Note on Coupled Line Cameras for Rectangle Reconstruction (ACDDE 2012)

Asian Conference on Design and Digital Engineering 2012 (ACDDE 2012)
Geometric Computing and CAD Workshop
Dec.6-8, 2012, Niseko, Hokkaido, Japan

Note on Coupled Line Cameras for
Rectangle Reconstruction

Joo-Haeng Lee

Robot & Cognitive Systems Dept.
ETRI

Outline
• Problem deﬁnition
• Outline of proposed solution
• Illustrative example
• Theory: coupled line cameras
• Experimental results
• Q&A

Joo-Haeng Lee (joohaeng@etri.re.kr)

QUIZ 1. You have some image quadrilaterals
taken from a camera. Which of the following
is the image of any rectangle?


(a) Rhombus (b) Parallelogram

Isosceles
(c) Trapezoid___ (d)
Trapezoid


Parallelogram

Isosceles
(d)
Trapezoid

Reconstructed Rectangle
Given Image Quadrilateral v0 v3
u0 u3
l0
l3
r f

l1
l2 Parallelogram
u1 u2
v1 v2

Isosceles
(d)
Trapezoid
Reconstructed
Projective Structure



Isosceles
(c) (d)
Trapezoid

Problem Deﬁnition
• Given: (1) a single image of a scene
rectangle of an unknown aspect ratio; (2)
a simple camera model with unknown
parameter values


Problem Deﬁnition
• Given: (1) a single image of a scene
rectangle of an unknown aspect ratio; (2)
a simple camera model with unknown
parameter values
• Problem: (1) to reconstruct the projective
structure including the scene rectangle;
(2) to calibrate unknown camera
parameters


Proposed Solution


Proposed Solution
1. An analytic solution based on coupled
line cameras is provided for the
constrained case where the center of a
scene rectangle is projected to the image
center.


Proposed Solution
1. An analytic solution based on coupled
line cameras is provided for the
constrained case where the center of a
scene rectangle is projected to the image
center.
2. By preﬁxing a simple pre-processing step,
we can solve the general cases without
the centering constraint above.


Proposed Solution

3. We also provide a determinant to tell if an
image quadrilateral is a projection of any
scene rectangle.


Proposed Solution

3. We also provide a determinant to tell if an
image quadrilateral is a projection of any
scene rectangle.
4. We present the experimental results of the
proposed method with synthetic and real
data.


Illustrative Example


l is a projection of a scene rectangle.
G
trate the performance of the proposed
h synthetic and real data.
Q
pa
a
1. Assume a simple camera model with di
Example what we can do! (ex) pinhole camera
unknown parameters:
C
mple camera •  Square pixel: fx= fy di
unknown •  No skew: s = 0 ca
•  Image center on the tw
principal axis sh

age
So

diagon
mple what we can do!
Constra
camera Illustrative= Example
•  Square pixel: f f
x y diagon
own •  No skew: s = 0 can be
2. Given an image quadrilateral Qg,
•  Image center on the two lin
principal axis share th

s given, Qg Solutio
solution
estimat
camera
uad Q
ng points Joo-Haeng Lee (joohaeng@etri.re.kr)

•  Image center on the two lin
principal axis share t
s given, Find a centered quad Q using the
3. Qg Solutio
vanishing points of Qg solutio
estimat
camera
uad Q
ng points
Q Exper
Synthe
e if the •  Determinant: D
! 100 ran
d Q is A +A ±2 A A Joo-Haeng Lee (joohaeng@etri.re.kr)

3. Find a centered quad Q using the
vanishing points of Qg

w1 w0

u3
u3
g Qg
u2
g um Q
u0 u0
u1


is given, Qg Soluti
solutio
Illustrative Example estima
camer
quad Q
ng points can determine if the the centered
4. We
quad Q is the image of a scene rectangle.Expe
Q

Synth
ne if the •  Determinant: D
! 100 ra
d Q is A0 + A1 ± 2 A0 A1 Gref; (
cene D± = F1(li ) = >0
A1 − A0 within
vertice
relativ
p , an

Q Exper

e if the Illustrative Example
•  Determinant: D
Synthet
100 ran
Q is !
A0 + A1 ± 2 A0 A1 Gref; (2)
ne 5. If so, ±weF1(li ) =
D = can reconstruct the scene >0
A1 inA0 metric sense within
rectangles, G and Gg, − a
before camera calibration. vertices
relative
nstruct pc, and
e
metric Real: (1
era Gg G
with a k
ratio is
desk: A

vertic
relati
construct Illustrative Example pc, an
ene
a metricFinally, we can calibrate camera
6. Real:
Gg G
mera parameters: (1) focal length f, (2) external with
params: [R|T] ratio
desk:
calibrate (2) in
ters: recon
calib
s: [R|T]


Line Camera


aint!
@etri.re.kr
! Line Camera
I, KOREA !
• Given: (1) 1D image of a scene line
denoted by l0 and l2; (2) the principal
pecial linear camera model!
axis passes through the center m of a
scene line.
pc
e of a
by l0 and y2 y
0
l2
axis l0
enter of s2 d s0
q0
v2 m v0

pc
(1) 1D image of a
Line Camera
ine denoted by l0 and
l2
y2 y
0
he principal axis l0
s2 d
through the center an analytic solution to the pose0
of s
• Solution: q0
line. estimation of a line camera
v2 v0
m

n: an analytic l0 − l 2
cos θ 0 = d = dα 0
n to the pose l0 + l 2
ion of a line camera

v2 m v0 c

pc
(1) 1D image of a
ine denoted by l0 and Line Camera l2
y2 y
0
he principal axis l0
s2 d
through the center an analytic solution to the pose0
of s
• Solution: q0
line. estimation of a line camera
v2 v0
m
mera model!
n: an analytic l0 − l 2
pc cos θ 0 = d = dα 0
n to the pose l0 + l 2
ion of a line camera
y 2 y0
l2
l0
s2 d s0
q0
v2 m v0 v2 m v0 c

Coupled Line Cameras


se l0 + l 2
e camera

• Given:v(1) a centered quad Q; (2) the
2 m v0 c
principal axis passes through the center
Cameras a φ"of G pin-hole camera model!
of a scene rectangle G; (3) a diagonal
angle
special
pc
u1
red quad Q
al axis l1 r
e center of u2 l2 l0 u0 v1 v0
G; (3) a Q
l3 G
m
G: φ u3 v2 v3

Cameras a special pin-hole camera model!
pc
u1
red quad Q
al axis l1 r
center• Constraint: (1)0 for each diagonal of Q, a
of u2 l2 l u0
v1 v0
G; (3) a line camera lcan be deﬁned; (2) these two
Q 3 G
m
G: φ line cameras share a2 principal axis.v3
u3 v

each cos θ 0 cos θ1
d= = = F2 (θ 0 ,θ1 ,li )
ne camera α0 α1
) these u0
u1 u3
hould
θ0
axis. u2 v0
v1 θ1

v2 v3 Joo-Haeng Lee (joohaeng@etri.re.kr)

Cameras adspecial =
G; (3) a
each cos θpin-hole θ1 G mmodel!
Q 30 cos camera
φ
G: camera = = F2 (θ 0 ,θ1 ,li )
ne α 0 u3 αv2 v3
pc 1
these
red quad
each
hould
d=
ucos θ
u0 1 0
cos θ1
= u1 Q=u3F2 (θ 0 ,θ1 ,li )
al axis
ine camera u l1 αr α1
θ0 0
axis. of u2 l v0
v1 θ1
center• Solution:2an analytic 1solution to the pose
2
u0
2) these u0 l0 v v0
G; (3) a estimation of3 coupled line 3camerasv
should v2 Q
l u1 G u
m
G: φ
3
θ0 u v0
l axis. u2
3 v2 v1 θ1 v3
ic θ0
tan θ = cos θ) = D±
cos 0 F1(li
eeach vd =
2 2 = 1
= F2 (θ 0 ,θ1 ,li ) v3
ne camera
led line α0 α1
ytic θ 0 → d θ 0 θ 1 → ψ i → si → φ
→
) these u0 tan = F1(li ) = D±
se
hould →G→ c 2 p u1 u3
pled line
axis. u2 θ → θ0 d →vθ → ψ θ1 s → φ
0 →
0 1
v1 i i
erformance of the proposed method!
→G→ p
v2 c v3 Joo-Haeng Lee (joohaeng@etri.re.kr)

Cameras a special
G; (3) a 3
Q pin-hole camera model! G
m
came
G: φ u3 v2 v3
d quad Q
pc
ucos θ cos θ1
hingquad
red points
each
1
Q= F (θ ,θ ,l )
d= 0
=
al axis
ine camera l1 αr
0
Qα1 2 0 1 i
Expe
center• Solution:l2an analytic 1solution to the pose
2) these of u2 u l0 u0
v v0
0
G; (3) a estimation of3 coupled line 3cameras
should Q
l u1 G u
Synt
ine if the u •  Determinant: D m
G: φ
l axis. !2 θ0 u v0
3 v2 v1 θ1 v3 100
uad Q is A0 + A1 ± 2 A0 A1 Gref;
scene
each D± = cosiθ 0= cos θ1
vd = F (l ) = > 03
2 1 = A (θ 0 ,θ1 ,li ) v
A1 − F20 with
ne camera α0 α1 verti
ytic θ0
) these u0 tan = F1(li ) = D± relat
se
hould 2 u1 u3
construct u2
pled line pc, a
axis. θ0 → θ0 d →vθ → ψ θ1 s → φ
0
v1 i → i
ene 1

a metric v2
→ G → pc v3

Cameras a special pin-holeu camera model!
should
0
u 1 3

θ0
l axis. u2 v0 θ1
pc v1
u1
red quad Q
al axis v2 l1 r v3

center• Solution:l2an θl0 u0 v1solution to the pose
tic of u2 analytic v0
G; (3) a estimation of3 2 = F1(li line cameras
tanl =
) G D±
0

e Q coupled m
G: φ line
pled u3 v2 v3
θ 0 → d → θ 1 → ψ i → si → φ
each cos θ 0 cos θ1
d = → G → pc = = F2 (θ 0 ,θ1 ,li )
ne camera α0 α1
) these
erformance of the proposed method! u3
u0
u1
hould
θ0
axis.
erated
u2 v0
v1 θ1
Qg Gg
ngles: v2 Q v3 Joo-Haeng Lee (joohaeng@etri.re.kr)

• Solution: an analytic solution to the pose
estimation of coupled line cameras
s0

Ψ0

x
k Β Θ0 pc

t0 Α0 d y z
Q
Φ G
t1 Α1
Ψ1

Θ1 s1

Ρ

pc
scene, which will be projected as a line u0 u2 in the line
camera C0 . Especially, we are interested in the posi-
Ψ2 Ψ
0 rectangle G in a pin-hole camera with the center of pro-
l2
tion pc and the orientation θ0 of C0 when the principal
l0 jection at pc . Note that the principal axis passes through
s2 d s0
axis passes through the center vm of v0 v2 and the center vm , um and pc .
Θ
um = (0, 0, 1)0of image line.
v2 v0 v
Using this configuration of coupled line cameras, we
m 2 m v0 c
To simplify the formulation, we assume a canon- find the orientation θi of each line camera Ci and the
ical configuration where Trajectory of the centerand vm is
(a) Line camera (b)
vm vi = 1, of projection length d of the common principal axis Line Camera C1
(a) Pin-hole Camera (b) Line Camera C0 (c) from a given
placed at the1:origin of the worldacoordinate system:
Figure A configuration of line camera quadrilateral H. Using the lengths of partial diagonals,
vm = (0, 0, 0). For derivation, we define followings: Figure 2: A pin-hole camera and its decomposition into
li = um ui , we can find the relation between the cou-
coupled line cameras.
d = pc vm , li = um ui , ψi = ∠vm pc vi , and pled cameras Ci from (1):
s i = pc v i .
tan ψ1 l1 sin θ1 (d − cos θ0 )
scene,this configuration, we can derive theu2 in the line
In which will be projected as a line u0 following re- = = (3)
camera C0 . Especially, we are interested in the posi-
lation: tan ψpin-hole camera (d − the center of pro-
rectangle G in a 0 l0 sin θ0 with cos θ1 )
l2
tion pc and the orientation cosof0C0 when the principal
=
d−θ θ
0 =
d0
(1) jection at pc . Note that the principal axis passes through
Manipulation of (2) and (3) leads to the system of non-
axis passes through l0 thed + cosvθ0 of vd1 2 and the center
center m 0v vm , um and pc .
linear equations:
um = (0, 0, 1)d − cos θ0 = s0 cos ψ0 and d1 = d +
where d0 = of image line. Using this configuration of coupled line cameras, we
β sin θ0 cos θ1 − cos θ0 sin θ1 cos θ cos θ1
cos θ0 simplify the . formulation, we assume a between
To = s2 cos ψ2 We can derive the relation canon- find the orientation θi of each line =
d= camera 0 i and the
C=
icaland d from (1): where vm vi = 1, and vm is
θ0 configuration length d of β sincommon θ1
the θ0 − sin principal axisα0 α1
from a given
(4)
placed at the origin of the world coordinate system: quadrilateral H. Using the lengths of partial diagonals,
where β = l1 /l0 . Using (4), the orientation θ0 can be
vm = (0, 0, θ0 =For(l0 − l2 )/(l0we ldefine followings:
cos 0). d derivation, + 2 ) = d α0 (2) li = um ui , we can find the relation between the cou-
represented with coefficients, α0 , α1 , and β, that are
d = pc vm , li = um ui , ψi = ∠vm pc vi , and pled cameras Ci from (1):
solely derived from a quadrilateral H:
si = pαvi = (li − li+2 )/(li + li+2 ), which is solely
where c i .
tan ψ1 l1 sin θ1 (d − cos θ0 )
derived from a image line ui ui+2 . Note following re-
In this configuration, we can derive the that θ0 and d = + A ± 2√ A A
=
θ0 ψ0 A0 l0 1 sin θ0 (d0− 1 θ1 ) (3)
are sufficient parameters to determine the exact position
lation: tan tan = cos
= D± (5)
l2
of pc in 2D. When α0 d − cos θpc is defined on a certain
is fixed, 0 d0 2 A 1 − A0
= = (1) Manipulation of (2) and (3) leads to the system of non-
sphere as in Fig l0 Once θi and d are 1
1b. d + cos θ0 d known, additional where
linear equations:
where d0 =can − cos θ0determined:ψtan ψi d1 sin θi /d
parameters d be also = s cos
0 0 and
= = d+
A0 sin B0cos2B1 , cos θ0 1 =θB0= cos 1 0 = cos θ1
β = θ0 + θ1 − A sin 1 − 2Bθ
cos θsi = s cos/ sinWe. can derive the relation between
and = sin θi ψ . ψi
0 2 2 d=
θ0 and d from (1): B0 = 2(α0 − 0 −(α0 θ1 α1 ) − 4α0 (α1 − 1)2 βα1
β sin θ 1)2 sin + 2
2 2 α0 2

2.2. Coupled Line Cameras (4)
(α0 − . 2 (α0 (4), the orientation θ0 can be
where1β== l1 /l0 1)Using− α1 )(α0 + α1 )
B
cos θ = d (l − l )/(l + l ) = d α (2)

Qg is given, Qg Solu
solu
estim
taken from a camera. Which of the following cam
d quad Q
hing points
Q Exp
(a) Rhombus (b) Parallelogram Synt
mine if the •  Determinant: D
! 100
uad Q is A0 + A1 ± 2 A0 A1 Gref;
scene D± = F1(li ) = >0
A1 − A0 Isosceles with
(c) (d) verti
Trapezoid
relat
p,a


D>0 D<0

Isosceles
(c) (d)
Trapezoid
D<0 D<0

of G: φ (1) for each
raint: cos θ 0 cos θ1 v
u3 d = 2
v = = F2 (θ 0 ,θ1 ,l
3

nal of Q, a line cameraθ
cos 0 cos θ1
α0 α1
ordeﬁned; (2) 2. You have some imageFquadrilaterals
e each QUIZ these d= = u0 = 2 (θ 0 ,θ1 ,li )
α0 α1
line camerashould camera. Which of the following
u1 u3
ne cameras from a
taken
(2) these the imageu0 anyurectangle? v0
is
the principal axis. of 2 θ0
v1 θ1
u1 u3
s should
θ0
pal axis. u2 v
v20 v1 θ1

on: an analytic Rhombus
v2
θ0
(a) tan Parallelogram = D±
= F1(li ) v3
on to the pose D>0
θ0 2
lytic of coupled linetan = F (l ) = D
ation
ose 2 θ 01 → d → θ±1 → ψ i → si → φ
i
ras
upled line → G → pc φ
θ → d →θ →ψ → s → Isosceles
0 1 i Trapezoid
i

→ G → pc
riments performance of the proposed method!

v0 v3

f

(a) Rhombus Parallelogram
D>0
v1 v2

Isosceles
Trapezoid

Experimental Results

• Synthetic Data
• Real Data


Real Data

•


Real Data


d; (2) these u0
u1 u3
ras should
cipal axis.
Synthetic Data
u2 θ0 v0
v1 θ1

v2 v3

nalytic θ0
1. Generated 100 random rectangles Gref and
corresponding image quads = refD±
tan = F1(li ) Q ;
pose 2
coupled 2. Get image θ → d →by adding noisesφto Qref
line quad Qg θ → ψ → s →
within dmax 0
pixels; 1 i i

→ G → pc
3. Reconstruct Gg from Qg;
ts performance of the errors between Gref and G
4. Measured proposed method! g.

generated Gg
Qg
ectangles: Q
d noises Gref G

0 1 i i

→ G → pc

Synthetic Data
performance of the proposed method!

nerated Gg
Qg
angles: Q
oises Gref G
s to the Error H%L
6
err; (3)
5
4
-vm|, φ , 3
2
1
dmax
1 2 3
|vi-vm| φ pc f

Aspect Ratio
gle 1.46 Ê
Ê

‡
Raw

Compensated
1.45 Joo-Haeng Lee (joohaeng@etri.re.kr)

Gerr; (3) 5
4
vi-vm|, φ , 3
2
1
1
Real Data
|vi-vm|
2
φ pc
3

f
dmax

Aspect Ratio
ngle 1. A rectangle with a known aspect ratio is
1.46 Ê
Ê

‡
Raw

Compensated

pect moving on a desk: (ex) A4-sized paper;
1.45
1.44
Ê Ê
1.43
on the 2. Take pictures to get 9 image quads;
Ê
‡
‡
1.42 Ê
‡ ‡
Ê ‡

φ = 1.414
1.41 ‡
Ê ‡ ‡
Ê

1.40 Ê ‡

y 3. Reconstructed and calibrated for each
1 2 3 4 5 6 7 8
Rect
9 ID

case.
Reconstructed aspect ratio: φ Merged frustums
d
cases. A moving
A4 paper
1 2 3 4

5 6 7 8 9


Gerr; (3) 5
4
vi-vm|, φ , 3
2
1
1
Real Data
|vi-vm|
2
φ pc
3

f
dmax

Aspect Ratio
ngle 1.46 Ê
Ê

‡
Raw

Compensated
1.45
pect 1.44
Ê Ê
1.43
on the
Ê
‡
‡
1.42 Ê
‡ ‡
Ê ‡

φ = 1.414
1.41 ‡
Ê ‡ ‡
Ê

1.40 Ê ‡
Rect

y 1 2 3 4 5 6 7 8 9 ID

Reconstructed aspect ratio: φ Merged frustums
d
cases. A moving
A4 paper
1 2 3 4

5 6 7 8 9


Summary
• We proposed an analytic solution to
reconstruct a scene rectangle of an
unknown aspect ratio from a single
image quadrilateral.
• Our method is based on novel
formulation of coupled line cameras and
rectangle constraint.


Acknowledgement

This research has been partially supported by
KMKE & KRC 2010-ZC1140 and KMKE ISTDP
No.10041743


Acknowledgement
This research has been first presented at ICPR
2012 (Int. Conf. Pattern Recognition), Tsukuba,
Japan. Nov., 2012.
Poster #5, Session TuPSAT2, ICPR 2012!

21st International Conference on Pattern Recognition (ICPR 2012)
November 11-15, 2012. Tsukuba, Japan Camera Calibration from a Single Image based on !
Coupled Line Cameras and Rectangle Constraint!
Joo-Haeng Lee joohaeng@etri.re.kr !
Robot & Cognitive Systems Dept., ETRI, KOREA!
Camera Calibration from a Single Image based on
Coupled Line Cameras and Rectangle Constraint Summary! Line Camera a special linear camera model!
pc
Given: (1) 1D image of a
Given: (1) an image of a scene rectangle of an unknown
scene line denoted by l0 and y2 y
Joo-Haeng Lee aspect ratio; (2) a simple camera model with unknown l2
0
l2; (2) the principal axis l0
Robot and Cognitive Systems Research Dept., ETRI parameter values: focal length, position, and orientation s2 d s0
passes through the center m
Problem: (1) to reconstruct the projective structure q0
joohaeng@etri.re.kr of a scene line v0v2. v2 v0
m
including the scene rectangle; (2) to calibrate unknown
camera parameters Solution: an analytic l0 − l 2
cos θ 0 = d = dα 0
solution to the pose l0 + l 2
Proposed Solution:
Abstract Several approaches are based on geometric prop- estimation of a line camera
erties of a rectangle or a parallelogram. Wu et al. 1.  Analytic solution based on coupled line cameras is
Given a single image of a scene rectangle of an un- proposed a calibration method based on rectangles of provided when the center of a scene rectangle is
known aspect ratio and size, we present a method to known aspect ratio [4]. Li et al. designed a rectangle projected to the image center.
v2 m v0 c
reconstruct the projective structure and to find camera landmark to localize a mobile robot with an approxi- 2.  By prefixing a simple pre-processing step, we can solve
parameters including focal length, position, and orien- mate rectangle constraint, which does not give an ana- the general cases without the centering constraint.
tation. First, we solve the special case when the center lytic solution [5]. Kim and Kweon propose a method to
Coupled Line Cameras a special pin-hole camera model!
3.  We also provide a determinant to tell if an image
of a scene rectangle is projected to the image center. We estimate intrinsic camera parameters from two or more quadrilateral is a projection of a scene rectangle. pc
u1
Given: (1) a centered quad Q
formulate this problem with coupled line cameras and rectangle of unknown aspect ratio [6]. A parallelogram
4.  We demonstrate the performance of the proposed Q; (2) the principal axis l1 r
present the analytic solution for it. Then, by prefixing constraint can be used for calibration, which generally method with synthetic and real data. passes through the center m u2 l2 l0 u0 v1 v0
a simple preprocessing step, we solve the general case requires more than two scene parallelograms projected of an unknown scene l3 G
Q m
without the centering constraint. We also provides a in multiple-view images as in [7, 8, 9]. rectangle G. (Diag. angle=φ )
determinant to tell if an image quadrilateral is a pro- Our contribution can be summarized as follows. Illustrative Example what we can do! u3 v2 v3

Constraint: (1) for each cos θ 0 cos θ1
jection of a rectangle. We demonstrate the performance Based on a geometric configuration of coupled line d= = = F2 (θ 0 ,θ1 ,li )
of the proposed method with synthetic and real data. cameras, which models a simple camera of an unknown (1) Assume a simple camera •  Square pixel: fx= fy diagonal of Q, a line camera α0 α1
model with unknown •  No skew: s = 0 can be defined; (2) these u0
focal length, we give an analytic solution to reconstruct u1 u3
parameters. •  Image center on the two line cameras should
a complete projective structure from a single image of θ0
principal axis share the principal axis. u2 v0
v1 θ1
1. Introduction an unknown rectangle in the scene: no prior knowledge
is required on the aspect ratio and correspondences. v2 v3
(2) When an image
Camera calibration is one of the most classical topics Then, the reconstruction result can be utilized in finding
quadrilateral Qg is given, Solution: an analytic θ0
Qg tan = F1(li ) = D±
in computer vision research. We have an extensive list the internal and external parameters of a camera: focal solution to the pose 2
of related works providing mature solutions. In this pa- length, rotation and translation. The proposed solution estimation of coupled line θ 0 → d → θ 1 → ψ i → si → φ
per, we are interested in a special problem to calibrate a also provides a simple determinant to tell if an image cameras
quadrilateral is a projection of a scene rectangle. (3) Find a centered quad Q → G → pc
camera from a single image of an unknown scene rect-
In a general framework for plane-based camera cal- using the vanishing points
angle. We do not assume any prior knowledge on cor-
ibration, camera parameters can be found first using of Qg. Q Experiments performance of the proposed method!
respondences between scene and image points. Due to
the limited information, a simple camera model is used: the image of the absolute conic (IAC) and its relation
Synthetic: (1) generated Gg
the focal length is the only unknown internal parameter. with projective features such as vanishing points [2, 10]. (4) We can determine if the •  Determinant: D Qg
! 100 random rectangles: Q
Then, a scene geometry can be reconstructed using a the centered quad Q is A0 + A1 ± 2 A0 A1 G
The problem in this paper has a different nature with Gref; (2) added noises Gref
the image of a scene D± = F1(li ) = >0
Error H%L
classical computer vision problems. In the PnP prob- non-linear optimization on geometric constrains such A1 − A0 within dmax pixels to the
orthogonality, which cannot be formulated as a closed- rectangle.! vertices of Gref; (3)
6
lem, we find transformation matrix between the scene 5
4
3
and the camera frames with prior knowledge on the form in general. relative errors between 2
1

correspondences between the scene and image points (5) If so, we can reconstruct Gref and reconstructed 1 2 3
dmax

the centered scene Gg: |vi-m|, φ , pc, and f. |vi-m| φ pc f
as well as the internal camera parameters [1]. In cam- 2. Problem Formulation
era resection, we find the projection matrix from known rectangle Gg in a metric Real: (1) a rectangle
Aspect Ratio
1.46
Ê Raw

Gg G
Ê

sense before camera
‡ Compensated
1.45
correspondences between the scene and image points 2.1. Line Camera with a known aspect 1.44

calibration.
Ê Ê
1.43
ratio is moving on the
Ê

without prior knowledge of camera parameters [2]. 1.42 ‡
‡
Ê
‡ ‡
Ê ‡

desk: A4 paper φ = 1.414;
1.41 ‡
Ê ‡ ‡
Ê

Camera self-calibration does not rely on a known Eu- A line camera is a conceptual camera, which follows 1.40 Ê ‡

(6) Finally, we can calibrate (2) independently 1 2 3 4 5 6 7 8
Rect
9 ID
clidean structure, but requires multiple images from the same projection model of a standard pin-hole cam- camera parameters: Reconstructed aspect ratio: φ Merged frustums

camera motion [3]. era. (See Fig 1a.) Let v0 v2 be a line segment in the reconstructed and
•  focal length: f calibrated for 9 cases. A moving
1 2 3 4
A4 paper
•  external params: [R|T]
5 6 7 8 9

978-4-9906441-1-6 ©2012 IAPR 758


Q &A

joohaeng at etri dot re dot kr


Note on Coupled Line Cameras for Rectangle Reconstruction (ACDDE 2012)

Note on Coupled Line Cameras for Rectangle Reconstruction (ACDDE 2012)

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