1. Geometric Camera Models and Camera Calibration
Computer Vision
Geometric Camera Models and
Camera Calibration

2. Coordinate Systems
Let O be the origin of a 3D coordinate system spanned by the unit vectors i, j, and k orthogonal to each other. i P O k j
Coordinate vector
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3. Homogeneous CoordinatesP O
Homogeneous coordinates
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4. Coordinate System Changes
Translation
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5. Coordinate System Changes
Rotation
where
Exercise: Write the rotation matrix for a 2D coordinate system.
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6. Coordinate System Changes
Rotation + Translation
In homogeneous coordinates
Rigid transformation matrix
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7. Perspective Projection
Perspective projection equations
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8. Intrinsic Camera Parameters
Perspective projection
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9. Intrinsic Camera Parameters
We need take into account the dimensions of the pixels.
CCD sensor array
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10. Intrinsic Camera Parameters
The center of the sensor chip may not coincide with the pinhole center.
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11. Intrinsic Camera Parameters
The camera coordinate system may be skewed due to some manufacturing error.
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12. Intrinsic Camera Parameters
In homogeneous coordinates
These five parameters are known as intrinsic parameters
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13. Intrinsic Camera Parameters
In a simpler notation:
With respect to the camera coordinate system
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14. Extrinsic Camera Parameters
Translation and rotation of the camera frame with respect to the world frame
In homogeneous coordinates
Using , we get
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15. Combine Intrinsic & Extrinsic Parameters
We can further simplify to
3x4 matrix with 11 degrees of freedom: 5 intrinsic,
3 rotation, and 3 translation parameters.
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16. Camera Calibration
Camera’s intrinsic and extrinsic parameters are found using a setup with known positions in some fixed world coordinate system.
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17. Camera Calibration Y Z X
courtesy of B. Wilburn
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18. Camera Calibration Mathematically, we are given n points
We want to find M
and
where
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19. Camera Calibration
We can write
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20. Camera Calibration
Scale and subtract last row from first and second rows
to get
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21. Camera Calibration Write in matrix form for n points to get
Let m34=1; that is, scale the projection matrix by m34.
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22. Camera Calibration The least square solution of is
From the matrix M, we can find the intrinsic and extrinsic parameters.
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23. Camera Calibration
Consider the case where skew angle is 90. Since we set m34=1, we need to take that into account at the end.
Notice that
Since R is a rotation matrix,
Therefore,
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24. Camera Calibration We get
See Forsyth & Ponce for details and skew-angle case.
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25. Applications courtesy of Sportvision First-down line
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26. Applications courtesy of Princeton Video Image Virtual advertising
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27. Parameters of a Stereo Systeml r P Ol Or Xl Xr Pl Pr fl fr Zl Yl Zr Yr
R, T
Intrinsic Parameters
Characterize the transformation from camera to pixel coordinate systems of each camera
Focal length, image center, aspect ratio
Extrinsic parameters
Describe the relative position and orientation of the two cameras
Rotation matrix R and translation vector T
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28. Calibrated Camera
Essential matrix
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29. Uncalibrated Camera
Fundamental matrix
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