Tensor Embedding of the Fundamental Matrix

We revisit the bilinear matching constraint between two perspective views of a 3D scene. Our objective is to represent the constraint in the same manner and form as the trilinear constraint among three views. The motivation is to establish a common terminology that bridges between the fundamental matrix F (associated with the bilinear constraint) and the trifocal tensor T jk i (associated with the trilinearities). By achieving this goal we can unify both the properties and the techniques introduced in the past for working with multiple views for geometric applications. Doing that we introduce a 3 3 3 tensor F i , we call the bifocal tensor, that represents the bilinear constraint. The bifocal and trifocal tensors share the same form and share the same contraction properties. By close inspection of the contractions of the bifocal tensor into matrices we show that one can represent the family of rank-2 homography matrices by [ ] F where is a free vector. We then discuss four applications of the new representation: (i) Quasi-metric viewing of projective data, (ii) triangulation, (iii) view synthesis, and (iv) recovery of camera ego-motion from a stream of views.