Intrinsic calibration matrix

The intrinsic calibration matrix contains the parameters intrinsic to the camera. It means that the matrix defines the internals of the camera that play a role in the Perspective Projection However, in most real-world cases, there are plenty of nuances to consider

• The image plane might be skewed
• The pixels might not be square
• There might be an origin offset of the camera frame To take all of these aspects into account, the intrinsic matrix $K$ should be modified. The modified matrix is shown below

$$K=\left[\begin{array}{ccc}f& 0& 0\\ 0& \gamma f& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& \frac{u_0}{f}\\ 0& 1& \frac{v_0}{\gamma f}\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& s& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}f& sf& u_0\\ 0& \gamma f& v_0\\ 0& 0& 1\end{array}\right]$$

• The first sub-matrix accounts for the non-squareness of pixels or different scaling on x and y. $\gamma$ is the aspect ratio
• The second sub-matrix accounts for the origin offset. $(u_0,v_0)$ is the principal point
• The third sub-matrix accounts for the skewing of the image. $s$ is the skew factor

Principal Point

The principal point is the point on the image plane onto which the perspective center is projected