Reconstruction from Projections Using Grassmann Tensors

Abstract

In this paper a general method is given for reconstruction of a set of feature points in an arbitrary dimensional projective space from their projections into lower dimensional spaces. The method extends the methods applied in the well-studied problem of reconstruction of a set of scene points in $\mathcal {P}^3$ given their projections in a set of images. In this case, the bifocal, trifocal and quadrifocal tensors are used to carry out this computation. It is shown that similar methods will apply in a much more general context, and hence may be applied to projections from $\mathcal {P}^n$ to $\mathcal {P}^m$ , which have been used in the analysis of dynamic scenes. For sufficiently many generic projections, reconstruction of the scene is shown to be unique up to projectivity, except in the case of projections onto one-dimensional image spaces (lines).