On Projective Reconstruction in Arbitrary Dimensions

Abstract

We study the theory of projective reconstruction for multiple projections from an arbitrary dimensional projective space into lower-dimensional spaces. This problem is important due to its applications in the analysis of dynamical scenes. The current theory, due to Hartley and Schaffalitzky, is based on the Grassmann tensor, generalizing the ideas of fundamental matrix, trifocal tensor and quadrifocal tensor used in the well-studied case of 3D to 2D projections. We present a theory whose point of departure is the projective equations rather than the Grassmann tensor. This is a better fit for the analysis of approaches such as bundle adjustment and projective factorization which seek to directly solve the projective equations. In a first step, we prove that there is a unique Grassmann tensor corresponding to each set of image points, a question that remained open in the work of Hartley and Schaffalitzky. Then, we prove that projective equivalence follows from the set of projective equations given certain conditions on the estimated camera-point setup or the estimated projective depths. Finally, we demonstrate how wrong solutions to the projective factorization problem can happen, and classify such degenerate solutions based on the zero patterns in the estimated depth matrix.