new geometric interpretation and analytic solution for quadrilateral reconstruction (icpr-2014...

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 joohaeng@etri.re.kr 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 (joohaeng@etri.re.kr)

• 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 (joohaeng@etri.re.kr)

• 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 (joohaeng@etri.re.kr)

• 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

∑