Let M = [[[x.sub.w] [y.sub.w] [z.sub.w] 1].sup.T] denote the homogeneous coordinates
of a 3D point and m = [[u v 1].sup.T] denote the homogeneous coordinates
of the corresponding image point.
The matrix maps a vector expressed in homogeneous coordinates
with respect of the OUVW coordinate system to the reference coordinate OXYZ system.
[c.sub.r] - homogeneous coordinates
of a point expressed in the C-frame
It was proved that results similar to those of the unit ball hold for projective spaces, but in this case the symbols that yield commutative [C.sup.*]-algebras are those that depend only on the radial part of the homogeneous coordinates
The homogeneous coordinates
of [B.sub.i] in the local-coordinate system oxyz can be described, respectively, as [sup.o][b.sub.1] = [(0 b 0 1).sup.T], [sup.o][b.sub.2] = [(-([square root of (3)]/2)b -(1/2)b 0 1).sup.T], and [sup.o][b.sub.3] = [(([square root of (3)]/2)b -(1/2)b 0 1).sup.T]; the homogeneous coordinates
of A; in the global-coordinate system OXYZ can be described, respectively, as [sup.O][a.sub.1] = [(0 a 0 1).sup.T], [sup.O][a.sub.2] = [(-([square root of (3)]/2)a -(1/2)a 0 1).sup.T], and [sup.O][a.sub.3] = [(([square root of (3)]/2)a -(1/2)a 0 1).sup.T].
Therefore, the transformational relationship of the space point P between homogeneous coordinates
[([x.sub.z], [y.sub.z], [z.sub.z], 1).sup.T] in grasping point coordinates [O.sub.Z][X.sub.Z][Y.sub.Z][Z.sub.Z] and homogeneous coordinates
[([x.sub.c], [y.sub.c], [z.sub.c], 1).sup.T] in the camera coordinate system [O.sub.C][X.sub.C][Y.sub.C][Z.sub.C] is
using the pinhole method of camera projection are
Hermes, "Homogeneous coordinates
and continuous asymptotically stabilizing feedback controls," in Differential Equations, vol.
Firstly, assume ([x.sub.f], [y.sub.f], 1) to be any pixel with homogeneous coordinates
in the panoramic coordinate system with a corresponding spherical homogeneous coordinate
([x.sub.Ps], [y.sub.Ps], [z.sub.Ps], 1).
To manage homogeneous coordinates
, DSD entails replacing the final row of the summed quadric matrix Q with [0 0 0 1] and determining the [Q.sup.-1] to obtain the optimal vertex position.
where d and o are the homogeneous coordinates
of the distorted and original input images.
Points in homogeneous coordinates
can be written as [x'.sub.i] = [Hx.sub.i], where H represents the transformation of 3x3 homography matrix: