Autocalibration & 3D Reconstruction with Non-central
Catadioptric Cameras
Branislav Micusık Tomas Pajdla
Center for Machine Perception, Czech Technical University in Prague,
�� � �� � �� � ��� �� � � � � � � �
CONTRIBUTIONS
• Precise non-central models for real catadioptric cameras.
• Central approximations for ransac-based estimation of epipolar geometry.
• 3D metric reconstruction from uncalibrated (slightly) non-central catadioptric images byhierarchical approach:
1. Auto-calibration using central model to validate tentative correspondences and to obtaininitial estimate of camera motion.
2. Bundle adjustment on inliers using the precise non-central model.
REAL CATA-DIOPTRIC CAMERAS
parablic mirror hyperbolic mirror spherical mirror+ perspective camera
PROBLEM FORMULATION
3D reconstruction from real para-catadioptric images using
a) central model b)non-central model
SKEWED CORRECT
Input images:
real para-catadioptriccamera
Fp
uP
Sfra
grep
lacem
ents
π
F
u
p
PSfra
grep
lacem
entsπ
π
NON-CENTRAL & CENTRAL APPROXIMATION
−800 −600 −400 −200 0 200 400 600 8000
100
200
300
400
500
600
700
800
1mm ~ 36.6 pxl, flens
= 70 mm
PSfrag replacements
‖u‖[pxl]
z[p
xl]
−600 −400 −200 0 200 400 600−200
−100
0
100
200
300
400
500
1mm ~ 24.4 pxl, flens
= 75.0 mm
PSfrag replacements
‖u‖[pxl]
z[p
xl]
−800 −600 −400 −200 0 200 400 600 8000
100
200
300
400
500
600
700
800
1mm ~ 36.6 pxl, flens
= 70.0 mm
PSfrag replacements
‖u‖[pxl]
z[p
xl]
5 10 15 20 250.63
0.635
0.64
0.645
0.65
0.655
0.66
0.665
PSfrag replacements
a
k
NON-CENTRAL MODEL
X
PSfrag replacements
xy
z
uv
u
p
F
π
γ γ
x
C
W ν
�
c, t
c
n
�
m, t
m
Perspective projection + law of reflection
xw = f (u, a,
�
m, tm,
�
c, tc,
�
)pw = f (u, a,
�
m, tm,
�
c, tc,
�
)
xw =
�−1m
(
λ
�>c
�−1u + tc
)
+ tm
pw =
�−1m
(
�>c
�−1u − 2
(
( �>c
�−1u)
·n
‖n‖
)
·n
‖n‖
)
λ, n depend on the shape of a mirror
AUTO-CALIBRATION WITH CENTRAL MODEL
PSfrag replacements x
z
pq X
g
u
S3
sensor plane
g/h
C
h
h(‖u‖)u
s
p =
(
g(‖u‖, a)uh(‖u‖, a)
)
for perspective proj.g(.), h(.) = 1
≈
px
py
pz
a=a0
+ (a − a0)
px
py
pz
a=a0
=
=
px − pxa0
py − pya0
pz − pza0
a=a0
+ a
px
py
pz
a=a0
=
= x + a s,
Using epipolar constaint
p>2
�
p1 = 0
(x2 + as2)> �
(x1 + as1) = 0
leads to the Quadratic Eigenvalue Problem (QEP):
(
�� + a
�� + a2 �� )f = 0
where f = [ F11 F12 F13 F21 . . . F33 ]> and�
i are functions of coordinates of point correspon-dences (we used WBS [2])
Solving the QEP gives: • mirror parameter a• essential matrix
�
• inliers/outliers (ransac)
3D REC. WITH NON-CENTRAL MODEL
PSfrag replacements
xi
w1
pi
w1
xi
w2
pi
w2
XW
CC
Since cameras are calibrated, then
RM = argmina,
�c,tc,
�m,tm,
�N∑
i=1
(
|(
xiw1 − xi
w2
)
·(
piw1 × pi
w2
)
|
|piw1 × pi
w2|
)2
RESULTS
REFERENCES
[1] Fitzgibbon A., Simultaneous linear estimation of multiple view geometry and lens distortion. CVPR, 2001.
[2] Matas J., Chum O., Urban M., and Pajdla T., Robust wide baseline stereo from maximally stable extremalregions. BMVC, 2002.
[3] Micusik B. and Pajdla T., Para-catadioptric Camera Auto-calibration from Epipolar Geometry ACCV, 2004.
[4] Swaminathan R. and Grossberg M. D. and Nayar S. K., Caustics of Catadioptric Cameras, ICCV, 2001.
This research was supported by the following projects: MSM 212300013, MSMT Kontakt 22-2003-04, BENOGO–IST–2001–39184 Presented at the Conference on Computer Vision and Pattern Recognition (CVPR), 2004, Washington D.C., USA
