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A Generic Concept for Camera Calibration Peter Sturm and Srikumar Ramaligam. Sung Huh CPSC 643 Individual Presentation 4 April 15, 2009. Table of Content. Introduction Calibration 2D Known and Unknown motion 3D Known and Unknown motion Discussion Conclusion Future Work. Introduction. - PowerPoint PPT Presentation
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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
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