Curvature and Continuity Control in Particle-Based Surface Models Richard Szeliski 1 , David Tonnesen 2 , and Demetri Terzopoulos 2 1 Digital Equipment Corporation, Cambridge Research Lab, One Kendall Square, Bldg. 700, Cambridge, MA 02139 2 Department of Computer Science, University of Toronto, Toronto, ON, M5S 1A4 Abstract This paper develops techniques to locally control curvature and continuity in particle-based surface models. Such models are a generalization of traditional spline surfaces built out of triangular patches. Traditional splines require the topology of the triangular mesh to be specified ahead of time. In contrast, particle-based surface models compute the topology dynamically as a function of the relative node positions, and can add or delete nodes as required. Such models are particularly important in computer vision and other inverse problems, where the topology of the surface being reconstructed is usually not known a priori. We develop techniques for both locally controlling the curvature of the surface (through additional state at each node), and for adapting the triangulation to surface curvature (by concentrating more particles in areas of high curvature). We show how the same ideas can also be applied to 3-D curves, which results in a flexible version of traditional dynamic contours (snakes). 1. Introduction Flexible 3D surface representations are an essential component of 3D computational vision, enabling the estimation of geometric shape from various types of visual data, including range and surface normal measurements. These surfaces can be used as an intermediate representation for object recognition, to guide robotics tasks such as grasping, to segment three-dimensional volumes (e.g., in medical applications), and to integrate different visual modalities such as stereo and shading. Existing surface representations have limitations—viewer-centered representations [3, 17, 21, 27] do not describe non- visible portions of object surfaces, while the object-centered representations [1, 16, 23] often make strong assumptions about object topology. Unknown object topology poses difficult challenges to surface modeling and reconstruction that have so far been largely ignored in the vision literature. Unfortunately, vision systems that must derive quantitative models of complex real-world objects from multiple views cannot avoid the issue of unpredictable topological structure. Unknown topology is also an important concern in the related fields of biomedical and geological imaging where there is a need to analyze three dimensional arrays of volumetric density or reflectivity data. One way to cope with unknown topological structure is to use a “patchwork” representation that describes the surface only locally in terms of planar, quadric, or cubic patches [15]. The lack of a global, continuous representation makes this approach cumbersome for surface analysis tasks such as area, curvature, and enclosed volume computations. More serious difficulties arise in the dynamic analysis of objects, including the incremental reconstruction of surfaces from sequential views around objects, or the reconstruction, tracking, and motion estimation of dynamic nonrigid objects such as a beating heart. A globally consistent surface model can provide powerful constraints for solving these dynamic estimation problems. In previous work, we developed a flexible surface model based on oriented particles which can model surfaces of arbitrary (and unknown) topology [19] and fit these surfaces to 3D range or volumetric data [20]. Our approach retains the topological flexibility of the local patch methods, while constructing globally coherent surface models that can evolve consistently with time-varying data. Our representation is three-dimensional and viewpoint invariant. It is also non-parametric, i.e., it does not represent the surface as a function of a two-dimensional parameter space. Instead, all information is represented in local differential geometric quantities (orientations, curvatures) which are used to form continuous surfaces. In our previous papers [19, 20], we concentrated on the discrete particle-based nature of our surface model. In this paper, we first focus on continuous models of curves and surfaces, and then show how to implement discrete oriented particle systems that approximate these models. Both the continuous and discrete models are based on collections of oriented trihedral coordinate frames which represent position and local differential geometric information such as orientation and curvature. For the discrete model, we design special interaction potentials which favor local arrangements of particles that are consistent with continuous