Geodesic distance evolution of surfaces: a new method for matching surfaces H. M. Yahia, E. G. Huot, I. L. Herlin INRIA-Rocquencourt BP 105, 78153 Le Chesnay Cedex France. Hussein.Yahia@inria.fr Etienne.Huot@inria.fr Isabelle.Herlin@inria.fr I. Cohen Institute for Robotics and Intelligent Systems University of Southern California Los Angeles, California 90089-0273, USA. icohen@iris.usc.edu Abstract The general problem of surface matching is taken up in this study. The process described in this work hinges on a geodesic distance equation for a family of surfaces embed- ded in the graph of a cost function. The cost function rep- resents the geometrical matching criterion between the two 3D surfaces. This graph is a hypersurface in 4-dimensional space, and the theory presented herein is a generalization of the geodesic curve evolution method introduced by R. Kim- mel et al [12]. It also generalizes a 2D matching process developed in [4]. An Eulerian level-set formulation of the geodesic surface evolution is also used, leading to a numer- ical scheme for solving partial differential equations origi- nating from hyperbolic conservation laws [17], which has proven to be very robust and stable. The method is applied on examples showing both small and large deformations, and arbitrary topological changes. 1. Introduction and previous work The problem of matching structures is a challenging is- sue in computer vision and graphics. It has a large number of applications and the definition of structures characteris- tics depends on the problem addressed. A general matching formulation problem can be stated as: given two structures and define a function which as- sociates to any point of a corresponding point on . The characterization of such function is an ill-posed problem in the sense there is no unique solution. However, introducing structures properties such as the geometry or the underlying image representation allows to characterize a unique match- ing function. Commonly used features are pixels grey level values for stereoscopic matching or optical flow [5], edges for token based approaches [21] and geometric properties of the structures [22, 23]. The latter properties are more ro- bust since they can deal with situations where there is no consistency of the image grey level value. Relevant geo- metric properties are selected on the basis of their ability to characterize a description of the structures which is in- variant to the actual deformation. In the case of rigid or small elastic deformations high curvature points [3, 16] or semi-differential invariants [14] can be considered as an in- variant description of the structure. Higher order geometric description can also be considered such as crest and ridge lines in order to characterize 3D structures properties [19]. These methods perform well but cannot deal large deforma- tion or when we are interested in studying the evolution of a structure whose topology evolves in time [4]. This situ- ation is very common in computer graphics where we are interested in defining a morphing function, ie. a set of paths departing from the source and ending at the destination structure . Here the matching criterion translates into ge- ometric constraints on the paths themselves [15]. Matching features in 3D images is also an important task in medical image analysis [8, 20]. In this paper the general problem of matching two arbi- trary surfaces and in is contemplated. The method proposed represents a generalization of the algorithm intro- duced by Cohen et al [4] for curves matching. This scheme has also motivated a generalization of the geodesic curve propagation method introduced by R. Kimmel et al [12] into a surface evolution framework. First, we briefly sum- marize the 2D case (section 2) in order to clearly state the objectives for 3D surfaces matching. As the curve match- ing is based on the curve evolution on a surface in , we set up a generalization by considering a geodesic surface evolution scheme on a hypersurface embedded in (see section 3.3). This theory makes use of the Hodge “star” ( ) operator (briefly presented in subsection 3.2). Subsection 3.4 is devoted to the level-set formulation of that geodesic surface evolution scheme, allowing for the computation of