Q&A : Homographies of the projective geometry

Vision List Digest: Article 1, Volume 18, Issue 43
From: "Chris"
Post-Followup: submission@VISLIST.com

Can anyone explain to me the homographies of the projective geometry
described below? Direct explanation or any reference would be
gratefully appreciated.

Let the projection matrix be P = [H | -Ht] = [H | e] where H is the
homography describing the projection of points from the reference plane
to the image plane, C = (the transpose of)[(the transpose of) t 1] the
optical center of the camera, ane e the projection of the origin (i.e.,
(the transpose of)[0 0 0 1]).

When the geometry is only determined up to a projective (or affine)
transformation, the first projection matrix can be chosen as follows: P1
= [I(3x3) | 0(3x1)]. This implies that the camera and scene coordinate
frames are aligned. In that case, the homographies also describe the
transfer from points lying in the reference plane from the first image
to the image under consideration. All these homographies are related
through the following equation: H' = H + e.(the transpose of)a with H'
is the homography of the plane [(the transpose of)a 1] M = 0 where M is
a homogeneous coordinates of the world points and a = (the transpose
of)[a1 a2 a3] a 3-vector.

Thanks a lot in advance,
Chris

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