Camera projection with the pinhole model

A camera is a mapping between the 3D world (object space) and a 2D image.

In general, the camera projection matrix P has 11 degrees of freedom: \[ P=K[R\ \ \ t] \]

Component	# DOF	Elements	Known As
K	5	\(f_x, f_y, s,p_x, p_y\)	Intrinsic Parameters; camera calibration matrix
R	3	\(\alpha,\beta,\gamma\)	Extrinsic Parameters
t (or \(\tilde{C}\))	3	\((t_x,t_y,t_z)\)	Extrinsic Parameters

3D world frame ----- R, t ----> 3D camera frame ------ K -----> 2D image

Explanation:

• P: Projective camera, maps 3D world points to 2D image points.

• K: Camera calibration matrix, 3 x 3, \(x=K[I|0]X_{cam}\), given 3D points in camera coordinate frame \(X_{cam}\), we can project it into 2D points on image \(x\).

  

• R and t: Camera Rotation and Translation, rigid transformation. \(X_{cam}=( X,Y,Z,1)^T\) is expressed in the camera coordinate frame. In general, 3D points are expressed in a different Euclidean coordinate frame, known as the world coordinate frame. The two frames are related via a rigid transformation (R, t).

Some other terms you may see

• P: 3x4, homogeneous, camera projection matrix, \(P=diag(f,f,1)[I|0]\). P is K without considering \((x_{cam},y_{cam})\) in the image. (In other words, it simplify \((p_x, p_y)=(0,0)\).