Projective Transformation


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Projective Transformation

 

a one-to-one mapping of the projective plane or projective space into itself such that collinear points are carried into collinear points (for this reason, a projective transformation is sometimes called a collineation). A projective transformation of a projective line is a one-to-one mapping of the projective line into itself such that the harmonicity of points of the line is preserved.

The simplest example of a projective transformation—and the one most important in applications—is the homology, which is the projective transformation that leaves invariant a line and a point not on the line. An example of a projective transformation of space is the perspective transformation, whereby a figure F in the plane Π is projected from a point S into a figure F′ in the plane Π′. Any projective transformation can be obtained by a finite sequence of perspective transformations. The projective transformations form a group whose fundamental invariant is the cross ratio of four points on a line. Theories of the invariants of groups of projective transformations that leave unchanged some figure are metric geometries.

The fundamental projective-transformation theorem for the projective plane can be stated as follows. Let A, B, C, and D be any four points in the plane Π such that no three of them are collinear. If A′, B′, C′, and D′ are also four points in this plane such that no three of them are collinear, then there exists one, and only one, projective transformation that carries the points A, B, C, and D into the points A′, B′, C′, and D′, respectively. This theorem is made use of in nomography and aerial photographic survey. A similar theorem is true in projective space. In this case, a projective transformation is determined by five points, no four of which are coplanar. This theorem is equivalent to Pappus’ theorem.

In homogeneous coordinates a projective transformation is expressed as a homogeneous linear transformation whose matrix has a nonzero determinant. Projective transformations of the Euclidean plane or space are also studied. In Cartesian coordinates, they are expressed by linear-fractional functions; the mappings, however, are not one-to-one in this case.

References in periodicals archive ?
Projective Transformation. Images have to be filmed vertically from the selected sea area for higher accuracy; however, it is not easy to obtain any vertical image.
Also, if [bar.E] is always associated by convention with the canonical projective point [mathematical expression not reproducible] on the celestial circle of E, then we deduce that [[tau].sup.[??]] corresponds by a projective transformation to 0.
When the cameras are uncalibrated and no additional geometric information of the point set is available, the reconstruction is determined only up to an unknown projective transformation. For any 3D projective transformation matrix H, [P.sup.j][H.sup.-1] and H[X.sub.i] produce an equally valid reconstruction.
[18] uses scale-invariant feature transform (SIFT) [21] points to obtain a crude estimate of the projective transformation, and identifies the features from the moving objects by the difference in moving velocities between objects and the background.
The points of equal loading are fixed points of projective transformation [m.sub.1] [right arrow] [m.sub.2], [J.sub.1] [right arrow] [J.sub.2], as it is shown in Figure 3.
In this paper 2D projective transformation brings all the images aligned to a common ground plane with overhead view.
The projective transformation type can be used in 2D/3D image registration as a constrained elastic transformation when a fully elastic transformation behaves inadequately.
The set of all invertible transformations of the form (2) constitutes the planar projective transformation group G.
A twisted polygon is a sequence A = ([A.sub.i])i [member of] z of points in the projective plane that is periodic modulo some projective transformation [phi], i.e., [A.sub.i+n] = [phi]([A.sub.i]) for all i [epsilon] Z.
The researchers propose a fuzzy spatial data warehouse, a layered architecture for cooperative work environments, an algorithm for identifying authors using synonyms, and an image registration method using a projective transformation model.
Constructing the projective transformation is straightforward [Morgan 1986a; 1986b; Watson et al.