(1) Assume a simple camera model with unknown parameters.

•  Square pixel: fx= fy

•  No skew: s = 0 •  Image center on the

principal axis

(2) When an image quadrilateral Qg is given,

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

(4) We can determine if the the centered quad Q is the image of a scene rectangle.  

•  Determinant: D  

(5) If so, we can reconstruct the centered scene rectangle Gg in a metric sense before camera calibration.

(6) Finally, we can calibrate camera parameters:

•  focal length: f • external params: [R|T]

Given: (1) an image of a scene rectangle of an unknown aspect ratio; (2) a simple camera model with unknown parameter values: focal length, position, and orientation

Problem: (1) to reconstruct the projective structure including the scene rectangle; (2) to calibrate unknown camera parameters

Proposed Solution:

1.  Analytic solution based on coupled line cameras is provided when 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.

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

4.  We demonstrate the performance of the proposed method with synthetic and real data.

Summary

Illustrative Example what we can do

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

Poster #5, Session TuPSAT2, ICPR 2012

D

±= F

1(l

i) =

A0+ A

1± 2 A

0A

1

A1− A

0

> 0

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 center m of a scene line v0v2.

Solution: an analytic solution to the pose estimation of a line camera

cosθ

0= d

l0− l

2

l0+ l

2

= dα0

Coupled Line Cameras a special pin-hole camera model

Given: (1) a centered quad Q; (2) the principal axis passes through the center m of an unknown scene rectangle G. (Diag. angle= )

Constraint: (1) for each diagonal of Q, a line camera can be defined; (2) these two line cameras should share the principal axis.

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

d =

cosθ0

α0

=cosθ

1

α1

= F2(θ

0,θ

1, l

i)

tan

θ0

2= F

1(l

i) = D

±

θ0→ d →θ

1→ψ

i→ s

i→φ

→ G → pc

Synthetic: (1) generated 100 random rectangles: Gref; (2) added noises within dmax pixels to the vertices of Gref; (3) relative errors between Gref and reconstructed Gg: |vi-m|, , pc, and f.

Real: (1) a rectangle with a known aspect ratio is moving on the desk: A4 paper (2) independently reconstructed and calibrated for 9 cases.

Experiments performance of the proposed method

Qg

Q

Gg G

cv0v2 m

u0

u1

u2

u3

l0

l1

l2l3

r

φ

φ = 1.414;

d s0s2q0

y2 y0

v0v2 m

pc

l0l2

φQ

|vi-m| pc f φ1 2 3

dmax123456

Error H%L

v0

v2

u0

u2 θ0 v1

v3

u1 u3

θ1

v0 v1

v2 v3

pc

Q

G m

Ê

Ê

Ê

Ê

Ê

Ê Ê

ÊÊ

‡

‡

‡

‡

‡

‡

‡

‡ ‡

1 2 3 4 5 6 7 8 9

RectID

1.40

1.41

1.42

1.43

1.44

1.45

1.46

Aspect Ratio

‡ Compensated

Ê Raw

Reconstructed aspect ratio: φ Merged frustums

A moving A4 paper

1 2 3 4

5 6 7 8 9

Gref Q

Qg

G

Gg