Abstract

poi We study the theory of projective reconstruction for resmultiple projections from an arbitrary dimensional proan jective space into lower-dimensional spaces. This probas lem is important due to its applications in the analobtysis of dynamical scenes. The current theory, due to theHartley and Schaffalitzky, is based on the Grassmann imatensor, generalizing the ideas of fundamental matrix, contrifocal tensor and quadrifocal tensor used in the wellbasstudied case of 3D to 2D projections. We present a anatheory whose point of departure is the projective equafactions rather than the Grassmann tensor. This is a betrecter fit for the analysis of approaches such as bundle soladjustment and projective factorization which seek to directly solve the projective equations. In a first step, imawe prove that there is a unique Grassmann tensor cortheresponding to each set of image points, a question that traremained open in the work of Hartley and Schaffalitzky. proThen, we prove that projective equivalence follows from damthe set of projective equations given certain conditions [4,on the estimated camera-point setup or the estimated tenprojective depths. Finally, we demonstrate how wrong camsolutions to the projective factorization problem can andhappen, and classify such degenerate solutions based on the zero patterns in the estimated depth matrix. len Sha pro