Geometric Camera Models and Camera Calibration 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 Bahadir K. Gunturk

Homogeneous Coordinates P O Homogeneous coordinates Bahadir K. Gunturk

Coordinate System Changes Translation Bahadir K. Gunturk

Coordinate System Changes Rotation where Exercise: Write the rotation matrix for a 2D coordinate system. Bahadir K. Gunturk

Coordinate System Changes Rotation + Translation In homogeneous coordinates Rigid transformation matrix Bahadir K. Gunturk

Perspective Projection Perspective projection equations Bahadir K. Gunturk

Intrinsic Camera Parameters Perspective projection Bahadir K. Gunturk

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

Intrinsic Camera Parameters The center of the sensor chip may not coincide with the pinhole center. Bahadir K. Gunturk

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

Intrinsic Camera Parameters In homogeneous coordinates These five parameters are known as intrinsic parameters Bahadir K. Gunturk

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

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

Combine Intrinsic & Extrinsic Parameters We can further simplify to 3x4 matrix with 11 degrees of freedom: 5 intrinsic, 3 rotation, and 3 translation parameters. Bahadir K. Gunturk

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

Camera Calibration Y Z X courtesy of B. Wilburn Bahadir K. Gunturk

Camera Calibration Mathematically, we are given n points We want to find M and where Bahadir K. Gunturk

Camera Calibration We can write Bahadir K. Gunturk

Camera Calibration Scale and subtract last row from first and second rows to get Bahadir K. Gunturk

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

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

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, Bahadir K. Gunturk

Camera Calibration We get See Forsyth & Ponce for details and skew-angle case. Bahadir K. Gunturk

Applications courtesy of Sportvision First-down line Bahadir K. Gunturk

Applications courtesy of Princeton Video Image Virtual advertising Bahadir K. Gunturk

Parameters of a Stereo System l 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 Bahadir K. Gunturk

Calibrated Camera Essential matrix Bahadir K. Gunturk

Uncalibrated Camera Fundamental matrix Bahadir K. Gunturk