Tensorial properties of multiple view constraints

We define and derive some properties of the different multiple view tensors . The multiple view geometry is described using a four-dimensional linear m anifold in R-3m, where m denotes the number of images. The Grassman co-ordi nates of this manifold build up the components of the different multiple vi ew tensors. All relations between these Grassman co-ordinates can be expres sed using the quadratic p-relations. From this formalism it is evident that the multiple view geometry is described by four different kinds of project ive invariants; the epipoles, the fundamental matrices, the trifocal tensor s and the quadrifocal tensors. We derive all constraint equations on these tensors that can be used to estimate them from corresponding points and/or lines in the images as well as all transfer equations that can be used to t ransfer features seen in some images to another image. As an application of this formalism we show how a representation of the mul tiple view geometry can be calculated from different combinations of multip le view tensors and how some tensors can be extracted from others. We also give necessary and sufficient conditions for the tensor components, i.e, th e constraints they have to obey in order to build up a correct tensor, as w ell as for arbitrary combinations of tensors. Finally, the tensorial rank o f the different multiple view tensors are considered and calculated. Copyri ght (C) 2000 John Wiley & Sons, Ltd.