A Generic Concept for Camera Calibration Peter Sturm and Srikumar Ramaligam

A Generic Concept for Camera CalibrationPeter Sturm and Srikumar RamaligamSung HuhCPSC 643Individual Presentation 4April 15, 2009

Table of ContentIntroductionCalibration

◦2D Known and Unknown motion◦3D Known and Unknown motion

DiscussionConclusionFuture Work

IntroductionDevelop a calibration method for any camera

model◦ Cameras w/o a single effective view point

General model of camera adopted:◦ Images consisting of pixels◦ Each pixel captures light that travels along a ray in

3D

Camera is fully described by:◦ Coordinate Of rays◦ Mapping b/w rays and pixels

Introduction – Related WorksExisting Calibration methods

R.I. Hartley, A. Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, 2000C.C. Slama. Manual of Photogrammetry. Fourth Edition, ASPRS, 1980

Calibration for the general imaging modelM.D. Grossberg, S.K. Naar. A general imaging model and a method for finding its parameters. ICC, 2001R. Swamminathan, M.D. Grossberg, and S.K. Nayar. Caustics of Catadioptric Cameras. ICCV, 2001

Introduction – Related WorksEpipolar geometry estimation and

modelingT. Pajdla. Stereo with oblique cameras. IJCV, 47(1), 2002.S. Seitz. The space of all stereo images. ICCV, 2001.Y. Wexler, A.W. Fitzgibbon, A. Zisserman. Learning epipolar geometryfrom image sequence. CVPR, 2003.

Motion estimation for calibrated camerasJ. Neumann, C. Fermuller, Y. Aloimonons. Polydioptric Camera design and 3D Motion Estimation. CVPR, 2003.R. Pless. Using Many Cameras as One. CVPR, 2003

Introduction – Related WorksThe special case of a linear calibration

objectP. Sturms, S. Ramalingam. A Generic Calibration Concept: Theory and Algorithms. Research Report 5058, INRIA, France, 2003

Camera ModelUse infinite extensions of half-ray

(Camera Rays)Non-central Camera

◦Camera rays correspond to different pixel does not intersect

Central Camera w/ optical center◦All camera rays intersect in a single

point

Calibration Concept – 2DKnown motion

◦ Object’s motion b/w image is known◦ Two object points can be mapped to a single coord.

Frame◦ Joining two points to compute pixel’s camera ray◦ Knowledge of point position relative to coord.

frames of object and the motion b/w the two coord. frame

Calibration Concept – 2DUnknown Motion

◦ Estimate unknown motion ◦ Let Q, Q’, Q” be the points on the calibration object◦ Common frame = coord. frame associated w/

object’s first position (relative motions are given by rotation matrix and translation vector)

Q T

R' ''

1

tQ

0 T

R" ""

1

tQ

0Collinear

Calibration Concept – 2DCoefficients of a trilinear matching tensorDepends on the calibration tensorTotal 27 coefficients ( 8 are always zero, 6

pairs of identical ones)

13

1

0i ii

CV

Ci = Trilinear product of point coordinatesVi = Associated coefficients of the tensor

Calibration Concept – 3DKnown Motion

◦ Two views are sufficient◦ Equivalent procedure as 2D camera

Unknown Motion◦ Start with Q, Q’, Q”◦ Same coord. frame as in 2D◦ Aligned points are collinear

Calibration Concept – 3DRank of matrix must be less than 3

◦ All sub-determinants of size 3 x 3 vanishes (4 of them)

◦ Corresponds to a trilinear equation in point coord.

34/64 coefficients are always zero

Calibration Concept – 3DEstimate tensors by solving linear equation

systemLet Vi’=Vi, Wi’ = Wi, i = 1…37

◦ Estimate factors ,

◦ Compute and , i = 1…37

◦ Compute R’ and R”

◦ Compute t’ and t” by solving a straight forward linear least square problem

'2 '2 '28 9 10λ V V V '2 '2 '2

1 2 3μ W W W

'λi

iV

V 'μi

iWW

15 16 17

15 16 17

8 9 10

R 'W W WV V VV V V

18 19 20

18 19 20

11 12 13

R"W W WV V VV V V

Experimental Result Result from Central camera Estimated motion parameters give rise to nearly

perfectly collinear calibration points Radial distortion modeled correctly

Experimental Result Result from fish-eye lens Aligned calibration points are not always perfectly

collinear Only the central image region has been calibrated

DiscussionAlgorithm for central cameras work fineNon-central catadioptric cameras give

unsatisfying result◦ Homography-based interpolation of calibration

pointsGeneral algorithm does not work for

perspective cameras, but for multi-stereo systems consisting of sufficiently many cmeras

ConclusionTheory and algorithms for a highly general

calibration concept proposedDifferent cameras will likely require different

designs of calibration object

Future WorksDeveloping bundle adjustment procedures to

calibrate from multiple imagesMotion and pose estimation and

triangulation from perspective to the general imaging model