Projective formalism
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Subsections

# Projective formalism

## Pinhole camera

The pinhole camera is the most common camera model in Computer Vision.

We represent a pinhole camera by a plane called retinian plane and a point C called optical center. Let M be a 3D point with the coordinates (X Y Z T)t in the projective space .In figure 1 we show how we project the point M in m ( T u, T v,T), its image on the retinian plane . The formula 1 gives us the relation between M and m, P being the projection matrix. The projection matrix P can be decomposed into a product of three matrixes as we show in formula 2.

 (1)

 (2)

where:

• , the 3D coordinates change matrix:    (3)
• P0, the standard projection :  (4)
• A, the 2D coordinates change matrix:

 (5)

The parameters of (i.e. rx,ry,rz,tx,ty,tz) are called the extrinsic parameters and the parameters of A (i.e. ) are called the intrinsic parameters. The parameter (the angle between the rows and columns of the camera) is usually set to [Vai90].

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