Accuracy, reliability and statistics in close-range photogrammetry - IDB

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-2
eral and flexjble
s e I f - € a 1 i b r a t i n g bundl e a dj u stmen t hav ing proved its efficjency
in aerial trianS!lation. For the further analytical treatnent reference is made
to the nodel of the existing bundle program 1,480P (14urich Bundle 0rientatjon with
Addjtional Pararneters), originally developed for aerial triansulätion problems:
- eB = Aldx * A2dxP * A3dt + A4dz - lB ; P- r-tl
i d,r P
Idz-)za Pz
(1)
= Vectors of true errors of image coordinates, additional parameters,
control point coordi nates
,dtrdr = Correction vectors of point coordinates, control point coordrnates,
elements of exterior orientation,
additional Darameters
= Vectors of observations of image coordinates, additional parameters,
control poi nt coordirates
I
= Identity natrix
Collecting the matrices and vectors of (1) in the following manner
(''
42 A3
00
IO
I
0
)
0
0
:,i,)
T
1,n
r ol
(dx', dx , dt', d2 )
oT rT rT
(e-, e'' e' )
(1) formally results in
,IT
l,n
and by using the operator
varlances dn0 covarrances
cvna.+erinn F :.1
(2)
rhe operatnr D wl^icf _eads to rhe
AE(x) = E{l ) ,
E(x) = !,x , E(l) = !t , E(e) = 0 ,
(3)
D{e) =."2p-1 , (o- = standard deviation of unit |reight
to be estimated)
Thereby the correction vect0rs of the grouid point coordinates dx and of the
elements of exterior orientation
_Pdas
ge'e"._ly to fjrea node's,
dt are always introdLrced as free !nknowns, what
System (l) is called a 6ene.alized Gauß-Markov f4odel (,{o.l /12/). For known expected
va_"es:, ard for P, = 0'r -1av be interp"eted as (__\ed) qe9-essior
'aL
Model
foy D_ I0. Frt, .0 ös ,-.)ed) coltocatjon l4ode_ (fr,"" ). I. !fe geod.t.c

-3
cortrol point coor.iinates are introdu.ed as free of errors (diag (Pp) ' -),like
it is often required in practice, ond if the additional parameters are treated as
free unkno!']ns (PZ - 0), the ordinary Gauß-l'4arkov j4odel , i.e. the Lrs!al cäse of
Adjustment of 0bservdtions is obtained.
The fiairix Pp allows to introduce a-priori known accuracy standards of th€ control
poirt "rrrdinatesi the same dpplies to the matrix PZ and the additlonal parameters.
Using Pp = 0 the system becomes singular,
Under certair
circumstances and with certain pärdmeter sets the treatment of systematic
errers as free unkno\,Jrs tPZ = 0) may leöd to remarlable deteriorations of
the condition of th€ system oi normal equations. Therefore it'is expedient to irtroduce
PZ 10, thus protecting to some extent against overparanetrization.
l{ith Ppl0lae obtain a vJeishted mjiinral fitting (with respect to a 3-dimensional
sinrjlarity transfofmation) of the phötogramretric poirts onto the cortrol pojnts.
l,Jith the set of adCitional paraLneteri from (1) yle are fully flexible, there is no
restriction
con€erning their nunber and type. This enables us to introduce blockrnvarrant
parameters, just as parameters vrhich belong to a single strip,
cer:a'. grouo of _mages o- Lv-ar to a s'r9le'nr!e.
In close-ran9e appljcatior,s this concept has to be extended by observation equations
for additional measurement elements as angles, dlstanc€s and exterior orjentaticn
parameters and by some special conditions as straight I ine conditions,
surface conditjons, af!1e conditions and so forth.
leads to the conditior
r -i
Q (I r^) ++ v:. ,
or if only the additional parameters are regarded
to a
For the purpose of the following
studies these equations can be omitted \1,ithout loss of generality.
The same applies to other image geometries l/lhich may replace the perspective relations.
For the determinabilitJ, of systeinatic errors and the related eigenvector problenl
see a!üt1 /1i /.
'I he problefi of a s!itable
choice of an additional parameter set vJas recently djscussed
in c"ün iia/. Thereby the functional, numerical and statisticdl ddvartöges
of bivariate orthogonal parameters,introduced by Ebne! /€/,
A sirategy in additional para,neter construction, i.e.
matrix A4 (see (1)) must considef two basic requirementsi
hare been emphasized.
the choice of the design
- The estifiation ot x and cf in 13) has to be unbiased, i.e. the systematic
deformrtions have to be model ed as wel I as possi bl e
- The variances of the additional parameters {and of the other parameters of
systen (I))should be as sfiall as possible.
To the fr'rst point -axper'r'ences, 9aineC over nrany years! have sho\,/n that polynomiäls
are a prop€r device ii Bodelling systenütic ifiage errors. Thereby the favourab_r
p-oDe-Ty J' oivar:are oo_y.ro,r'als co'sists ii its cBDabil'ty to comDe'sare
rre roral sysre_atic effec- at al
poinL, u- a p, esumeo "eg,lar imaqe Doint scree
lELneT /€/).
The second requirenent, wfich includes equally the demand for minimal covariances,
{4)

-4
0-- = {A;P^4,) -Min
(s)
As it is proved )n F.;. liE,/, the minimun of 0.. is obtained l,lhen
4..!,4.. r 0 :j t,...,i_I, i t,...,2
z = nLrnber of add, parameters
(6)
i.e. for a diagonal vJeight matrix PB the optinral choice of A4 is obtained, if the
columns of A, are orihogoral, not necessarily orthonormal . (Nence orthogoral ity i s
defined as : A'PA = D, D= diagonal matrix).
Although the concept of orthosonality is strongly valid only in the case of extremely
regular network arrangements, orthogonal sets provide in practice for the
hc (i n.(

o"@
-Fb--+
Fi gure ir Regul ar irnage point screens
@...
o...
3x3 di stribution
5x5 di stribution
For vertical photography and flät object this set i s outl ifed to be even independent
on the exterior orientation elements, lJnder very general conditions (large
rotdtion e'lements, space character of the object) the corresponding sets can be
eltended by some of the 6 coefficierts
which are highly correlating with th€
6 exterior orientation elements in the case of vertica'l photography and ilat obip.r
-1 1-t>-. ' -1r lF.. sFr /hp-6 rhF o,-a-eTe-q of inrerior orierTdtion
are also incl!ded). l,lith the creation of the complete set one has to pay
Ettentron to the sequence of the rotation elements. For exampl€, if r is chosen
dependert on ( and o ther one of the two ljnear terms |,{hich lead to image torsjons
has to b€ rejected a-prr'ori.
Eecause of th€ possibly high correlations the extended sets have to be tspplied
very carefully. To protect a!ainst overparametrj2ation, j.e. to avoid the introduction
of remarkable instabilities into the linear systefi (l) sophisticated statistica'l
test methods nust be used for ädditional paraneter sjgnificanc€ testing
ir any case (see G?üa /1t/, /ii,/ and also sectior Dl of this paper).0ne has to
note thät the 1-dinensional Student test leads to vrrong concl usions concerning
t.e €(Leptdlc€ o- yelect'or o' irai"'d"dl
roo'Ttoral Dora-elP"s 'f dBngero's corYelations
do occur, Thus a more comprehensive treatment of statistical
hypotheses
C. Accuracy and rel iabil ity of close-range bundle systems
Nitherto in photogranmetric close-ran9e systems the accuracy was almost exclusi!el
jn the center of interest. In ttris conneciion sometim,.s d lot of |./ork was invested
to derive "accuracy predictora" v/hich should represent a k'ird of accuracy models
for special, often appearing net\,,ork arrargements. Nevertheless, among all accuracy
predictors the inverse of the nornal equatiors of a bundle system is still
the best one, It acconmodates €very changing netl,vork configuration and evefy model
väriation. So the design of a proiect should be conneci:ed with the deterfiination
of the corresponding inve.se for sophisticated accuracy studies. This becones important
more than ever, since the introduction of additioral parameter sets complicates
an empirical accuracy appraisal, Besides the acc!racy of a systen its
reliability
should become a mair objective of future studies.
The reliability of a model descr'ibes its sensitiveness with respect to model
errors. If a highly developed s e I f - c a I i b r a t i n I bund le model js used \,ve maJ restrict
our considerations on the problen of gross errors.

-6
5o b, je'ni o1 tle'€r- "-e'do'.i'y' sho" d oesc/toe rhe syste- s q,al'ties
for gro ss error detecti on.
It was laa?ja /2/, /3/ wha developed a rather complete reliability theory. Nis
dpproach enables the treatment of the gross error detection problem on a statistical
basis. tidrdr's theory maJ equally be applied to theoretical reliability
s Ld'es o' b,ndle ad'-srne.T sys:e-s as to rre derecr or or grass erro-s ' O.aLtical
projects. ior detaill€d information abaLrt the whole theory see ,/2/, /3/, 1tl
the following only a short extract is presented, €s far as it is necessary to
-.de-sLard t'e p-"cL'cd corp.rd'ior c pyererred i-r tfls paper.
Starting fron system (1) with the statistical
2
-e=Ax-l ; P
D(e) = D(r) = oo2p-1 Ete) = 0
t -l r
(A'PA) A Pl = qlxA Pl
nodel
variance unbiased estinators for x and
(8)
reounaancY r = n-uJ
I tol-t ttoror-, t = $r
and for th€ residual s we obtain
and with
v-A,-l -- 1-A0\^!'P\l
(10)
vle get
Quu = P-1-AQrrAT
v = - QvvPl
(11)
(\2)
Regarding nrodel (8)
the test criterion
o2, th"n if Ho is true
as null-hypothesis Ho with E(i02 /Ho) = (13)
is distributed as the central F-djstrib!tion F(r,*). Thus e is used as a global
test criteriorn tottest
I (l ! N(Ax,c.'P -)).
the presum€d multi-dinensional normal distribLrtion of
1f Hc isreiectedone has to set up one or more alternative hypotheses HA.
Since the systemat'ic errors are widely compensated by our self-calibrating model
v/e may confine oLrrselves on the investigation of gross errors. Then a set of p
€_re"raT ve hyoorl^eses rA_ car be "o--ulated as
P = 1,..,P alternative
tOO, o,O = oo.O.O nyp0rneses
(14)
c" = vector representing the
r proportions of the
gr0ss err0rs
If HAh is accepted, then o is distributed
F(r,-:lp)
l,9ith the noncentral jty parameter ip.
p
as the foncentral F-distribution

-7
HA^ I eads to
'
tl;:/AA'J = Eta:/H") - "".2p
(15a)
,"2 = !-L"2
(15b)
).p depends on the type I error size d, on the type II error size F(),p)
(E(rp) = 1-P(II), P(II) = probability of acceptins
degrees of freedom r, -. For a given d,ß(ip), r we can find rp by usirg lcc?Ca,s
nomograms in /3/, pp. 2I-23.
If NAp is true, ther the estimators i a,rd i"2 are no mor€ unbiased. If the effect
of a gross erroT vector nlp in the observations upon the residual vector v has to
be computed then we obtain
A'lthough in practical
E(v/HAp ) = l(v/H
vvo = - Quu P rl p
projects we normally don't kfot anythirg about the vector
cp - it indicates the relationship of the gross errors - eq!ation {17) enables
us to study the effect of an a-priori supposed qross error vector nlp upon th€
residual vector v. From (14), (15), (17) and from the idernpotency of the natrix
product QvvP it follows for small gross errors
(16)
(17)
rO . Noro2
no = cf rouurc o
(18)
The vector !l
p contains all
orl y one gross
error in the
pt h al ternati ve hypothesi
we ge!
HA1:
rlg(r) = o-.tot
(re)
and l4i th
T
(0,
.0,1,0,.,.,0)
t1A1 :
!t 1(i
) -
(20)
combinirg (18) and (20) we
r,e. !11\'/
tl^.,
rl'{i)=
r1
rs the minimal gross error !./hich car be det€cted wr'th the probaility
B(r1). Thus for a constant rt for all possible i !11(i) represents a suitable individual
reliability indicator ("fieasure for internal reliabjlity" in 8dcüa /3/)
Furthermore, taa!ia /3/ has developed a t€st criterion for testing the residuals
of an adjustment (" d a t a - s n o o p i n g ' ) . If Ho is rejected, we get
_.I
",,
t' n
- =-:-.- 2? )
!iith one of the possible alternative hypotheses (c1,..., cp) = I from (14) and by
using a weight matrix of diagonal forri \{€ obtain
for diagonal P:
' N1 = Pi2 qvivi ' Gr)
Qvivj = i!r diagoral
el ement of Qvv

-8
*1 --i-
-o \qvivi
t,z3 )
The acceptance i nterval for wr i s then
- ril, 1r -.", r,-1.,r.. r'/, t,t -'.,t,*)
\24 )
\Jhere oo is the type I error si:e for the test cr'r'terjon wi.
'Tevel
Tire significance oo for th€ test of individual residLräls (23) is dependent
on the significanc€ l€ve1 d of the global test (13). ;acrda harßronizes o l'vith sö
by putting t(r-) = p()r), i.e. by using equal type II error sizes for the global
test and the individual iest. For the relations between d and öc see the nomogranrs
itt /3/, pp. 2I-23. As far as the author could !nderstand thi s, the harmonizing
between o and o
was based on the assumption that the single pvents wi are independent
on each otirer. Nence a most
rel iable system is required for ar effective
applicatior
of the data-snooping techniquei otherwise the danger of false
inference is too great. At least we have to exclude those observations from the
test procedure - or better: lre have to change the network arrargement in a ray
that those observations don't occLrr - vlhich are correlating too much |lith others,
This is valid especially for "spur' observations, i.e.
for observations which
lead to o,, = 0 (zero variance problem), sr'nce in the extrefie case of spur observations
the test criterion lar= 9 is not defined.
lil 'vras
Irl (21) tlj
denoted as an individual reliability indicator. The amount for
rne compurdrron or dr rnoivraual N1= Plevivi (if P is diagonal and if only one
gross error is assumed) may sometimes prevent of individual conpLrtations. However
r/e are able to make some statements on the global reliability
of a system (see
Fi)rstne! /71, '.!rr /1i/, Pope ,/17/).
1n Gvür /11/ the average diagonal term of Qvv was proposed to serve as g'lobal
rel iabil ity indicatori
tr10 )
Rl 1 = ---!l{ =; .-5"
nr1,';= 2tI(qiu) ,
tr"raJ
xtty J = ---r
t
(25b)
r = redundancy, n
= number of observations
0f course global reliability
indicators can be constructed whjch do more refer to
the "exterral" r€liability (Ba. e /3/), that means to the effect of gross e.rors
upon the final
prod!ct of the ddjustnent - th€ obiect point coordifates. Those
would correspond even more with the follovJin! accuracy indjcators.
But here th€
term "reliabjlity' should be stronger connected with the detection ihän with the
effect of gro ss errors.
rr o1d,o9ou- 9i0od. o.(u'd.) rc_cat0r ,orra u" t'.ll'*), yet we Lrse the average
standard deviations of the point coordinates (l,vith co = 1):
Ar (T) =
Ar(x) =
tr (
! k: number of points, including (26a)
the control poi nts

Ar(Y) =
,.('\fd; )
--- t26b)
Ar \7)
In the follo!'ling a sinrple practical e,ample is introducted to clarify the above
mentioned problens and to show the practicdl potential of the datö-snooping
techniq!e.
Although in C?ü, /ii/ tdee's approach of the gross
emofasized, n 1l's paDe.:-.:: s opp.odcr 's ucea because the values of the
Tau - distributiort lP.pe /i?/) are not available for the author by no\1,.
Another interesting method was praposed hy Eer-r';Lg/4./ whjch shoul d al so recei ve
future atterti or.
for '-rITe. de-o sr.a-io' rhe p"36a',u, e,d"ple o'
on synthetic data.
z=ro-r
z= 5+
.//,/
4... control points
o... points to b€ determined (ne\,v pojnts)
a.,, camefa stations
Figure 2: Net!'lork arrangement for the denonstration
of acc!racy and r€'l iability problems
lle suppose a cube which contains 27 regularly distributed points;8 of them serve
-'le
as corrro- pcirts. The orrers aye nek po"Is. camerä sta!ions a'€ de'loled by
nos. 0, ...,8.
Table l shows the orientatior elements of the individLral inages.

10
Table 1:
0rientati on el ementsof the images of the netl,Jork arrangement of
Figure 2
Images
0
I
2
3
4
5
6
7
8
c
["] [']
100
100
100
100
i00
100
100
100
100
0
-1
l
5
0
0
0
0
.Yr
LfiI
0
0
0
0
0
0
0
0
0
z.
['] tir ral
a
5
5
5
5
4
6
0
10
0
0
0
0
0
0
0
0
0
0
0
0
-24.483
20.483
0
0
0
0
0
0
0
0
0
0
0
20.483
-24 .483
l,Jhile the object poirts and the control distribution
renains unchanged in the
fol lol,ing computational versions, the images are grouped as seen froßr Table 2,
For simpl ification al I versions are conrpLrted ur'thout additional paralrreters and
with fixed control points (diaS (Pp) + -).
Table 2: Inrage arrangements and global accuracy and reliability
irdicators
AI(X), AI(Y), AI{z), AI(T), RI(x'), RI(y'), RI(T);.o = t um
I
c
D
E
l-2 t08
3-4 108
3-0-4
1-2-5-6 216
3-4-7 -8 2r6
69
69
75
81
8i
39
39
81
AI(X)
t"t
4.21
0.09
0.08
0.18
0 .07
AI(Y)
[.*]
t.r7
0.29
0.28
a .92
0. 21
i cato
AI(Z)
["]
a .27
0.11
0.10
0.18
0 .07
s
Relia
AI lT]
LnrmJ
RI(x'
0. 57 0.19
0.16 0.20
0.15 0.43
0 .43 0 .62
0.1? 4.62
ility
RI(Y'
0.s3
0.52
0 .64
a .62
a .62
indic
Rr(r)
0.36
0.36
0.54
0.62
0 .62
number of observations
number of u n knov/n s
r -u = redundancy
R I(T) Figure 3:
i.." ri. v ih/li.:t^hc.f tha
n/:.t1.nl a$mnl !( .f T:hlp

11
Table 2 together llith Fjgure I show the global reliability
Some interesting conclLrsions can be drawn from these inv€stigations,
Comparing
the Slobal irdicators
änd acc!racy indicators.
RI(l) and AI(T) of the individual network versions
l,le notice t höt good accuracy doesr t correspond necessarily with good rel iabil ity.
Though version B provides for fairly Sood accuracy, the reliabilitJ is poor. This
refers mainly to the reliability of the r'-coordinate observations. Here we have
to state that if only two images ar€ available (see al so versior A) a gross error
of those inage coordinates lvhich belong to the epipolar plane cannot be detected
at all (zero variance problem - 'sp!r' observations).
0f course this is not trLre for control points, so that RI(x )of version A, which
Tfe ) _-es.dua-s (w'Ti o .I:r) ts ro! eo_a_
to zero. Though the x'-coordjnates of version B don't belons exactly to the epipolar
plane (d,=-{,. = 20.4819J, their reliability is still poor (that yields for
the non-cortrol points), 0n the contrary the reliability
of the y'-cöordinates of
versions A,B is s!fficient (A:RI(y') = 0.53,8;RI(y')= 0.52), i.e. a gross error
ir those y'-coordinates has a good chal1ce to be detected (see the subsequert
exanples of data - snooping); bLrt here another problem reve!ls:
gross el.ror local ization.
the problem of
E.ianrple C (l images) provides for better reliabjlitJ, especially in the x'-coordr'-
rEtes (ql('') - 0.41). Ir rFis case we dre .b p Lo deiect ä gross e-ro" ever 'r
the x'-coordinates of non-contrcl points.
Theexanples D and E sho!J an equally good reliability
behaviour in x'- and y - coordinates,
what is due to the symmetric network arrängement, |,/hile the accuracy
of \ersron L r s fEr more berter
Summarizing the experiences gained by these investigations and indicated by the
applied accuräcy önd relidbrlity indicators \'e have to prefer definitely the
arranSement E (images 3-4-7-8), because only ir this case the x'- and yr- observations
are of sufficient
object poi nts.
lvoreover. tlese
cLrrdcy and reliability
reliabil ity together !,,r-th a satisfactory öccuracy of the
ir /es+iSa'_ons mdy s ou !!e necess_Ty of oe"ormirg d-Dr o_: a(-
studies for a proposed proiect, This becomes the more nec-
:'F1 rrn-ir:'t ( "'P Ar a rLle ofe
should aspire reliabil ity indicators whjch ar€ at least equal or even greater tlran
0.6 (RI(x') :0.6, RI(y') :0.6) - in accordanc€ with the values Sained by the investigation
of aerial triangulation systems (trrr /ill) .
Natural'ly the global indicators don't provide in each practical case for sufficiert
accuracy / relr'abiljty of all individual object points / observations, e.9. if
sofie observations have to be cancelled or if points cannot be observed from certain
camera statiors,
whatever the reasor may be for that. So the global investisation
has often to be replaced by an individual checking (see (21)). fo get further
insight
in detail problems and to become famil iar with the data-snooping technique
the arrang€nrents of Fjgure 2 and Table 2 will be subject to the data-snooping
procedure, To leep as close äs possible !o practic€ the synthetic imaSe coordinate
observations of al I vers'ions are superinrposed by a random generator l'itil neans
.0 ard s.arddrd devtariors , - ..,.
r t--1.
Then in all versiors A - E different gross errors are irtroduced as {the gr0ss

12
errors do exclusively appear at images no, l resp.
3 and at points no. 1 resp. 2)l
Ca se a
0ne gross error at the jmage coordinate
of inrage 1 resp. 3: rxi = - 30 lnr
of control point no. L
Ca se b
Ca se c
Ca se d
Two gross errors sim!ltaneously at the image coordinates x', y of
of inage I resp. 3r ryf = - 30 unr
control point no.1 of image 1resp.3: txi =ryi = -30unr
j.>-o
.l, adtnaIe l' of non-control point no.2
of image 1resp. 3: txi = - 30 unr
0ne gross error dt the image coordjnatey' of non-control point no.2
Lirl -espect to case o tre act as i or_\ one c/oss e-ro. e)jsrs, d Lho,gr Lwo
ir!r1:
-ad.
no = 0 '001 {0,Ii)
E(ip) = E(),1) = Bo = 0,8 (801;)
io = lz.r ,\n-= 4.14
Critical
value for data-snooping (with o
Hence the minimal gross error \4hich can
lwith P- = I)l
",1i)_ li==
" \'vivi
= 0.00I) from -n-q -): c{w.)= j.29.
rF'o"e- Llrh 'rF -v.hrfili-r js
(27 )
val ues
obtain for the different arrangemenls an0
of Table 3 for the ninimal gr0ss err0rs.
Table 3
:[linimal
gross errors to
be detected wi th
the Drobabil itv B = 0,8.
Arrangemen!s
rxi(min)
Ir']
Ca
txi('nin)
Iu']
eb
L!t
[a se c
rxi(nrin)
[u']
Ca se d
!y2(min)
lu'l
B
c
D
E
28 .3
29.6
25 .5
27 .2
28 ,3
3r .4
29.6
25.5
27.2
27 .2
29.6
27 .7
25.2
26.8
48.5
3?.4
31.3
32.0
32.4
27 .3
26 .2
From Tabl e 3 Vle see that the minimal gross errors are grouped around 30.m,
the cases Ac, Bc, Cc. That's th€ reason l'hy !i.e have introduced for all
cance I evel oo for data-snooping was kept corstantly,
excePt
cases a-d
a gross error of -30 !m,
After the conp!tatjon of all error cases wjth all netlaork arrangements we have got
rne.esLlrs o{ tao'e + -o- .1e s oDo- Les LFo: E'61. =,1, Rar: L (a +h.

13
the global test - taken from tccnc'a,s
tit 0.45.
nom0granr s 1n /';/ - is varyin9
f ron A .22 rn-
Table 4: Global test of the null
arrangenents and gros s
-hypot hesi s N!:
E(t2) = c2 for atI
Arrange-
A
B
c
D
E
6 for gross error
2.06
4.92
"i.]c
1.30
0.85
2 .51
1.07
1.38
i.58
0.98
r.40
0. 57
1.13
1.28
0.83
1.75
0. 99
7 .23
L 34
0.94
c(e)
1.18
1.18
1 .02
l.a2
a .22
0 .22
0. 37
0.45
gross ef r
1.40
0. 57
0. 96
L,2L
o.72
. i3
a6
c(€) fron F (1-a,r,-)
Regarding the results of Table 4 r€asonab'l e doubt is p!t föruärd with respect to
the practical application of the -olobal test.
In the gross error cases both versions
I and E don t lead to the correct statenent, i.e. to the rejection of the
n u I 1 - h y p o t h e s i s , Though there are fairly good p re s Lrmpt i on s, bec au se the expecta -
ti0n E(i;) = c: is a-Ffi0ri fiown by the utilization of a random generator even
under the null-hyfothesis {|,{hen no gross errors are iicluded) the critical vä'lues
are exceeded tvJice. 0f course the random generator produced di screte random
val ues, whose standard deviat'ion differs fiore ore I ess fron the a-priori choser
value of 5 rfi, bLrt this probl€m is ever intersified in real practrcal projects:
llhat is ther the expectation of ä2? ThEt is probably the reason why ?ape /1?/
doesn t use the global test - generally its pot,er of assertion is not sLrfficient,
Nence it is not reconmended in practjce to make the setup of individual tests
( d a t a - s n o o p i n ,o) dependent on the resul t of the 9l obal test. The procedure of datasloot''g
'lould te applied ir an) case i' -easo'rab'e srsp'c_o e,jsls Ltso g_oss
errors may appear, v/hat is the standard situdtior
In this way vre contin!e \{ith the irdividual
at least partly the disturbing effects of correlations
VJ
l4i=-tsmaxrmat.
"vi
As a result it l4as found out that alI a-priori
knovrn gross errors have been detected,
€xpect in the follo\4ing cases;
in prdctical projects.
checking of the residLrals. To avoid
bet\'een the residuals the
those observations, for \,rhich the test criterion
1, Gross error case c in arrangement A and B: zero variance situation
2, Failure vJith the gross error case a of arrangement 8:
the xr-observation of point no. 3 at image 3led
thus to the prior r€jection,
to flrax (\{i) and
although not jncluding a gross error
3. Failur€ with the gross error case b of arrangemert B:
the xr-observatr'on of point no. 9 at image 3led to max (\,ri),
although not includjng a gross error

- 1.4
4. Failure \,Jith the gross error case c of arrangement D,
v,here the test criterjon wi = 3,15 of the x -observation of
point no.2 at inrage l remained slightty urder the crjtical
value of c(1,.lj) = 3.29
Analysing these fail!res it becones evident that in 1. there is a-prjori no
chance for detection at all (zero variance problem). In 4. the situation |ias net,
l,]here th€ actual gross error (30um) is s'naller than the ninimal gnoss error \ahich
can be detected wr'th the Frobability Eo = 0,8 (see Table 4). Setting rxi = - 32.4fßr
in version D the corresponding test criterion exceeds the critical value, thus
leading to a correct rejectjon.
The false r€iection of x! at image 3 jn case Ba afd of xb at inaSe 3 ir case 8b
is probably a r€sult of shiftings caused by correlations. Actuatty the preserce
of correlations distLrrbs significantl] the statisttcal foundation of the gross
error problernr so that correlations
should be subject to future studies, The
correlations becone particularly of interest in a doLrble sense:
- For the rejection procedLrre of data-snooping. The ans\1,er to the question
vJhether orly max (\1i) or all \,i v/hich exceed the critical value should
be rejected depends on the size of correlations.
- For the locali?ation of gross errors the correlatr'ons are of decisive
jmportance. Consr'der our practical e{amples: ln the cases Ad, Bd the
localization of vyi is impossible since the residLrals vyi (image i)
and vyä (ima!e 2) resp. vy, (image 3) and vy, (imaSe 4) are perfect'ly
correlating,
x'-residuals
Stronq correlätions exist also lr case Cc bet!/een !ne
of a non-control point jn all three jmages, Here |,/e obtain
for the correlation coefficr'ents of the xr-residuals of point no.2:
rv1 f:lvr, fOl=
- 0.960 , rvx , l3l vx , l+l
= o. sso , ru" ' lol vy' l4l = - 0.960
(ifirage nunber in brackets)
with the result that the gross error rxä cannot be locdlized.
Experien.es gained so far Nith the data-snooping technique are very enrouraging,
expecially if its results äre compared vrith corventional , non-statistical
of gross error detection.
methods
Fron all 23 gross error cases of our examples (the 2 zero variance cases are excluded)
\,ve obtaired by corventional methods and by data-snooping:
So it becomes ar ir"jportant and promi sing task to do further investjgatiofs in
_are_r
th's "eld o'"esea"ch ro a"-ive
ör slsre':s for point determiration pro-
!_dj.9 to. both öccurac) and -e_ iat'_'T\.
lvi> 3c.: 15
lvl,3ä": t0
t9
da!a-sn0oprng: .^hrar + roia.tr.n<
T2
i ncorrect re jet ions
incofrect rejectiofs
i ncorrect rejecti on s

15
D. Statistical test criteriors for parameter testing in photo!ranrmetric point
deteimination pro b1 em s
Recertly photogramm€try uses mor€ and more the methods of statistical
interval
estination and htpotheses testrng. Thereby statr'stical test criterians ar€ som€trries
applred in d little
car€less manrer, vrhat maJ often lead to wrong decisions
concernin! the acceptance or rejection of a hypothesi s.
So ihe problem of statistical
hypotheses testing is treated ir a nore general way
and sone test criteriors are derived whr'ch ar€ often useful for the analysr's of
'es- ts oDIa,leo or ^eI'ods o' oro'o9Yom-et_'c oo;nI detevrr'äIjor.
At any rate one should alvJays keep ir mind that a stati stical test cannot be an
end in itself, it cannot serve as an jnevitable standard, but m!st orly be regar_
dec as an aid for decision,
For hypotheses testir!
adj u stment, derived iron (1)
\le suppose the I inear mod€l of self-cal ibrating bundle
E(l) = Ax ,
r(e) - 0 , D(e) = D(r
(28 )
wjih the general I inear nulI -hypoth€si s of full
rank
where B is a bxu- matrix rvith rank(B) = b: u.
The nrinimunr variance unbiased estimators for x and
- -t -
i = (A'PA) -A'Pl
r-=:rAi-tlplA/-t)
T=:
b ä:
-1 - -1
R = {Bi -w) (B(A PA) B') (Bi-w)
oi from (28) ar€
For statistical analysis th€ distribution of I is assufied as
i.e. a n-dinensional nornal distribution.
l!N(Ax,
d.'P
(2e)
(30a )
(30b)
Testinq H reourres the derivation of a test function T such that the distribution
of T is knolJn l./hen Bx = !'r(r',e. the null-hypothesis is tru€).
Funciions 11ith optrnral properties are the Likelihood ratio functions. Th€ir appli_
cai'ior leads to the test criterion
For the derivatior
of T see
If fo is true, then T is di
1o obtain the po!rer of the
distribution
of T must also
1!a!b-;1,i ./-Q/, i.ach /131.
stri buted as Snedecor's F(b,r)
test Ho against an alternative
hypoth€sis HA the
-),
(31 )
132)
be known, if the alternative hypothesis HA: Bx I t1,
(Brx = vi') is true. Then we fifd
the noncentral ity parameter
T distributed as the noncertral F' (b,r,r) wr'th
TT -tuTl-11a,x_w)
r = -! lB)(-wl ' lBlA PA\
2ct
(33)
The power 8(i) of the test
beta di st|ibution (Cra.lbaLL,/:)/),
'.'_es of The norLe.t_al

16
The general ljnear hypothesis (29) respectively some of its simplifications can
be dpplied to quite different p.oblems of photogrammetric point detefmination.
lr this paper vre \rill restrict or three test problems, often äppearin! in closerang€
photogrammetry, as significance testin!
add'itioral parameters for systematic error estimation
residual s at
empirical accu)acy studies
check points and control pojnts for
coordinat€ differences in deformation neasul.ements
(movement studies).
of
l. S g i,=carce testin.,' o'.dd''oral oa.a.ete-s
-o c!ecl rfe srol_sri(a_ s'gr_rrcä Le o'a' .ddiL'o1öl pa-ometer set. ,e. to test
wrer.er a srsre-a'ic
_ _lue.ce coes oc ," dl a . !r'e tave to corsrrucl a global
test for the whole set of additional parameters. For that purpose l4e put with
'. :t io the nomenclature of the unkno ns ir {1):
Thai is, v,e corrrpare simultaneously each individ!al
(34)
additional paraneter dzk
(k = l, ...,p) lvith a supposed !aluedz't. If ro further rnformations are
available,
dz
p,t
e.g. from p r e c a I j b r a t i o n s , dz' = 0 r'Jill be chosen.
Thus the te st cri teri on T becomes
T -1
- - --, dr-d: A?, di-dz
I = Identi ty nat|ix
F = number of additional parameters
(35)
According to (L) the symfietric matrix N of the
norrna I equations is
rlrrr, elrre,
./. rlrrrr*ro
.t. ./.
AJP-Ar]p^r.
nlp^n,
n]p^n"
e] n ^r'., rleoro
3"--t..:.'::.::_
. /. irltrro*r,
(
N", N'
t l, t,,
(35)
Then
. = N - N N -N (37 )
is the partl y red1r c ed matrix of the normal eq!ations
The noncentral i ty param€ter for the test Ho i s
N for the set of additional
td-a-eter(. 1''hp addi'io'al p.rar€te.s a_e arrd"ged ri +h. ch.l ^+ tha..nnl.ta
vector of unkno\,/ns as it r's supposed in (1), the'r a;] is obtained withoLrt addilt,
tjoral effort during the trian!ular factorization of
(38a)
, = J- G,-ar, )Tr,rrr qar-dr, r]1 rcr-ar' yTrT,lt-1lt*, {a, -a,' )
(38b)

17
The power Bli) is an increasing furction of ),, so i should be as lar-oe as possi bte.
i is naximal if N*, = 0 lcre'!b':ii ttt/),:..e. if dz is orthogonal to dx, dxP, dt
;hLrs a statr'stical advanta!e of the ortho!onal additional parameter concept becomes
evident: it jncreases the po\./ef of this test.
Eeside this it is clear that th€ power of a test depends mainly on the alterlrative
hypothesi s dnd on the type I error size d.
If fl. 1s rejected, i,e, if a significani systenatic influence is apparent, the
detection of individual significant components becones necessary.
Ther€fore the sei of hypotheses
rji) , a.t = Iti (i = 1, ..p (J9)
i s commonly used to test the individu6l
lience v,/e get the test criterions
pa ramete r s on signifi cance,
T(i )= td2j-dz )2 qzi zi
= vari ance of the i th {40)
'.'rzizi
additr'onal parameter
lnaer rjl) tre J- values are distributed ds St!dent's t with r degrees of
the m o r e - d i n e n s i o n a I T-test (31) is reduced tö an one-dimensional
In the case of independence of the individual
paraneters the type I error size d
0f the individual test is related to the type I error sjze o of the global test
l- q = tr
- a/ (41)
Tf- main problem in testjnc subs€is or ever sin!le parameters estimated in nultie
sjo.a_ mooc s a. oc wirl'.e depe cence o'rrese suosers.s'rgle oa-öTete,s)
- the other model paranet€rs. Ir the case of si!nificant correlations the probabrlity
E = P(T{i)'c" Hji)) of rejection of a trLre null-hypotnesls lli):
dz. = dz, of a sirqle event (parameter) is no further independent. It should be
noii€ed that the appl ication of the one-dimensionaI t-test
with the usual limjts
and corfidence irtervals leads:he more to rrrong decisions the more the correlatjons
do increase. -ihus we have another important argument for using the c0ncept
of orthogonal additiöna'l parameter sets.
,he-eas r'e or..ogonal co.cepTs are -e9a'deo as veyJ use.L rhe p-a(ti.a gFonreirical
conditions ho\,{ever often do not provide for sufficient orthogonal ity.
Then $e have to set up sinultaneous corfidence intervals for the single everts,
This can be done by the a-posterjori o r t h o 9 o n a I i z a t i o n of the additional paramet€r
set, j.e, the transforFatr'on of the additional paraneter vector (or, if
necessary, even the vector of all !nknowns of the bundle systen (1))into orthoqonal
components, which ca'r be tested independently (see .rog, Brs" /i3/, teize(
,.1,/),
So_et'_res rrgr co-relör1o.s oo aDpear orly uJ"f 'n a s-bser o{ The aoa irio'1a_ pa -
rameters. Then these parameters can be tested together on common significance.
The corresponding test criterion may analogously be derived from the general I i -

1B
near hypothesi s (29).
Here a bivariate test c)"iterion is derived, which can
nal parameters are highly correlating.
The null-hypothesis is formulated as
I dz = dz'
2,2 2,t 2,7
two additio-
(42)
Suppose dz'
= 0 and dz'= (dzi, dzk), then T resLrlts in
- i ^a ^'" I-z1t' c7121
-,:. \".i..I/ l
\92'zi az"z"
)'(l;,)
The application of varioL.is test criterions and a possjble strategy for additional
parameter rejection l,as denonstrated ia a!Ln /1a,/ l'Jith the use of a practical
(43)
aerial triangulation exanple. Because of the extensiof ofaerjal triangulation
systems one has to work with some neglectings concernin! the application of completely
rr'gid statistical tests; thus the concept of a-posteriori orthogonaliza_
tionoftheWho]esystenishard]ytorea]ize.|]oVlever,c]ose-rangesystens0pena
vlide dnd irteresting field of application of sophisticated statistical tests. So
this subejct should be a nain objectjve for further studies.
2. Sjgrifjcance testing of fesiduals at check points and control po'ints
It is a vlidespread and usef!1 procedure in empirical accuracy studies to c0mpare
the set of coordinates of a photogrammetric adiustmert vJith their "tr!e" values,
mostly obtained by more precise observation fiethods. Then the r.m.s values of
the residuals serve as absolLlte accuracy 'neasures to check the photogrammetrical
ly achi eved accuracy.
If we denote the estimated residual vectors at check p0rnts as
,iT
.iT
t7'
(rxt'...
(rY1,...
,rXpL )
(121,. . . ,A2",)
PL = nunrber of Pl arinetric
cnecK p0rrrs
pZ = number of height
check pornts
(44)
\,vith the cofiponents
'"i
aZ t
"i
-
"i
.. s .)P h
xt,
i = 1, .,.,
iPh aPh iPh -
PL
Pz
esti mated photogramme- (45)
tri c coord i nates
'true " coord i nates
t hen t he
ifr
r.m.s. val ues
oiToi
pL
"2
,ir.i
F1
t2
, tz
- o2ri -\
(46)

19
!1.Leiaa! /'i5/ has proved that in the case of onl.v random inftuences the estlmators
p !i, p, ti, p,ü:, 2p r"i ,, a.e indeed urbr'ased, but nat sufficient estimators for
the corr€spondins parts of !he trace of the v a r i a n c e - c o v a r i a n c e riatrix of the
-.. 5,
'ihe r.m. s. errors lestißators) are often Eccepted to be uibiased without any statistjcal
investigation or from tre plots of the residuals a sJsternatic deformation
is stated ar nat. r:.i.elddr,,/1.c,./ has shown, that a sophisticated statistical treatment
is possible to get a better decr'sion basis. He interprets the photogrammetric
adjusiment as an adjustment ir steps and is thus deriving the test criterjons.
ir the following the corresponCing test critefions
are derived vJith help of the
concept of the Seneral linear hypothesrs (29), Ylhich seenls to be a little
clearer and which makes evident at the same time a suitab'le conputational strategy.
lJe supoose the sta srlca_ iooo: (28) ano srarr 'ro. Tle n,ll-h/porhes_s rha'lhL
common residual vector dX = (IX , r.Y ,12 ) is m u I t i - d i m e n s r' o n a I norrnally distrib!ted
with dX-N(dX, o:4....). 0..., is the l,leight coefficient matrix of the esti-
bit
Thus Nc becomes
f-:
'
I ax = xS - xN
P'p P'l p,1 p,1
D = 2!, + p_ = total
r...rirri
cs
\17 )
d . s,b-q"arr_ry for rhe checl oo'.ts o'd^ (see (l))

-2A-
;o obtair a better insight in the character of a systematic defornration it is
necessary to test individual qroups of residuals or even sirgle residuals. For
+öc+ --it€rions can al so be derived as it !'Jas demonstrated
for additjonal parameters in secrion D1. But notice the difficLrlties arising
with high correlations with respect to the confidence interval setup.
l:
's an unp_easarr e)per_e'ce rfar :le radeoJd'e o-e"'. ior o4 Lo'l.o oo'ats
does oft€n disturb the photogramnetrically obtained coordinates, thus leading
to rvrong conclusions with respect to the p h o t o g r a m nr e t r i c accuracy potential, The
disturbinq inf'l uence of control points may be caused randomly or systenatically.
flolJever. a statistical
treatrient of the resrduals at control points is a h€lpful
device to arrive at better conclusions \4ith respect to the control accuracJ in
0f cours€ it is not eäsy or often inpossible to seperate the photogrammetric
€ffect from the cortro'l influence and to seperate the random control errors from
l p syslema :- ror t.ol e"-or\, BLl - a s e I ' - c a I i o r a r i r g ddjusLment 's peY{ormed
vre can assune that the nain photogrammetric systenratic influence is compensated,
so that the residuals at control points mtsy iidicBte at least d part of the
origi nal systematic control deformation,
: clobal test on th€ si!nificance of control residual s can be based on
. P D'
: I dx = dx' , d = rußber of control (50)
(10,t - P - n,
Y 0,' l,l I ll 0e aSSUre0 aS -^
li!n.a u,. nat +h! rc

2t
3, Significance testing of coordjrate differences in deformatjon measurenents
The determination of point movements requires repetitive
object measuremerts to
various periods. If the observation progran is exöctly identical durirg the
different periods the fiul tivariate concept r s an appropriate model for point
estimation, interval
Generally the corresponding repetitive
estimation and hypotheses testing.
observations are not irdependent. In thr s
case the ccvarjance rnatrix of these observations can be estinated immediatly - on
tr€ contrary to the univariate adjustment, |./here usually only the variance of
urit
l]reight is estinated.
For multivarinte
estimation problems dnd hypothesis testins see Ande!san./1/,
!..4 ,/:a/. Statistically seen, the multivariate concept is th€ most general one,
but it requr'res design matrices lahich are identically for each observation period-
Since this is an essential practical restriction for photogranmetric problens -
observations can only be rejected if this is done for those belongin! to!ether
ir all periods, the camera stations and the rotation elemerts are not allolved to
be changed - Ne use an univariate firodel , adapted to the special situation of
deformation neasurenents.
Uithout loss of generality \ae restrict on two observation periods I, IL Thus we
get the model
- e = AIxt + AIIxtI - l
"=(:1,
D(e) = D
"l
r)
ll
lrt
I ttI
p-L
;',:,)
t(e) = 0
(54 )
Model (54), vrhich i s an
t he unkno\'rns. For exampl e, ihe solution vector of the control points can be refor
both periods. The same may yield for the vector garded as a joint vector
of
:arli+i.n:l
Accordi ng to the purpose of novement determination w€ set up the globel null -
hypothesi s (wi th model ( 54) and the notation of (1)) to
N-: B dx = l]v
" 3m,6n 6nr,1 6m.1
extension of model (28),
perfiits various arrangements of
f no coflrmon !nkno\,vns are !sed, system (54) can be divided
n = h"mhar.f n^inrc
+. ha i.intl ! ra(+ca
(s5)
The structure of B depends or the seqLrence of the unknov,ns in nrodel (54)
."cror of po r! coordi.ares ro be resred 's a.ranged _ ile
If the
axT = (dNrt' dxttt, dYI1, dYIIl
d7Y,' dzrn,
d x t l r
' d Y I r ' d Y I I r
. d z t r ' d Z u r )
then B is structured as
3r,6m
(i
-l 0.
i)

-22
dx = vector of coordinates
to be jointly testei.
bel ongi ng to!ethe. in period I and IT
'i he
T
test criterr'on -I resul ts ir
-R
Jm d;
(57 )
R
1ai, -ai, , -rv;T
T'^
(B QrrB ) (d/r-d,rr-'i)
(58)
d'I T T
dxtt
1dlr1' dYrl '
(dxut'dYrlr
d7t1, ..., dxtm, dYIn, dzlm) '
'dZttt'
...! dXIln, dYltr'dZIIr)
Q"* = joint u/ei9ht coefficient ßratrix of the point coordinates to
DE I.ASIEC
lilostly \a' = 0 has to be chosen (especjally if equal approximate values are used)
ASain each adfiissible sub-hypothesis can be t€sted. Nere it becofies particularly
important to test for significant deformations at individual points or ever at
special coordinates of indr'vidual poirts. After all vrhat has been shown before
it should be an easy exercise to derive those special nul l-hypotheses and the
c0rresponding te st criterions.
0n acco!nt
of the purpos€ of deformation studies it becomes very important in
this case to apply correct confidenc€ rntervals. Hence the a-posteriori orthogonaljzation
is ar indispensable demand to el irninate the disturbing effects of
hi!h correlations.
E, Concl!ding remarks
The relatively s'rall extension of the linear equations of close-range systems
often enables the application of sophisticated lineär models, estimation procedures
and statistical techniques. This fact shoLrld be utilized more than it was
done by nor{ to obtain best possible results with respect to accuracy and reliability.
Like in aerial triän9ulätion systems the s e I f - c a I r b r a t i n g bundle method
can provide for sigrjficant model improvements and thus for renarlable öccuracy
enhancementsr if suitable and highly developed additional parameter sets are
applied, as presented in this paper vrith the concept of ortho0onal bivarjate polynomial
s. Practical experiences with aerr'al triangulation
systems snow nowever,
that the functional extension of a bundle nödel by additional parameters must be
attended by comprehensive statjstical test procedures to protect a-oainst overp
a r a m e t r i z a t r' o n . Actual ly statr'stical methods sho!ld fi nd more attenti on in cl oserange
appljcatjo.s, especjally for test field analysis and in defornation measure-
A really virgin area but nevertheless a very important subject in close-range

- 23
photogrammetry js the stötistical treatment of retiability probtems.
As it yr'as shown in this paper ä statistr'cally founded strategy for qross error
detecti0n provides for a remarkable efficiency.
Suitable approaches for gross error detection on d statistical
and should obtain more attention in the near future.
base are availabl
ihus photogrammetric close-range systems |.lill reach an accuracy standard and a
reliability level \,rhich proviCe for outstanding results in each practical projec
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/1/ LnCexeor, T"t.: An introd!ction to mLrltivariate statjstical analysis.
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/2/ Eaa!Aa, it,: Statistical concepts in ceodesy. ttetherlands Geodetic
Comnission. Publications or ceodesy. Vol!me 2, Number 4, Delft 1967.
/3/ Baaldd, V.: A testing procedure for use in geodetic nett,lorks. Netherland
üeodetr'c Comrnission. Publications on 6eodesy. Vglume 2, llumber 5,
lel ft i968.
/4/ Eern1,\g, .: The analysis of Seodetic measurenents laith automatical
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/a/ Ebae!, I-: A posteriori v,eight estinatjon in general iz€d least sq!ares
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N eumä i er, l,lr €ir i978.
/6/ Ebne?) E, i Self calibratinE block adjustment, InviteC paper to the
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Lorq-ers of tre IS.. Co-:r-ss'or III, hets'n

- 24 -
/18/ Rd., a.P.: Linear statistical inference and its appljcations. John ltliley
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