Matching constraints and the joint image

Abstract

The paper studies the geometry of multi image perspective projection and the matching constraints that this induces on image measurements. The combined image projections define a 3D joint image subspace of the space of combined homogeneous image coordinates. This is a complete projective replica of the 3D world in image coordinates. Its location encodes the imaging geometry and is captured by the 4 index joint image Grassmannian tensor. Projective reconstruction in the joint image is a canonical process requiring only a simple rescaling of image coordinates. Reconstruction in world coordinates amounts to a choice of basis in the joint image. The matching constraints are multilinear tensorial equations in image coordinates that tell whether tokens in different images could be the projections of a single world token. For 2D images of 3D points there are exactly three basic types: the epipolar constraint, A. Shashua's (1995) trilinear one, and a new quadrilinear 4 image one. For images of lines, R. Hartley's (1994) trilinear constraint is the only type. The coefficients of the matching constraints are tensors built directly from the joint image Grassmannian. Their complex algebraic interdependency is captured by quadratic structural simplicity constraints on the Grassmannian.>