XIV . K()NGRESS
DE~
INTERNATIONALEN GESELLSCHAFT
FUR PHOTOGRAMMETRIE
HAMBURG 193o
Kommission :
III
Arbeitsgruppe
1
Presented Paper
Dr . GUnther Kupfer
&
Dr . Landolf Mauelshagen
Mitteilung
aus dem Institut ftir Photograrnrnetrie
der Unsiversitat Bonn
Bundesrepublik Deutschland
CORRELATIONS AND STANDARD ERRORS IN BUNDLE BLOCK ADJUSTMENT
WITH SOME EMPHASIS ON ADDITIONAL PARAMETERS
Abstract :
The bundle adjustment program BOBUE provides the possibility
for computation of the whole standardized inverse of the
normal equations or of standard errors only . With this software blocks of synthetic data with generated randomly
distributed errors were investigated. Standard errors and
correlation coefficients resulting from bundle solutions are
presented numerically and grap hically. An attempt is made
to deal with the influence of point density per photo and
variation of parameterization . Some t y pical findings from
real block adjustment are also given .'.
KORRELATIONEN UND MITTLERE FEHLER IN DER BUNDEL- BLOCKAUS GLEICHUNG UNTER TEILWEISER BERDCKSICHTIGUNG ZUSATZLICHER
PARAMETER
zusammenfassung :
Das Btindelausgleichungsprogramm BOBUE gibt die Moglichkeit ,
wahlweise die gesamte standardisierte Inverse der Normal gleichungen oder nur die mittleren Fehler der Unbekannten zu
berechnen . Mit diesem Programm werden Blocks aus synthetischen
Daten mit zufallsverteilten Fehlern untersucht . Mittlere
Fehler und Korrelationskoeffizienten aus der Blindellosung
werden numerisch und graphisch dargestellt . Es wird versucht ,
den EinfluB der Punktdichte pro Bild und der Variation der
Parametenvahl zu behandeln . AuBerdem werden einige t yp ische
Ergebnisse aus tatsachlichen Blockausgleichungen mitgeteilt .
398.
CORRELATION ET ERREURS DANS LA CO~PENSATION EN BLOCS
DE FAISCEAUX DANS LE CAS CLASSIQUE ET APR~S INTRODUCTION
DE PARAM~TRES SUPLL~MENTAIRES
Resume :
Le programme de compensation en blocs de faisceaux BOBUE
permet de choisir alternativment entre le calcul d ' une
inversion norme d ' equations normales , et le calcul des
erreurs moyennes des inconnues . Dans notre cas on a employe
ce programme pour l ' analyse des blocs se composant des donnees
synthethiques auxquelles sont supe~posees des erreurs
alleatoires . Les erreurs moy ennes et les valeurs des corre lations sont listees et representes graphiquement . On a
etudie l'influence de la densite des POints oar imaqe et
du choix des parametres utilises . En plus on ~ presente
divers resultas typiques des compensations en blocs effectues
avec donnees reelles .
1)INTRODUCTION
The first systematical investigations in accuracy after bundle
adjustment were published b y KUNJI /4/ and TALTS / 1o/ twelve
years ago . Their concern mainly were the standard errors of
coordinates of tie points , consequently ski~ping those of
orientation parameters and as a matter of fact all covariances .
Physically induced cov ariances of image coordinates a posteriori have been dealt with to some extent by ELLENBECK /2/
and KUPFER /5//6//7/ , whereas ~UELSHAGEN /8//9/ investigated
algebraica l
corr~lations , using the bundle adjustment prog ram
BOBUE . It was ortginally written b y him and is capable of computing standard errors of unknowns as well as the whole standardized inverse of normal e q uations .
As there is a lack of insight into the behaviors of correlations wit0 ~ ~ blocks of different s i zes and control distribu t i on , this shall be the main object of this paper .
2)STANDA~D
ERRORS AND ALGEBRAICAL CORRELATIONS IN SYNTHETIC
BLOCKS .
2 .1
PATTERN OF BLOCKS COMPUTED
To avoid long CPU times , computations have been restricted
to comparatively small blocks . Their oattern is defined in
Table 1 . All imag e coordinates and control data have been
computed synthetical l y with random errors attributed to image coordinates with a standard error of + 5 ~m at an image
scale of 1 : 1o , ooo .
Control point coordinates are assumed to be error free . The
blocks are without any height differences .
399.
Table 1
Synthetic Block Patterns
Strips
Photos
per Strip
3
4
Full
7
9
Control
Elevation
4
12
12
4
4
21
16
9
Side
Lap
2o %
single
or
crosswise
5
7
7
9
4
12
12
4
4
21
16
9
6o %
Besides this the whole investigation - partly under performance
shall cover the following variations (the now available standard
version always given first)
1. Parallel and crosswise strips with 6o% end lap and 2o%
side lap.
2 . Nine and twentyfive regularily distributed points per
fully covered photo .
3 . Bundle adjustment without and with use of additional para~eters. Additional parameters will however in this paper
only be dealt with in context with practically performed
flights .
4. WA-, NA- and SWA-photography shall be simulated . In this
paper only the first camera type is dealt with.
2.2 STANDARD ERRORS
As the simulated flight patterns as well as point and control
distributions are fully symmetrical , resulting standard errors
need only be given for one quarter of the whole block. They shall
be presented in image scale Mb and referring to a standard error
of unit weight m0 = 1 ~m, i.e.
m
m
X
X
m
0
According to Table 1 one gets for one block size six sets of
normalized standard errors , which will be presented for the
l:I:OO.
parameters of outer orientation and coordinates of object points
separately. It seems however prohibitive to give all data of all
variations of procedures in a paper like this. A proper documentation will be given elsewhere. Thus reference is made only to
blocks of 4 x 9 strips and flight patterns for this block size.
The smaller version is only verbally dealt within comparison to
the ones exposed. So is the procedure with others then the standard version.
STANDARD ERRORS OF OUTER ORIENTATION
It goes
with
without saying, that accuracies of these elements grow
- increasing number of well distributed control;
- increasing number of intersections of rays per
object point, i.e. increasing side lap etc.
- increasing number of points per photo;
alteration of camera type and block size excluded.
Fig.1 thru 3 show the amounts of standard errors with some explanations concerning Fig.1. The uppermost stations are at the
ends of the strips, thus standard errors are at a maximum. As
the pattern of elevation control is invariant for all flight
patterns, there is no large variation of the mz . Crosswise flight pattern gives overall most homogeneous resBlts .
As may be expected, the smaller block size (i.e. 3 x 7 photos)
gives numerically better results, at least partly due to the
relatively higher density of control.
STANDARD ERRORS OF OBJECT POINTS
The general statements of the foregoing para hold also true in
this context. As far as comparable there is good agreement with
the results given by KUNJI /3/ and TALTS /9/. Besides that the
homogenizing effect of dense peripheral control is obvious, when
comparing Fig.4 thru 6. This is most clearly seen for crosswise
flight. Also here standard errors decrease with decrease of block size.
Point densification from 9 to 25 per
to an increase in accuracy of better
ct of roughly doubling flight lines
in accuracy (Fig.4 against Fig.5 and
fully covered photo leads
than ten percent. The effeleads to considerable gains
6).
2 . 3 ALGEBRAICAL CORRELATIONS
All values are given in per cent as integers .
ELEMENTS OF OUTER ORIENTATION
Fig.7 thru 9 show for different flying patterns and control
distributions at the chosen block size the notorious correlations between Y and w as well as those between X and ¢ over
0
flat terrain. All correlations are quite high . TRere is however
a marked difference between lower values in case of sparse horizontal control and higher amounts with dense control at the
block perimeter. Changes of flight patterns do not result in
considerable changes of correlations.
40:1..
4. 1
4. 3
2.3
1.6
1.2
0.7
I
3.9
4. 9
2. 2
1. 6
1 .1
o.7
3.5
1.0
0.7
0. 4
I
j
2. 8
1 - - - --
0.8
2. 8
1. 1
--- - - - --- -- · ·-
o.8
!
2.
----1-----
o.4
1
3 1 2 9 1 2
-· ....,.- -o-·.· 9_- - -· o ·. 4--
l---
___2 .
2
o.7
- t --
2
~
-o
Y -.·--7~ - --o~ .· -7~- _oo___-__3~ A
~- ~~_o_J :_- ~
o.7
- - -·-- -- ·---
o.4
--- y - -(m =H~ SJ-lm)
_2 .
o.6
s
o.6
o.8
o.3
.L
0
-
/
I
2. 8
o.8
~ ~-~--9--~ -~-~ .-~-~:- :~ : --6--1
Controll : 4 Full, 21 Elevation
8
1.2
o.4
.6
3.3
1.3
3.4
1. 1
1.6
_12 __~ · 6
o.5
o. 5
o.2
4. o.5
o.5
o.2
2. o
2. 4
o.3
2.1
1. 9
o.8
1
3.2
1. 3
3.1
1.o
1.1
o.4
4r-
~ -~ ~- --·-1·. ? -~: ~ _____i__
1
__ __l __
.~ : 8.. o:s- -c; :3-
0 .
7
0 .
6
'
I
-Q
0: 2·-- ,-
t-::
(m- =-=+4. 9
I
-~:+ ~: : -~:: - ~<+ . :::--1
2.o
o.7
2.4
o.8
0
-
)Jill)
~-6- 2 ~- -= - - . - ~--- ~ ___o_. s__ A
o_:
o.3
o.7
o.6
0 2
1
'j
Control : 16 Full, 9 Elevation
Fig . 1
Normalized Standard Errors , Outer Orientation .
4 Strips of 9 Photos, 9 Points ner full y covered
~hoto,
X , Y ,
0
0
z0
are given in
)Jill
in Photoscale ,
<t,, w, K are given in mgon , arr angedl in this
order at the place of the exnosure station .
402.
J . L.
3. 3
1. 6
3.2
3.5
1. 6
1. 3
1.0
o.5
1.2
o.9
0.5
2.1
2. 1
0.8
1. 8
2. 4
0.8
0.6
o.6
o.3
0.6
0.6
0.3
2.4
2. 1
o.8
2.0
2.1
o.7
o.7
o.7
0.3
o.6
o.6
0.3
2. 4
1. 8
0.8
1. 9
1. 9
0.6
o.6
0.6
o.3
o.5
o.5
0.2
2.5
2. 1
0.9
2.0
1. 9
o.7
o.3
0.6
o.6
0.2
------
- ----
0.6
0 .7
(m =+ 5 . 1 ]Jill)
0 -
r
Control : 4 Full , 21 Elevation
2. 9
2.8
1. 2
2.8
2.6
1. 2
1. 2
o.9
o.4
1. 1
0.8
o.4
1. 5
o.5
1. 5
1. 6
0.5
o.5
o.5
0.2
o.5
o.5
0.2
1. 7
1. 9
o.7
1. 8
1. 7
0.6
1•5
------·--
------------
0.6
0.6
0.2
o.6
0.6
o . 2 (m =+ 5 . 1 ]Jill)
0 -
1. 6
1. 5
0.5
1. 6
1. 6
0.6
0.5
o.5
0.2
o.5
o.5
0.2
1. 7
1. 9
o.7
1.8
1.7
0.6
0.6
o.6
0.2
0.6
0.6
0.2
Control : 16 Full , 9 Elevation
Fig . 2
Normalized Standard Errors , Outer Orientation
2 x 4 Strips
(Crosswise)
of 9 Photos , 9 Points/
Photo
For details c . £ . Fig . 1
There are only 2 strips disnlayed , as there
exists full symmetry .
£4:03.
1:\ 35
.
38
.
1 . 8138
1. 4
1. 1
o. 5 I 1. 5
.
I
.
.
35
1.0
....
.
2.o l 3 3
35
I 1. 3
I
0.9
0.5
2. 4
o.9
2. 2
2.6
o.7
o.7
o.3 I o.7
0.7
0.9 I 1.9
o.3 I 0.6
2.6
2.6
1.o I 2. 2
2.3(~
o.8 1 2.1
2.1
o.7
o.8
o.4 I o.6
o.7
o. 3
I 0.6
0.6
I
,..,
I
I
2.2
1.6
2.0
o.9
0.6
0.6
o.3 I 0.6
.\ 2 . 7
o.7
2. 5
1.o
1
2. 3
o.7
o.4
1
0.6
1
I
2. 4
0.6
o.8
1.9
1.9
0.
o.3 I 0.5
o.5
1
I
2 . 2/'~ o . 8 I 2 . 1
o.6
o.3 1 o.6
2.0
0.6
Control : 4 Full , 21 Elevation
1\
~
...
3o
•
32
.
1 2136
1. 2
1.0
o. 4 I 1. 4
1. 6
0.5
1. 6
o.5
.
.
I
0. 6 I 2.0
0.2 I 0.7
26
.
0 . 9.. ~
1.8
o.6
I
.....
1.9
2. 2
o . 8 I 1. 8
o.7
o.7
0.3 I 0.6
I
O. 6 1 1.7
1. 6
1•6
0.5
o.5
1. 8
2.2
o.2 I 0.6
I
o . 8 1 1. 8
o.7
o.7
o.3 I 0.6
1
1 . 9,....'\.
o . 6Y
.
0.3
2.2
0.8
0.6
0.3
/. . 1
0.7
0.6
o.2
0.7 I 2. 1
2.1
:'\
o.2 I 0.6
0.6
o.7
I
o.8 I 2. 1
0 . 3 I 0 . 6'
I
0 . 6 I 2. 0
o.2 I 0.6
I
0.6
.
2 6
.
1 8
0.9
0.4
1.8
o.7
1.6
o.6
····- -
1
- - - - --
1.9
---
.
o.2 I 0.6
o.6
0.2
1.7
o.7
o.6
o . 7 I 1 . 8/~ 1 . 8
o . 2 1 o . 6Y 0 . 6
1. 6
o.6
0.5
1.8
o.2 I 0.6
0.2 I o.6
I
o.7 I 1.6
o.2 I 0.5
o.7
1
I
(6o % side lap)
Photo
For details c . f . Fig . 1
o.2
1
1.7
1 •8
o.7
o.2
1
o.6
0.6
o.2
1 •7
o.7
1
1 •8
o.7
o.6
o.2
1
1 . ~ ....
o.6
o.6
o.2
( rn
I
0
= -+
5 . o JliTI)
Normalized Standard Errors , Outer Orientation
7 Strips
.
o.5
I
1 . 9 .\.
0.6
5 . o JliTI)
0.9
I
o.2
= +
0.4 I 1.1
I
1.3
0
1 . 2135
o . 4 1 1 . 4.. j-o
I
0.5
o.7
.
27
0 .7 I 1.5
0.2 I 0.5
I
0 . 7 I 1.8
.0
0.9
.
18129
~
2
2.5
(rn
Control : 16 Full , 9 Elevation
Fig . 3
o.8 I 2. 2
0.3 I o.7
I
2. 1
f)
.
I
2.3
1
.
.
1 7137
3 5
1 • 4 , .,..,
o.5 I
0.9
of 9 Photos , 9 Points/
XXX
XXX
XXX
1.(
1.7
2.7
1. .S
2. 1
2.5
2. 1
2. 2
2.2
2.4
2 . 1)
2.9
XXX
2.4
2.8
2.7
XXX
1. 5
1.6
1. 8
1.7
1.9
1.0
1.8
2.2
1.6
2.0
2.2
1.0
1.7
1.8
XXX
1.7
2.0
1.4
1.7
1. 9
XXX
1. 9
1.8
1.4
XXX
2.4
2.9
2.3
2.1
2.0
1.7
1.7
1.9
0.9
1.7
1.9
1.5
1.7
1.8
0.9
2. 6
2.7
2.1
2.1
1. 4
1.7
1.9
1.7
1. 9
1.4
1. 6
1. 8
XXX
XXX
4 Full ,
Control
XXX
XXX
XXX
1. 2
1.0
2.7
XXX
XXX
XXX
0.9
1.5
2. 4
0.8
0.8
1. 8
o.6
o.7
1.0
0.8
o.8
1.6
o.7
0.6
1.0
0.7
0.8
1.4
o.6
0.7
o.S
o.8
1. 4
o.7
o.7
XXX
XXX
(m
0
o.S
1. 4
2.2
o.8
o.S
1.7
o.7
o.7
0.9 ,
o.8
o.8
1. 5
o.7
o.7
0.9
XXX
XXX
XXX
o.7
o.S
1. 4
o.7
0.7
o.S
o.8
1.4
0.7
o.7
XXX
=+ 4 . 8 l'm)
-
21 Elevation
1. 2
1.0
2.7
I
0
XXX
XXX
XXX
XXX
XXX
XXX
XXX
(m
= + 4.9
XXX
Control : 16 Full, 9 Elevation
Fig . 4
Normalized Standard Errors of Object Point
Coordinates
4 Strips of 9 Photos , 9 Points per fully
covered photo
mX , my , mZ aiven in ]..lm in photo scale
J
XXX depicts controll at the
coordinate concerned .
405.
~lace
and
wm)
XXX
XXX
XXX
1.2
1.0
1.6
1.4
1.4
XXX
1.7
1.6
1. 5
1.7
1. 7
XXX
1.0
1. 2
1. 6
1.0
1.0
1.1
1.1
1. 2
o.8
1.2
1. 3
1.0
1.3
1.3
o.8
1. 4
1. 4
XXX
1.2
1. 1
o.B
1.1
1.1
XXX
1. 1
1. 2
o .7
1.1
1.1
XXX
1.6
1. 7
1. 5
1. 3
1. 2
1.0
1. 2
1. 1
o.7
1.1
1.1
1.0
1.1
1. 1
o.7
1.7
1. 7
XXX
1.3
1.3
o.8
1.1
1.1
XXX
1. 1
1.1
o.7
1.1
1.1
XXX
.
(m
= + 5 . 1 wm)
(m
= + 5 . 1 wm)
0
Control : 4 Full , 21 Elevation
XXX
XXX
XXX
0.9
0.6
1.6
XXX
XXX
XXX
0.9
o.6
1.5
XXX
XXX
XXX
0.6
o .9
1. 6
XXX
XXX
XXX
o .5
0.5
1.1
o.5
o.5
0.8
o.5
o.5
1.0
o.5
o.5
0.8
o.5
o.5
0.8
o.S
o.5
XXX
o.5
o.5
0.7
o.5
o.5
XXX
0.6
o.9
1. 5
o.5
o.5
1.0
o.5
o.5
0.7
o.5
o.5
1.0
o.5
o.5
o.7
XXX
XXX
XXX
o.5
o.5
0.8
o.5
o.5
XXX
o.5
o.5
o.7
o.5
o.5
XXX
0
Control : 16 Ful l, 9 Elevation
Fig . 5
Normalized Standard Errors of Object Point
Coord inates 2 x 4 Strips (Crosswise) of 9 Photos
9 Points/Photo
For details c . f . Fig . 4
406.
XXX
XXX
XXX
1.3
1. 4
1. 8
1.5
1.7
XXX
1. 8
1.7
1. 4
1.8
1.8
XXX
1.3
1. 8
2.3
1.2
1.3
1.3
1. 3
1. 3
0.9
1.3
1. 4
o.9
1.4
1.4
0.8
1.7
1.7
XXX
1.3
1.4
1.0
1.2
1. 3
XXX
1.3
1.3
o.8
1.3
1.3
XXX
1. 8
2 .4
2.1
1.5
1.5
1.2
1.3
1.3
o.8
1.2
1. 3
0.8
1.2
1.2
o.8
1. 9
2.0
XXX
1.5
1.4
1.0
1.3
1.3
XXX
1.2
1.2
o.8
1. 2
1.2
XXX
(rn 0
= + 5 , o 11m)
Control : 4 Full , 2 1 Elevation
XXX
XXX
XXX
0.9
o.9
1. 8
XXX
XXX
XXX
o.7
o .7
1.4
XXX
XXX
XXX
0.8
1. 4
2.2
0.6
o .7
1.3
o.5
0.5
0.9
0.5
o.5
0.9
0.5
o.5
0.8
XXX
XXX
XXX
0.6
0 .7
1.0
o.5
0.5
XXX
o.5
0.5
o.8
o.5
o.5
XXX
0.8
1.4
2.0
o.6
0.7
1. 2
0.5
o.6
o.8
o.5
0.5
o.8
o.5
o.5
o.8
XXX
XXX
XXX
o .7
o.6
1.0
o.5
0.6
XXX
0.5
0.5
0.8
0.5
0.5
XXX
(rn0
= + 5 , o 11m)
Control : 1 6 Fu ll, 9 E l evation
Fig. :
6
Normalized Standard Errors of Object Point
Coordinates
7 Strips (6o % side lap) of 9 Photos , 9 Points/Photo
For detai l s c . f . Fig . 4
£!07.
95
871
95 T 8 4
85
84
87
63
79-r77 -+ -:: -t - ~:69--0---71- 6- 81---6--78--~
57
I
69
1
i
X -¢- Correlations
0
0
96
4 Full , 2 1 Elevation
96
I
94
-97-1---95-+
Y -w-Correlations
0
Control
Fi g. 7
98
98
95
95
. ·-·-9s -
97- -t - 9 5
95
I
I
Y -w-Correlations
Control
!
96
95
94
-95
95
X
0
-~ -Cor~elations
16 Full , 9 Elevation
4 Strins With 9 Photos ,
9 Points/Photo
6o % end lap , 2o % side lap
l!OB.
96
9o
- 95
- 88
89
86
- 89
- 75
86
84
- 87
- 82
75
81
- 86
- 81
So
82
- 85
- 84
Y - w- Correlations
X - ¢-Correlations
0
0
Control : 4 Full , 2 1 Elevation
Y 0
w-Correlations
- 97
· - - 98 - ·
- 96
- 96
- 96
- 96
- 96
- 95
- 96
- 96
X - ¢ - Correlations
0
Control : 1 6 Full , 9 Elevation
Fig . S
2 x 4 Strips (Crosswise) With 9 Photos ,
9 Points/Photo
6o % end l ap , 2o % side lap
As there is full symmetry in both flight
directions , only one direction of lines
is shown .
409.
9 6 - 9o ~ 88 - 87-
.
I
I
88
86
82
I
83
I
86-83-8o-8 1 -
I
71
8o
76
- 9 5 - - 92 - -8 7 - - 8 9 -
I
- 88
- 83
I
I
I
l
I
8o-8o-79-8 1 -
Y - w- Correlation
0
- 78
I
- 8 5 - - 8 2 - - 8 o - -8o - -
I
81
I
- 72
1
- 82
- 85
- 78
- 8o
I
I
1
I
- 84- - 83- - 82- - 82-
X - ¢ - Corre l ation
0
Contro l: 4 Fu l l , 2 1 E l evation
98-96-98 -
95-
I
I
I
97-96-96-96-
- 98 - - 98 -
I
1
96
96
96
1
96
96
95
96
I
98
I
I
1
I
· -97-96-95-96-
Y
0
- w- Corre l ation
- 96
- 97
- 98- - 95 - 96
I
I
- 96
I
- 9 5 - - 96- - 9 6 - - 9 6 -
I
- 95
- 95
- 95
I
I
- 98
I
I
- 95- - 96- - 96- - 98-
I
X -¢ - Correlation
0
Con t rol : 1 6 Full , 9 Elevation
Fi g . 9
7 Str ips With 9 Photos
(6o % end lap , 6o % side lap)
q:l.O .
There are significant correlations between Z and ~ at the
0
ends of the strip s, where on l y halves of the photos are cove red with points . The amounts are about 7o in case of dense co ntrol against 6o in case of sparse control .
OBJE CT COOR DI NA TE S
Cross correlations between ground coordinates , that are of any
consequence , do not occur after adjustment . Hence one can re strict onese l f to auto cor relations . These are displayed graphically for typical s i tuations of points within a given block
and flight pattern , c . f . Fig .1 o . Cor relations between different
reference points and the remaining points are shown by iso li nes with 1o % intervals . The heavier lines are those of 0 % and
So % respectively .
Largest auto correlations occur with reference points which
are at largest distances to control points . Thus sparse control leads to strong correlations . In the cases at hand with
dense elevation contro l, X- and Y- correlations are dominant .
Both coordinate directions (flight direction is in X- direction)
give only sma l l differences in results . For the block center
as reference po i nt , half the block size is roughly correlated
with larger amounts than 7o %, which is commonly thought to be
significant .
Sparse control given , larger blocks show this effect more dis tinctly .
Auto corre l ations collaps at once , if at least dense peripheral
horizontal control is at hand . Changes of flight patterns do
not alter correlation patterns significantly .
The change from 9 to 25 points per fully covered photo dimini shes the different amounts only s l ightly . This effect was howe ver only studied with the 3 x 7 block .
CROSS CORRE LA TI ONS BETWEEN OUTER ORI ENTAT I ON AND OBJECT
COO RDINATES
Strong correlations as explained in the forgoing chapters do
not occur i n this domaine . There are however typical patterns
which are observed in well def i ned areas of a block . Correlat i ons between Z at the last photo of an oute r strip and
Y- object - coo r dina~es are found as shown in Fig .1 1
There is a quite similar correlation between K and X- Objectcoordinates .
While these correlations occur typical l y only in connection
with photos at the perimeter of a block , there e x ist cross correlation s of a s i milar kind in the middle of a block be tween X - refe r ence- positions and X- object- coordinates . The
same apBlies to y and Y at this posit i on . F li ght patterns
with more strips ~t t he same b l ock size lead to a s l ight de c li ne of these typ i cal cor re l ations .
As was found in other contex t t hese cross cor relations dwin de l wi th densification of control . Thus a Z - Z- Cor r elation does
not come to any amounts because of the dens~ contr ol in e l e vation .
411.
21
31
34
37 - - - -
21
37
41
41
42
39
46
48
46
47
42
51
53
51
So
54
57
58
55
53
48
51
6o
6o
59
56
49
57
65
65
64
58
48
4o
68
74
69
58
43
85
7o
52
33---
6---[g]
Fig . 11
---6I
9-
I
I
I
r
----8
Cross Correlations between~
at Reference Position
0
And Y-Object-Coordinates
4 Strips of 9 Photos , 6o % End. Lap, 2o % Side Lap
9 Points/Photo
Control : 4 Full , 21 Elevation
Cross correlations are not exhaustively described so far .
The remaining ones are however of lesser consequence .
2 . 4 CONCLUSIONS
Dealing only with the algebraical aspect of &andarderrors and
correlations after bundle adjustment, one finds that sparse control gives rise to poor inhomogeneous standard errors as well as
to relatively high correlations . An adverse effect is only
discernible with cross correlations of elements of outer orientation . Hence dense perimeter control should be recommended as
a remedy to these shortcomings . To a certain extent flight patterns with 6o % side lap and - even better - crosswise flights
are able to procure homogeneous and fairly good results .
A carr · ~ ned effect will be reached by procuring both , good control distribution as well as a properly chosen flight pattern .
A further enhancement will be achieved by enlarging the number
of points per photo .
All findings hold true for normally distributed observations
which may however be ascertainedby removing systematic errors
by field or selfcalibration technics .
3) REAL BLOCK ADJUSTMENTS INCLUDING ADDITIONAL PARAMETERS
The test area of Rheinbach , as discribed in /3/ has been chaos en as an example for demonstration of practical circumstances .
The block was flown with a Zeiss RMK A 15/23 WA camera . Four
parallel strips of 1 1 photos each at a scale of ~pL1 : 1 1, ooo
from one block of cppr8~m x 1o Km . The version chosen in this
context is with sparse planimetric control at the corners ,
a densification of elevation control with 2 b distance at the
4:12.
perimeter and one additional elevation point at the center
of the block. Thus there is less control, at least in elevation, as was chosen for the synthetic block . On the other hand
the blocks are of similar size .
There exist three computational versions of bundle adjustment:
-one without additional parameters (version. 1),
- one with twelve additional parameters (version 2)
- one with a set out of that beforementioned one
(version 3)
The additional parameters do ( or should) from orthogonal sets:
X
y
2 2
(+ a22x y )
2
2 2
(+ b12 xy 2 )+ b21 X 2 y + b
X (+ b22x y )
20
= a o 1 y (+ a11 xy) + a12 xy 2 + a21
b
o1
y
(+ b11xy)
X
2
2
y (+ a2 y )
The expressions in brackets were removed after a T-te st , which
proved them to be not significant under any circumstances . As
there were about 11 to 16 points per fully covered photo there
was theoretically a good reason for choosing the whole parameterset . Adverse point distributions led to the effect mentioned .
The weight zero for all additional parameters may have added to
this end.
·
The following Table 2 shall give at least an insight into correlations which exist between elements of arter orientation themselves and object point coordinates on the other hand . Table 3
shall depict correlations between additional parameters and object point coordinates .
In this case object coordinates directions and flight line directions cut at an angle of appr . So gon . Thus coordinates X
0
and Y as well as angles w and ~ are not handled separately,
but m~an values for both of them will be shown in Table 2 .
2
It appeared that the term b
xy was not significant inver 12
sion 3 which it had been in version 2.
A comparison of versions 2 and 3 at Table 2 reveals the follo wing . The full set of additional parameters led to high autoand cross-correlations between elements of outer orientation .
These diminish after deletiqy the non significant terms . The
phenomenon is also to be observed with correlations between outer orientation elements and object point coordinates . These
effects are on the other hand restricted to X /Y and ~/ w
0 0
respectively and do not occur with z and K.
0
High correlations between elements of outer orientation and
additional parameters do not occur . Root mean square errors
at check points prove version 2 to be better than version 3,
which may however be quite accidental.
Table 3 at least reveals that high correlations between addi tional parameters and object coordinates are also in version 2
rather seldom .
The fact that some terms of version 2 are insignificant does
not seem to be a good criterium for omission or to avoid an
ill conditioned system . Orthogonal ity of terms (which was not
at all the case in version 2) and 3) of parameters is no
radical medicine . Restricted number of terms does not (or at
least seems not to) guarantee best results .
Standard errors of all l unknowns do not clearly alter their
amount from one version to an other .
Correlations between object point coordinates remain relatively stabel without changing from version to version .
These few remarks show to some extent that we may get good
results when applying additional parameters in field or self
calibration . There remain however some crucial lacks of insight into the criteria which have us on the safe side of
their application .
ACKNOWLEDGEMENTS
This paper would not be presented without programming and
computational work carried out by Miss Dipl .- Ing . Gertrud
Terbrack and Mr . Dipl .- Ing . Harald Schuh concerning synthe tical blocks .
All computations were carried out at the IBM 37o/168 of the
RHRZ Bonn .
5) REFERENCES
1)
Brown , D. C .
Evolution , application and potential
of the bundle method of photogrammetric
triangulation . Symposium Comm . III , ISP ,
Stuttgart 1974
2)
Ellenbeck K. H.:
Untersuchungen zum stochastischen Ver halten von Reseauaufnahmen . Presen ted paper , XIII Congress of Photogrammetry , Comm . III , Helsinki 1976
3) Ellenbeck H.
and G. Kupfer
Bundle adjustment with field and self
calibration - recent results .
Symposium Comm . III , ISP , Moscow 1978
4) Kunji B .
The accuracy of spatially adjusted
blocks . IAP XVII , Part 8 , Lausanne 1968
5) Kupfer G.
Zur Geometrie des Luftbildes , DGK ,
Reihe C, Heft 17o , Mtinchen 1971
6)
II
Improvement of the geometry of aerial
photos . Phot . Eng . 1972 PE·479 - 486
7)
II
8) Mauelshagen L .
Image Geometry as obtained from Rheidt
Test Area photography . OEEPE Official
Publication No . 8 , pp 255 - 281
Frabkfurt 1973
Teilkalibrierung eines photogrammetri schen Systems unter Verwendung von Test feldern . XIII . Congress of Photogramme try , Comm . III ., Helsinki 1976 15 pp .
9)
Mauelshagen L .
1o) Talts J .
Teilkalibrierung eines photograrnrne trischen Systems mit variabler PaB punktanordnung und unterschiedlichen
deterministischen Ansatzen . DGK ,
Reihe C , Heft 236, Mlinchen 1977
On the theoretical accuracy of rige rous block adjustment in plan i metry
and elevation . I A P XVII , Part 8 ,
Lausanne 1968
415.
Table 2
Correlation of Elements of Outer Orientation Between Themselves
and Between These Elements and Object Point Coordinates
There are the numbers of Correlation given in per cent which
surpass a certain amount
Un knowns
xo
y
I
0
Correlation Exceed the M20nitude Given Below For
E l ements of Outer Or ientation
Object Point Coordinates
9o
8o
7o
6o
So
9o
8o
7o
6o
So
6
26
98
64
17
-
2
7
28
36
2
2
-
6
4
-
8
12
11
16
3
27
88
78
2o
-
-
2
18
43
-
-
-
-
4
-
-
-
-
s
2
-
2
2
11
-
-
-
6
11
2
2
-
6
8
-
8
12
11
17
-
-
3
3
14
-
-
4
6
1o
-
-
-
-
2
-
-
-
-
-
N
zo
~
0
·rl
(I)
~
J-l
m
<P
(JJ
I
H
(J)
>
K
X
I
0
y
0
zo
(")
~
0
·rl
<P
I
(I)
H
(J)
>
Whole Number
of Unknowns
K
(JJ
26 4
- - - -- -
·
- - -- --- -
- --
The given va l ues a r e related to a photo , which is situated
o f the block , th us avoiding peripheral ef f ects .
4 3S
-
-
--
-
with i n t h e cente r
Term
Version 2
8
Version 3
Pl
tJ
1--'
9o
~0 1
y
?.1 1 X V
·-
2
@.1 2 XY
2
~2 1 X Y
a
y2
Corr elations
So
7o
6o
F.xceP~
So
-
-
-
39
42
-
2
27
4o
43
-
-
-
-
-
-
-
-
(])
6o
So
-
-
-
32
24
-
-
-
-
-
-
-
-
-
-
-
-
w
..-..PJO
0 ~ 0
• p, 11
H1
11
• 0 (])
tJ I--'
8 LJ. Pl
Pl (]) rt
tJ 0 I-'·
1--' rtO
(])
1
16
2S
21
2 2
9.22 X y
-
-
-
-
-
y
-
~I-' ·
~
bo 1
b 11 XY
b 1 2 XY
2
2
b2 1 x Y
x2
b 2o
2 2
b22 X y
-
-
-
6
ttl
(])
rt~
oro
0 (])
0 ~
11
P,::t:"
~
1-l
..J
~
rorn
NO
-
2
T.he Maani t ude of
9o
So
7o
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I-'· p,
~
I
p,
Pl I-'·
rt rt
(]) 1--'-
rn o
~
Pl
1--'
ro
Pl
11
-
-
-
-
-
~
-
-
-
-
-
rt
(])
(])
11
rn
'
' '.
""'~"'"~
·"""
'·"
t
I
"
I
/1
'
.__ _. _ L .~ ~ - 4.
CJ F-RELRC'0\5
CORRELRTIONS
1!1
REFERENCE:
l't 'FEREWf PO!"<T
(!)
POINT
b)
a)
I
/1
--~----~
j
-----~-4'~::~ i
-X
X-X - CORRELRTIONS
0
C!J
CORRtLRTiONS
REFERENCE
P:Jl~T
REFERENCE P:JJNT
d)
c)
Fig . 1 o
X-X
(!)
CQRRELR'ION5
~
Full Control
0
Elevation Control
X
Exposure Station
•
Object Point
REFERENCE PQJNT
e)
q:J..B.
Y- 1
CORRE LA TION S
llEF ERE~CE
(!)
Y- Y
PO j NT
(!)
CORREL R T! O~ S
REf lRf"CE
PO:~
I
f)
I
I
~
•(
\
\
\.
~·
." .
o...:>-~'"--"'-----"-..::;,.. __
Y- Y
(!)
.
g)
t
-~ 1
/ ~
.. ..
\v-
.'
~'< .//.11
__ _. _ -
/_~.l
--
- -==----~·-
Y- Y -
C ORR[LATIOr~ s
REF E~E NC E
CaR~EcRTION S
PO:"~
h)
i)
Fig. 1 o
4•5
'
~~ CTCS
.,. _. I / --
~•6 0 ~
C=20 ·.
~/
Cont inued
!
.i
, ·~
I:
~- - ~--~1
j)
'-!:I. 9.
~ - X
(!)
X-X - CORRELRTIONS
CORRELR~I ONS
REFERE NCE POINT
(!)
REFERENCE PO !NT
k)
Y- Y -
X- X - CORRELATIONS
(!)
REFERENCE
1)
POINT
~E
(!)
CORRELRTION~
FfRfNCf PO I NT
n)
m)
7•9 PHO TOS P=60% 0=60%
\
\
.
~- X
(!)
·"""'--,
\
\
.
Y-Y - CORRf Li1 T; ONS
CORRE LRTI OhS
(!)
REHRENCE POINT
REFERENCE P;J;NT
o)
p)
Fig.1o continued
420.