Experiences with self-calibrating bundle adjustment - IDB

ACSII-ASP CCNVENTION
I^IASHINGTON, D,C,
FEBRUARY/l4ARcH 1978
PRESEIITED PAPER
EXPERlENCES 1.1ITH SELF-CALIBRATING BUNDLE ADJUST|,IENT
ARMIN GRÜN
INsT I TUTE oF PF]oToGRAMNETRY
TEcHr'.u cAL UNTvERSITy oF iluNlcH
ARC I SSTRASSE 21
D-8000 f]üNcHEN 2
FRG

EXPtRIiNCIS IIITN S i L F - C A L I B R A T I I C B UJIDL t A!JUSTI'1INT
1. Introduction
As a result of the coapensation of systematic ima!e errors mcdern photo!rammel.ic
methods of point deternination have achieved an accuracy level, ,rhich Fakes then
competjtive |,/ith traditjonal g€odetic irethods even ir the fleld of densificaticn
of control netHorks and cadastral survevlng.
Hovever, a successful application of photo_arammetric nethods to hiqh precision
point determination requires a conprehensjve and efficient conrpensation of the
systematic image errors. D!ring the last feI.l years various technics of treatment
of systematic errors have been p.oposed. Thereby !he slfi!ltaneo!s self-calibration
techniq!e by using additional parameters ltsauer /3,/, prcüa /3/, Ebret /t/,
clin ,/13/) becdne the ncst promising procedure, rdther than the ä-priori
field correction lrtpfe! /16/) and the conpensarion by flight arrangenent
(rhcücs /23/).
iire advanta!es of s e I f - c a I i b r a t 1 o n are quite evidentl
ihe extension of the mathenatical mod€l !f bundle adjustment allows for the
c0fnpensation of the actunl systematic errors, whjch may vary f.orTt project to
pr0Ject; tnere is no ddlitional efforr necessary for es!abljshing and ftyin! a
test field,
for supplementnry ima!e point |rreasurefxents and cofipLrtatjons.
8y applying the methods of test field calibratlon and self-calibratior to a part
of the '0bersch!/dben' tes! block the superiority
rs derf0nstrated. iloreover, !re results of sorne other practical
projects are refered
and conpared ,{jth the theorelical
test
of s e I f - c a I i b r a t i o n technique
accurdcy e^pecta!ions of bundle adjustllokever,
by !sjn!
the s e I f - c a I i b r a t i o n techniq!e some prob I ens arise \ritn respect
to the choice and handling of the aiditional
parameters. lniending the compensation
of the rhole systematic errors requires d hignly general and flexjbte
parameter
nodel. This may increase the danger of overparametri:ation. So one has tc
take cdre of the stability of th€ system of equations (e.9. normal eq!ations) jn
eacn partrcular case because of changing geometrical condjtions jn Cjfferent Droiects
(control point distrjb!tion, fli!ht directr'on, image oveftap, image point
djstribution, shape of the terrain). Therefore the q!estion of a Droper choice of
tyDe and nu rber cf additiondt pardneters ls disc!ssed and suitabte statisti.dl
!ests are pf0posed to clreck the sjqnificaice
of the conputed Dararneter amounts,
.rurte r3cently piotograrnetric research Lrnderrakes !r--at efforts io stabirize the
reliability of protoJranretric systems. It is !/r'dety known thar grcss errors are
rery hdrdly to ietect sjnDte r€t1oCs after
"ith
th€ perforrßan.e of brnJle adjust_
nent. io get mostly objective decisrons for the rejection of observations re_
ference is nad€ !o B.rdrC. r retiability theory (i.iri. /1,/, /2/ t.
All ccmDLrtaiions presented Ii thjs paper !ere perforFed djth the bLrndle pro_cran
il80P ( 'Iu'rr'ch Cuidle 0rientation . jth Adcitional para,neters,), !lhich the authcr
developeij i, I9ll/Ja at the Iistitute of photo!rannetry, Technical Universr.ty of
llirni ch (ar:ir ./tr-l,).

2
In the follo\{ing chapter the rTrathematical model of this program is described
together !lith some remarks or gross error detectr0n.
2.
The b!ndle program li80P. FLrnctional and stochastic model.
The detection of gross errors
The linearized mathematical model of the s e I f - c a I i b r a t i n g bundl e progran
yields the follov/ing:
Z
+ A2 dx' +
I dx P
Ldz
tB r
]Z ;
0;
P8 (=r )
Pz
(1)
,2, ,P
Yectors of residuals of image coordinates
-h.r,-t .... - ...rrindtes
, äddi tional
dt, dz -
Correction yectors of point coordinates,
coordinates, j riage pdrameters, additional
control
poi nt
A1' 42, 43, A1
= C 1.r-r ' 1. rrp'.icierrS
vectors of corstants of image coorainares. aodi! oral
..n!.11 ^. in. . ,, -.rinates
B''Z''P
= L/r'eiqht matrices (diaqonal form) for ima!e coordinates
additional
pararneters, control point coordinates
= ldentity natrix
The matrix Pp allows to inirod!ce ä-priofi knawn accuracy standards of the control
point coordinates; the snfie dDplies uith resDect to the matrix PZ and the
ddditional parameters.
Under certain circunstances and with certain pararietef sets the treatment of
sJrtemetic errors as free !nKno\rns (PZ = 0) nay lead to renarkable deteriorations
of the conoition of the system of nornral equations. lherefore it is expedient
to introd!ce PZ+ 0, thus protectrng against överparanetrization.
(cltn /13/).
l,lith
Pp + 0 He obtain a weighted minimal fitting (l/lith respect to a 3-dimensional
sr'uildrity transfarmation) of the p h o t o q r a nr m e t r i c points anto the control pojnts
Collecting the matrices and veciors of (1) jn the foll
owi ng
At Az A: A,t
000I
0I00
)
P
(
Pa o
a Pz
00
a\
ol
x
1,u
= {dx'
"T
-
t)
,
t,n
(IBT
.tr
(r)
1,n

the system (l) farnally results in
v = Ax - I i P (2)
and by using the operator of expectati.n t and th: operatcr ! vhich leads tc the
,dridrcer
d d c va. e1!es ?e 9et
At r - rll F{.(l =:
^"-l'
c(r ) = d: P ', (.^ = standard deviation of unit (3)
weight to be e5timated)
E(v) = 0
Thereby the correctlon vectors of the grorrnd point coordinates Cx and of th€
orientaticn
parameters dt are all!ays introduced as free !nknowns,
System (3) is cdlled a generalizen Gduß-Markov llodel (.{ocl /1s/). For known
e^pected values tr- and for P7 + 0 i t may be interpreted as Reqression l'lodel
\:o:h.;:/), far Pa + I, E(lr)= 0 as Collocatior, lladel (xble? /t/).
Using Pp - 0 the system becomes srf!uldr. If ihe geodetic control point coordi_
naies are introduced as errorfree quantiti€s (diaq (Pp)r -), like it is often
required in oractice, and if the additronal parameters are tred!ed as free unknor'rns
(PZ : 0), the ordinary GauS-llarkov Model, i.e. the usual case of Adjustment
of 0bservations is obtaineC.
Llith the set of additional paraireters dz fron {l) we are fLrlly flexible, there
rs no res!riction concerning th€ir type and number. Tiis enäbles us to jntroduce
blockinvarjant
parameters, just as parameters l1hich belonq to a sinqle strip,
to a ce"roin grouo of |a jes or evel to a silgle
All practical
r_aq".
comoLrtations presented in this Daper were performed by applyinq
b I o c k i n v d r i a n t .r a r a n € t € r s only.
Besides a sophisiicated funcrional and stochastic model a high precision pretending
and operational bundle orogrdm requires the aLrtomatical determination of
the approxinate val!es of the aaj!sLment dnd the autonatical detection and localisation
of gross errors to cone out with a nininrun of program runnjngs and to
oatain re 1i abl e resul ts.
0ne imporiant Droblem consists in the procedLrre of detection of small qross
errors. A decision concerninq the rejection of an obser!dtion rs forndlly
uprn the dndlysis of the resjdudls of adj!stments. If !e try to objectify
based
decisions Lve have to refer to st.tistical concepts. Il r/as ,rrrdc /1/, /2/, rtho
de!elooec e starisr.cal corcept of e.ror derpctron, !.r'c s dl.ead/ dip'ied
-o
qeodetic netvork olanning and adjustflent (tca"Ca /2/, Bailncf;n /4,/, larita /2a/),
Furthermore some inv€sti!ations vrere performed concerning photogrammet.lc systeris
- the block adjustnent of inderendent nroCels (.1örstr?, /i1/, !!6Leüt /1!/).
!a:rid'E theory nust be regirded as a "reliability" theory. This ternr ha! to be
seperat€d very strictly
from the term riacc!racyii. A meäsurjn! system may be
accurata, \,rithoLrt bejn! reliable at all.r,le consiCer a system as not reliable',
lf there is a risk that one cannot letec! mocel errors, for exdriole gross errors
or systemetjc error!, \rithin..€rtain statistical securjty. For example, if in
b!ndle ddlusiment tho p\otoqrtsmnetric coordinates of a ground poin! have to Je
deri ved fror only ivo image point neasurefrents (tro ray5), then these coordinates
may be of moderate acc!racy, i./hereas their relidbiliry is !rorse, beca!se d !ros5
cur

error in those image coordinätes, whi.n belon! to the epipolar plane, cannot be
:,rdrda's theory ma,v be equally apDlied to theoretical reliability studies of
bLrndle adjustnent systefis as to the detection and localization
of qross errors
0n€ resLrl t af ?aa!d3'
mÄrn. fho .hp.l inn ^f
For that the tes t cri
.
'nvesticatio.s is rFe 'd"ta-sroopino' !ecl^nioue. hlicl'
the inCividual resiCuals for gross error detection.
-vj
'lvivi
q-rvr ! j are rne vdrrdrces o';i a.d oo is tre standard oevration of unit Lergtt.
(4)
i s proposed, vlh€reby the
ro< i., ,l c ,,-,, \., J.e oe.]0Ied urr!n vt;
fo. Ur, i.e. tqe reo or krere Lhe r"ll hyoorhesis (the
s nornally distrib!teJ) is valid is then
- F I , 2 \ 1 - , o , I . co J = \ . = Fl 2rt-,o,',ooy (5)
vhere do is the type l.error siz€ {.0 - P(l)) for the test criterion Vi.
Tire essential restrictjon for lar!e systems concernrng the conrputinq anount in
(4) are the values Qv.v.. Thereby t\.io oossjbilities for a !./ay cut arise. The first
is to modify the photogranrmetric systems in d \,lay which allows to use equal vat!es
for al I Qvivj sucn that
-p vi
(6)
This nay not be feasible jn most of the practj€al
pfojects.
The other possibility is to cornnLrte the dctudl values for Q",u tlen on€ has to
subdivide the Nhole system into subsystems to reduce compLrtati;n tin,e. This is
also required from a theoretjcal
point of view, oecduse ihe derjvation of (4) is
Ldsed upoir tlre presLmption that only one _aross error appears, so that rith
!se of subsysiens !Je are closer to theory,
The procedr,rre öf coaputation of approxi ate val!es (coordinates of qround points
carameters of exterior orr'entatlon) for the bundle adjustnrent in ll80P (relativ
model ori€ntation, strip forflration , absol!te orientdtion of strips) is a yrellsult€d
tecfnique for the introd!ctron
the
of subsJstems for a p.eliminary cnecking,
l/hereas the fjnal ciecking has to be oerforfired with bundle subsystens.
l' ',
r^, cn.hir.j..-o4 s-1-rs-- prlr. ,---ction
are nol i'-
cluCed in [180P, because sone further investiqations
have to be performed,
:Lcording ta other ccncepts as published by ?ote /19/ drd 9ezt":vg /5/.
Finally it rnust be stated, that the Dhotogramnretri c systefis, nowddafs L,sed for
highly precise point determinatior as occuring in densifjcation of control networks
or caaastral surveyin!, nostly are qui te !/ell sLrited for gross error
detection and lokalization.
This is due to the great nunrber and approprjat€
distribution of image points, to the dpplicörion of 60i sidelrp and to the !se
of groLrps of pojnts i2,3 or even rore targets per qroup) instead of determinin!
isolated, \.reak poi nts.

Conrpali9or of ä-Driori test field calibrdtion and s e I f - c a I i b r a t i o n
It was neniiofed in tte introduction that block adjustnent Hith s e I f - c a I i b r a t i o n
is consjdered as rhe nost effective
ard economic nethod for the conpensation of
systenetjc jriaqe errors, This ho'rever, is not !enerally acceDted dno above all
jn dispute ir lJest Gerrnany. EsDecially the test field calibratron
persation b"v ili!ht arranqefient L.rere proDosed as competjtive nethods (i!pr-r! /.tdl,
r.!eLskdEen /J7/, ritont, ,l2J/). Ihe.eby the advanidge of the tes! fjeld catibration
consists in the possibilitT
tion terms than it may be feasible by s e I f - c a I i b r a t i o n .
of applyjn! a hi!her deqree of inage corre.-
To get an iCea of the efficiency of possibly alternative nrethods as ä-priori
test field calibratian and sirnultareous s e I f - c a I i 5 r a t i o n both methods !rere
appljed to a comnon practical project. For this purpose d part of the ,0berschwaben"
- testblock, as it lras used jn rlaer /t/ and Grü4 /1;/, 'las selected.
It ls a bl0ck consisting of 4 strips \rith 26 photos each (totat 101 photos),
6Ci1 forward overlap and 20; sidelap. Soth yrere como!ted, the !Jide dn!le (HA)
and tne;!per |lid€ angle (SliA) flignt.
The test field fli!ht Darameters arei
iiA: RirK 15/23,4
strips with 3 photooraphs each in preftight
and costfliqht, image scale 1:10,,100
SllA: RllK 3.5/23, L strip with 3 photoqraphs in nreftiqht and
4 strips !]ith 3 photoqraphs each in postflight, inraSe scale 1j10,400
The polynomial coefficieits
for the Sl.lA systematic errors,
computed as mean.,
irofl or€ and post test field fljqnts (test 3rea ,,Rher'dt,') Nere taken over from
tfe Irstitute of Plrotogrammetry, University of B.nn (trare jsnc:,1ek /17/).
All other compLrtations \./ere exec!ted by the author. The coefficients for the t{A
syst€natrc errors were oL\tained by apelyjng an orthogonal bivaridte polynonial
for a reqular 5 x 5 inäqe point screen (see chapter 5)t the results from preand
Dostfljqht are sho\rn separately. The results of self-catjbrdrron were ootained
by applyinq a blockinvariant orthogonal bivariate potynom for d regutar
I r 3 inaqe point screen lEbse! /3,/), due to the jmage point distfibutr'on
0berschr/aben test bl ock fl i qht.
of the
Fil
tl
pranrder ry
Fiqure ldi Co.trol point distribution of rfe
(strips nos. 5,7.9,11), i = bridqi
'tberscir!aben' tes t
.n rll(t:nra ,n ,i

Table I contains the resul ts. eoth |'€re
(P4) control point dlstribution as it is
ve ry dense (P1)
in Fj!ure 1d.
and a very sparse
r0
rao
Fi !aure 1b: Test
Iach
of 3
area 'Rheidt".
pornt represents a gr0up
gr0un0 p01nrs.
550
Tdble 1:
Cofioarison of d-priori test field calibration and s e I f - c a I i b r a t I o n
Test block
'0berschwaben" (strips nos.5,7,9,1.1), 104 photos,
20X sideldp, imaqe scale - l:28,300.
Cariera
Coitrol
distrrbutior
wi thout correction
of systemati c
self-calibration
d /., ,, !,
o n,I
l"l t'.1 [,'] t,.l t,.l
,z
t"'l
with test field
prefliqht
I
cal ibration
postfli-oht
oo ux,Y uz
l"'l [,'] t,-l
P1
5.3 8.8 15.8 : .3 4 .2 L2.2 4.7 4.3 8.8 9.4 17.1 lB.2
acy i nprovenent 1.6 1.7 1.3 Ll L.l 1.0 1.9 1.9 0.!
ii | 4.r 24 e '!4 2 1.2 8,0 18.9 3.6 3.5 22. t 22.2 31.4 56.3
s;l
11
sll
Accuracy r nprovenent 1.3 3.1 2.3 1,1 1.2 1.1 l.l 1.1 0.3
I s.: to.: t;.0
|
6.0 9.8 14.3 i.2 12.6 15.7
Acc!racy ii:provefien t L-L L.] 1.1
I s. e :e.q q,1.2 5.9 t!..1 r7.6
Accuracy i nprovement 1.1 2.4 2.5 1.0 1.4 1.6
Analysing tl're results we recotni.e significantly different accuracy levels in
\'/ide angle (llA) and super wid€ anqle (5liA) fliqhts.
iloliever, th€ accuracy imorovenent
by s e I f - c a I i b r a t i o . is simjlar
in botir cases. A decreasing control distrjbu-
!ior results in an increasinq accuracy irnprovenent (up to a factor 3.1) This ls
due to the fict
that -.he propaqation of systenatlc errors is the rnore Lrnfavourable
the le5s control points are Lrsed. Thus s e I f - c a I i b r a t i o n yields I arger a.curacy
imDrovements in the case of poorer control distribLrtions. The;-priori test
field calibratr'on acts quite Ciff€rently. ljjthin the SllA fli!ht r'/e obtain a re-
'iarkable improvem,ent ccnrpared ,iith the ccrioutation frithoLlt correctian of systefiatjc
errors. This is nrt true in the case of llA flight, fere an inpr.venent {as
achieved only 5y ti.e pre'_liqht and the sparse control distrjbution (iactor 1.4
in heiqht). vhereds tne postfliqht calibration larqely deteriorates the results.

7
This clearly sholJs the unreliability of the ä-oriori test field calibration
technjque. Though test fi€ld cdlibration oDens the possibiljty of the applicatjon
of highly developed inrage correction terms this advanta!e m!st nct be
overvalued. I' decisive dlsadvanta!e results from the fact that lhe sJ/stematrc
errors can chenge !ihen the act!al pr.ject fli!it is perforned. This may lead to
a remarkable insec!rity ,,rher apDlyin! test field calibration. There is no real
indicator that perflits lhe conclusion whether a test field calibration is representative
for the actual project or not.0nly s e I f - c a I i b r a t i n I block adj!stment
is able !o coripensate the actLrdl systematic errors kith sufficient reliability.
Therefore the use of additional parameters in a modern bundle adjustnent progran
is an j|ldi:pensab]e demand.
4. The accuracy potentjal of s e 1 f - c a I i b r a t i n q bundle adjustment. Theoretical
expectations and oracti cal resLrl ts
Asking for the theoretical
dccuracy expectations of an actual photogrammetric
block requires the determinatioi of the standard deviations of the point coordinates.
Generally thjs is dn expensive paocedure beca!se tt presu'nes the inversion
of the matrix of the nornal equatjons. Ho,rever, photogrammetric point
d"-".".r,-j i. r,.(p - ^a.-rltstton sets.
iherefore th€re is no need to coflrp!te the accul.acy exoectations for each project
ane\r, but we can lrse accuracy moCels, which 1,rere derjved frorn synthetic
da!4. In a.ecent pJblrca-ro1 E.aer) t?ack, lc.Lbe?: I nave p.ese.ted sucr
dccuracy models for bLrndle adjustnent, \,rhich !Jill be appljed to the follo,'{ing
inyestigations.
If the efficiency
of s e I f - c a I i b r a t i n g bunCle d d j Lr s t L , e n t rr a s to be checked we have
to compare rts.esults,iith those theoretical öccLrracy nodel:. This comparjson
Coes nol yield e(act cri lerions,
becaLrse sometimes rerial"kable differences betreef
theoretical block conditions dnd practical test circumstances irith respect to
image point Cistribution,
inage overlap, position of control points, do occ!r.
l!evertheless the aoplication oi those accuracy nodels yields d good saanCard of
conrparjson to check the practically
obtained accrrracy values,
Table 2: Slock specifications
of test blocks
Block
pnotoqra0ns
0verl ap
plan.lhei!ht
ol an.I hei ght
testblocl (Pl
itA + st,iA
t04 1 : 28 ,300 60 20 3l 17 225 111
"Steinbergen"
top.height block 109 1 8,000 60-90 50 2i 55
''i,loosach'
cadastral bl ock 93 1: 3,300 6C 60 25 10 113
cadastral block Lq I '1 , ,101 7C 60 10 tl 15

ln the followinq the results of sone blocks are presented, adjLrsted tl,lice vrith
blockinveriant additiondl parameters and wtthout addrtronal paraneters. The aajn
redson oi this representaticn is to sholl the extent of the accuracy iriprovement
\{nen appl_vin! sinrultaneo!s s e I f - c a I i b r a r i o n technique.
Tabl€ 2 incicates the block specifications,
lable 3r Results of p h o t o g r a n fir e t r i c accuracy tests
Block
oo ux,Y rz
t,'l
ith
while Table 3 sho,r'Js the results.
ddd.pdr.
Appr0x.th€oret.
accuracy €xoect
lu "l I, rttl r,;t t;;i t";r l"'l [u']
PI 5.3 8.8 14.8 3.3 5.2 12.2 3.0 a.l
P1
5 'riA
Accuracv irrrov€n€nt 1.6 1.7 1.1
8.2 16.3 17.6 6.0 9.8 14.3 5.4 5.2
AccLrracy improverien t t.4 1.7 r.2
5.6 - 14.3 5.4 - 8.0 - 5.9
Accuracy r nrpr0vemen! 1.0 - 1.3
5.0 10.6 4.2 5.1 2.6
AccLrracy i mproYement 1.2 2.1
4.2 I .7 4.0 1.8 2.1
Accuracy inproveoent 1.1 1.5
These results allo!] some interesting
conclusions.
First we have to state th€ large acc!racy improvements, obtained by self-calib.aiion
techniq!e (!p to fdctor 2.1). As jt f/as nentjoned alreadv ln the fore-
!oing chapter, the extent of inrorovement depends on the con!rol distribLrtion,
i,e. the lar!es! improvenent usually is achjeved lrith a very sparse control distribution.0n
the other hand the tneoreticdl accurdcv e(Dectalions are not fnet
completely. Firstly
this can be explaineC bv the fact that the enrployeC theoretical
.ccuracy rnodels do not correspond perfectly !./ith the practical
arrangements.
Arother djsturbinq effect is exerted by the i ..-e..rr4 1..r.r. ip

9o t'loL Lhe corcl Jr'o
is allo.eL
deformations are to ir!rute tc the
cadastre block M00SACH
that wi th great
!eodeti c point
n..h:hi I i fw thr

10
satisfy this clajm b!t there is no secLrrity at all that these model concepts of
systematic defornations cofrespond ,rith the actual sy!ternatic inflLrences in any
practi cal pro ject.
Therefore, tbaep /9/ !lent another rjay, asking for oarameters, which cornpensate
the total systenatic effect in a presumed ind!e fornt screen rrithoLrt expldining
the errar sour€es. He developed a bivariate
orthogonal oaranreter set in relation
to a regula. l,{J image poinl distribution. lhereby the orthogonality is not only
valrd l{ith respect to the addrtional parameters themselves but also in relation
to the elements of the extericr orientation.
In /1i/ the author proposed a redLrced third de!ree polynomial, Hhich fits to the
Jrain conponents of the systenatic deformation's model concepts,0ther arrangeirents
are found in Pduer /3/, Salneape?c a.o. /21/, schtt /22/.
polynomials have prcved to be a suitable mocel for st/stenatic imaqe err0rsi
the treatfient of polynomials is cansidered fron a more !eneral point of vie$.
ihereby the favourable pl.ooerty of bivariate polynomials nust be emDhasi.ed:
They dllolv for the compensation of the t.!dl systeflaLic effect in all points
oi d presurned inage pornt distribution.
th!s
The natnenatical backqroLrnd for the construction
of bivariate polynonial sets of additional Daraneters is the follolring
The displacenent of inrage coordinates, modeled by bivariate
AX - f(x,y)
c-v = g(x,y)
- ,vTAx
= yT3x
polynomials is
!here A,B denote the pdraneter matrices and the ve€tors x,-v include the po,/ers
of i nage coordi nates as
(6)
I
(L x x2
(tvv2
xn)
vn )
(7)
Tne bi vari a!e transfornation
i
^ie-v"i_:5
1 r ,' l. ,.)/t, r" y
^ 1,.- | ,
y ^v ^zv,(^'v l,ev ',...
z: -. -l
,2 ^r2.42r2t, r) rz t,^a r2l...
-',-1t----t-!)
t) xY' ,'Y'
"Y' "Y'1..
,.4 .l 2l r 4 .r .rl
:::;:
yn *yn *tyn "tyn
*"yn. . .
-
l
nz
., n,,3
.. n.,4
:
(3)
!hereby the li|lits of a 3x3 ina,qe point distributicn (S9) dnd a 5x5 inaqe poini
distributian
(S2S) are narked.
If th€ ina!e points are not disrributed eq!all,v in x- and y- direction a proper
reLLdroL-er se-ectior car oe !orr'.C o.r ds rfdrcr_eC . { .
lf someone r?an!s to confirfi hinself
diaaonal lines have to.\e
to a certain deqree of polynomials, then the
follo..red. Thouqh this concept is occasionally applied,
it rusl be reqarded as a quito drbirrary proceedin-o, beca!se it leeCs to en inco
Dlete bivariate polynomi dl .

11
In
this cage rne c0rresponcr ng
parafneter malrices are not ful
ly popLrlateC, as
aLl a12 a13
'2t "22 "?.,
'31 o32
a41 | 0
I
0
34
"11 -12 "13
b2t hz2 lzs
b3l b3z c
b,t1 0 0
ot,
\
ol
,l
ol
Th!s
in 9
.1nans !l-,rt possibl/ si!rr"-"r L
t.]e ser of ddoitrolal pd.dr.r!ers
inage points is
parümeters are let o!t of consideration.
c0rnpensatinq the whole systematic effect
13(x,y)
-o3(x,yl
, t0,,
,tu,"
(e)
wl tn
A3
I'11 '12
a2t uzz
a3t 432
"13
423
u3:
"3
(
"11
b12
"2r -22
":l
D:z
ot3
ozz
o3:
)
(r0)
rx = d11+ar
22
2x+d2ly+ar :x2+a zzxy+a:t
y2 r u yr 2312 u 3zry2
t u 33,
r,y = b + ,12
1I
b +b l2x 21v+
b,. x 2+b
(11)
r rxy+
b,, y2+t: y +t.
rrx2 rxy2+a rrx
For vertical photoqraphy and flat terrain the 6 parajreters of
tation are included in (11). To avojd hiah ccrrelations
ted. Hence ve obtain for exanrle
u rrf, ^ ,r*!,
u rr!2
*u z: "
2y*
u::ry 2n.:: * 2 y 2
brrr+o. rx2+brrxy+
t ra^?y
ra
"
r"v2 ra r.*2v2
these
extert0r ort ennave
!0 !e reJec-
(12)
ilLrnerical aid statistical considerations lead to the idea to construct orthoqonal
parameter sets, i.e. additional paraneters, which are independent amonq
each 0ther and with respect to the elements of exterjor orientation. At this
point it must be emphasized tirat orthoqonality amon! the addjtional pa!"aneter:
js strongly valid only eith resp€ct to a certain reqLrlar imaqe point djstri5utron,
aFong idditional paraneters anC the exterior orientation elements of the
individLral imaees only in tfe case cf exact vertical Dhotoqraphy änd flat terrain.
F'rrthermore one has io notjce the fac! that even ir thos€ extrenel_v regular cases
correlation between the ddditional par3fieters and the ccorCinate Lrnknc!ns of tne
qroun0 pornts aay 0ccur. l_rp t0 no,v tnjs t/as not sufficiently t3ken jnto consideration,
?hat fiay result r'r serious conseq!ences as jt wjll be sho,{n later on,
The o r t n o _o n a I i ? a t i a n of {12) leads to ttre follo!,irng oa.emeter set, firs!t_v
publ i shed by Et,te! /-a/.
l2
l

12
rr = f{x,y) = blx+b2y-b32k+b4xy+b5l+ 0 +b7xl+ 0 +b9yk+ 0 +bIlll+ O ,
-- (lrl ,,,,
,1y = S(x,y)
=-b1y+bZx+b3xy-b421+ 0 +böl+ D +b8yl+ 0 +510(l+ 0 +b12kl;
Thus, for edch reqLrlar screen of imaqe poinLs on orthogoral set of parameters
can be constr!cted and nodified i:n hF if.lp.endenr with.ecnp.t
to the orientation.
elerrients.
Sesrdes !he l,(l image point screen tne 5x5 i age point screen (Fig!re 3) is of
greaa l nportance.
Fi gure 3: 5x5 1flage point
distribution
L
I
buti on starts from
3y
f5 (x 'y)
ss(x,y)
,too*
,tuu"
(14)
_'e
utt '
As ' Bs =
:
ust '
)
o . L _t ^ g o n o I I z J r i o n o.C tre
r"jecrion
the 6 exterior orientation elements lead
(15)
!{hi ch col. respond
f. fha f^ll^!inn '14 paremerer ser;
to
n:rimEtp.< fn. fhi< reg!lar i maqe point distri-
'-*
-Tt2
, 1 -,2
2 t7 2
t7
o.12-
2A
2A
. = r2('*2 - , = ? 31
,2(, 12) ' -! ca
2A
2A 1n
2
Äx = at2x
a21y +a 2rxy
+ a31l
0 -b ^^i:k+
rv =-4. -v
urr" -urrllt*
o
(r,(.
(rY.
)+ ari.(f+ a.ryk
){ 0 + 0
a32xt+a41yq
0+0
0
b l
Z3yk+bj2x

L:
a^..1rd..r -',1 ra,^^vo-r..5 - 0 r 0 .1
trl..tr 0 - 0 - 0 | 0 .ol;r -D2I,yorl L]+bII yq-b5.s'
1;r..) ra".vr +r."^lo +a,.vlo+a""xs + 0 + 0 + 0 + 0 i
(ly..) + 0 + 0 + 0 + 0 +b2./r + b r I p + b : 4 .t I y I q + b 52
j( s +
,:^..) + a " . I r ' a , , x v o c ' a . . I s + 0 ' 0 0
irl..) + 0 + 0 r 0 +oJ-" rb41\yoo-b531's -
{r{,.) -a../rr-a,^os ! 0 r 0 -d..rs r 0 ,
,.y..)r 0 ' 0 rb..yortF "
- (16)
,, 5l "' " -55 - '
li(.,

Therefors the probler oi statistjcal testina !/ill be treateC unio.
(.d. n.nr.:l
SupDose the linear roCel cerived from (1)
v = Ar - I i
or in statistical
P
nötation
L\r/ r 2"-I
.( , -(.) . '
oo
i f(v) = 0
(17)
vrith the general linerr
hypothesis
Ho: 3x = l"l (1s)
llhere B is a bxu matrix \"lith rank (e) = b=u.
The minimum variance unbiased estimators lor r,.l from (1i) are,rith r = n-u
r = (q Pq) ,A Pt (19.)
':
= - 'r -Ai, 'P, t -Ai ) (20)
LJnder the addi tional constrain (i8) they becorne
i = i-1rroa1-ls- s,,'"o11-lnrr-lqei-nt , (/1,
-2 L .. "^. T^. - .-.
.; - : -A

15
i nfl uence does occur
unknoLlns in (1):
I dz = dz'
alto!ether, Ee set vrith resDect to the sLrbdivisioi of the
(l = I den ti ty ratrix) (?6)
then T becones
T=+(d?
"";
-dz
tTl;l {a: - a'')
(m d number
of addi tionä l Daranreters )
127 )
Accordina to (i)
the symrietrr'c matrix it of the normal equatjons is
qlp"r,
oJp^r^
./. rln re r* e,
n] e rr,
'r lr un,
"1'B^4
AiPBA4
t,zej
rl n nr, r]rrno
\ ttT ::)
AqPAA,l*Pz
'zz - 'zz -ll
l-ll
is the 0artly reduced |l.atfix of t'1€ norral equations for thp se! of addr'tionäl
paraneters. If the aCditional Daraneters are arranqed at the end of the cornolete
vector of unkno,r'/ns as it is s!pposed in (l). then 0;l i! obrained
additioral effort during the factorizati.h of .
The test criterion T (27) cofresDonds to potelljrg's T2-tesi, i.e. the muttivarjate
eriension of the univarlate t-test.
reduced to a t-test.
:.t i th
Thjs imDticates that (27) can be
-'i
-_ i
(30)
' |,2
-ozi J
(31)
where T is distributed as St!de11t's t-test,rith r deorees of freedo,n.
The global test (27) on all additionat paraneters atto,rs the ascertajnnent Df d
global siqrificänt svstenatjc influerce, Bhereas to check the siq,lifjcance of a
subset of ödCitional pararneters one has to nodify Ho in a suitable manner,
If only ä single parameter has to 5e tested and if only sliaht corr-"lations of
this param€ter trr all otirer unknovrns do occur {:l) can be aoDlied. So the corj.elatjons
bet!/een Daraneters are at least of the Same inDortance !nan ahe
But since the corneutation of a renarkable nuFber of €orrelations (.ddjtjonal paraneters
to Doint Coordirates, ddditjonal para.retefs to orientation elejnentS)

16
becones a very expensive orocedure, tfe advanta€e in usin! orthogofal parameter
sets is qui te obvious.
lf 5!fficient independency of a set of addjtional garameteI.s is ensured \,rith
resoect to t e orienLdL'o' ele-e.ts ano a so to t\F pojnt coo"dindt.s tlen 0;]
from (29) incl!des all necessary irformation for a sophisticated testing of
additional parameters. If a correlation coefficient for tvo additional pdrarßeters
exceeds a certain limit,
tested tosether.
This I eads to
let us say 0,75, then those two parametera have to be
IT
2,2
2,1 2,7
(32)
- 0 dnd d2 , (d!. .lz, ) then T resul !s
I
26t
(däj d2k )
Q,i.;
Q,*.1
Qrl.1
qrg,1
)'(,,,)
(33)
l.ih i le for a type I error si:e o we mostly use
power I depend on the al ternati ve hypotheses.
To de-onst.ate Lhe aoove -enElored Droolens a
tr'ca1 example is introd!ced.
Thereoy lle DraLe use of a test field flight abo1,'e the test field
Fi_oures 4a änd 4b).
- 0.01 or s = 0.C5, I and the
Rheidt (see the
Fi!!re 4a: Test field fli!nt "Rheidt 74" Figure 4b: Inraqe point distribution of
(tlay 13, r974). R:ll( AR l5123, test flight "Rheidt 74'
3 strips ur'th 5+6+5 = 16 images,
60X forvrard overlap, 60.1
sr'delap, alternative fl i qht
di recti on, i mage scale 1 :5,300
The testfield Rheidt consists of 4l much re-culary distributed groups of ground
points \aith 3 p0i nts each group
Tne fli!ht arrangement leads to the imaqe point dlstrjbution of Figure 4b.

I]
In the follovrin! demonstration He \vill use only 4 control points (nos 110' 15c'
510, 550), all the other points serve as check polrts.
lleglecting all other !eometrical corditions the image ooint distribution seens
to allovr the introductjon of the (nonorthogonal) paranret-'r set derivec frcm (8)
a21y+a22xy+a3ly2+uru*3*ur,r2Y*u,rtY2*"0.V3
(34)
aY = !lx'YJ brrv+br,x2+urrxv+orox3+lr,x2v+l,rxv2+bntv3
It must be emphasized, lhat this set of a reduced bivariate
being regarded as representative of a suitable parameter set. It
the demonstl^ation of some inconvenient effects \rhich result fronr
of an unqual i fi ed set.
Table 4 shows the results of vario!s prograrn runnings rith
presufrptlons.
fne medrrng o' tne oifferenL ruf ings 1s:
Al1 paraneters of equal wei ght pi =0 ( free unkno!]ns )
paramelers
rmage p0rn!
1*l = lvl "
of equdl weighi, !Jhich is equivalent t0
di.nl).amcnr.f 1q m fn. prrh n;.rE)far
100 mm
t-vpe is far allay from
is only used for
the i ntroducti on
changinq parameter
B, but with oo (parameters al4, b41 are excluded)
As B. but wilh o : o-
'22 dll
a.,,
b,, are e)(cluded)
+.o rDdra''leLers a-2. ell
J41
iatrle a: Pi.anerer
(4 coiüol pol^s, L1s.n..( 'oln!3).
(r4), dpprled
,i:, :;,1 .',']
13.r r,0 .2
'F
9.I -d l. 1.9
--_-t-
rortiiieirv
reiqiil

IB
case A, vrhere all parämeters are regarded as free unkno!,/ns, jndicates qLrite in_
ferior res!lts, especially in lre jqht (rrZ = 150.2rm). This is mainly due to the
fact that the qeometrical conditions (only 4 perimeter control points) don't
al'lo!J a seperation of the paraneters a14, b23 and a32, b41 (correlation coefficrents:
r- * = 0.995, r" = 0.987). This becomes evjdent if the effects
o 14' 23
d32"41 "
of these pdrameters are st!died. As it was sho\'/n for a sjnOle strip in schtt /22/,
t h e d i s p I a c e m e n t s Äx = a,,x", Äy = b,, r'y lead to a lonqi tudindl hei ght curvature,
L1e dlsDlacererts a^ = a3zry' \ ay r o4lyJ le"o lo d heiort curvoLLre across sl''D
direciion.
For n block this is valid in like manner, so that these deformations
cannot be determjned by only 4 perineter heiqht control points
To avold the larqe and unreliable paranreter amounts of those parameters, Ylhich
cannot be determined \vith sufficient reliability a sultabl€ strateqy mdy consist
in introd!cing
small keights. This has been done in Case B _ for each pdrameter
a vreight \ras introduced whjch js eq!ivalent
to dn i'nage point displdcenrent of
15 rrn near the imaqe corners ( l*l = lyl
= 100 mm). This proceedinq leads to m!ch
better resLrlts (accuräcy improvement of factor 1.7'in planimetry and factor 4.3
in heiqht) and cecreases the dangerouS parameter afiounts toqether wr'th thei r root
fiean square errors considerably. The critical correlations stil1 remain large,
t,hereas they are decreasing sli!htlyr
= 0.981
"uralr,
,
Secause of these high correlatiors the 1-djnensjonal t-test nust be reDlaced by
the 2-dimensionai F-test
(33). Applyinq this test to a14/b23 and to d32lb41 leads
to the statenent that bath critical sets are siqnificant even on the 99: d_level.
flence one pardmeter of each par r must remain in the adjustfient. It was decided to
reiect at4 and bat and the computatjon of Case C \,as performed. T!re accuracy in_
provenent against case ^ is of factor 1.6 in planimetry and of factor 9.5 in
rei!ht. The hidhest correlation no!r appearing is bet\,/een a2l dnd b." : r. ,
= -0./t llhrch is not be regerded as inconvenient. "' "21")2
The correlation matrjces of Case A and Case C are shown in Figure 5.
Case A
Fi!Lrre 5l Correl äti on matri ces nonorthogonal Darameter set (34)

19
Co'nparing the signifjcant pdraneters received from t-test in Case A and Case C,
a reirarkable different beha!iour of third degree parameters is noticed, what is
dLre to the fact thal t-lest application in Case A leads to wronE conclusions
concernjn! the paranreters al4,a3?, b23, O,tt.
This yields the requirement of mostly independent oarairreters, ühich can be satisfied
by an orlhoqondl set as described above, as practicall,v applied to o!r Cenorstration
nrojecL drd prese.ted ir Tdble:-.
3LL lhere is arotl'er ohe.c enor recessdr/ to Dut'rto con-tde.dtior, up Lo ror'
Ne had restrjcted oLrr inverstigations to correlations bet'Jeen additional parameters
and between additional parametert and orientation ele'nents, !rhpreby Ädmissible
statistical tests of uni- or nultivariate tyDe cdn be performed vrith
"^
^ _oTeters a^. dro a-r on tre qaourd poifrs one frrds
out that a., leads to a Dldnimer.r. ^. -ach strip in strip
direction, only to be determined r'/ith at least one additional control point in

2A
iä5le 5: Prrahe!?r aho!i:5 iid
):r:matar ser (1ll
-l
rrn j
iLr I
r 0.6 I
la-7
a
!a L-.1a-,1_ L:- . !:l_
L.i
^ltl.
i:_1
ß T- L.s
ro rits 1pl 'iinetrv
helgIi)
Fig!re 6: Correlation matrix of
Case A of the ortho!onal
l2 pa rapie te r ser ,I3).
0nly if the critical parameters b4, b5 are reiected, vlhich corresponds !lith the
reiection
of a22, a1\ in the set (34) ar accuracy i$provement appears (factors
1.5 in planimetry and height) !rhereas a bad behavioLrr of the critical paraPreters
is not visi5le if only the correlations betl/een additional paraneters and betv/een
additional parameters and orientation elerlents are checked'
Tnis leads to the irportant conclusion that high correlations between additional
paraneters anJ grcund point coordinates can occur, so that an exanination of
correlations nust incluCe the ground points to get nost reliable results As it
is seen from the correlation ratrices publi5hed ir flcueLshagen /17/, these 'orre'
latjons firay jncrease up to a value of 0.96 or even more' provid€d that an !nproper
choice of ndditional parameters \'/as flrade !lith regafd to some of the !eometrical
conditions (image overlap, control point distribution).
So a drastic protection a!ar'nst h/orse parameter effects must incl!de the examination
of the correlations betreen additional parämeters and ground point coordrnates.
Jo avoid an excessive computational expense one has to restrict on a fe'{
ground goints at most. Probably the conputation anC exanination of the c0rrelations
of ä single, vrell_locat€d groLrnd point will be sufficient For that a
perineter point should be selected !lhich has to be as far a'tay irom the aciacent
control points as possible. lle!ertheless, some further investiqations are
necessary to obtain a better insight in the problems concerninq the correlations
bet"reen additional parameters and !round points, especially \.lith respect to the
!eneration of an operational procedure for detection of those !ndesired effects

2l
6. Concl udinq remarls
Through the conpensation of systenatic image errors the photo!rdmmetric methcds
of point determination, especially the bundle adj!stment |1ith s c I f - c a 1 i b r a t i o n ,
have achieved such a hiCh accuracy level, that new oossibjljties of apoljcation
arise, e.g. in control network denslficätion and cadastrdl surveyinq.
l,{owadays the s e I f - c a I j b r a t i o n te€hnique offers the poss i bi 1i ty te meet the theoreticdl
accuracy expectdtjons. As it wds shown in this paper by several test
projects, thes€ expectations have not been met conpletely. 8ut it has to be
pointed o!t that on those high accuracy levels (up to l-2cn) as presented in
chapter 4, an inrportant restricticn consists in the insufficient accuracy of geodetically
deteririned contro I pojnts.
The superiority of s e I f - c a I i b r a t I o n technjqLre co mpa re d !'/i th test field cal j bration
|'ith respect to accuracy and reliability hds been sho\,ln in chaDter 3, so
that no\.,/adays s e I f - c a I i b r a t i o n technjque m!st be re!arded as an essent.ial part
in a modern b!ndle adjustnent program.
lhereby the whole accul.acy potential of photogrammetry js to be utjlized only
jf a suitable selection of additional parafi€ters js nade. This requires a higirly
developed, flexr'ble parameter set 1/hrclr äl lovrs for the totat co'npensation of the
systeoratic inage errors and which avoids the danger of o v e r p a r a n e t r i z a t i o n . An
overpararnetrization is iniicated by the appearance of hieh correlations. Ceneral
ly those may occuf ainong additional
paraneters and bet|leen additionat param€ters,
€xterjor crientation.elenents and ground point coordinates. Then the 1-djßensional
t-test appljcation yields vrong concl!sions and th!s \,rrono rejection decisions
are fidde.
Startin! from a general linear hypothesjs it has been sho!,/n in which wa_v stätistical
test criterion5 can be Cerived providing for sophisticated testjng.
Ey applying orthogonal parameter sets tre correlations among additional paräfteters
and betl.reen alditiönal oarat,reters and orientation eleirents can be reduc-.d
to a certain minimun. This is one reason for the proposal of the concept oi bi_
varrä!e ortho!onal polynomials for addjtional
parafneter sets.
From an operational ooint of vietr an inportant restriction consists if the
necessity of cornD!tätion of the correlations betr/een adCjtjonal parameters and
Sround point coordinates. Though further investigations have to be performed in
thjs field, a possjble strategy may consist in coflputin! the correlations to one
specially selected qround ooint only.
The results of !he practjcal examFles oresented jn thjs paper have been comruted
by usinq blockinvarr'ant paranreter sets exclusively. Sonre previoLrs tests dith
stripinvariant (project "l.tcosach,') and even iirageinvariant (project Rheidt 71 )
0!Tarneters indicated no further iilr!rover,€nt. This wculd allo|, the conclusion thai
the systenaijc indge errors renajn largely stable durinq a fliqft performance,
!/hat h01/ever slror.rld be confirmed by additional tests.
llhile the achievenrent of hiah photoqranf0etric accuracy levels (3-5!nr) is no lon!er
problenratic, ricre attention must be pajd !pon the stabjlity
of rhotoqrannretrlc
systems with reso€ct to reliability problefis. In the fjeld of !eodetic net!lork
adj!strnent several investigations have been perforfied dealinq rith those

22
llok the time has cone to utilize lhe !urpcaed statisticdl test rnethods for photogrammetric
systems, This is vdlid |!ith resoect to investiqations
nLrmber ani distributl-on of points and the flight
concerning the
arran!ement to obtain systems as
.Fl1:hla ä< n.q

23
/12/
/13/
/14/
/15/
/16/
/17/
/18/
/1-a/
/2C/
/ 21/
/22/
/23/
/ 21/
Cra!!bi1i, I.,{.r An introduction to ljnear statisticat models. Voturie
Ic Graw-Hill Book Compdny, Inc., tiew york, Toronto. London 1961.
-lhe
C.üa, A.: sinultaneoLls conoensation of systelratic
llunich bundle proqram 11B0P. Ir cerman. prese;teC paoer to the XllIth
Congress of the ISP, CommissioF III, Helsinkj l_076.
i(aeh, K.R- i DistribLrtions of probabiljti€s for statistical reviews
of adjustment results. In ce ian. r,litteitun!en aus den InstitLrt für
Theoretische Geodäsie der Universität Bonn,'rir. 39, Bonn 1975.
Xoch, it.F. i ilodel formation for parame
ter estimaiion. In Gernan.
Allg. Vermessungsnachrichten, Heft 7, Juli 1977, Dt. 272 - 277.
Xupie!, c... Impl.ovemext of analytica I aerial triangulation by field
calibration. Proceedi nqs of the ASP, Phoenix, 26.-31.10.1975,
pp. 51 - 67.
l,laue l s hagea, r.I Partly caljbration of a Dhotograrfretric system with
variable control point distribution and different deteririnistic irodels.
ln Gernan. DGK, Reihe C, Heft llr. 236, '1ünchen 1977.
i,!ol,etaa!, 1. r Error detection in planinretric block adjüstment. Presented
Paper to the XIIIth Conoress of the I5P, Conmission ill, felsirki 1975.
Pape, A.J-: The statistjcs of residuals and the detecti on of out
PaDer presented to the XYIth General Assefibly of the IAG, Crenöb
Pabe"ts, F.G. r -lests of bundle block adjustment for s Lrrvey coord
Presented PaDer to the XIIIth Congress of the I5P, Commissjon II
Helsinki 1976.
Salnenpe!ö, t:., Ande!Eos, .t., SaraLd":ner, ..1. j Efficiencv of |re
extended mathematical nodel in bundle adjustment. prese'nted paper
to the Symposium of Commission III of the ISp, Stuttgart 1974.
3chut, c.il. r 0n correction terfis for systenatic errors jn bundle
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