NETWORK OPTIUIZATION
IN
INDUSTRIAL PHOTOGRAUUETRY
A. Chibunichev
Photogrammetry Departmant of Moscow Institute of Geodesy, Aerial Surveying and
Cartografy (MllGAfK). Gorochovsky By-str .• 4 .103064, Moscow, RuuAa.
Commission V
Abstnd:
The desiner formulates an initial network geometry. the precision of tbe image coordinate measurements and
precision of 1be object coordinates. A solution is obtained for the optimal geometry and tbe number of camera
stations. An ideal matrix of normal equations (tor required preclsion) h used as a target funcHon in the
optimlzatlon process. The optimlzation h earried out by a least squares method wich mlnlmi:zes the
discrepancies between the real normal equations matrix of a given initial networ};, and tbe corresponding ideal
matrix. Tbe constrains for values of tbe exterior orientation elements for camera $t~tions are introduced inlo
Ihe optimi2ation proeess by means 01 substitution of variahles.
KBY WOllDS: analitie. industrial. photogrammetry. design. network. optimization. erHellon. matrix.
o
INTRODUCTION
Ae1ualy, photogrammetry h widely applyed, Ior
investigation of various industrialo bleets. One of lhe
main problems in tbis cue is photognmmetry net·
worl design, whicb must provide requind pncision
o t 0 bleet po in t eoo rdtnates and a minimum Rumber 0 t
eamera stations. The design is eommonly sohred by
means of an interactive method through the process
of netwokr simulation and bundle block adjustment
(C.S.Fraser, 1982, 1984. 19(1).
Tbis paper deals with the algorithm of lhe analytieal
solution of the optimintion problem. wbicb parmHs
to solve the networl design automaticaly. The main
idea is to provide file required precision of file objed
point! coordinates by shUting the eamera stations. A
criterion (ideal) matrix is und as a target funcHon.
Thh matrix b formed the way an ideal eovarianee
estimated matrix of the objud point coordinates is.
K.R.Koch (1982) was (be first to uss a crilarion
matrix as a target tunetion for geodetie network
design. The similar approch tor photogrammetric
networks was uud by S, Zinndorf (1989), A.
Chibunichev(1990) and D.Pritsch. P. Grosilla (1990).
U a Ou 1ft a He a I 10 r 1ft u la Ho n a f Uu
phololrammetric networl opUmbaUon
Ki ..
O.
To solve eq. (2) it is to be expressed in a Taylor
nrias restrichd to linear hrms:
BA + L .. Y
(4)
with
iEV .....
j€.{l ..... ~;
k}; s is tbe number of pbotognphs
in a block; k is the number of abjeet points; n is the
number of unknown parameters A . m - 6lr. is the
n um be r 0 f e qua ti 0 n $; b,.) =~ a Te par ti Cl 1 der i va H ve s
of eq.(2) witb respect to unknown elements of
photographs orientation (X-!~. y1j ....
,;p;
nx '
Ni
[
flxYl
n~, 1
ny:, ny, n y2..
nZJ(~
nz\,; n z .:
Sinee N, h Cl symmetrieal matrix. any objeet point
g!ve 6 independent equations wilh 6q unlr.nown.
Here q is tne number of photos on whieh the point i
appears.
Optimization proceS5 is based on solving tbe following target funcHon:
K,: -
coordinates. Nondiagonal elements of matrix NiaTe
egual to uro. In more detail this problem h described in A.ChLbunichev (1990).
( 1)
Taldgn info account. thai
w her e Ki is are a I co v a r 1a n c e m a tri x 0 f t h e e s ti m a ted
unknown (0 object point coordinates,
is Ihe
co r re s po nd in g i d €I a 1 (c r it er ion) m a t r ix:. Ki i $ 0 b te in d
from phototriangulation using fhe results 01 network
Ki
N '" AT, A.
the
partial derivatives
(5)
of N
witb
respect
ta
the
~imulation.
( 6)
(2)
Tbe anaHlieal expressions of the coefficients uf the
matrix Aare very !limple for the collinearity equaUons. Therefore analitieal expressions of eq.(6)
an simple as weiL
wh e Te Ni. an d N~ are <: 0 rr es p 0 nd i n 21 y re a } a n d
criterion matrices of normal equations.
Eq. (2) can be sohed with respect to photouaphs
orientation elements whlen will correspond to the
required precision of lile object points. Tbe diagonal
elements of tbe <:rihrion matrix of funcHan (2) may
be computed in the tollowing expresslons:
o'pi!..
nx,,· m::'
p.<!.
()
n y.
'"
The optimization problem is solved by muns of tbe
least-squares method (yT y .. min).
pi!.
nz," mol •
(3 )
Xi
'm",,.'
z:
where JU 15 standard error of unH welght. whieh
caracterizes the image point eoo rliinates prechion; m)(
• mv • mz' is t he Te q 1.1 ire d p re ci 5 ion cd t h e 0 b j e e t po in t
SR
-./.'
683
The parameters of photographs to be determined may
be restricted to cerhlin intervals (becaun of tbe
camera format. the inability to move tbe camen to a
eertain position and so on) ean be expressed by the
ineq ualities
Xmin~ X"
deslng proceu of lhe photogrammetric network with
$Uch panmeten, The fint 1itep (starting conti·
iuntion of eamen station) is perfomed automaticly
on the buh of the approl(lmate tormula rehling the
'Iahe of ihe standard eUOH of XYZ coordinattu. to
tbe sc<!le of pbotoiraphy <!nd the prechion of the
image coordinatu measurements (C.Pruer, 1984).
The location 01 file intermediate tamen shitions Wili
obtatned as a result or lhe solution ot lhe target
function (2). In this case tue optimintion proeus
was interrupted because of the overlap restrietion
(the pncision of the objectpoint wu m~= 2mm, m~=
Z m m, m~ = 4 m m a t t his mo me n t). T heHl tu re t h e s ix t h
camera station was added and the optimlzation
(1)
X mQx
in whicb Xmin and X/)')(.\x are known vectors.
Ta avaid iOlving Il qUildnHc programming problem
we ean substitute file variables accordlng to following expusion:
X
= Xmin +
(Xmt\)( - Xm ,,.,) S i n2.tp
(8)
In this cue lhe formulu for the derivatives of
normal equations with uspect to the new variables
are:
(9)
~~
This approach permits to solve ahe optimization
problem without constrains ilnd in Cl more simple
way.
Bi ::
ff-
procl!ss
_...
vlJ.lues;.
CO r r ~ l; p n j) d Hi g Pi u ~ r iun In i 0 I II pli mi laU [I n I) t ,;." m e I (l
station confie:uration tor persolul computer (BM PC
was elaborated (0. Kortchaglna, 1991,. Chlbu'
n iche v. O. Ko rteh agina. 1992). T his pro gra mm permits to
fuUil lhe foUowing tuk,.; 1) The duinl of stuting
configuration of file totognmmetric network. This
proceu h pelfomed automatiealy tor simple shapes
o t the objech (cylinder. sphere. cube and so on) and
in interaetive mode tor lhe objecU otcomlex stupes.
2) The photogrammelrlc network optimiution with
ilutomatie change of photoguphi number in block. 3)
The bundh block adjustment tor checkine: rasults of
optimlzation proceu.
WJ;J!:
:i~re
IUt'
f.>.+..
c/
3
200 0.005
5
3
2000.0010.10.1
100
0.01
5
3
100
0.005
5
3
5
5
0.1
5
\
final eamen station desiJI1
Ir+-
1.2
0.9
2.1
11.2
0.6
0.4
0.9
0.1
0.1
0.1
5
5
11.2
5
5
11.2
1.2
0.9
1.5
6.9
0.1
0.1
0.1
0.16
--0
intermediate camen. 1itatio n durine: opUmizaHon
process
Then examples demoßstnite that this method may
be used tor photogrammetric network design. But it
h necessary to pertorm lU comprehensive Inves'
tigafion.
2.3
1.7
REPERENCES
Chibunichev A .• 1981. Conditionality or normal
equations in bundle block adjustment. ('iJlltiYIIR'leB
A., 06YCJI08JIeRJlOCT& 1I0pMaJI&JI&IX ypa8Rellli, 103'
2.9
DBlaJODJ,JU
1000.0010.10.1
--a
Pie:. 1 Step1i of design procus.
11.2
17.2
<I"
0
,
--t>
D-+
(m) (mm) (mm) (mm)
5
tlnlsh'fd
requind
starUng configuratio n of camen station
0-+-
simulation
5
'N"!~·
!.he
t
D
5
n
~
Table 1. Objed points preclsion obtalned atter opUmizaUon proces!
200 0.01
.,:t.E'
f
Some eKamples of photognmmetric network duing
is iUustrated in table 1 tor the eylinder 01 10'm
diameter. Here t.}I,mx.my.mz are the initial data. and
s.D.m~.m~.m~ are the ruultut the design (f h the
heus of the camen; s iI; fhe number of photos; D is
lhe objeet·to·camen distace; ~>m~,~ an standud
erron of object points determination after bundle
block adjustment using simulated photognphs wich
cOHespond to U18 optimal configuntion of camen
s ta Ho ns.
(mm) (mrn) (mm)(mm)(mm)
prt'd<.I'.:-n
I
EXAMPLES
req. precisio n
pr()Cf!~'l
conllnueo. Thi$
Opll
ypalllBllaJlBB
4'oTOrpaMM8TplI'le·
eeTel 110 ciIoeotiy e81130lL 14 3B. BY30B. "reO,ll;e'
11 a ~ po. 0 T 0 C'U Mr. 8 ", 4: 8 1 . 9 3 ) .
C1UIll
:3 IU
IN this table we can see that in some cases lile
prechion 01 the networl after optimhaUon h higher
t h a n t h e re q u Ire d p re ci s ion to r 5 p hot 0 S . T h e
programm reducu auto maHely lhe number 0 t photos
to 4, becouse the required precision permits to do it.
But in these cases fhe overlap restriction works,
theretore these varianU cf nelwork are considered
optimal.
Chibunichev A., 1990, Optimization of pbotognm'
metricnetwork duigl1 of industrialobjech ('iB6YIIII'
'leB A. OUTBMBSUUUI IIpoelTBpoBaJUU .OTOr-paMMeT·
PII'UCI.IIll C'IoeMOr. JlRlleSepSLIK coopYllesJIII. H3B.
8Y301 'TeO.lle31111 11 a3potoTOct.eMla", 5: 87'95).
Chibuniehev A .• Kortchagina 0,.1992, Algorithm 01
optimal design ot photogrammetric network tor in'
dustrialobfects.
(fiBcYIIBIIleB A .• Koplllauu8 O.
Let's consider steps of fhe designing proceu for one
of the examples. Suppose that f = IOOmm, p. =
o.0 Olm m. mx " my'" mz " O. 1 m m (t hel a tt er e l( a m pie in
h.ble 1). Figure I cleuly demonstrates stups of the
AJIrOpllTM
UPOUTlIPOBIIDBlil
M6TPB"I8C10i
C1>eM11I
BYSOB. ·TeO.l{e:3BI
684
11
OIlTlIMIUl>uoi
II11lieR8pllloIK
tDTOrpCIIM-
06t.elT08.
1438.
83po4!oTOC1.eMKa", 1992, N 4.).
Fraser C.S .• 1982. OpHmiution of Predsion in
Clou'Rang, Photogummetry. Phot. Eng. Rem. Sens .•
48:561'510.
Fruer C.S .• 1984. Network design considerations
tor non·topographie photogrammetry. Phot. Eng.
Rem. Sens., 50(8): 1115-1125.
FUSElr C.S., 1987, Limiting error propagation in
network design. Phot. Eng. Rem. Sens .• 53(5):
481-493.
FrHsch D .• CrosiHa F .• 1990. First order design
strategien; for industrial photogrammetry. In: ClosRRenge Photogrammetry Meeh Machine Vision .• SwHzerland. VoL1395. 1'1'_432'438.
Kortchagina 0., 1991, Some problems of the
optimization method of photognmmetric network
design tor industrial objecu. (KOp"t8UR8 O. Heu'
TOP&U BOIIPOCW 118TOMaTII'IeClOrO cIIoco6alIpoelTII'
POIUBBli
OIiTJiUIlaJlLBWX
Tel:
C1aeM1B
.ll:JlI
B Y 3 0 8.
'T e 0 .ll:e 3 11 11
4!oTorpaMMeTpJi'UCXBX
IIIlm.ellepIlWK
11
ce-
coopym.elllill.
H3B.
3;
10 7 '
IU po. 0 TU C 10 e M l a .. ..
1 10).
Zinndorf S .• 1989. Optimization of imaging con'
figuration. Photogrammetria. 43(5): 211·285.
685