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