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## Image formation

Image formation involves the projection of points in (the world) to points in (the image plane). The perspective projection equations with which we are familiar,

where the point (X,Y,Z) in the world is projected to the point (x,y)on the image plane, are inherently nonlinear. Converting to homogeneous coordinates, however, makes them linear:

where , , and the perspective projection matrix T is given by:

The entire image formation process includes perspective projection, along with matrices for internal and external calibration:

(4)

where and are the scale factors of the image plane (in units of the focal length f), is the skew ( for most real cameras), the point (u0,v0) is the principal point, R is the rotation matrix, and is the translation vector. The matrix A contains the internal parameters and perspective projection, while D contains the external parameters.

It is sometimes convenient to decompose the projection matrix into a matrix P and a vector p

so that

 (5)

Next: Essential and fundamental matrices Up: Projective Geometry Applied to Previous: Projective Geometry Applied to
Stanley Birchfield
1998-04-23