  Correspondences estimation or feature matching is a key step in the
image-based 3D reconstruction problem. In this paper, we propose two algebraic
properties for correspondences. The first is a rank deficient matrix construct
from the correspondences of at least nine key-points on two images (two-view
correspondences) and the second is also another rank deficient matrix built
from the other correspondences of six key-points on at least five images
(multi-view correspondences). To our knowledge, there are no theoretical
results for multi-view correspondences prior to this paper. To obtain accurate
correspondences, multi-view correspondences seem to be more useful than
two-view correspondences. From these two algebraic properties, we propose an
refinement algorithm for correspondences. This algorithm is a combination of
correspondences refinement, outliers recognition and missing key-points
recovery. Real experiments from the project of reconstructing Buddha statue
show that the proposed refinement algorithm can reduce the average error from
77 pixels to 55 pixels on the correspondences estimation. This drop is
substantial and it validates our results.
