Calibration parameters

Intrinsic parameters describe the property of camera itself and don't depend on the scene viewed, they are:

• Focal length (fc): measure camera's ability to converge/diverge light.

• Principle point (cc): where optical axis intersect with image plane.

• Skew coefficient (alpha_c): defining the angle between the x and y pixel axes.

• Distortions (kc): normally include radial and tangential distortions for colour camera.

Let P be a point in space of coordinate vector XX_c = [X_c;Y_c;Z_c] in the camera reference frame. Let x_n be the normalized (pinhole) image projection:
 
Let r² = x² + y². After including lens distortion, the new normalized point coordinate x_d is defined as follows:

where dx is the tangential distortion vector:
 

kc(1), kc(2), kc(5) are radial distortion coefficients, kc(3), kc(4) are tangential distortion coefficients. For most colour camera, this distortion model is enough. However, for the depth camera, disparity to depth values may be considered.

Once distortion is applied, the final pixel coordinates x_pixel = [x_p;y_p] of the projection of P on the image plane is:

Writ the equations in matrix:

where KK is known as the camera intrinsic matrix, and defined as follows: 

It is suggested that we usually ignore the skew factor (alpha_c=0) in practice as currently manufactured camera used rectangular pixels. Furthermore the 6th order distortion model may not be necessary for standard filed of views (non wide-range camera), so the kc(5)=0

Extrinsic parameters denote the transformation from 3D world coordinate to 3D camera coordinate.

For rigid transformation we have:
Rotation matrix (Rc): 3x3

Translation vector (Tc): 3x1

Let P be a point vector XX = [X;Y;Z] in the world reference frame
Let XX_c = [X_c;Y_c;Z_c] be the coordinate vector of P in the camera reference frame.
Then XX and XX_c are related to each other through the following rigid motion equation:

XX_c = Rc * XX + Tc