Front Propagation and Level-Set Approach For Geodesic Active Stereovision Rachid DERICHE, Christophe BOUVIN and Olivier FAUGERAS I.N.R.I.A, 2004 Route des Lucioles BP 93, 06902 Spohia Antipolis Cedex, France email: der@sophia.inria.fr Abstract Given a weakly calibrated stereo system and a virtual 3D surveillance plane specified by any 3 points given by an external operator, we describe a framework for match- ing complex 2D planar curves lying at the intersection of the 3D surveillance plane and the 3D scene being observed. This important information may then be used to know which parts of the objects being observed are between the stereo system and the virtual 3D surveillance plane, and which parts are behind the 3D virtual surveillance plane i.e out- side a security zone specified around the stereo system, Us- ing an energy minimization based approach, we reformulate this stereo problem as a front propagation problem. The Euler Lagrange equation of the designed energy functional is derived and the flow minimizing the energy is obtained. This original scheme may be viewed as a geodesic active stereo model which basically attract the given curves to the bottom of a potential well corresponding to pixels having similar intensities. Using the level set formulation scheme of Osher and Sethian [11], complex curves can be matched and topological changes for the evolving curves are natu- rally managed. The final result is also relatively indepen- dent of the curve initialization. Promising experimental re- sults have been obtained on various real images KEY WORDS :Stereo, Front propagation, Level-set ap- proach, Curve evolution, Deformable contours, Geodesic Active contours, PDE 1. INTRODUCTION Given two different views of a 3D scene, and a virtual 3D surveillance plane, the method we propose allows to re- cover the 2D projections, in the two images, of the 3D pla- nar curves corresponding to the intersection of the virtual surveillance plane with the different objects in the scene being observed. This important information may then be used to know which parts of the objects being observed are between the stereo system and the virtual 3D surveillance plane, and which parts are behind the 3D virtual 3D surveil- lance plane i.e outside a security zone specified around the stereo system, The 3D surveillance plane is specified just by the knowledge of the 2D projections on the two images of 3 points lying in this plane. An arbitrary curve is first initial- ized in one of the two images. This curve, and its associated homographic curve in the second image are then designed to move under the influence of internal and external image de- pendent forces while minimizing an energy functional. Fol- lowing the work on geodesic active contours by Caselles et al [3, 4] ( see also the work by Malladi et al [9, 10] and Kichenassamy et al [8]), we then transform the prob- lem of minimizing this energy functional into a problem of geodesic computation in a Riemannian space, according to a new metric. The Euler-Lagrange equation of this new functional is derived and its associated PDE is then solved using the level set formulation scheme of Osher and Sethian [11] by viewing it as a front propagating with internal and external image correlation dependent speed. The curves to be matched are therefore modelized as geodesic active con- tours evolving toward the minimum of the designed func- tional which basically lie at the bottom of a correlation po- tential well. Using the level set technique, the propagating curve is viewed as the zero level set of a time dependent surface. The resulting equation is then solved using tech- niques borrowed from hyperbolic conservation laws. With this technique, complex curves can be matched, and the fi- nal result is relatively independent of the curve initializa- tion. Topological changes for the evolving curves are also naturally obtained in this setting. Hence, during their evo- lution, the matched curves may change connectivity, split and merge, allowing the simultaneous detection and match- ing of all the planar curves. This numerical scheme has been implemented and many experimental results of apply- ing it to real stereo pair of images including different objects demonstrates its power. The problem addressed is very im- portant from the point of view of the applications. Match- ing 2D planar curves provides useful 3D information as the relative positioning of any point in the scene with respect to this plane. This is an important application in robotics 1