Rochester Institute of Technology
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Theses
Thesis/Dissertation Collections
1969
Polychromatic MTF of a Pinhole Camera
Arthur Messenger
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Recommended Citation
Messenger, Arthur, "Polychromatic MTF of a Pinhole Camera" (1969). Thesis. Rochester Institute of Technology. Accessed from
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ABSTRACT
Sayanagi are developed
for a pinhole camera using standard 5500K daylight and a black and white,
and a color reversal film.
On and off -axis MTF were developed for t.ie
radial and tangential cases.
Conclusions are drarai about the depth of focus
Polychromatic MTFs based
"best"
focal
both
the
showing
and
on
the
work
of
Hopkins
and
A monochromatic depth of field study
to focus and the infinite character of the
done,
position.
was
need
pinhole
camera.
INTRODUCTION
An interest in the
Newman
and
and
ftible have
Hitterdal
used
a simulator
Neglected
reported
a pinhole
for
using
space
Pinhole,"
to
an
docking
uses
of
array
a
replace
step
and
system when
J. H. Waddell in his
the
of
to
Gallas, Gilbert,
focusing
automatic
in the literature.
shown
pinholes
circuits.1
exercises.2
other
gives
an
replace
device is
imaging
an
for making integrated
camera
repeat
as
pinhole
making
article,
"The
3
camera.^
pinhole
OBJECT
IMAGE PLANE
PINHOLE
IMAGE
Figure One. Diagram
Figure One
Experimental
evidence
is too large
or
The
major
a
shows
too
problem
diagram
and
small
of
the
There
are
no
optimizing a pinhole
in the tenth
century.
^6
image
pinhole
papers
simple
pinhole
considerations
a
very
fuzzy
then is
camera
a pinhole
and
camera
show
one
is
imaging
that is
all
that
the distance s
taken
under
Petzval
Sayanagi.
and
records
the
Rayliegh wrote
Recently, Selwyn, Sayanagi, Young,
oldest
papers
and
Swing
object.
pinhole
obtained.
What is the
optimization:
picture
an
that if the
produces
conditions.
optimum
in the literature that discuss the
camera.
century.^
or
Pinhole Camera
a
diameter, d;
Figure Two is
many
a
theoritical
relationship between pinhole
the best image.
of
of
problem
reference
in the
and
as
of
being
nineteenth
Rooney
have
writ-
ten
excellent papers
Hopkins,
Steel, have
and
standing the
In
pinhole,
that
imaging
finding
infinity.1
3
f,
this
When
as
produced
in
Young
on
used physical
that have
of pinhole
to
infinity
lenses.
Petzvai
equal
image
of
s
is
used
a point
a.
the
intensity
central
reasoning from
the
notably
bearing
the
with
on under
as
pinhole
the
and
his optimizing
of
presence
physical
the
of
diameter
criterion.
Selwyn
defocus.
spread
point
at
object
the focal length
criterion of
source
geometrical
direct
started
called
of
Others,
earners.11'12
Fraunhof ar diffraction in the
of
'^>
pinhole,?'
written articles
smalles*
the maximizing
the
optimum relation most papers
other
the
optics of
characteristics
is
s
Rayleigh based his
used
the
on
function.
optics.
Sayanagi
function"
used
the
figure
one
central
of merit
intensity
and
the
dimensional figure
leijh's,
and
Where k is
is the
a
Selwyn'
s
constant
of
the
central
point
intensity
of
the line
Sayanagi has
of merit.
optimization
relatioship
depending
the
on
as
spread
shown
can
criterion
be
a
spread
that
two dimensional
function
a
as
his, Petsval's,
Ray-
into the form d^=kyf.
put
taken for
optimization
and
y
wavelength.
IMAGE. PLANE
O
PINHOLE
Pinhole
from P
Figure Two. Image of
pinhole
off-axis
as
seen
Pinhole
from
P
as
seen
The
each
discussed
off -axis,
'
pinhole
(radial)
as
field.
Noting
an
physical
as
the
the
the
These differences
ellipse.
showed
optimised
the image
as
moved
diameter
of
diameter in the horizontal
the field angle.
in
result
is
pinhole
that
Sayanagi
and
(tangential;
vertical
effective
of
cosine
the
when
reasoning
7"<*>o,
the same; while,
Selwyn, Young,
imaging,
off -axis
in Figure
decreases
plane
imaging.
on-axLs
happens to
shewn
remains
appears
is for
Selwyn thru
is
as
-kyf
what
on-axis.'-'*1"*1
the
d
equation
Thje
astigmatism
pinhole
and
thus
curvature
of
*
that the distance from the
to cose,
proportional
2
to d Sayanagi's
and
best
radial
is the field
where
for the
equations
.0
monochromatic
that there
correct
curves
be
should
are minimal
The
MTF
over a
equation
d
for only
for 30
a pinhole
relatively
=kyf
is
with
spectrum
out
a
broad
for 00
nm
and
spectrum.
to be 400 to
600 nm,
then the
size
aberration.
while, Young thought its "effect is
sentation,
showing that if the
inversely
proportional
focus, f^.=fcos3,
Sayanagi reported
Young,
curvature
Most
sources
of
and
field
Selwyn
mentioned
and astigmatism
are not
size
monochromatic;
consequences
of
the
Varies only by
pinhole
about
length f
varies with wavelength
Selwyn
that this is
small.
noted
"
Sayanagi
diameter
of
assuming the working visible
Young,
and
therefore is strictly
investigated the
that if the design
pinhole
is
19 * 20
each
optimum pinhole
longitudinal
18
of wavelength and
Selwyn
optimum
best tangential
some work.
that
plane
that f is
off-axis.10
field.
noted
and
o
such
Sayanagi
the
chromatic
_,
sources.
If d is held constant,
after
hS
function
monochromatic
therefore, Selwyn, Young,
imaging
a
,
.
and
wide
angle
position of
fr=fcos e follow
focus,
to the film
pinhole
and
gave
a
1 0%+
worked
uas
21
*
22
leading
to
"excessive;"
detail
focal length
ed repre
are picked
according to his
"loss in
The
5%
could
the
of
The MTF
of
conjugates
worked
typical
and
wavelengths, 400 to
"2^
distance'
based
Selwyn,
extreem
meddle wavenuraber.
the
on
Young
object
showed
being
in his paper, he did
put
an
infinite
+ 1/s1
that 1/s
Although Sayanagi did
in the literature
develop
the
panchromatic
black
polychromatic
and
white,
for
are
Standard 5j?00 K daylight is
receiver.
for both
then the
d2=3.8yf,
merit, yielding
l/f
=
include
not
in information
that it
so
MTF
used
and
a
for
curves
as
source
a
a
can
be
over a given
The
focus.
finite
a
made.
field
second
object
map
of
the
on
This map
of view as
objective
distances to
and
gives
for
receivers
a
film
are used.
With
and
as
well
is to
examine
off-axis polychromatic
information
use
MTF,
on
Sayanagi 's information
depth
of
as
on optimization of
furnish information
and
source
specific
color reversal
Thus it is
light.
monochromatic
'
tbis information
changed
discus
a
out.
curves
interest to
also
camera. 2425
pinhole
of
of merit
to the
compared
pinhole.
finite
be
as
figure
equation d2=kyf is
holds for the
sion of
dimensional figure
one-dimensional
700 nm, is only
in front
one
s
'
the
the depth
on
is
responce
of
of
effect,
field.
PROCEDURE
Bi figure Three COD is
eatering the
pinhole
the
a wavefront when
shape
point
P1
of
an
e2q?anding
with on-axis
in the image plane.
the
point
wavefront
at
on-axis
0.
point
from
A0B is
is
Departures from this
at
a point
a reference
0 that
reference
source
at
sphere
converges
sphere
are
P
showing
to
a
known
IMAGE PLANE
POINT
SOURCE
Figure Three. Diagram showing
as
wavefront
When the
function.
the
as
a
pinhole
def ocused
this
was
camera
case and
the MTF
of
the
which
has
free
an
f/#
optical
the MTF.
pinhole.
of
l-IIF-
A
about
system.
Sayanagi
at pinhole.
describes it is
function in terms
pupil
aberrtion
function for this
calculate
the
of
the function
aberration
function is known the
pupil
autocorrelation
Since the
of
and
aberrations
wavefront
can
of
200
calculated
it
axis
Hopkin's
computer
pupil
using
coordinates,
suitable
on
the
by
can
pupil
equation
program
?7
28
'j11
be thought
Hopkins developed the
modified
monochromatic
be
called
to
based
on
written.
Barn's has
shown
that the
polychromatic
MTF P(S)
can
be
2
using the
following
equation.
Pis)
-
00
i
RnsuTyMu(s>dVJ
JTRuSuTudy
y^y'y
calculated
by
Hy
is the
is the
spectral responce
MTF.
a
finite
This
sum
the lfy(S)
calculates
summ
seventeen
using
r
then
equation was
to-
them
form the
Off-axis both
radical
case,
as
weights
radial
the
pinhole
Using
one program
the
calculate
To
d =kyf
After
evaluated
for each, the
was worked
MTFs for finite
monochro
integrals with
appropriate
that
written
computer program was
weighting factor
OP1
the distance
OP1
angle
polychromatic
MTF
inversely
For tangential
the
as
For the
considered.
increases
.
wary
be
must
of
cosine
program was
case
both
the field
generalized
angle.
to
MTF.
MTF's for finite
thru using the
d2=kyf (s+f )/s.
doing so,
l-fy(S) is the
and
a
of
case
MTF.
the field
of
on and off-axis
calculate
A
tangential images
the distance
and
source
them according to the
and
in the
one
Ty
wavelength.
of
by approximating the
intervals.
Figure One shows,
diameter
function
a
the
of
polychromatic
proportionally to the cosine
as
the optical system,
of
is the energy distribution
Sy
pinhole.
matic
transmission
spectral
the detector
of
(\
object
distances, Sayanagi's
for finite
coefficient
aberration
.monochromatic
derivation
program was
of
conjugates.
modified
to
calculate
conjugates.
DATA AND CONCLUSIONS
Figure Four
black
and
matic
MTF for
white
shows
the weighting function
panchromatic
a pinhole
of
film.
diameter
and
Figure Five
equal
to 0.2
its
shows
mm
conponents
on
and
and
off-
for
a
axis
focal length
typical
polychro
of
20.9
.o
using this weighting function.
.off
-axis
are
equations
essentially
for
position
In Figure Five the MTFs for
identical.
of
This is
best tangential
as
on-axis
and
15
indicated from Sayanagi's
and radial
focus.
When
s'
equals
mm
nm
Spectral Sensitivity
of
Receiver
100
700
Energy Distribution
of
nm
Source
1200
700
300
Weighting
Figure Four.
Factor
Weighting
Function
and
Components
nm
CJ
in
TZ.
G)
-P
-P
O
ia,
ti
H
Pi
0>
Xi
CQ
fin
CO
H
00
oo
00
=h
O
a
ti
d
o
o
O
rO
H
f-.
a
H
C\J
LO
y<
0)
-P
LO
P
o
y
S
rH
Rj
H
ti
CD
CQ
H
00
<\J
<A
h
o
C
cd
5
O
m
rO
>
cu
P=4
H
P=H
to
Pn
E-i
s
3
CM
O
c
cd
ti
o
o
>
H
0)
P*H
00
00
CM
OJ
o
LO
<B
LO
-P
-P
O
ca
H
cd
cd
Pi
-fr
^
11
"co
ti
CQ
6
CQ
H
oo
00
OJ
OJ
&
ch
o
c
o
o
ti
o
o
c
cd
o
>
<D
I
rl
fr
00
OJ
cx>
OJ
o
LO
o
LO
T3
0)
p
p
o
XZ)
cd
H
TZ1
cd
Pi
co
cv
ti
Xi
3
<Vh
ch
O
ti
a
a
o
pq
H
fr
CD
U
g,
H
fr
15.7
mm
on-axis, the
KTFs for this film
on-axis
lane
higher
MTF is the
position give
millimeter
and
prediction
that
KEF
middle wavelength
of.
the
a
such
a
cut
the polychromatic MTF.-
has the bes MTF
Sayanagi's
for
This
taken
over
25
Figure Five
axis
shows
is the
which
of
shows
the
optimum since
the
off-
in- axis
per
that the 20.9
MTF
mm
approxima
film
the
versus
depth
of
shows
greater modula
If
nm
using
that
the film
acceptable
of
plane.
focus is
focus is
of
the
on-axis
gigher
cut
off
side
near
ion
position
shows
of
millemeter, then the depth
the
for
Figure Seven is
position
focus.
on
50U
at
Figure Six
criterion.
the
uion
plane
per
Young's
and
predicted
for best focus
that it is better to be
axis
good
good approximation.
on-axis
themaximum lines
Five
mm.
of
a
a
lines
equal
Sayanagi
be
off -axis
all
Selwyn 's
confirms
exists.
position
is,
for
modulation
would
of merit
per millimeter
indication
an
$0%
as
source
the monochromatic MTF is
the lines
gives
the
of
that
casej
This
plane position
dimensional figure
one
on-axis
a plot of
on-
higher
frequency.
off
film
limiting
.Jid
frequencies.
Figure Eight
shows
color reversal
film.
color reversal
and
value
to, be
for the
color
Figure Ten is
identical
back
film.
a
Figure Nine is
All
one.
the weighting function
and
results
of
except
Figure Eleven
compares
length
equal
.focal
with
s
after
for the
comparison
its
of
panchromatic
for
a
typical
thw weighting functions for
by
normalizing
components
considering the largest
black
and
white
film hold
Differences feflect differences in the weighting functions.
comparison
conditions
rhite
a
and
fo
a
.
black
MTF
with
a
color
MTF
under
film.
KTFs
to $f
and white
a pinhole
of
0.2
20f.with the MTF based
on
an
calculated
and
for
mm
and
infinite
20.9
mm
14
Figure Six.
Comparison of Polychromatic MTF
28
with
MTF
of middle
wavelength, 500
nm.
20
10
Firsure Seven.
On-axis depth
off.
field.
30
60
mr
700
Spectral Sensitivity
of
Receiver
700
300
Energy
Distribution
of
nm
Scarce
700
Figure Eight.
nm
Weighting
Functior
and
Color Reversal Film.
Co.vpoents for
nm
700
Comparison of Black
Color Reversal
with
Figure Ten.r
Comparison
MTF
of
of
Black
MTF
and
of
White
Weighting
Weighting
Function.
and
Color Reversal Film
.Ehite
nm
Filn.
(dots).
Function
with
(dots)
00
H
H
'a
a
cd
ch
O
<N
P
cd
CQ
p
o
CD
""a
Xi
CD
rA
O
o
u
o
ch
CQ
fr
cS
rH
<D
IO
H
fr
<+H
O
^5
P
Ph
CD
P3
CD
>
CD
0)
H
fr
H
S3
H
ch
ti
rC
ti
cd
ch
u>
P
0}
m
-P
A3
<D
>-a
O
ri
O
ch
CQ
fr
S'
distance.
object
equals
considered practical! in
for the
camera
20.9 'mm.
finite.
particular object
Yet,
depth
of
of
length
as
162
mm was
focus,
the
for 20.9
Outside
made.
same
general
field
of
it showd that it is best to
shown
of
that the depth
shows
distance if
Using the weighting function
length
.This
s
is
under
in Figure Four
less lines
conclusions
can
can
optimize
be
the
10f.
a
a
map using
per millimeter
and
focal
larger
be drawn for this focal
mm.
CONCLUSIONS
In this
was
paper
developed.
It
the
on
and
off -axis
polychromatic
demonstrated that there is
was
a
MTF for
film
apinhole
camera
position
plane
such
MTF'
that the
on
and
that the depth
it is
best'
to
off-
of
have approximately
axis
field
optimize
can
be
for this
considered
object
8
similar cMTF's.
infinite j yet,
distance in
order
when
s
It
is
was
shown
under
10f
to increase modulation.
BIBLIOGRAPHY
1. R. A. Newman and V. E. Hible,
2. A. H. Gallas, C. A. Gilbert,
Aopl.,
and
A. B.
Telev. Eng.,7lj., 321 (1965).
3. J. 31. Waddell, Res. /Development, Ijf, 26
k. K. Sayanagi, J. Opt. Soc. Am., 57, 1091
5. J. Petzval, Phil. Mag, XVII, 1 "j\ 659).
5-
1225 (1966).
HLtterdal, J. Soc
Opt..
Motion Pict.
(1963).
(1967).
6. Loard Rayleigh, Phil. Mag. XXXI, 87 (1891).
7. E. W. H. Selx-iyn, Phot. J. 90B, kl (1950).
8. loc. cit. k
9. M. Young. Appl. Opt. 10, 2763 (1970.
10. R. E. Swing and D. ?. Rooney, J. Opt. Soc. Am., 58, 629 (1968).
H. Hopkins, Proc. Phjrs. Soc. (iondon), A231 , 91 (1955).
11. H
12. W. M. Steel, Opt, Acta, 3, 65 (1956).
.
13.
Ttiis
work
bd.seu
n
work
in k.
loc. cit. k.
loc. cit. 7.
loc. cit. 9.
loc. cit. km
18. loc. cit. U.
19. loc Cit. 9.
20. loc. cit. 7*
21
loc. cit. 7.
22. loc. cit. 9.
Ik.
15.
16.
17.
.
23. loc.
cit.
k,
2k. loc.
25. loc.
26. loc.
27. loc.
28. loc.
cit.
7.
cit.
9.
cit.
U.
29.
K.-
cit.
11.
cit.
k.
p.
109&.
Barns, The Optical Transfer Function,
Company, Inc., New York (1971).
R.
American Elsevier
Publishing