Prediction of cut-off frequency from the maximum slope of a transmittance scan of an ideal edge

(1)

Rochester Institute of Technology

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5-20-1982

Prediction of cut-off frequency from the maximum

slope of a transmittance scan of an ideal edge

Dale E. Ewbank

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(2)

PREDICTION OF CUT-OFF FREQUENCY FROM THE MAXIMUM

SLOPE OF A TRANSMITTANCE SCAN OF AN IDEAL EDGE

by

Dale E. Ewbank

A thesis submitted in partial fulfillment

of the requirements for the degree of

Bachelor of

Scien~e

in the School of

Photographic Arts and Sciences in the

College of Graphic Arts and Photography

of the Rochester Institute of Technology

Signature of the Author Dale E. Ewbank

Photographic Science and

Instrumentation

Division

Certified by..JameS J. Jakubowski

lfIesilS Advisor

(3)

THESIS RELEASE PERMISSION FORM

ROCHESTER INSTITUTE OF TECHNOLOGY

COLLEGE OF GRAPHIC ARTS AND PHOTOGRAPHY

Title of thesis PREDICTION OF CUT-OFF FREQUENCY FROM THE

MAXIMUM SLOPE OF A TRANSMITTANCE SCAN OF A IDEAL EDGE

I,

Dale E. Ewbank

hereby (grant, deny)

permission to the Wallace Memorial Library of R.I.T. to

reproduce my thesis in whole or in part.

Any reproduction

will not be for commercial use or profit.

Date

May

20,

1982

(4)

PREDICTION

OF

CUT-OFF

FREQUENCY

FROM

THE MAXIMUM

SLOPE OF

A

TRANSMITTANCE

SCAN

OF

AN

IDEAL EDGE

by

Dale

E. Ewbank

Submitted to the

Photographic

Science and Instrumentation Division

in partial

fulfillment

of the requirements

for

the Bachelor of Science degree

at the Rochester Institute of

Technology

ABSTRACT

This

_study

has

empirically shown that the maximum

slope of an ideal edge scan in transmittance versus postion

is an excellent indicator of the cut-off

frequency

at which

a microdensitometer system's modulation transfer

function

goes to zero. It was found that numerical aperture of col

lection objective, scanning slit width and _alignment, system

focus,

and _sampling interval are parameters which must be

carefully chosen and _set-up when

implementing

or _using this

slope technique to check system performance.

(5)

ACKNOWLEDGEMENTS

The

author wishes to thank

James

Jakubowski

for

his

time,

ideas,

and encouragement in

helping

to complete this

thesis

.

A _very special thank you is to go to Martha

Leahy

for

her

typing

assistance and for her love and patience which

may have past unnoticed

during

the last year. I

hope

that

she will be more than repaid in the years to _come; which we

shall spend together.

(6)

TABLE OF

Modeled

MTF(f)

and theoretical

MTF(f)

4

Example

plot of transmittance versus position.

6

Edge response curve

11

Edge response

for

.10 NA 12

MTF(f)

for .10 NA scan

H

Edge response

for

.08 NA

15

MTF(f)

for

.08 NA scan 16

(8)

INTRODUCTION

Microdensitometers

are used _extensively to gain in

formation

about photographic systems.

They

are used to

digitize

images which are on

film

or _glass, and to _study

such parameters as _granularity and modulation transfer.

When using a microdensitometer_, it is necessary to

know

its performance characteristics. One important char

acteristic is

the

system's modulation transfer

function

(MTF(f))

and its

limiting

frequency.

Edge gradient analysis

may

be

done

to

determine

a system's

MTF(f)

by

either of

two

different

methods. In the conventional _method, the

edge scan is

differentiated

to give the line spread function

of the system.

This

line spread function is then Fourier

Transformed;

and the modulus of this transform is the

MTF(f)

of the system. ' ' ' The second _way of _obtaining the trans

fer function of a system is Tatian's method. In Tatian's

method the transfer function of the system is calculated

by

a trigonometric series whose coefficients are proportional

to the sampled values of the edge response. The series is

also modified

by

added terms to take into account the

known

asymptotic behavior of the edge response.

For

a

diffraction

limited optical system with a _circular

aperture, the

frequency

at which the transfer

function

goes to zero is predictable.

This

limiting

frequency

(f

)

(9)

f

= ₂

NA

o ₎

where NA is the numerical aperture of the collection _system,

and

)\

is the wavelength of radiation. '

Many

microdensitometers are designed to work at the p

diffraction

limit of their microscope objective. Thus the

frequency

at which the transfer function of the system goes

to zero is predictable

from

this objective's numerical

aperture. The conventional method and Tatian's method of

edge gradient analysis are two ways of _checking the system's

MTF(f),

and also _estimating the system's cut-off frequency.

Both of these require computation which is usually done on a

computer .

Swing

suggests that the maximum slope of a scan of an

ideal edge in transmittance versus position is equal to the

9

cut-off

frequency

of the system. This would be a good

check on the system and _only requires minimal calculations.

For the diffraction limited case on a normalized

frequency

scale, the line spread function of the system

(l(x))

can be modeled _using one dimensional analysis as:

l(x)

=

fQ

sinc^(fox).

The function

has

area of unity. The Fourier Transform

of the line spread function

(L(f))

is:

(10)

Thus

the modulation transfer

function

of the system

(MTF(f))

is

(Figure

1

)

:

tri ( "~f

)

MTF(f)

=

The modulation transfer

function

goes to zero as its

argue-ment goes to one;

MTF(f

)

= _tri (' '

)

o

f

=_f f=f

Therefore,

the

limiting

frequency

is f^for "the line spread

given .

The scan of an ideal edge with a finite slit on the above

optical system

(e(x))

can be mathematically represented

2

as a convolution of the line spread

(fQsinc

(f

_x)) with the edge

(step(x))

and the slit (-1- rect

(--)).

e(x) = f sinc2(f x)*step(x)-*-i-rect(--).

If the slit is made _very small so as to appear as a delta

function to the system;

lim

a->0

JL

rect(-^-)=

<f(x).

The

edge response

(e,(x))

becomes:

e.(x) =

f

sinc2(f^x)^step(x)^cT(x). I v '

o o

Since

f(x)*cf(x)

= f(x);n

the edge response

(e^x))

(11)

1.0

.5

Q O

0.0

4

modeled

MTF(f)

diffraction

limited

MTF(f)

FREQUENCY

Figure

1.

Comparison of modeled

MTF(f)

to the theoretical

(12)

. 2

ej(x)

=

f0sincZ(fox)-X-step(x)

The

derivative

of

this

_convolution _is

equal to the slope of

the edge scan.

Knowinc

11

gf-(f(x)*g(x)) =

f(x)*

^X9(X));

the

derivative

of the edge response (e,1

(x))

can be written

as :

1

ej'

(x)

=

fosinc2(fdb)?f-<((x)

Thus

the slope is equal to:

e,'

(x)

= _f _sine

(f

x),

and the maximum slope is at x=0

e,'

(0)

= _f _sinc(f

x) I v '

o v o '

x=₀

The

maximum slope

(f

)

yields the same results as the

limiting

frequency

(f

)

for

the line spread

function.

Using

the

maximum slope to indicate the cut-off

frequency

for

an optical system will

be

a quick and _easy check on system

performance.

The

transmittance values of a scan can _easily

be plotted and the maximum slope

drawn.

The slope then _only

needs to

be

normalized to reflect the value of the cut-off

frequency,

(Figure

2).

(13)

whether the maximum slope of an ideal edge scan is equal to the

cut-off

frequency

of the system as predicted

by

the modulation

transfer function and

by

the numerical aperture

(for

diffraction

limited system), and show the

limitations

of

implementing

and

using this technique to check system performance.

max

LU

vo

<

min

*

x X *

x X

* * X * *

relative

distance

Figure 2. Example plot of transmittance versus position

for

an edge scan.

normalized slope =

(14)

EXPERIMENTAL

The

determination

of the cut-off

frequency

_using the

numerical aperture of the system is straight

forward.

The

limiting

frequency,

(f

),

is related to the numerical aperture

(NA)

and the wavelength

(

h

)

of radiation:

S

Since the work was

done

with incoherent _radiation, the radiation

was

broadband;

and a median wavelength of .500 micrometers

was used

for

the microdensitometer system.

Determining

the cut-off

frequency

from the

MTF(f)

is

more involved.

The

MTF(f)

can be

determined

using an edge

gradient analysis

(conventional

or Tatian's method). An

ideal edge is scanned in transmittance versus position on the

system, and the edge response

function

is transformed to give

the system MTF(f). The cut-off

frequency

is where the

MTF(f)

goes to zero.

Tatian's method of edge gradient analysis was used to

obtain the MTF(f)s of the edge scans because it proved easier

to use and to be influenced less

by

system noise in the

data.

When using the conventional edge gradient analysis _method, the

effect of noise is to introduce a random error and a positive

1 2 bias to the modulation transfer function.

The maximum slope of the edge scans was evaluated

graphically

by

plotting the transmittance values versus

(15)

The

maximum slope can then

be

drawn

for

the edge response

The normalized slope is calculated

from

the slope (

)

and the minimum and maximum

transmittances

(t

, and t

)

v

min

max'

maximum slope = At

AX

normalized slope =

(

) (t

-t .

max min

)

The

MTF(f)s

calculated

by

Tatian's method do not go

identically

to _zero,

because

of noise in the discrete data,

Therefore these

MTF(f)s

are _visually matched to the theor

etical

MTF(f)

for

a

diffraction

limited system of known

cut-off

frequency

(f

),

(Figure

1).

theoretical

MTF(f)

= k

(cos

(

r-)

- r

J

1- (

p)

)

o o '

o

The

image scanning was

done

on a PDS microdensitometer

1 3

using incoherent irradiation. The scans were run of an NBS

edge to give transmittance versus position. The smallest

sampling interval

during

scanning is 1 urn on the PDS. This

sampling interval sets the maximum

frequency

for which the

transform can

be

calculated. This is the

folding

frequency

(f

)

for

the interval

(

ax

)

:

v max'

max 2Ax

Therefore the frequencies which could be evaluated were

(16)

fmax

~

2(.

001mm)

=

500

cycles/mm

The

efflux objective of the

PDS

needed to

have

a cut-off of

less

than

500

cycles/mm.

Assuming

a mean wavelength of .5 mm

for

the radiation.

Numerical

_apertures of .10 and .08

were _used,

having

theoretical

cut-off

frequencies

shown.

f

=

2(NA)

f

= 2.(.,1.) =

400

_cycles/mm

O tD

fo

= 2(.'0ft.), = ₃₂₀ _cycles/mm

Scans

were run of the NBS edge in transmittance versus

position on the PDS microdensitometer. Multiple

focus

pos

itions

(at

10 urn steps

from

best visual

focus)

were used to

obtain a

focus

series for each of the microdensitometer set

ups in table

1

.

Set-up

Efflux objective

Efflux ocular

detected area

.10 NA

"4X"

Zeiss

10X Zeiss

. 17um X .145 mm

system magnification

45.

IX

13

image _scanning yes

sampling interval

1

urn

focus series yes

B

.08 NA

"2.5X"

Zeiss

6.2X special

.49um X .439mm

15.

4X

yes

1

urn

yes

Table

1

: PDS microdensitometer set-ups used in

data

(17)

10

Data

from

the

PDS

microdensitometer scans is stored on

magnetic tape. The records on the magnetic tape are read

and converted _using the programs in

Appendix

A.

The

final

output of these programs is a plot of transmittance versus

position

for

the region of the scan abound the edge.

(Figure

3)

The

plot is then hand smoothed to eliminate the noise in

the tails.

Then

the

discrete

data points are transformed

using a program

(EGA,

Appendix

B)

on the Apple II computer.

The

program uses

Tatian's

method to calculate the transfer

function

of the system. The cut-off

frequency

was then deter

mined

by

_matching the

MTF(f)

with theoretical

MTF(f)s

of

known cut-off

frequencies.

The maximum slope is drawn on the plot at the center of

the edge scan.

(Figure

4)

This

maximum slope is then normalized

by

the range of transmittance

(t

-*,)

to Qive the norma

(18)

11

\

~m O

-?

H 10 O

in

3

V)

1-1 (0

"rsi >

V

u

c o

E

vt

C CO O

o *>

o 0)

in c O Q.

V) <1>

o "O

LU

o

<u u 3 D

H

00V '0 OSE'O OOE'O 0SZ"0 002 "0 0SI "0

3DNVllINSNVa'i

001 "0 00 JA9

(19)

12

E E "v.

-o in

<u <D

N r*

H II O .-1 >s

O <u o

o

ro *

CD

> H +

. _u

fSi V

-l

SI O

< "* z

c

o u

in

u

O

o

in

c o a

m <D U

<0 en

"O

u 3 CO

00* 0SE"0 ooe-o ose -o ooz -o osi-o

BDNVllIWSNVcIl

00T "0 00 JA9

(20)

13

RESULTS

The plot in Figure

4

shows the transmittance values

from one of

the

scans

from

a

focus

series done with a .10 NA.

The maximum slope is _.047; when this is normalized

by

l/(t

-^min^

^ne

va^-ue is -1ST. The normalized slope is

.181 urn ; this is

181

cycles/mm in

frequency.

The transform

of the

discrete

data

from

Figure

4

is shown in Figure 5.

This

transfer

function

has the same

form

as the theoretical

MTF(f)

with a cut-off

frequency

of 181 cycles/mm

(Figure

5)

down to _{approximately} 30%.

The

plot in Figure 6 shows the transmittance values

from one of the scans

from

a

focus

series done with a .08 NA.

The maximum slope is .024; when this is normalized

by

l/(t - _t

.

)

the value is .071. The normalized slope is

v

max min

.071 _; this is 71 cycles/mm

frequency.

The transform

of the discrete

data

from Figure 6 is shown in Figure 7.

This transfer function has the same

form

as the theoretical

MTF(f)

with a cut-off

frequency

of 71 cycles/mm

(Figure

7)

down to approximately 20%.

In both of these cases it is seen that the normalized

slope is a _very good indicator of the cut-off

frequency

at

which the

MTF(f)

of the microdensitometer system

does

go

to zero. It is also apparent that both the slope method and

Tatian's method are not good indicators of the cut-off

fre

quency as calculated from the numerical aperture.

Possible

(21)

14

1.0

z

o

< .5

Q

O

.3

0.0

MFT(f)

.10 NA

theoretical

MTF(f)

f

=

181

o

_

181

500

FREQUENCY

(cycles/mm)

Figure

5.

Comparison

of

MTF(f)

for

edge scan with .10 NA

(22)

15

x o E

E E *.

-o w

<u 9)

N t

H II u

-H X

O d) o E Q.

u O ^

o <-i r^

c in

c

H

E

%

\

00V "0 OSE;0 00E "0 OSZ '0 OOS"0 OSI "0 001*0

3DNVllI^SNVai 00 JA9 I'HM

cr

t .V)

>*

'O

X

0>

>

u <u

"I .>

O

CO

o

c o O

in

u

o

<u

in

c O a

in

V

u

<u

UJ

0S0"0 000

(23)

16

MTF(f)

.08

NA

theoretical

MTF(f)

fo

=

7'

-500

FREQUENCY

(cycles/mm)

Figure

7.

Comparison

of

MTF(f)

for

edge scan with .08 NA

(24)

17

limited

are :

1

)

The objective is not used at the proper tube length

and magnification.

2)

The

microdensitometer is not

focused

on the object.

3)

The

_scanning slit is not aligned with the edge.

These

reasons

for

which the microdensitometer would not

be a

diffraction

limited

system were investigated _using the

.08 numerical aperture objective. The objective tube length

and tube

length

at which it was used were checked.

They

were close enough to _only give a 2.3% change in magnification

in conjunction with a 6.3X ocular. This would _only have a

very slight effect on the

MTF(f)

of the system.

Microdensitometer scans were done with a through

focus

series at 10 urn increments. The change in edge profiles

was _only slight as compared with the predicted rise or

drop

in

MTF(f)

do

to

defocus

for an optical path difference of

7\/2 or more. '

The transfer function of a misaligned slit

C"

|<ew(f))

is :

t /*\ - sinfarLSff)

W

where L is the slit length and

0

is the angle of

misalign-15

merit.

(25)

18

limit

the system cut-off _; the angle of misalignment

{j6)

must be less than:

o

where

f

is the maximum

frequency

to be transfered and L

o ^ '

is slit

length.

When _using a .08 NA with a detected area of .49 urn X

.439 _mm;

the

misalignment must be less than .0071 radians or

.41

degrees.

The

slit length can be shortened to increase the

angle

tolerance,

but then the signal to noise

for

the system

goes

down.

Even with a 100 urn slit

length,

the slit must be

aligned to within

1.8

degrees.

This is a difficult

task,

and

the slit alignment

(misalignment)

is believed to be the reason

for

which the microdensitometer system is not diffraction

(26)

19

CONCLUSIONS

The maximum slope of a normalized scan of an ideal edge

is an excellent indicator of the cut-off

frequency

at which

the system's

MTF(f)

goes to zero.

When

some

factor

(i.e.

defocus,

slit _{misalignment)} causes the

MTF(f)

to cut-off

lower than the

diffraction

limit of the _system, the maximum

slope of an ideal edge scan predicts the

MTF(f)

of the system

to 30 percent modulation or

less.

The

difficulties

of _verifying the slope technique for

the diffraction limited case are:

1

)

using a

low

numerical aperture objective

2)

setting up the objective at the proper tube length

and magnification

3)

setting microdensitometer focus

4)

aligning detector slit with ideal edge

(skew)

5)

using a slit width small enough to give good

modulation out to cut-off of the objective

6)

using sampling interval small enough to allow

calculation of the cut-off frequency.

The slope technique proved to be an excellent check on

microdensitometer performance.

Knowing

the slope technique

works even when the microdensitometer system is not

diffrac

(27)

20

does

provide a quick check of system performance

for

high

numerical aperture objectives.

For the

future,

it would be of value to continue the

empirical investigation to

determine

the conditions under

which low numerical aperture objectives can

be

made to work

at their

diffraction

limit.

Also a _study _using

high

numerical

aperture objectives could lead to a better _{understanding} of

system capabilities and could indicate the

limiting

compo

(28)

21

REFERENCES

1.

Edward C.

Yeadon,

"Edge

Gradient

Analysis",

The

Perkin-Elmer

Corporation,

Norwalk, Connecticut,

Mar.

1968,

rev.

Apr.

1969.

2. R.A.

Jones,

"An

Automated

Technique

for

Deriving

MTF's

from

Edge

Traces",

Photogr.

Sci.

Eng.,

1

1 : 1 02

(

1967)

.

3.

F.

Scott,

R.M.

Scott,

R.V.

Shack,

"The

Use of Edge

Gradients

in

Determining

Modulation-Transfer

Functions",

Photog.

Sci.

Eng.,

7:345(1963).

4.

J.C.

Dainty

and R.

Shaw,

Image

Science,

Academic Press

Inc.,

London

Ltd.,

1976.

5. Berge

Tatian,

"Method

for

Obtaining

the Transfer Func tion

from

the Edge Response

Function",

J.

Opt. Soc.

Amer.,

55:1014(1965).

6. W.J. Smith. Modern Optical

Engineering,

New

York,

McGraw-Hill Book

Company,

1966,

p. 319.

7.

Dainty

and

Shaw,

Image

Science,

Academic Press

Inc.,

London

Ltd.,

1976,

p. 329.

8 . Ibid. _, p. 331.

9. R.E.

Swing,

per personal conversation with J.

Jakubowski,

information to be published at a later date

by

Swing.

10.

Dainty

and

Shaw,

Image

Science,

Academic

Press

Inc.,

London Ltd,.

1976,

pT 331.

11. Jack

D.

Gaskill,

Linear

Systems,

Fourier

Transforms,

and

Optics,

New

York,

John

Wiley

&

Sons,

Inc.,

1978,

p. 161.

12.

Dainty

and

Shaw,

Image

Science,

Academic

Press

Inc.,

London

Ltd.,

1976,

p. 255.

13. R.E.

Swing,

"Microdensitometer Optical

Performance:

Scalar

Theory

and

Experiment",

Opt.

Eng.,

15:

555(

1976)

14. William R. Smallwood and R.E.

Swing,

"An

Improved

Photo

(29)

22

APPENDIX

A

1.000 C*****************************M

2.000 C* *

3.000 C* HAL IS A FORTRAN PROGRAM WHICH READS THE INFORMATION *

4.000 C* FROM A DEBLOCKED FILE OF PDS DATA. THE FILE SHOULD *

5.000 C* BE IN TWO BYTE RECORDS. THE OUTPUT OF THIS PROGRAM IS A *

6.000 C* FILE WHICH CAN BE READ BY PLOTDAL AND BE PLOTTED ON THE *

7.000 C* ZETA PLOTTER. TO RUN THIS PROGRAM USE THE FOLLOWING *

s.ooo c* commands: *

9.000 C* *

10.000 C* IFORT DAL OVER * *

11.000 C* EXT. FORTRAN IV, VERSION F02 *

12.000 C* OPTIONS >NS *

13.000 C* *

14.000 C* !SET FtlOO/SOURCEFILEJIN *

15.000 C* !SET F:i06/0UTPUTFILEJ0UT *

16.000 C* *

17.000 C* !RUN * *

IS. 000 C* *

19.000 C* *

20.000 C***#**###****##***#****************#

21.000 C

22.000 C

23.000 C************* CHANGES IN LINES 37 AND 38 *****#**###**#*#**

24.000 C* *

25.000 C* N = _< _THE _NUMBER _OF _DATA _PIONTS _IN _EACH _SCAN ₎ _*

26.000 C* ZZ = ₍ _THE _NUMBER _OF _SCANS _IN _THE _FILE ₎ _*

27.000 C* *

23.000 C**#****#********************^

29.000 C

30.000 C

31.000 C

32.000 IMPLICIT INTEGER (A-Z)

33.000 DIMENSION BUF<2144)f TBL(256> 0UT(83)r YC1024)

34.000 REAL Yr Tls> T2

35.000 C

36.000 C

37.000 N = ₁₂₈

38.000 ZZ = ₁₉

39.000 N2P96 = _N*2 ₊₉₆

40.000 C

41.000 C

42.000 DATA TBL(32)/'

'/

43.000 DATA TBL(160)/'

'/

44.000 DATA TBL(171)/' +'/

45.000 DATA TBL(173)/'

-V

46.000 DATA TBLC174)/'

.'/

47.000 DATA TBLU75)/'

/'/

48.000 DATA (TBL ( I ) , 1=176, 185)

/' 0'' l'' 2'

r ' 3'?' 4'

49.000 * ' 5'r' 6't' 7'r' B'r'

9'/

50.000 DATA TBL(192)/'

(30)

23

51.000 DATA (TBL(I), 1=193,218)/' A',' B','

C ,

' D'

,

' E'

52.000 * ' F',' G',' H',' I',' J',' K'' L',

53.000 * ' M',' N',' 0',' P',' Q',' R','

S', 54.000 * ' T',' U',' V'!-' W'i' X',' Y','

Z'/ 55.000 C

56.000 C

57.000 WRITE(108,1) 58.000 1 FORMAT(IX,'

DAL IS RUNNING*****') 59.000 C

60.000 C

61.000 DO 999 Z=1,ZZ 62.000 C

63.000 C

64.000 READ (100? 2? END=50 ERR=60 ) (BUF ( X ) ,X=l,N2P96 )

65.000 2 FORMAT (2A1) 66.000 DO 10 I=1,N2P96

67.000 BUF(I)=ISL(BUF(I) , -24)

68.000 10 CONTINUE

6".000 C 70.000 C

71.000 IF ( .NOT. < (BUF(l) .EG. 63) .AND. (BUF(2) .EQ. 63))) GOTO 70

72.000 C 73.000 C

74.000 DO 20 1=3,81,2

75.000 C= _{BUF<I)*64+BUF(I+1)}

76.000 20 OUT(I) = _TBL(C)

77.000 C 78.000 C

79.000 DO 30 1=_{1, N}

80.000 Y( I )=_{(FLOAT (BUF( 1*2+95 >*64+BUF( 1*2+96) ) )/4000.0}

81.000 30 CONTINUE 82.000 C

83.000 C

84.000 DO 40 1=1, N~l 85.000 C

86.000 Tl = _ABS( _Y(I+1)

-Y(I) ) 87.000 IF (Tl .LT. T2) GOTO 40

88.000 T2 = _Tl

89.000 INDEX = _I

90.000 C

91.000 40 CONTINUE

92.000 C

93.000 INDEX = _INDEX

-15

94.000 INDEXP = _INDEX ₊₃₀

95.000 C

96.000 C

97.000 WRITE (108,3) ( ( OUT ( X ) ,X=3,81 ,2) , INDEX? T2 )

98.000 WRITE (106,3) ( ( OUT ( X ) ,X=3,81 r2) , INDEX, T2 )

99.000 3 FORMAT ( IX, 40R1, 15,F10.4 )

100.000 C

(31)

24

101.000 c

102.000 WRITE (106,4) (Y(I), I=INDEX, INDEXP)

103.000 4 FORMAT (16F8.4) 104.000 C

105.000 C

106.000 999 CONTINUE 107.000 50 STOP 108.000 C

109.000 C

110.000 60 WRITE (108,61)

111.000 61 FORMATC ERROR IN READ') 112.000 STOP

113.000 C

114.000 C

115.000 70 WRITE (108,71)

116.000 71 FORMAT ('

BAD RECORD') 117.000 GOTO 999

118.000 C

119.000 C

120.000 END

(32)

25

1 .000 c****************************************************************

2.000 C* *

3.000 C* PLOTDAL IS A FORTRAN PROGRAM TO PLOT THE DATA SET IN A * 4.000 C* FILE BY DAL. THIS PROGRAM MUST BE COMPILE AND THEN RUM * 5.000 C* ON BATCH WITH A CARD DECK. TO COMPILE THIS PROGRAM USE #

6.000 C* THESE COMMANDS? *

7.000 C* *

8.000 C* !FORT PLOTDAL OVER DPLOT *

9.000 C* EXT. FORTRAN IV, VERSION F02 *

10.000 C* OPTIONS >NS *

11.000 C* _%

12.000 C* ! _*

13.000 C* *

14.000 C* WHEN THE SYSTEM COMES BACK WITH A ! THE PROGRAM IS * 15.000 C* COMPILED IN DPLOT. IT CAN BE RUN ON BATCH WITH A DECK OF *

16.000 C* CARDS WHICH HAVE THE FOLLOWING COMMANDS? *

17.000 C* *

18.000 C* !JOB ACC0UNTNUHBER(NAME),7.NAME *

19.000 C* ! LIMIT ( TIME, 1 ) , < CORE,24 ) , ( LO,50 ) *

20.000 C* (ASSIGN F \ 100,(FILE,SOURCEFILE),( IN ),( SAVE) *

21.000 C* ILYNX DPLOT, .R0MLIB1 *

22.000 C* !RUN *

23.000 C* !E0D *

24.000 C# *

25.000 C* *

26.000 C****************************************************************

27.000 C

28.000 C

29.000 C*************** CHANGES IN LINE 41 *************************

30.000 C* *

31.000 C* ZZ = ₍ _THE _NUMBER _OF _SCANS _IN _THE _INPUT _FILE ₎ _*

32.000 C* *

33.000 C***************************************************************>i

34.000 C

35.000 C

36.000 INTEGER FILE (10), NX(2), NY(2)

37.000 REAL X(33), Y(33), DX(1),DY(1)

38.000 N= 31

39.000 C

40.000 C

41.000 ZZ = ₁₉

42.000 C

43.000 C

44.000 45.000 46.000 47.000

48.000

49.000

50.000 C

DX(1)= _1.0/3

NX(1)= ₃₀

NX(2)= ₀

DY(1)== _.2

NY<1)= ₄₀

(33)

26

51.000

C

52.000 C

53.000 DC) 99 M=_1,ZZ

54.000 C

55.000 C

56.000 READ (100,5) FILE 57.000 5 FORMAT (10A4)

58.000 C 59.000 C

60.000 READ (100,10) (Y(I),I=1,N)

61.000 10 FORMAT (16F8.4) 62.000 C

63.000 C

64.000 DO 20 1=_{1, N}

65.000 X(I)= _{FLOAT (I)}_-1.0

66.000 20 CONTINUE 67.000 C

68.000 C

69.000 C

70.000 X(N+1)= _0.0

71.000 X(N+2)= _3.0

72.000 Y(N+1)= _0.0 73.000 Y(N+2)= _.05

74.000 C 75.000 C 76.000 C

77.000 CALL BEGPT (3,11) 78.000 C

79.000 CALL AXIS( .5, .5, ' DISTANCE',-10 ,10.5, 0.0,X ( N+l ), X (N+2), 0) 80.000 CALL AXIS( .5, . 5,FILE,40P8.5,90.0,Y(N+l ) ,Y(N+2) ,3)

81.000 C

82.000 CALL GRID ( .5, .5,DX,NX,DY,NY )

83.000 C

84.000 CALL PLOT ( .5, .5, -3)

85.000 C

86.000 CALL LINE(X,Y,N, 1,0,2 )

87.000 C

88.000 CALL FINPT 89.000 C

90.000 C

91.000 99 CONTINUE

92.000 C

93.000 C

94.000 STOP

(34)

APPENDIX

B Eh

g

03 27

e

a

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8

E

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E

85

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B .._{D M}

0= 0- OS

s z a.

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(x,HP

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ow -1_W

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_w OS *9"9-s < ..E-i 2 l-l w 2* M 3 Id

1

s

fe

w : 2 E-i 2 t-H

2

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SCC2!

W

W 2>

HH&p03 Cn

r-cm mw itivd

Sh II

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d S

5

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2p < -cm ro.=r so-w

lilii1!!

l-l2OhQX

E-i H 33 > ? E-i =T = s E z

9

g

-to- ,.

< SOh 2 P r-H

-^-' .. ta E .. Cd

IS

69-H

ro w 2

w SE-i n

%> _OS

-52 PIS'

8

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ssr

fe

M H 8iSE~ MCO 8s

BC= <s

OS CO 2H E

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CJ Ed

M E-i-te-H

~

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E-i=3>=> >JOh Oh II

UZbZ

2 2 II

M OSM

s&

ZHHHetibtiHosoas 2

* z

PI

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E-i =

2eh

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E = _QCM H H ITl

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o-a-OhOh

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[1

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^-cm

ma-in

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vOt_CO~

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C--Cd

iniao

E-ita W 03

r-o o t-00CT> 33 OhEh Cd E-"- --,

CO Oh [b ! W ta n'->

fa*^2

co ~v.E n

fe,-^CM

II , E-i -+ [tH O OK, II SClH II M r* oH o mooi-Si

ON*-r- --

w-2 + 2 W ai CM Eh CM CM< Eh^ -^2 PM OS CO *. II 2 CM l-l CO-^

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(35)

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sg

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S^S9- '

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a-p

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CM CM II33 II II

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(36)

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CMCMCM CMCMCM O O O O O O OVO

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cm

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(37)

30

H

g

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cMroLnSggS82

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gsssi-,2

asos os os os os Oh Oh Oh Oh Oh Oh

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inininminininininininininin

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c-wmairnot-ooonO'-cm

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(38)

31 O ar

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II E

691

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3

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fc.*~~.s-* coco

vOO_; a-j inm co o o in

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(39)

32

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p

0H2 _S, 33ZH J 3H0S-69-E = =

g0-^

HH *

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oinomoino i

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P0S03 00000000000 inoinoinoinoinoin cocogNQNOO'-^iMwmm ifvjcMCNjCMcocororororororo

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(40)

33

VITA

Dale E. Ewbank is currently an undergraduate student

in

Photographic

Science and Instrumentation at the Rochester

Institute

of

Technology

and is anticipating his B.S. degree

in May.

1982.

hie was raised in

Orlando,

Florida and attended

Colonial

High

School. He received a Fuji

Award

Scholarship

while at

R.I.T.

After

graduation and

his

_wedding with

Martha

Leahy

in

July,

1982,

he will

be

employed

by

American

Theses

Thesis/Dissertation Collections

Prediction of cut-off frequency from the maximum slope of a transmittance scan of an ideal edge

Rochester Institute of Technology

Recommended Citation

of the Rochester Institute of Technology

MAXIMUM SLOPE OF A TRANSMITTANCE SCAN OF A IDEAL EDGE

SLOPE OF

TRANSMITTANCE

function

ideas,

TABLE OF

CONTENTS

LIST OF FIGURES

Modeled

function

known

Swing

function

(fQsinc

FREQUENCY

f0sincZ(fox)-X-step(x)

drawn.

distance

EXPERIMENTAL

determination

function

drawn

frequency

folding

focus

Appendix

(Figure

00V '0 OSE'O OOE'O 0SZ"0 002 "0 0SI "0

RESULTS

focus

frequency

theoretical

00V "0 OSE;0 00E "0 OSZ '0 OOS"0 OSI "0 001*0

theoretical

FREQUENCY

Figure

focus

0

limit

frequency

Knowing

limiting

REFERENCES

Yeadon,

Science,

Academic

APPENDIX

3.000 C* HAL IS A FORTRAN PROGRAM WHICH READS THE INFORMATION *

10.000 C* IFORT DAL OVER * *

45.000 DATA TBL(173)/'

103.000 4 FORMAT (16F8.4) 104.000 C

6.000 C* THESE COMMANDS? *

21.000 C* ILYNX DPLOT, .R0MLIB1 *

APPENDIX

.-CM COST CM CM CM CM CM CM CMCM

OOOOOOOOOOO*-'-'

!lwS2i5E

" E E E IIH

W -69-69-HCOH H H H

Photographic