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EE1411
Computer Vision Geometric Camera Calibration 1S.M. Abdallah
Computer VisionComputer Vision
Geometric Camera CalibrationGeometric Camera Calibration
Samer M Abdallah, PhD
Faculty of Engineering and ArchitectureAmerican University of BeirutBeirut, Lebanon
September 2, 2004
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Computer Vision Geometric Camera Calibration 2S.M. Abdallah
Series outlineSeries outline
Cameras and lensesGeometric camera modelsGeometric camera calibrationStereopsis
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Computer Vision Geometric Camera Calibration 3S.M. Abdallah
Lecture outlineLecture outline
The calibration problemLeast-square techniqueCalibration from pointsRadial distortionA note on calibration patterns
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Computer Vision Geometric Camera Calibration 4S.M. Abdallah
Camera calibrationCamera calibrationCamera calibration is determining the intrinsic and extrinsic parameters of
the camera.
The are three coordinate systems involved: image, camera, and world.
Key idea: to write the projection equations linking the known coordinates of a set of 3-D points and their projections, and solve for the camera parameters.
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Computer Vision Geometric Camera Calibration 5S.M. Abdallah
Projection matrixProjection matrix
M is only defined up to scale in this setting.
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Computer Vision Geometric Camera Calibration 6S.M. Abdallah
The calibration problemThe calibration problem
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Computer Vision Geometric Camera Calibration 7S.M. Abdallah
Linear systemsLinear systems
A x b=
A x b=
Square system:Unique solutionGaussian elimination
Rectangular system:underconstrained: Infinity of solutionsOverconstrained: no solution
Minimize |Ax-b|2
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Computer Vision Geometric Camera Calibration 8S.M. Abdallah
How do you solve overconstrained linear equations?How do you solve overconstrained linear equations?
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Computer Vision Geometric Camera Calibration 9S.M. Abdallah
Homogeneous linear equationsHomogeneous linear equations
A x 0=
A x 0=
Square system:Unique solution = 0Unless Det(A) = 0
Rectangular system:0 is always a solution
Minimize |Ax|2 under the constraint |x|2 = 1
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Computer Vision Geometric Camera Calibration 10S.M. Abdallah
How do you solve overconstrained homogeneous linear equations?How do you solve overconstrained homogeneous linear equations?
The solution is the eigenvector e1 with least eigenvalue of UT U.
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Computer Vision Geometric Camera Calibration 11S.M. Abdallah
Example: Line fittingExample: Line fittingProblem: minimize
with respect to (a,b,d).
• Minimize E with respect to d:
• Minimize E with respect to a,b:
where and
EE14112
Computer Vision Geometric Camera Calibration 12S.M. Abdallah
Estimation of the projection matrixEstimation of the projection matrix
The constraints associated with the n points yield a system of 2n homogeneous linear equations in the 12 coefficients of the matrix M,
When n ≥ 6, homogeneous linear least-square can be used to compute the value of the unit vector m (hence the matrix M) that minimizes |Pm|2 as the solution of an eigenvalue problem. The solution is the eigenvector with least eigenvalue of PTP.
EE14113
Computer Vision Geometric Camera Calibration 13S.M. Abdallah
Once M is known, you still got to recover the intrinsic and extrinsic parameters!
This is a decomposition problem, NOT an estimation problem.
Intrinsic parametersExtrinsic parameters
ρ
Estimation of the intrinsic and extrinsic parametersEstimation of the intrinsic and extrinsic parameters
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Computer Vision Geometric Camera Calibration 14S.M. Abdallah
Estimation of the intrinsic and extrinsic parametersEstimation of the intrinsic and extrinsic parametersWrite M = (A, b), therefore
Using the fact that the rows of a rotation matrix have unit length and are perpendicular to each other yields
and
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Computer Vision Geometric Camera Calibration 15S.M. Abdallah
Estimation of the intrinsic and extrinsic parametersEstimation of the intrinsic and extrinsic parameters
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Computer Vision Geometric Camera Calibration 16S.M. Abdallah
Taking radial distortion into accountTaking radial distortion into account
PMz
p⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
100010001
1 λλ
Assuming that the image centre is known (u0 = v0 = 0), model the projection process as:
where λ is a polynomial function of the squared distance d 2 between the image centre and the image point p.
It is sufficient to use low-degree polynomial:
),,1( tscoefficien distortion theand 3 with ,11
2 qpqd p
q
p
pp K=≤+= ∑
=
κκλ
222 ˆˆ vud +=
This yields highly nonlinear constraints on the q + 11 camera parameters.
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Computer Vision Geometric Camera Calibration 17S.M. Abdallah
Calibration patternCalibration patternThe accuracy of the calibration
depends on the accuracy of the measurements of the calibration pattern.
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Computer Vision Geometric Camera Calibration 18S.M. Abdallah
Line intersection and point sortingLine intersection and point sorting
Extract and link edges using Canny;Fit lines to edges using orthogonal regression;Intersect lines.
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Computer Vision Geometric Camera Calibration 19S.M. Abdallah
ReferencesReferences“Computer Vision: A Modern Approach”. D. Forsyth and J. Ponce, Prentice Hall, 2003“Introductory Techniques for 3-D Computer Vision”. E. Trucco and A. Verri, Prentice Hall, 2000“Geometric Frame Work for Vision – Lecture Notes”. A. Zisserman, University of Oxford
