Points-based Visual Servoing with Central Cameras - Visual Servoing via Advanced Numerical Methods - page 286

16.2.1 Generic Projection Model

Central imaging systems can be modeled using two consecutive projections: spher-
ical projection succeeded by a perspective one. This geometric formulation called
unified model has been proposed by Geyer and Daniilidis in [9] and has been inten-
sively used by the vision and robotics community (structure from motion, calibra-
tion, visual servoing, etc ).
Consider the virtual unitary sphere centered in the origin of the mirror frame F m

as shown in Figure 16.1 and the perspective camera centered in the origin of the
camera frame

F c . Without lost of generality, a simple translation of − ξ , along the

Z axis of the mirror frame, between F m and F c is considered. Let X be a 3D

point with coordinates X =[ XYZ ] in is projected in the

image plane into the point of homogeneous coordinates x i =[ x i y i 1] .Theimage
formation process can be split in three steps given in the following:

F m . The world point X

• the 3D world point X is mapped onto the unit sphere surface:

XYZ

1
ρ

,

X s = (16.1)

= √ X 2 + Y 2 + Z 2 ; where ρ = X

Fig. 16.1 Unified central projection and two views geometry