Dynamic P n to P n Alignment
We introduce in this chapter A generalization of the classical collineation of P n. The generalization allows for a certain degree of freedom in the local-ization of points in P n at the expense of using multiple m > 2 views. The degree of freedom per point is governed by an additional parameter 1 ≤ k < n which stands for the dimension of the subspace in which the indvidual points are allowed to move while the projective change of coordinates take place. In other words, the point set is not necessarily stationary allowing the configuration to change while the entire coordinate system undergoes a projective change of coordinates. If we denote a change of coordinates as a " view " of the physical set of points, then in this chapter we discuss the multi-view relations that can be determined from observations (views) of a dynamically changing point-configuration —-the underlying transformations and how they can be recovered. For example, for a point set in P 2 (planar configuration) undergoing linear motion the multiple views of the point set generate a multilinear constraint across three views governed by a 3 × 3 × 3 contravariant tensor H. The tensor, referred to as homography tensor, can be recovered linearly from 26 observations (matching points across three views) and once recovered can be unfolded to yield the global coordinate change (the individual pair of homography matrices). A point set in P 3 (3D configuration) can undergo motion in a plane or along a line (each point independently). For the line motion, the multi-view constraints are governed by a 4 × 4 × 4 family of contravariant tensors J that capture the dynamic 3D-to-3D alignment problem. More generally, the family of homography tensors is captured by three parameters: the dimension n of the observation space, the dimension k < n of the subspace along which each point of the point set is allowed to move, and the number of " views " m. Formally , the homography tensors form a GL(V) module, denoted by V (n, m, k), defined by the set of all tensors v 1 ⊗· · ·⊗v m ∈ V ⊗m where v i are n-dimensional vectors and dim Span{v 1 ,. .. , v m } ≤ k. We will be interested in the structure and dimension of V (n, m, k). The notion of using mutli-view analysis for non-rigid …