INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Algebraic Relations among Matching Constraints of Multiple Images

Given a set of n 2 2 uncalibrated views, for any corresponding point across n views, there exist three types of matching constraints: bilinear constraints (for n 2 2), trilinear constraints (for n 2 3, [12]) and quadrilinear constraints (for n 2 4, 16, 14, 31). The exact algebraic relations among these multi-1inea.r constraints have not been elu~ida~ted by previous authors. This paper examines the relations between these matching constraints by singling out the degenerate view and point configurations. The key result that will be established is that for generic view configurations and generic points, all multi-linear constraints may algebraically be reduced to the algebraically independent bilinear constraints. In other words, all matching constraints are contained in the ideal generated only by the bilinear constraints for generic views and points. As a consequence, 2n 3 algebraically independent bilinearities from pairs of views completely describe the algebraic/geometric structure of n uncalibrated views for generic views and points. For degenerate points of generic views, each type of constraint reduces differently. The exact reduced form of the matching constraints are also made explicit by computer algebra. Key-words: geometry, invariant, epipolar geometry, bilinearity, trilinear ity, uncalibrated image.