By Smirnov V. A.
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Extra resources for A-Simplicial Objects and A-Topological Groups
X~,'] hence the appellation p o i n t a t i n ~ n i t ~ . The projective space P" can be viewed as the union of the usual space R" (points [~1. ' The neat thing about this formalism is that points at infinity are not special and are treated just like any other point. 6: Three images of a sequence taken from a helicopter: Courtesy Let's go back to the projective plane. There is one point at infinity for each direction in the plane: [l,' ] O , is associated with the horizontal direction, [0, 1,01' is associated with the vertical direction, and so on.
15) where e P is an arbitrary projection matrix (11parameters) 11is the projective equa- tion of an arbitrary plane (3 parameters), p is an arbitrary constant (1 parameter), which is in fact the common scale of P and II in the matrix %. Together, these 15 parameters represent the projective ambiguity in reconstruction: the arbitrary choice of the projective basis in 3-D, or, equivalently, of the matrix 3-1. 0 The remaining elements in P’ are: the epipole e’ of F in the second image and the hornography H, compatible with F and generated by the plane II.
It makes it possible to describe naturally the phenomena at infinity that we just noticed. 4). Let’s start with a point of Euclidean2 coordinates [U, vir in the plane. Its projective coordinates are obtained by just adding 1 at the end: [U, v, 1IT. Having now three coordinates, in order to obtain a “one-to-one” correspondence between IThe only difference between displacements and similarities is that the latter ones allow for a global scale factor. Since in the context of reconstruction from images, such an ambiguity is always present, we will designate by abuse of language Euclidean.