  Key points, correspondences, projection matrices, point clouds and dense
clouds are the skeletons in image-based 3D reconstruction, of which point
clouds have the important role in generating a realistic and natural model for
a 3D reconstructed object. To achieve a good 3D reconstruction, the point
clouds must be almost everywhere in the surface of the object. In this article,
with a main purpose to build the point clouds covering the entire surface of
the object, we propose a new feature named a geodesic feature or geo-feature.
Based on the new geo-feature, if there are several (given) initial world points
on the object's surface along with all accurately estimated projection
matrices, some new world points on the geodesics connecting any two of these
given world points will be reconstructed. Then the regions on the surface
bordering by these initial world points will be covered by the point clouds.
Thus, if the initial world points are around the surface, the point clouds will
cover the entire surface.
  This article proposes a new method to estimate the world points and
projection matrices from their correspondences. This method derives the
closed-form and iterative solutions for the world points and projection
matrices and proves that when the number of world points is less than seven and
the number of images is at least five, the proposed solutions are global
optimal. We propose an algorithm named World points from their Correspondences
(WPfC) to estimate the world points and projection matrices from their
correspondences, and another algorithm named Creating Point Clouds (CrPC) to
create the point clouds from the world points and projection matrices given by
the first algorithm.

Computer Vision and Pattern Recognition