intrinsicsmatrix

The intrinsic matrix, often denoted K, is a 3x3 matrix that encodes the internal parameters of a camera within the pinhole model. It relates 3D camera coordinates to 2D image coordinates and contains information about focal lengths, the optical center, and any skew between the image axes. In most formulations the matrix is used together with the extrinsic parameters (rotation and translation) to form the full projection of 3D points into the image.

The standard form of the intrinsic matrix is:

K = [ fx s cx

0 fy cy

0 0 1 ]

Here fx and fy are the focal lengths expressed in pixel units along the x and y axes, cx and cy are the coordinates of the principal point (the image center), and s is the skew term that expresses non-orthogonality between the image axes. In many cameras, s is zero.

In practice, the intrinsic matrix is used in the projection equation x ≈ K [R|t] X, where X is a point in world coordinates, R and t are the rotation and translation aligning the world with the camera, and x is the resulting image point in homogeneous coordinates. The intrinsics thus map normalized camera coordinates to pixel coordinates.

Intrinsic parameters are typically estimated during camera calibration, alongside distortion coefficients. Calibration uses images of known patterns (such as checkerboards) and methods like Zhang’s or those implemented in OpenCV’s calibrateCamera. The resulting K is camera-specific and can change with zoom, focal length, or sensor configuration. In many software libraries, K is referred to as the camera matrix or intrinsic matrix.