The Study of 3D-from-2D Using Elimination

This paper uniies most of the current literature on 3D geometric invariants from point correspondences across multiple 2D views by using the tool of Elimination from algebraic geometry. The technique allows one to predict results by counting parameters and reduces many complicated results obtained in the past (reconstruction from two and three views, epipolar geometry from seven points, trilinearity of three views, the use of a priori 3D information such as bilateral symmetry, shading and color constancy, and more) into a few lines of reasoning each. The tool of Grobner base computation is used in the elimination process. In the process we obtain several results on N-view geometry, and obtain a general result on invariant functions of 4 views and its corresponding quadlinear tensor: 4 views admit minimal sets of 16 invariant functions (of quadlinear forms) with 81 distinct coef-cients that can be solved linearly from 6 corresponding points across 4 views. This result has non-trivial implications to the understanding of N-view geometry. We show a new result on single-view invariants based on 6 points and show that certain relationships are impossible. One of the appealing features of the elimination approach is that it is simple to apply and does not require any understanding of the underlying 3D from 2D geometry and algebra.