TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.6, Jun
e
201
4, pp. 4648 ~ 4
6
5
3
DOI: 10.115
9
1
/telkomni
ka.
v
12i6.544
7
4648
Re
cei
v
ed
De
cem
ber 2
9
, 2013; Re
vi
sed
March 3, 201
4; Acce
pted
March 18, 20
14
Automatic Selection for Optimal Calibration Model of
Camera
Xuecong Li*,
Yonghua Wang, Qinruo
Wang
Schoo
l of Auto
mation, Guan
g
don
g Univ
er
sit
y
of T
e
chnolo
g
y
, Guan
gzh
ou, Chin
a
telp: +
86-20-
39
322
55
2
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: lee
x
u
e
co
ng
@12
6
.com
A
b
st
r
a
ct
W
hen c
a
li
brati
on
mode
l ex
pr
ess ca
mer
a
st
atus w
e
ll,
c
a
li
b
r
ation
resu
lt w
ill b
e
accurate.
How
to
select
optim
a
l calibr
a
tion m
o
del for
differe
nt camera
by pr
ogra
m
is
sig
n
ifi
c
ant to ca
libr
a
te auto
m
atica
lly
. A
meth
od
of sel
e
cting
opti
m
al
cali
bratio
n
mode
l w
a
s pr
o
p
o
sed
in th
is p
aper. F
i
rst, the mod
e
ls i
n
cl
u
d
in
g
eno
ug
h poss
i
bl
e status w
e
re selecte
d
from
physic
a
l
mo
del
s and C
h
e
b
ysh
ev mode
ls. T
hese
mod
e
ls w
o
u
l
d
be taken as th
e inp
u
t of our meth
od. Seco
nd, cand
idat
e calibr
a
tio
n
mo
dels w
e
re obta
i
ne
d by varia
n
c
e
detectio
n
. T
h
ir
d, opti
m
al c
a
li
bratio
n
mo
del
w
a
s extr
acted
by uti
l
i
z
i
n
g
d
e
tection
of
mi
ni
mu
m
descri
p
tio
n
len
g
th. Exp
e
ri
me
ntal
resu
lts
show
that c
a
li
b
r
ation
re
si
du
al
s is
low
e
st vi
a
opti
m
a
l
mo
del,
co
mp
utatio
n e
rror
of princi
ple d
i
stance is n
o
mor
e
than 0.2 p
i
xel
s
and vari
anc
e lies in th
e bo
un
ds of 0.5 pixe
ls
.
Ke
y
w
ords
: ca
mer
a
cali
brati
o
n, calibr
a
tion
mode
l, va
rianc
e
detectio
n
, mi
ni
mu
m d
e
scri
p
tio
n
len
g
th
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Whe
n
came
ra was fixed
o
n
the m
e
ter, t
he d
a
ta of th
e mete
r
can
be di
re
ctly re
ad [1]. A
lot of image
pro
c
e
s
sing
techn
o
logi
e
s
have
be
e
n
used i
n
these mete
rs su
ch
as
g
r
ey
transfo
rmin
g, edg
e dete
c
ti
ng [2], profile extra
c
ting,
segm
entation
[3] and te
m
p
late mat
c
hi
ng.
Ho
wever,
pre
c
isi
on
of read
ing d
a
ta d
o
e
s
not
sati
sfy th
e requi
reme
n
t
s of
use
r
s. In o
r
de
r to
re
ad
data of the meter a
c
curately and au
tomatically
, camera cali
bration wa
s n
eede
d. Cam
e
ra
calib
ration i
s
the key
step i
n
ma
chine
vision. T
he result of image
pro
c
e
ssi
ng
wi
ll be influe
nced
greatly by
cal
i
bration
re
sult
[4]. There
are many
g
ood
method
s fo
r
came
ra
cali
bration such a
s
traditional
cali
bration, self-calibratio
n
and
calibration b
a
se
d on a
c
tive vision [5].
In traditional
calib
ration, al
l param
eters are
comp
ute
d
by using re
lation betwee
n
points
in calib
ration
body and homolog
ou
s p
o
ints in imag
e [6].
In calib
ration ba
se
d on active visi
on,
came
ra pa
ra
meters
a
r
e
o
b
tained by
controllin
g ca
mera do so
me
spe
c
ial movement
s[7
,
8].
In
self-calib
ratio
n
, plenty of r
e
stri
ction info
rmati
on a
r
e use
d
in com
puting ca
mera param
eters[9].
Becau
s
e
the
ca
mera o
n
meter is li
mited by
op
erating
spa
c
e an
d fixing
co
ndition, t
hese
calib
ration m
e
thod
s are h
a
r
d to be appli
ed to dire
ct-readin
g
meter.
Another p
r
o
b
l
e
m is that op
timal calib
rati
on
mod
e
l sh
ould not b
e
exclu
s
ive to different
came
ra.
Wh
en
con
d
ition
have
bee
n
cha
nge
d in
si
de a
nd
outsi
de, optim
al
calib
ration
m
odel
sho
u
ld be different even if the same ca
mera. On
ce
meter wa
s u
s
ed in locale,
came
ra fixed
o
n
meter
wa
s dif
f
icult to be a
d
j
usted
again.
So, accurate
and a
u
tomati
c calibration
method
sho
u
ld
be develo
ped
. In this paper, a method of sele
cting o
p
timal calib
ratio
n
model was
prop
osed.
2. Automatic Selection of Optimal Calibration Model
2.1.
Camer
a
Projection Mod
e
l
Came
ra p
r
oje
c
tion can be repre
s
e
n
ted b
y
a collinea
rity equation sh
own a
s
form
u
l
a (1).
c
c
c
image
O
object
CCD
y
x
z
c
P
P
P
R
P
)
(
(1)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Autom
a
tic Selection for
Op
tim
a
l Calibrati
on Model of
Cam
e
ra (X
ue
con
g
Li)
4649
Whe
r
e,
object
P
is the coo
r
din
a
te of object poi
nt in the worl
d.
CCD
P
is the coo
r
dinate o
f
obje
c
t poi
nt i
n
the
ca
mera
syste
m
.
R
is
t
he
rotation matrix.
image
P
is th
e
c
o
or
d
i
na
te
o
f
o
b
j
ec
t
point in imag
e.
c
is the prin
ciple di
stan
ce of the camera. Plane
c
z
c
d
enote
s
image
plane.
CCD
P
is de
scrib
ed
by the 3D
co
ordin
a
tes
)
,
,
(
c
c
c
z
y
x
de
scribe
s.
image
P
is d
e
scrib
ed by t
he 3
D
c
o
or
d
i
na
te
s
)
,
(
y
x
descri
b
e
s
.
Duri
ng a
c
tual
proje
c
tion
p
r
ocess, ide
a
l
instan
ce i
n
formula
(1
) i
s
di
sturb
ed b
y
many
factors. The
s
e disturban
ces ar
e described by image disto
r
tion.
Consi
d
e
r
ing
this conditi
on,
formula (
1
) tu
rn into formul
a (2).
c
c
c
image
image
y
x
z
c
P
P
(2)
Whe
r
e,
image
P
describe
s
disto
r
tio
n
by
)
,
(
y
x
.
Ho
w to d
e
scribe im
age
d
i
stortion
in
calibratio
n
mo
del a
c
curatel
y
is the
key
duri
ng
calib
ration
p
r
oce
s
s. Radial
disto
r
tion, ta
ngential
di
st
o
r
tion a
nd tilt d
i
stortion
a
r
e i
n
clu
ded
in m
any
method
s.
2.2. Ph
y
s
ical
Model
All kinds of di
stortion a
r
e cons
i
dered in physi
cal mod
e
l. This mode
l is sho
w
n as
formula
(3):
H
j
j
O
j
j
H
i
O
i
i
i
y
c
y
y
x
D
y
x
D
y
x
D
xy
B
y
r
B
y
r
r
A
y
x
c
x
y
x
D
y
x
D
y
x
D
y
C
x
C
xy
B
x
r
B
x
r
r
A
x
/
))
(
2
)
(
(
2
)
2
(
)
(
/
))
(
2
)
(
(
2
)
2
(
)
(
4
4
3
2
2
2
2
2
1
1
2
2
2
3
1
2
2
4
4
3
2
2
2
2
2
1
2
1
2
2
2
1
2
2
3
1
(3)
Whe
r
e,
i
A
and
j
A
descri
be
radi
al disto
r
tion.
1
B
and
2
B
descri
be
radial
-a
symm
etric a
n
d
tangential
di
stortion.
1
C
and
2
C
de
scribe
tilt distortio
n
.
1
D
,
2
D
and
3
D
describ
e gl
obal
deform
a
tion o
f
image.
H
x
and
H
y
are the coordinate
s
of the princi
ple poi
nt. And
2
2
2
y
x
r
.
When item
s
of physical model was a
ccepted or
reje
cted according
to different instan
ce,
physi
cal mod
e
l in formula (3) wo
uld turn into many cali
bration m
odel
s.
3. Cheby
s
he
v
Model
Solution to p
h
ysical mode
l is very com
p
lex. Compa
r
ed to physi
cal model, sol
u
tion to
Che
b
ysh
e
v model is
simple
.
Che
b
ysh
e
v
calib
ration model can be
expr
esse
d by the n
o
rmali
z
e
d
orthogon
al
polynomial.
Coeffici
ents
of polynomial
reflect co
rre
c
tion of imag
e disto
r
tion. High
correl
ations
betwe
en the polynomial
coefficient
s ca
n be avoided
by estimation in Cheby
shev calib
ratio
n
.
Che
b
ysh
e
v calibratio
n
mo
del is sho
w
n
as form
ula (4
).
M
m
N
n
y
n
x
m
mn
M
m
N
n
y
n
x
m
mn
y
k
K
x
k
K
y
y
k
K
x
k
K
x
00
00
)
(
)
(
)
(
)
(
(4)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 6, June 20
14: 4648 – 4
653
4650
Whe
r
e,
mn
and
mn
are con
s
tant coeffici
ents.
)
(
x
K
m
is define
d
like
formula (5).
)
1
1
(
))
arccos(
cos(
)
(
x
x
m
x
K
m
(5)
In formula (5),
x
k
and
y
k
can m
ap co
ordi
nate
of image to range [-1,1].
In formula (4
), Che
b
yshev
model woul
d also
tu
rn i
n
to many cal
i
bration m
o
d
e
ls when
different valu
es are given to
m
and
n
.
4.
Iterativ
e Sol
u
tion of
Calibration Model
Whe
n
image
P
)
,
(
y
x
is ta
ken i
n
to formula (2), int
a
c
t
pr
o
j
ec
tion
mo
de
l is
bu
ilt. B
y
comp
uting un
kno
w
n p
a
ra
m
e
ter in formul
a (2), came
ra
calibration can be a
c
com
p
lish
ed.
Formul
a (2
)
can b
e
loo
k
e
d
on a
s
a M
a
rkoff model.
So its soluti
on ca
n be
re
alize
d
by
least squa
re
s method. Markoff model is
sho
w
n a
s
formula (6
).
1
2
0
)
(
T
u
D
X
v
u
(6)
Whe
r
e,
X
is a
matrix of
q
p
ran
k
.
is an un
known
ve
ctor of
1
q
d
i
me
ns
io
n.
u
is
an observatio
n
vector of
1
p
dimensi
on.
)
(
u
D
is a covari
an
ce matrix of
p
p
ran
k
.
T
is a
weight matrix.
Whe
n
rel
a
tio
n
amon
g all
observation v
e
ctors i
s
un
known,
)
(
u
D
in formula (6)
can
be
predi
ge
sted i
n
to
I
2
0
.
I
is an u
n
it matrix. When
covari
an
ce i
s
obtain
e
d
, formula
(6
) ca
n be
solved.
All observati
on vecto
r
s
are
co
rrelative in ca
mera cali
bratio
n. This
co
rrel
ation is
descri
bed by
T
.
The solution
of
is like formula
Tu
X
TX
X
1
)
(
ˆ
. When
formula was chan
ged a
little, corre
cti
on item of observatio
n
vector can b
e
sh
own by form
u
l
a
u
X
v
ˆ
ˆ
. So es
timation
of difference is sh
own as fo
rmula (7).
q
p
v
T
v
ˆ
ˆ
ˆ
2
0
(7)
Then, covaria
n
ce of un
kn
o
w
n pa
ram
e
ter
can be e
s
timated by formula (8
).
1
2
0
)
(
ˆ
)
ˆ
(
ˆ
TX
X
D
(8)
Take e
a
ch item of calibrati
on model in formul
a (2) in
formula (6) a
nd ca
rry out iterative
cal
c
ulatio
n like from formul
a (6) to form
ula (8
).
Then,
each p
a
ra
m
e
ter of cali
bration model
can
be solve
d
.
5.
Automatic Selection of Optimal Calibration Model
To ap
ply ca
mera to
dire
ct-readi
ng m
e
ter, came
ra
calib
ration
should
be o
p
timal and
automatic. So
, this work sh
ould han
d to prog
ram.
In o
u
r program, this work ca
n divided into two
st
ep
s.
(a)
Cre
a
tion o
f
candid
a
te calibratio
n
mo
del
Select a gro
up of model
s from phy
si
cal mod
e
ls a
nd Ch
ebyshe
v
models en
ough to
inclu
de po
ssi
b
le condition
s in
calib
ratio
n
pro
c
e
s
s, an
d take th
ese
model
s a
s
in
put. Then,
so
lve
each of these
models by th
e method in 2
.
4 sectio
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Autom
a
tic Selection for
Op
tim
a
l Calibrati
on Model of
Cam
e
ra (X
ue
con
g
Li)
4651
After every model
wa
s solved, calib
ra
tion
paramet
ers
we
re ta
ken into form
ula (2
).
Then,
worl
d
coo
r
din
a
te of
calib
ratio
n
p
o
int wo
uld
be
com
puted
a
c
cordi
ng im
a
ge coo
r
din
a
te of
c
a
libration point.
After thes
e work
, differenc
e
2
0
would be
compute
d
a
gain. If the result was diff
erent
with a
n
terio
r
differen
c
e, thi
s
calib
ration
model
wo
uld
be a
band
one
d. Co
ntra
rily, this mo
del
was
taken a
s
can
d
idate calibra
tion model.
(b) Sele
ction
of optimal cali
bration m
odel
If there
we
re
two
or plu
r
al
mod
e
ls in fi
nal
con
c
o
u
rse of
can
d
idat
e calibration
model
s,
minimum de
scriptio
n lengt
h rule would
be used in se
lecting o
p
timal model.
Minimum de
scriptio
n lengt
h can b
e
de
scrib
ed in form
ula (9
) in cam
e
ra calib
ratio
n
D
M
L
B
B
B
(9)
Whe
r
e,
M
B
are bits of model p
a
ram
e
ter.
D
B
are bits of mod
e
l descri
p
tion
.
Whe
n
ob
serv
ation vector d
e
fers u
n
iform
distrib
u
tion, formul
a (9
) turns to formul
a (10
)
.
)
2
(
log
2
ˆ
ˆ
)
(
log
2
2
e
L
v
T
v
p
q
B
(10)
3. Experimental Re
sults
and An
aly
s
is
3.1.
Design o
f
Ex
periments
In order to validate the method in this pape
r, two cameras (cam
era 1 and
ca
mera 2
)
had be
en
ca
librated. In
calibratio
n
, ten calib
ratio
n
model
s were
taken a
s
in
put of sele
cti
ng
optimal mode
l from physical model
s an
d Cheby
shev
models. Parameters confi
guratio
n of these
model a
r
e list
ed in Table 1.
Table 1. Para
meter Config
uration of 10
Model
s
Model Parameters
confi
guration
Number
Ph
y
s
ical model
H
x
、
H
y
、
1
A
、
1
C
3
H
x
、
H
y
、
1
A
、
2
A
、
1
B
、
2
B
、
1
C
、
2
C
H
x
、
H
y
、
1
A
、
2
A
、
3
A
、
1
B
、
2
B
、
1
C
、
2
C
、
1
D
、
2
D
、
3
D
Cheb
y
s
hev mod
e
l
mn
、
mn
、
)
6
,
,
0
(
N
M
i
k
i
7
3.2.
Automatic S
e
lection Exp
e
riment of O
p
timal Calibr
a
tion Model
Optimal
calib
ration m
odel
for two
came
ra
were
sele
cted from
10
model
s in ta
b
l
e I. The
results a
r
e sh
own a
s
Figu
re 1 and Figu
re 2.
Ser
i
al
num
ber of
m
o
del
s
L
B
L
B
Figure 1. Selection
Re
sult of First Ca
me
ra
Figure 2. Selection
Re
sult of Second
Ca
mera
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Vol. 12, No. 6, June 20
14: 4648 – 4
653
4652
In Figure 1
and Fig
u
re
2, absci
ssa
of
X axis d
enote
s
3 p
h
y
sical
model
s an
d 7
Che
b
ysh
e
v model
s for first came
ra in turn. Forked
model is ab
a
ndon
ed be
ca
use they do not
pass
differe
nce
che
ckin
g
. Integer o
f
Y axis d
e
s
cribe
s
mini
mum d
e
scri
p
t
ion length
L
B
.
Arro
whe
ad
m
odel i
s
th
e o
p
t
imal mod
e
l b
e
ca
us
e it h
a
s pa
ssed
difference
che
c
kin
g
an
d its
L
B
is
least.
4.
Performan
c
e
Experiment
for Op
timal Model
The optimal
calib
ration m
odel i
s
obtain
ed by the me
thod in thi
s
p
aper. By com
parin
g
resi
dual
s of
10 calib
ration
model i
n
Ta
ble 1,
we
ca
n find that
re
sidu
al of the
optimal mo
de
l is
least. Re
sid
u
a
ls of third m
odel an
d opti
m
al model
for first came
ra
are sho
w
n a
s
in Figure 3.
(a)
Re
sidu
al in 3
rd
model
(b)
Re
sidu
al in 9
th
model
Figure 3. Re
sidual
s in Diffe
rent Mod
e
ls o
f
First Cam
e
ra
In Figu
re
3(a) is
re
sid
ual
of third
mod
e
l a
nd
(b) is resi
d
ual of
optimal
mod
e
l. It is o
b
vious
that perform
a
n
ce of optima
l
model is bet
ter. Some
cal
i
bration
re
sult
s of optimal model a
r
e list
ed
in Table 2 for
came
ra 1 a
n
d
came
ra 2.
From Table
2,
we
can
se
e
that com
p
u
t
ation
e
rro
r
of pri
n
ci
ple
dist
ance i
s
no m
o
re
than
0.2 pixels an
d varian
ce lie
s in the bou
n
d
s of 0.5 pixe
ls.
Root mea
n
square e
rro
r(RMS) is a
key
para
m
eter be
tween ob
serv
ations and
m
odel
in
image pla
ne. RMS in Table
2 also sho
w
s that selecte
d
model is opti
m
al.
4. Conclusio
n
In orde
r to cal
i
brate
came
ra
fixed on the di
re
ct-readi
ng
meter autom
atically, a method of
sele
cting
opti
m
al mo
del
was fo
und
ed.
Some mo
del
s
were
sele
ct
ed fro
m
p
h
ysical m
odel
s
a
n
d
taken a
s
inp
u
t. Then, optimal model
wa
s obtai
n
e
d
by difference che
c
king a
nd dete
c
tion
of
minimum d
e
scriptio
n len
g
th. By analyzi
ng qualitat
ive
and qu
antita
t
ive experime
n
tal re
sults,
we
can find that
optimal mod
e
l
had bee
n selecte
d
and
calibratio
n
met
hod in this p
aper h
ad a hi
gh
pre
c
isi
on.
Ackn
o
w
l
e
dg
ements
This
work h
a
s
bee
n supp
orted by Developi
n
g
Fun
d
for Innovative Re
sea
r
ch
Team of
Guan
gdo
ng
University of Tech
nolo
g
y (No. GDUT
20
11-1
0
).
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TELKOM
NIKA
ISSN:
2302-4
046
Autom
a
tic Selection for
Op
tim
a
l Calibrati
on Model of
Cam
e
ra (X
ue
con
g
Li)
4653
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