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### Transcript of Computer Vision Geometric Camera Models and Camera Calibration

• Computer Vision Geometric Camera Models andCamera Calibration

• Coordinate SystemsLet O be the origin of a 3D coordinate system spanned by the unit vectors i, j, and k orthogonal to each other.ijkOPCoordinate vector

• Homogeneous CoordinatesnHOPHomogeneous coordinates

• Coordinate System ChangesTranslation

• Coordinate System ChangesRotation

whereExercise: Write the rotation matrix for a 2D coordinate system.

• Coordinate System ChangesRotation + TranslationIn homogeneous coordinatesRigid transformation matrix

• Perspective ProjectionPerspective projection equations

• Intrinsic Camera ParametersPerspective projection

• Intrinsic Camera ParametersWe need take into account the dimensions of the pixels. CCD sensor array

• Intrinsic Camera ParametersThe center of the sensor chip may not coincide with the pinhole center.

• Intrinsic Camera ParametersThe camera coordinate system may be skewed due to some manufacturing error.

• Intrinsic Camera ParametersIn homogeneous coordinatesThese five parameters are known as intrinsic parameters

• Intrinsic Camera ParametersIn a simpler notation:With respect to the camera coordinate system

• Extrinsic Camera ParametersTranslation and rotation of the camera frame with respect to the world frameIn homogeneous coordinatesUsing , we get

• Combine Intrinsic & Extrinsic ParametersWe can further simplify to3x4 matrix with 11 degrees of freedom: 5 intrinsic, 3 rotation, and 3 translation parameters.

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

• Camera Calibrationcourtesy of B. WilburnXZY

• Camera CalibrationMathematically, we are given n points

We want to find M

andwhere

• Camera CalibrationWe can write

• Camera CalibrationScale and subtract last row from first and second rowsto get

• Camera CalibrationWrite in matrix form for n pointsto getLet m34=1; that is, scale the projection matrix by m34.

• Camera CalibrationThe least square solution of is

From the matrix M, we can find the intrinsic and extrinsic parameters.

• Camera CalibrationConsider the case where skew angle is 90. Since we set m34=1, we need to take that into account at the end.

Notice thatSince R is a rotation matrix, Therefore,

• Camera CalibrationWe get

See Forsyth & Ponce for details and skew-angle case.

• Applications courtesy of SportvisionFirst-down line

• ApplicationsVirtual advertising courtesy of Princeton Video Image

• Parameters of a Stereo SystemIntrinsic ParametersCharacterize the transformation from camera to pixel coordinate systems of each cameraFocal length, image center, aspect ratio

Extrinsic parametersDescribe the relative position and orientation of the two camerasRotation matrix R and translation vector T

• Calibrated CameraEssential matrix

• Uncalibrated CameraFundamental matrix