14 th Congress of the International Society for Photogrammetry
Hamburg
1980
Comm i ssion I
W.G .
1 I 2
Image Geometry with Camera Calibration
Presented Pape r
J.Hakkarainen
Finnish Geodet i c Institute, Helsinki
K. - J.Rosenbruch
PTB, Braunschweig
Studies on Image Evaluation Methods and Lens Aberrations
Abstract
The first-mentioned author was with Alexander von Humboldt-fellowsh i p in the
PTB in Braunschweig in 1978-79. The author had an aerial wide-angle lens
Wild RC-8 15Ag from Finland with him for measurements . The focal length,
spherical aberration, coma , field curvature , astigmat i sm , influence of defocus i ng on the axis, po i nt images, and MTF for seven fie l d angles were deter mined from this lens in the PTB. The geometrical errors, resolving power and
MTF were determined from the same lens with the Helsink i University of Technology (HUT) gon i ometer. MTF values in Braunschweig and Helsinki were the
same with an adequate accuracy . The method used i n Hels i nk i gives on an average from 3 to 10% greater values than the EROS method used i n the PTB
depending somewhat on the field angle .
11
Introduction
The lens qua lit y can mainly be described with l ens aberrations, resolving
powe r or MTF, or also with a criteria connected with the discerning of some
i mage details. In this study the relation between the resolving power and
MTF, and the lens aberrations is dealt with. A comparison between two different MTF determination methods has also been made .
The f ir st-mentioned author was with the Alexander von Humboldt - fellowship
in the Department of optics of the PTB in Braunschweig in 1978-79 having an
aerial wide-angle lens RC-8 15Ag No.102 from Finland with h i m for measurements . For studying the above- mentioned subject the focal length, spherical
aberration, coma, field curvature and astigmatism, point images and the influence of defocusing on the RP on axis were determined in the PTB in winter
1979. Also MTF measurements for 7 field angles were accomplished. The lens
cone was detached from the camera body for measurements in the PTB. Additionally the geometrical calibration of the camera was made by Wild Heerbrugg on May. The aim was to clear up wheather the detaching of the lens
from the camera body had had any influence on the image geometry . In the
summer 1979 the first-mentioned author finally made a complete calibration
and MTF measurement for the camera with the gon i ometer in Helsinki. Because
all these measuring methods are wel 1-known, the interest is directed mainly
on the results in this paper.
Focal length and aberrations
1.1 Focal length
The focal length was measured with an equ i pment constructed in the PTB for
the measurement of lenses with a long focal length . The measu ri ng pr in c i p l e
is as fo 11 ows:
f =
yI
tan w
w
i s a sma 11 ang 1e ( see fig. 1 . )
The determination was made with y 1 values of 4, 5, 6 and 8 mm. The angles w
were measured with a theodolite. The result was that the effective focal
length of the 15Ag No.102 lens was 153.36 mm. The calib r ated focal l ength
determined in Helsink i was 153.35 mm and that by Wild 153.34 mm .
1.2 Spherical aberrat i on and coma
The spherical aberrat i on and the coma were determined with the Hartmann
method /1/ . A para ll el bundle of rays coming from a two - meters-col] imator
was exposed on intra - and extrafocal photo plates . The star i mages formed on
these were measured wi th an Abbe - compa rator . The aberrations were calcu l ated
as i ntersections of the straight l in es between intra-and extrafocal points .
Because the aerial lens could not be turned in the opt i cal bench due to i ts
b ig s iz e, its center i ng was not good enough . As a consequence the po i nts
near the opt i cal axis remain very uncertain . The results are shown in the
figures 2 and 3.
12
1.3 Field curvature and ast i gmatism
The sagittal and tangential f ield curvatures were determ i ned in the Wetthauer bench by measuring the places of the best focus wi th a microscope i n
19 field ang l es from 0°to 45° . In the fig . 4 . we can see that the f i eld cur vatures are rather symmetrical in the d i rect i on A- C. The ast i gmat i sm is very
great when the f i e l d angle is greater than 38° ( r > 120 mm) and great when
r = 40 - 90 mm .
1 .4 Distortion
The radial and tangential distort i ons were measu red in Hels i nk i wi th the go niometer. The results are in the f i gures 5 and 6 . The came ra has a very good
symmetry i n the d i rect i on B-D . The asymmetry i n the d i agona l A- C i s caused
by the decentering of one or more l ens elements i n th i s d i rection . The axis
of the maximal asymmetry of radial distort i on and the 0- ax i s of the tangen tial distortion are about the same . The rat i o between the rad i al and tangen tial components of the decentering is 3 . The determinat i on of the rad i al
distortion i n the corner of the i mage wi th the HUT gon iometer i s uncerta i n
because of the phase shift phenomenon . Therefore the radial d i stort i on has
been not i f i ed only up to the va l ue of r = 100 mm . Up to th i s va l ue the place
of the gr i d cross i n an inclined position could be dete r mined wi thout diff i culties .
1 . 5 Lateral ch romatic aberration
Lateral chromatic aberration was measured with the gon i ometer in the spec tral r eg i on from 450 nm to 650 nm . A cont i nuous monochromat i c f il ter was be hind the eyep i ece of the observing te l escope . The results are in the fig. 7 .
1 . 6 Point images
The i mages of a br i ght point wereobserved and drawn in the Wetthauer bench
using a mi croscope . The s i ze of t he poin t was 0 . 08 mm i n the middle po i nt of
the i mage . The obse r vat i ons were made i n the plane of best def i nit i on dete rmi ned on the optical axis and wi th defocus i ng of± 0 . 5 mm and ± 1 . 0 mm i n
fie l d angles from 0° to 45° . The results drawn are in the f i g . 8 . In some
cases i t is d i ff i cult to determine the weight point of the i mage of the
bri ght po i nt even in the plane of best defin i t i on .
2
Resolution
2 .1 Reso l v i ng powe r i n gon i ome t er
The opt i cal reso l v i ng powe r of the camera was dete r mi ned wi th the gon i omete r
us in g a test p l ate on the frame p l ane . The r e we r e 9 gro ups of test figu r es
i n the plate i n one rad i us . The greatest spat i al f r equency was 220 L/mm . The
contrast at the order of 100 L/mm i s 1 . 6 and at the o r der of 200 L/mm 1 . 0.
The RP va l ues are g i ven i n the f i g . 9 . The f i gures can be photog r aphed
through the Jen sand obse r v i ng telescope . When the f i gures 4 . and 9 . a r e compared , the r e can be seen c l early that the f i e l d c ur vatu r e of 0 .1 5 mm has a
strong decreas i ng i nf lu ence on the RP .
13
The dependence of the rad i al RP on the aperture with the values 1 : 5.6,8,11
and 16 is shown in the fig. 10. The determination was made visually . The
trend of tangential 1 ines is about the same .
2.2 Resolving power on axis determined i n the Wetthauer bench
The influence of the aperture and the relat i on between the RP and the diffraction limit are shown in the fig . 11.
2.3 Influence of defocusing on the RP on axis
The influence of the defocusing on axis i s presented in the fig . 12. The observations were made in the Wetthauer bench wi th square wave high contrast
test targets . The maximum RP with the aperture of 5.6 was about 270 L/mm.
Phase shift of 180° or spurious resolution occurs both inside and outside
the best focus. About 0.3 mm outside the best focus there i s also a region
of normal image formation, but wi th a weak modulation .
3 MTF measurements
3 . 1 MTF methods
Nowadays many MTF methods with different principles are used. When comparing
the MTF values of the same lens measured in different l aboratories it has
been verified that the agreement of the results on axis is good , but outside
the axis, especially with great field angles, large differences may be found.
3.1 .1 Method in the PTB
In the PTB the EROS method is used. The test target i s in principle a parallellogram. The lens to be studied is stationary on the optical bench. The
transmitter is on a carriage on an optical bench which is perpendicular to
the primary bench at a distance of 6 m from the lens . The receiver i s moving
on the horizontal diameter i n the image plane respect i vely. So i t i s possible to measure the MTF's of field angles up to 45° outside the axis. The
output signal is measured with a slit . The right receiver field angle is ad justed with the phase of a gross spatial frequency. The EROS method gives
both the MTF and the phase transfer. Plus-minus 4% is given for the pre cision of the method. The difference betweenooand 6 m causes an error of
about 1 % on the MTF .
3.1 . 2 Method in Helsinki
In the method used in Helsinki the test target is on the frame plane of the
camera to be studied (see section 2.1) . The f i gures are photographed through
the observ i ng telescope with an enlargement of about lOx on the film of the
register camera . The MTF is calculated from the measurements made from this
film . The main steps of the method are :
1)
2)
Goniometer photography .
Microdensitometer measurements .
14
3)
4)
5)
6)
7)
8)
Sens i tometric exposure of the reg i ster film .
Microdensitometer values to effect i ve exposure values with the
characteristic curve .
Calcu l at i on of the square wave modulat i on .
Square wave modulat i on to the MTF with Fourier analys i s .
Corrections caused by the microdensitometer sl i t and object figures .
MTF of the register film and the adjacency effect influence in the
oppos i te d i rect i ons . The errors caused by them and the goniometer
optics r emain as error factors of the method.
3 . 2 Reference plane
The place of the plane of best def i n i t i on was determ i ned in the Wetthauer
bench wi th the wave legth of 548 nm . Its d i stance from the last lens surface
was 58.44 mm . The corresponding distance of the frame plane measured mechani cal l y was 0 . 05 mm longer.
When the plane of best def i n i tion was measured wi th 6 d i fferent wave lengths
the diffe r ence between the distances measured wi th the f il ters of 548 nm
and 585 nm was 0 . 06 mm . Th i s means that the plane of best def i n i t i on was i n
Hels i nk i very near the frame plane with th i s ill um i nat i on . In orde r to have
the reference planes in Braunschwe i g and Hels in k i as c l ose to each other as
possib l e , the 585 nm f il ter was chosen for the MTF measurements i n the PTB ,
because the spectral weight point i n He l s in ki was 580 nm .
3 . 3 Compar i son between the PTB and Helsinki methods
MTF for the radius A of the RC - 8 15Ag No .1 02 camera was measured i n the same
7 field ang l es in Braunschweig and Helsin ki. The results are i n the f i gures
14 . We can see that in general the measurement in Helsinki has g i ven about
from 3 to 10 % greater va l ues tha n the measurement in Braunschwe i g . The d i f ferences i n % are in the table 1. The d if ferences are l argest at sgat i a l
frequencies from 10 to 30 L/mm when the field angle is 27 . 5° , 33 .1 and 37 . 6?
I t can be concluded from the results that the accuracy of the method deve loped in Hels in ki is sat i sfactory . The method i s suitable for the comparison
measurements of d i fferent cameras .
3 . 4 Symmetry of the MTF
The d i stort i on measurements show that the lens e l ements have moved in the
d ir ection A- C. The MTF measurement i n the PTB gave some % better values fo r
the radial 1 i nes i n the d i rection A and for tangent i a l li nes in the d i rec t ion B. The symmetry was stud i ed i n He l sink i by measur i ng the MTF for a ll
four rad ii A,B , C and D in the field angles of 27 . 5° and 37 . 6° . Differences
between diffe r ent rad ii were small (See figures 15 . and 16 . ) .
3 . 5 Influence of d ifferent factors on the MTF
3 . 5 .1 Defocusing
The MTF was measured i n the PTB wi th the defocus ing of± 0 . 05 and ± 0 .1 0 mm
fo r a ll 7 f i e l d angles . The results of angles 0° , 27 . 5° and 37 . 6° a r e in t he
figures 17 , 18 and 19 . Because of d i fferent f i eld curvatures the e ff ect of
the defocusing is changing much . The i nfluence of the field cu r vatu r e
15
appears as differences between the MTF's of the plane of best def ini tion and
MTF
of the reference plane.
3 . 5 . 2 Wave length of light
The MTF curves of 6 different wave lengths in the plane of best definition
are in the fig. 20. We see that the MTF of the blue is remarkably worse than
that of the other colors.
3.5.3 MTF's on the plane of best definition in different field angles
The results for the different field angles with the 1 ight of 585 nm on the
plane of best definition are in the figures 21. The d i fference between the
MTF of a diffraction l i mit ed system and ax i s of the 15Ag No.102 expressesthe
influence of the spherical aberration. The differences between the MTF's on
axis and in different field angles are caused mainly by the coma. The influence of the coma is much greater on the tangential MTF than the radial MTF.
4
Phase shift phenomenon
The phase shift phenomenon appears in the 15Ag No.102 lens both in racial and
tangent i al 1 ines. The phenomenon arises when the field curvature is great
enough. When the curvature is outwards, as with the va l ues of r = 40 - 80 mm
(14 . 6°- 27 . 5°) in the sagittal lines, the figures become grey after the RP
1 imit. When the curvature is inwards a strong phase shift which develops
a spurious resolution is in the corner of the image. The outwards-d i rected
field curvature by tangential 1 ines results in a rather strong ones i ded image . The harmonics with d if ferent spatial frequences move clear l y by different amounts of defocusing from the original place .
The fig. 12. shows that the spurious resolution can be generated in both i ntrafocal and extrafocal region, in the extrafocal reg i on by very high fre quencies. A rather strong region
of normal image formation has been generated in a wide spat i al frequency region with a defocusing of 0.3 mm outwards in the case of this camera . It was
ver i fied with the MTF measurement that the modulat i on value there was 12 %.
The psu do-MTF inside the focus was 16 % in the maximum. The visual observ ation showed that the spurious resolution is generated by a very 1 itt 1e
amount of defocusing after the grey region, which is between the normal and
spur i ous resolution. The phase sh i ft causes also zones of illegal image fo rmation . The above-mentioned pseudo -phenomena occur only rarely in the aer ial negative. Instead they are a remarkable drawback in the geometrical laboratory calibration.
5
Stab ilit y of the 15Ag No .1 02 lens
First the lens was kept one month on the optical bench in the hozontal pos ition s.othat the diagonal A- C was horizontally . After this it was turned
180°. In this position it remained 2 months, after wh i ch i t was removed onto
an other opt i cal bench where i t stayed 1.5 months in the same posit ion. One
half year thereafter i t was measured in Helsinki. In the measurement it was
ver i f i ed that the decenter i ng component of the rad i a l d i stort i on and the
16
tangential d i stortion were the same as earlier with an accuracy of~ 1 ~m .
The calibrated focal length had changed only 6 ~m because of the dismounting of the camera in the PTB . So the stability of the 15Ag No . 102 lens seems
to be excellent.
Conclusion
The field curvature mainly gives the form for the RP curve. Some tenth parts
of mm of field curvature are needed to generate the spurious resolution. It
can be inside or outside the focus. The MTF method developed in Helsinki
seems to be accurate enough for comparison measurements of cameras. The RC-8
15Ag camera can also be kept in the horizontal posit i on for long times without a danger that the lens elements move.
References
Flugge,J.:
Das photographische Objektiv
Wien 1955
Hakkarainen,J.:
Image Evaluation of Reseau Cameras
Photogrammetria, 33 (1977) pp.115-132
Rosenbruch,K.-J.:
Considerations Regarding Image Geometry and Quality
Photogrammetria, 33 (1977) pp.155-169
17
h{-J
12
s[- J
Fi g. 1 . Mea s ur ement of the
foca 1 1en gt h.
Fi g . 2 . Sphe ri cal aber r at i on .
s'
[nom]
Fi g . 3. Coma and sphe ri cal aberrat .i on at f i eld a ngles 15 0 , 25 0 , 37 . 50 and 42 . 5.0
a) Ri ght - Left
b ) Up -
c
Down
A
I
Ll.Z t:.
II
[mmJ
0 .(,
0.'1
--...
"tS" 'tO'f
I
I
I
I
3()'
.zo•
.......
.._
to•
0 . 2.
/
/
/
lO"
10.
-O.l.
\
-M
\
- 0.6
I
• t e.s t fi~ure
I
In
9 o niometer
Fi g . 4. Fi e l d curva t u r e .
1d
l(s<'
"'"\
\
\
\
-
[Prn]
,1r
0
·~
·
'
''
D
-·-·
-·-·
50 \
\\
\
\ ··. ..
C
""
r [mmJ
\\. .
\
\
"
0
" ~'~.
\
"
\
- 12
\... ...
"
\
\
- 14
\
""
\
""'
"""'
s.
"
...-
.........
".,
'
._
4
Fig .
""'\
"""'
\
\
0
...._
.--
~
'\.·-·
\
"'
'
·
\
-to
"
0
2
0
MAX =_..7 ~m
Fi g. 6. Tangent i al d i stort ion .
Radial distortion ,
or i g i n at PPA .
De f ocu.sm g {mm}
ll
(llr)
[I'm]
'"
...
- 1.0
Anglt!
r [mm)
\\
-o 5
0
+ OS
... 1.0
o·
\
\
15"
.!>
20.
' ./
25"
i
.A [nm]
30"
Fi g . 7. Late ra l chromat i c
aber ration.
..
35"
'-irJ
45"
Fi g . 8 . Images of a br i ght po i nt ,
s i ze 0. 08 mm on the ax i s .
19
t.
J;
"
I
'
-
Dir•dion oulw-c.rd:s
~1
')
'.
-5.6
8
II
---- 16
I rv.o.Ps .•,
\~i
~
of(l
Fig.
60
eo
100
118
132
'I
I
m
I
I
I
I
I
OAD.PS.RP]!
I
I
I
"'D.PS.OPI
1-41
r
I
I
[mm]
9. Resolving power.
Fig. 10. Radial RP with apertures
5.6, 8, 11 and 16.
,
RP
~z
[mm]
t1Tf,"f%
HTF"-'20~
O.'i
0.:)
O.l
0.1
Lincs/mm
0
-o.r
-0.1
-<l.~
-o.'l
PSI
~ re>j zone
-<).~
Ot.;cc.t
PSI. PSI
Fig. 12. Influence of defocusing on the axis.
20
MTF
MTF
MTF
1.
Tangential
~
~
~
\
~
~
\
::,..
''
........
100
MTF
cycles / mm
1.0
~
'
~
......
100
MTF
100
cycles / mmMTF
c y c les /mm
1.0
1.0
'
I
Rad ial
\
\
\
\
\
\
''
\
' -......
\
10 0
100
146°
cycles j mm
MTF
275°
MTF
c y c les;mm
3 3.1°
MTF
1.
1.
10 0
MTF
100
cycles; mm
100
100
c ycles/mm MTF
cycles / mm MTF
cycles / mm
''
...... ._ -
100
100
100
cycl e sjmm
cycles; mm
4 2.6
°
cycles; mm
Fi g . 14 . Compar i son be tween t he MTF va l ues ach i eved i n Bra unsc hwe i g
a nd He l s i nki
Braunsc hwe i g
21
----
He l s i nk i
i MTF
MTF
·~·
= JJ
~
"'
''
'.·· '·
~
D
.. \
'
·.
''
···.. ~
·.. ~
'
····"'·\ '
· .~
100
MTF
''
.....
cycles/mm
1.0
= II
C
100
MTF
cycles / mm
'
'
MTF
100
MTF
cycles / mm
1.0
de t erm in a t i ons
100
100
cyclesjmm
275
0
100
cyclesjmm
3 7. 6
Q
cycles; mm
Fig. 14 . (continued) .
Fi g . 16 . Symme tr y .
Fig . 15 . Symme t ry .
MTF
MTF
MTF
1.0
1.
- ·-
focus
0
- - - +0.05,0.10
- 0.05 , 0.10
MTF
1.0
MTF
100
cycles / mm
100
cyclesjmm
Fi g . 17 . Ef f ec t of
defocus in g .
0
0
0 .0
14 .6 ---- - 37.6
2 75
4 0.7
33.1 42.6
100
cyclesjmm
1.
100
1
1
3 76 °
cycles/mm
Fig . 18. Effe c t of
de f oc us ing.
22
MTF
1.0
100
cycles / mm
100
cyclesjmm
Fi g . 21. MTF 1 s of dif fe ren t fi e l d an g le s
on t he p l a ne of
best de fi n i t i on .
MTF
MTF
1.
nm
---------
S8S
654
.541
491
701
438
100
100
cyclesjmm
cycles/mm
Fig. 19. Effect of defocusing.
RP
Fig. 20. MTF's with different wave
lengths on the plane of
best definition.
548nm
{L/,m
Dilf
RP'
300
limif
.o. z'
HTF~ 4%
clear RP
HTI=~20%
"'"'
0.1
2.00
0.1
0 r-----~---T----------~~------~~~
100 nm
/00
I
-0.1
II
'I
:su
,a,..
SIO •onlorncriar
S.b
B
It
N
16
Fig. 13. Plane of best definition
with different wave lengths.
Fig. 11. RP on the axis with
different apertures.
cycles/mm
T
R
T
R
T
T
T
R
T
R T
R
10
2
-4 . -6
20
2
6
5
1
30
9
40
-3 3
-3
9
so
60
70
80
-1
-1
90
-3
-2
0
Table 1. Differencies Helsinki-Braunschweig
in the MTF measurement in %.
23