Camera Matrix - Normalized Camera Matrix and Normalized Image Coordinates

Normalized Camera Matrix and Normalized Image Coordinates

The camera matrix derived above can be simplified even further if we assume that f = 1:

where here denotes a identity matrix. Note that matrix here is divided into a concatenation of a matrix and a 3-dimensional vector. The camera matrix is sometimes referred to as a canonical form.

So far all points in the 3D world have been represented in a camera centered coordinate system, that is, a coordinate system which has its origin at the camera focal point. In practice however, the 3D points may be represented in terms of coordinates relative to an arbitrary coordinate system (X1',X2',X3'). Assuming that the camera coordinate axes (X1,X2,X3) and the axes (X1',X2',X3') are of Euclidean type (orthogonal and isotropic), there is a unique Euclidean 3D transformation (rotation and translation) between the two coordinate systems.

The two operations of rotation and translation of 3D coordinates can be represented as the two matrices

and

where is a rotation matrix and is a 3-dimensional translation vector. When the first matrix is multiplied onto the homogeneous representation of a 3D point, the result is the homogeneous representation of the rotated point, and the second matrix performs instead a translation. Performing the two operations in sequence gives a combined rotation and translation matrix

Assuming that and are precisely the rotation and translations which relate the two coordinate system (X1,X2,X3) and (X1',X2',X3') above, this implies that

where is the homogeneous representation of the point P in the coordinate system (X1',X2',X3').

Assuming also that the camera matrix is given by, the mapping from the coordinates in the (X1',X2',X3') system to homogeneous image coordinates becomes

Consequently, the camera matrix which relates points in the coordinate system (X1',X2',X3') to image coordinates is

a concatenation of a 3D rotation matrix and a 3-dimensional translation vector.

This type of camera matrix is referred to as a normalized camera matrix, it assumes focal length = 1 and that image coordinates are measured in a coordinate system where the origin is located at the intersection between axis X3 and the image plane and has the same units as the 3D coordinate system. The resulting image coordinates are referred to as normalized image coordinates.