95 = ' 066 04 4

021 94 +/'- 0 6 7 •j -r 022 42 STO 068 39 11' 023 04 04 0 6 9 95 = 024 43 RCL 0 7 0 42 STD

025 04 04 071 05 05

1j 'ci o 35 + 072 9 H RDV

027 43 RCL 073 43 RCL

023 02 02 074 01 01

029 95 = 0,5 99 PRT

030 42 STD 076 43 RCL

031 0 3 0 3 077 05 05

032 43 RCL 073 99 PRT

0 3 3 02 02 079 43 RCL

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References

1. J. Co Dainty and R. Shaw, Image Science , Academic Press, 1974.

2. To H. James, The Theory of the Photographic Process , Macmillan Publishing

Co., 1977.

3. K. Rossmann, "The Spatial Frequency Spectrum," Radiology 90, 1, (1968) 1.

4. G. U. V. Rao, The Modulation Transfer Function in Radiology & Nuclear

Medicine, 14th Annual Meeting of Amer. Assoc, of Physicists in Medicine,

1972.

5. 0. H. Schade, "A New System of Measuring and Specifying Image Definition,"

NBS Circular 526 (1954) 231.

6. J. W. Goodman, Introduction to Fourier Optics , McGraw-Hill Book Co.,

1968.

7. R. A. Schrack, "A MicroChannel Plate Neutron Detector". Nuclear Inst, and

Methods (in press).
N3S-114A (rev. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

4. TITLE AND SUBTITLE

The Modulation Transfer Function for Two-Point and Periodic Objects Using Gaussian

and Lorentzian Resolution Functions

5. AUTHOR(S)

Roald A. Schrack

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Report & Period Covered

Interim

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Department of Energy

10. SUPPLEMENTARY NOTES

This work was done in connection with the development of the MicroChannel Plate

Neutron Detector.

I | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (-4 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

This paper presents an analytical study of the effect of Gaussian- and Lorentzianshaped line spread functions in non-coherent noise-free imaging systems. A mathematic

development is given for the calculation of the Modulation Transfer Function (MTF).

This technique is used to calculate the MTF for two-point and periodic objects using

Gaussian and Lorentzian resolution functions. Figures and graphs are used to illustrate the comparison of the results. Relationships between the results obtained are developed that are useful in the interpretation of experiments used to determine the

resolution of experimental systems. The development covers only noise-free, incoherent

one-dimensional systems.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by -.r- c Gaussian; imaging systems; Lorentzian; modulation transfer function; resolution;

visibility

13. AVAILABILITY

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