Using projective geometry to recover planar surfaces in stereovision

Our purpose is to build the planar surfaces [called three-dimensional (3D) faces] of the objects of a polyhedral scene using a stereovision system composed of three cameras. In each image, our system builds structures known as 2D faces (which are 2D polygons), which are supposed to be the projections of 3D faces, and then matches the 2D faces of each given triplet of images. Due to the large errors caused by the stereo reconstruction, it is difficult to check for planar surfaces after the 3D process. Instead, we prefer to search among the set of triplets of matched 2D faces for the triplets which really correspond to planar 3D surfaces, before the reconstruction step. For this purpose, we propose a new approach to validate the 3D coplanarity of a set of 3D segments, working only on their corresponding matched 2D images and using principles of projective geometry. For each tripler of matched 2D segments, new information, known as trace, is computed and added to the triplets. Then using this new characteristic related to each triplet of matched 2D segments, 3D geometric properties such as the planarity and even coplanarity of a 3D structure are established in 2D space. All these 3D properties, 3D planarity and 3D coplanarity, checked in 2D space, lead to an accurate reconstruction of the surfaces. In this paper, we first introduce the required mathematical tools which explain our approach. Then we present our system for building 3D faces which includes a new unit, in its stereovision part, based on projective geometry. Some experimental results are presented at the end of this paper showing the efficiency of our method.