3x4 Projection Matrix

There are three coordinate systems involved --- camera, image and world.

1. Camera: perspective projection.
   

   This can be written as a linear mapping between homogeneous coordinates (the equation is only up to a scale factor):

   

   where a projection matrix represents a map from 3D to 2D.

2. Image: (intrinsic/internal camera parameters)
   

   

   is a upper triangular matrix, called the camera calibration matrix:

   

   where , .

   • provides the transformation between an image point and a ray in Euclidean 3-space.
   • There are four parameters:
     1. The scaling in the image x and y directions, and .
     2. The principal point , which is the point where the optic axis intersects the image plane.
     The aspect ratio is .
   • Once is known the camera is termed calibrated.
   • A calibrated camera is a direction sensor, able to measure the direction of rays --- like a 2D protractor.

3. World: (extrinsic/external camera parameters)
   The Euclidean transformation between the camera and world coordinates is :

   

Finally, concatenating the three matrices,

which defines the projection matrix from Euclidean 3-space to an image: