new geometric interpretation and analytic solution for quadrilateral reconstruction (icpr-2014...
DESCRIPTION
Poster presentation for ICPR 2014 paper. Title: New geometric interpretation and analytic solution for quadrilateral reconstruction Author: Joo-Haeng Lee (ETRI)TRANSCRIPT
ds0
s2q0
y2 y0
u0
u2
v0v2 vm
pc
l0
l2
m0m2
Coupled Line Cameras a special pin-hole camera model
(1) For an unknown scene quad, a set of image quads Qg from uncalibrated cameras is given.
(2) Find a centered proxy quad Q by perspectively translating off-centered quad Q. A vanishing line should be available for each image.���à Contribution #3
(3) Find the diagonal parameters mi of the scene quad using numerical optimization. à Contribution #4
(4) We can reconstruct the scene quad G in a metric sense using the analytic solution based on generalized coupled line cameras (GCLC).���à Contribution #2
(5) We can also calibrate unknown camera parameters for each image: - focal length: f - external params: [R|T] à Contribution #1
Given: (1) An unknown scene quad Gg ; (2) A set of image quad Qg ; (3) A vanishing line in each image; (4) A simple camera model with unknown parameter values: intrinsic (focal length), extrinsic (position and orientation)
Problem: (1) To reconstruct the scene quad from given images and a prior knowledge; (2) To calibrate unknown camera parameters for each image
Contributions:
1. Basically, we generalize the solution based on coupled line cameras (CLC) of [Lee:2012:ICPR, LEE:2012:ETRIJ] developed for a single-view reconstruction of a unknown scene rectangle.
2. An analytic solution based on generalized coupled line cameras (GCLC) is given for single-view reconstruction of a quad when diagonal parameters of the scene quad is known and its center is projected to the image center.
3. A geometric method for perspective translation is given to handle the case of an off-centered quad assuming a vanishing line is available.
4. A numeric solution is given for a completely unknown scene quad when sufficient number (i.e., at least for for a genera quad) of images are given.
Summary
Illustrative Example what we can do
New Geometric Interpretation and Analytic Solution for Quadrilateral Reconstruction
Joo-Haeng Lee [email protected] Intelligent & Cognitive Systems Dept., ETRI, KOREA
Poster #7, Session ThCT1p, ICPR 2014
Line Camera a special linear camera model
Given: (1) 1D image of a scene line denoted by l0 and l2; (2) The principal axis passes through the scene line v0v2 with the division ratio m0 and m2.
Problem: Can we estimate the pose of a line camera when l0, l2, m0 and m2 are given?
Solution: An analytic solution exists.
Given: (1) A centered quad Q; (2) The principal axis passes through the center of a scene quad G; (3) Known diagonal parameters mi of G; and (4) Unknown diagonal angle of G.
Formulation: (1) For each diagonal of Q, a line camera can be defined; (2) Two line cameras should share the principal axis; (3) Three unknowns in three equations.
Problem: Can we estimate the pose of a camera when li and mi are given?
Solution: An analytic solution exists.
Off-Centered Quad: (1) Using a vanishing line, perspectively translate the off-centered quad Qg to get the centered proxy quad Q. (2) Then, apply CLC reconstruction to Q.
n-View Reconstruction: (1) We need to know mi to apply CLC; (2) With n views, the unknown mi can be approximated by optimization:
(3) Then, we can apply CLC.
Quadrilateral Reconstruction handling a real-world problem
Qg
φ
arg
mi{ }min cosφ
j− cosφ
j+1j=0
n−1
∑
cv0v2 vm
cosθ
i= α
g ,id
αg ,i=
mi+2
li− m
ili+2
mim
i+2li+ l
i+2( )where
u0
u1 u2
u3r um
Q
d =cosθ
0
αg ,0
=cosθ
1
αg ,1
= F(mi,θ
i,β )
d = Ag ,0
/ Ag ,1
= Fd
l{0,1,2,3}
,m{0,1,2,3}( )
cosθi= α
g ,id
cosφ = cosρ sinθ0sinθ
1+ cosθ
0cosθ
1
p
c=
d
sinφ(sinφ cosθ
0,cosθ
1− cosφ cosθ
0,sinρ sinθ
0sinθ
1)
QgQ
omug,0 ug,1
ug,2ug,3 um
u0 u1
u2u3
w0w1
wd,0
wd,1
wm
Joo-Haeng Lee ([email protected])
• When the diagonal ratios mi of a scene quadrilateral are known, we can find the diagonal angle j using G-CLC.
Known mi’s
known mi of Gd = A
g ,0/ A
g ,1
Ag ,0
= l02l22(m
0+m
2)2m
12m
32 − l
12l32m
02m
22(m
1+m
3)2
Ag ,1= l
02l22(m
0+m
2)2(l
1m
3+ l
3m
1)2 − l
12l32(l
0m
2+ l
2m
0)2(m
1+m
3)2
known li and r of Q
G-CLC
View 0 Scene Quad G
φ
Joo-Haeng Lee ([email protected])
• First, find mi from n views!• Then, apply G-CLC for each view
n-View Reconstruction
n Views Scene Quad G
known li and r of Q
inferred mi of G
G-CLC
d = Ag ,0
/ Ag ,1
Ag ,0
= l02l22(m
0+m
2)2m
12m
32 − l
12l32m
02m
22(m
1+m
3)2
Ag ,1= l
02l22(m
0+m
2)2(l
1m
3+ l
3m
1)2 − l
12l32(l
0m
2+ l
2m
0)2(m
1+m
3)2
Joo-Haeng Lee ([email protected])
• In practice, the diagonal ratios mi of a scene quadrilateral is unknown as well as the angle j.
Unknown mi’s
?View 0 Scene Quad G
known li and r of Q
unknown mi of G
G-CLC
d = Ag ,0
/ Ag ,1
Ag ,0
= l02l22(m
0+m
2)2m
12m
32 − l
12l32m
02m
22(m
1+m
3)2
Ag ,1= l
02l22(m
0+m
2)2(l
1m
3+ l
3m
1)2 − l
12l32(l
0m
2+ l
2m
0)2(m
1+m
3)2
cosφ
0= cosφ
1= cosφ
2= cosφ
3
φ
i= Fφ l
i ,{0,1,2,3}, m
{0,1,2,3}( ) where m0= 1
arg
mi{ }min cosφ
j− cosφ
j+1j=0
n−1
∑