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# 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:

### Bob Fisher
Wed Apr 16 00:58:54 BST 1997