Tensorial properties of multiple view constraints

Abstract

We define and derive some properties of the different multiple view tensors. The multiple view geometry is described using a four-dimensional linear manifold in R 3m , where m denotes the number of images. The Grassman co-ordinates of this manifold build up the components of the different multiple view tensors. All relations between these Grassman co-ordinates can be expressed using the quadratic p-relations. From this formalism it is evident that the multiple view geometry is described by four different kinds of projective invariants; the epipoles, the fundamental matrices, the trifocal tensors 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 transfer features seen in some images to another image. As an application of this formalism we show how a representation of the multiple view geometry can be calculated from different combinations of multiple view tensors and how some tensors can be extracted from others. We also give necessary and sufficient conditions for the tensor components, i.e. the constraints they have to obey in order to build up a correct tensor, as well as for arbitrary combinations of tensors. Finally, the tensorial rank of the different multiple view tensors are considered and calculated.