Certain and Uncertain Triangulation in Multiple Camera Vision Systems via LMIs

Abstract

Triangulation is a fundamental problem in computer vision that consists of estimating the 3D position of a point of the scene from the estimates of its image projections on some cameras and from the estimates of the projection matrices of these cameras. This chapter addresses multiple view L2 triangulation, i.e. triangulation for vision systems with a generic number of cameras where the sought 3D point is searched by minimizing the L2 norm of the image reprojection error. The authors consider the standard case where estimates of all the image points are available (referring to such a case as certain triangulation), and consider also the case where some of such estimates are not available for example due to occlusions (referring to such a case as uncertain triangulation). In the latter case, it is supposed that the unknown image points belong to known regions such as line segments or ellipses. For these problems, the authors propose a unified methodology that exploits the fundamental matrices among the views and provides a candidate 3D point through the solution of a convex optimization problem based on linear matrix inequalities (LMIs). Moreover, the chapter provides a simple condition that allows one to immediately establish whether the found 3D point is optimal. Various examples with synthetic and real data illustrate the proposed technique, showing in particular that the obtained 3D point is almost always optimal in practice, and that its computational time is indeed small.