New Rank Deficiency Condition for Multiple View Geometry of Point Features
Yi Ma, René Vidal, Huang, Kun, Shankar P. Sastry · Illinois Digital Environment for Access to Learning and Scholarship (University of Illinois at Urbana-Champaign) · 2001
In this paper, a new rank deficiency condition for multiple images of a point is presented. It is shown that a set of m images correspond to a unique 3-D pre-image point if and only if an associated 3(n-1) x 2} matrix, the so-called H matrix, is of maximum rank 1. If the rank is always 0 (i.e., the matrix is zero), then the pre-image is only determined up to a line on which all the camera centers must lie. This condition is shown to be equivalent to all the multilinear constraints, but it tremendously simplifies the derivation and proof of all the algebraic relationships among bilinear, trilinear and quadrilinear constraints. Since rank deficiency is a purely linear algebraic condition, it gives rise to a set of natural linear algorithms for purposes such as matching feature points, mapping images to a new view and motion estimation from images of multiple points. These linear algorithms use all available data simultaneously without specifying a particular choice of a set of pairwise, triple-wise or quadruple-wise images. Hence, to a large extent, such algorithms allow us to bypass the use of trifocal or quadrifocal tensors for similar purposes. The proposed rank deficiency condition is believed to be a more concise and universal way of describing the algebraic relationship among multiple images. Although only point features are discussed in this paper, a similar condition is studied in a companion paper for line features, as well as how the duality between points and lines are reflected in such rank deficiency conditions.