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Computer Vision

Computer Vision . Geometric Camera Models and Camera Calibration. 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. Homogeneous Coordinates. n. H. P. O.

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Computer Vision

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  1. 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

  3. Homogeneous Coordinates n H P O Homogeneous coordinates

  4. Coordinate System Changes • Translation

  5. Coordinate System Changes • Rotation where Exercise: Write the rotation matrix for a 2D coordinate system.

  6. Coordinate System Changes • Rotation + Translation In homogeneous coordinates Rigid transformation matrix

  7. Perspective Projection • Perspective projection equations

  8. Intrinsic Camera Parameters Perspective projection

  9. Intrinsic Camera Parameters We need take into account the dimensions of the pixels. CCD sensor array

  10. Intrinsic Camera Parameters The center of the sensor chip may not coincide with the pinhole center.

  11. Intrinsic Camera Parameters The camera coordinate system may be skewed due to some manufacturing error.

  12. Intrinsic Camera Parameters In homogeneous coordinates These five parameters are known as intrinsic parameters

  13. Intrinsic Camera Parameters In a simpler notation: With respect to the camera coordinate system

  14. Extrinsic Camera Parameters • Translation and rotation of the camera frame with respect to the world frame In homogeneous coordinates Using , we get

  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.

  16. Camera Calibration • Camera’s intrinsic and extrinsic parameters are found using a setup with known positions in some fixed world coordinate system.

  17. Camera Calibration Y Z X courtesy of B. Wilburn

  18. Camera Calibration • Mathematically, we are given n points • We want to find M and where

  19. Camera Calibration • We can write

  20. Camera Calibration • Scale and subtract last row from first and second rows to get

  21. Camera Calibration • Write in matrix form for n points to get Let m34=1; that is, scale the projection matrix by m34.

  22. Camera Calibration • The least square solution of is • From the matrix M, we can find the intrinsic and extrinsic parameters.

  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,

  24. Camera Calibration • We get See Forsyth & Ponce for details and skew-angle case.

  25. Applications First-down line courtesy of Sportvision

  26. Applications Virtual advertising courtesy of Princeton Video Image

  27. P Pl Pr Yr p p r l Yl Xl Zl Zr fl fr Ol Or R, T Xr Parameters of a Stereo System • 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

  28. Calibrated Camera Essential matrix

  29. Uncalibrated Camera Fundamental matrix

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