Estimating principal curvatures and principal directions of a surface from a polyhedral approximation with a large number of small faces, such as those produced by iso-surface construction algorithms, has become a basic step in many computer vision algorithms. Particularly in those targeted at medical applications. In this paper we describe a method to estimate the tensor of curvature of a surface at the vertices of a polyhedral approximation. Principal curvatures and principal directions are obtained by computing in closed form the eigenvalues and eigenvectors of certain # # # symmetric matrices defined by integral formulas, and closely related to the matrix representation of the tensor of curvature. The resulting algorithm is linear, both in time and in space, as a function of the number of vertices and faces of the polyhedral surface. 1

In this paper, we propose a novel polygonal remeshing technique that exploits a key aspect of surfaces: the intrinsic anisotropy of natural or man-made geometry. In particular, we use curvature directions to drive the remeshing process, mimicking the lines that artists themselves would use when creating 3D models from scratch. After extracting and smoothing the curvature tensor field of an input geometry patch, lines of minimum and maximum curvatures are used to determine appropriate edges for the remeshed version in anisotropic regions, while spherical regions are simply point-sampled since there is no natural direction of symmetry locally. As a result our technique generates polygon meshes mainly composed of quads in anisotropic regions, and of triangles in spherical regions. Our approach provides the flexibility to produce meshes ranging from isotropic to anisotropic, from coarse to dense, and from uniform to curvature adapted.

The problem of obtaining a mathematical representation of the cortex of the human brain is examined. A parametrization of the outer cortex is first obtained using a deformable surface algorithm which, motivated by the structure of the cortex, is constructed to find the central layer of thick surfaces. Based on this parametrization, a hierarchical representation of the cortical structure is proposed through its depth map and its curvature maps at various scales. Various experiments on magnetic resonance data are presented. I. Introduction The problem of finding and parametrizing boundaries in two- and three-dimensional images is often an important step toward shape visualization and analysis, and has been extensively studied in the image analysis and computer vision literature. Several methods have been proposed, basedboth on bottom-up and top-bottom procedures. One very promising model which combines robustness to noise and the flexibility to represent a broad class of shapes is base...

We are considering the problem of recovering the three-dimensional geometry of a scene from binocu-lor stereo disparity. Once a dense disparity map has been computed from a stereo pair of images, one of-ten needs to calculate some local diferential proper-ties of the cowesponding 3-0 surface such as orien-tation or curvatures. The wual approach is to build a 3-0 reconstruction of the surface(s) from which all shape properties will then be derived without ever go-ing back to the original images. In this paper, we de-part from this paradigm and propose to w e the images directly to compute the shape properties. We thus pro-pose a new method extending the classical cowelation method to estimate accurately both the disparity and its derivatives directly from the image data. We then relate those derivatives to diferential properties of the surface such as orientation and curvatures.

A real-time vision system is described that can recognize 100 complex three-dimensional objects. In contrast to traditional strategies that rely on object geometry and local image features, the present system is founded on the concept of appearance matching. Appearance manifolds of the 100 objects were automatically learned using a computer-controlled turntable. The entire learning process was completed in 1 day. A recognition loop has been implemented that performs scene change detection, image segmentation, region normalizations, and appearance matching, in less than 1 second. The hardware used by the recognition system includes no more than a CCD color camera and a workstation. The real-time capability and interactive nature of the system have allowed numerous observers to test its performance. To quantify performance, we have conducted controlled experiments on recognition and pose estimation. The recognition rate was found to be 100 % and object pose was estimated with a mean abso...

We present an approach for recovering surface shape from the occluding contour using an active (i.e., moving) observer. It is based onarelation between the geometries of a surface inascene and its occluding contour: If the viewing direction of the observer is along a principal direction for a surface point whose projection is on the contour, surface shape (i.e., curvature) at the surfacepoint can be recovered from the contour. Unlike previous approaches for recovering shape from the occluding contour, we use an observer that purposefully changes viewpoint in order to achieve a well-de ned geometric relationship with respect to a 3D shape prior to its recognition. We show that there is a simple and e cient viewing strategy that allows the observer to align the viewing direction with one of the two principal directions for a point on the surface. Experimental results demonstrate that our method can be easily implemented and can provide reliable shape information. 1

We present a novel approach to reliable and efficient recovery of part-descriptions in terms of superquadric models from range data. We show that superquadrics can directly be recovered from unsegmented data, thus avoiding any presegmentation steps (e.g., in terms of surfaces). The approach is based on the recover-andselect paradigm [10]. We present several experiments on real and synthetic range images, where we demonstrate the stability of the results with respect to viewpoint and noise.

This report describes research partially done at the Massachusetts Institute of Technology within the Artificial Intelligence Laboratory. Support for the laboratory's artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N00014-85-K-0124. *Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139 tDept. of Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

We introduce a new surface representation for recognizing curved objects. Our approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object. The mesh is computed from the data by deforming a standard shaped mesh, for example, an ellipsoid, until it fits the surface of the object. We define local regularity constraints that the mesh must satisfy. We then define a canonical mapping between the mesh describing the object and a standard spherical mesh. A surface curvature index that is pose-invariant is stored at every node of the mesh. We use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene. We show how the similarity between reference model and observed data can be evaluated and we show how the pose of the reference object in the observed scene can be easily computed using this representation. We present results on real range images which show that this approach to modelling and recognizing three-dimensional objects has three main advantages: First, it is applicable to complex curved surfaces that cannot be handled by conventional techniques. Second, it reduces the recognition problem to the computation of similarity between spherical distributions; in particular, the recognition algorithm does not require any combinatorial search. Finally, even though it is based on a spherical mapping, the approach can handle occlusions and partial views.

In contrast to the traditional approach, the recognition problem is formulated as one of matching appearance rather than shape. For any given vision task, all possible appearance variations define its visual workspace. A set of images is obtained by coarsely sampling the workspace. The image set is compressed to obtain a low-dimensional subspace, called the eigenspace, in which the visual workspace is represented as a continuous appearance manifold. Given an unknown input image, the recognition system first projects the image to eigenspace. The parameters of the vision task are recognized based on the exact position of the projection on the appearance manifold. The proposed appearance representation has several applications in visual perception. As examples, a real-time recognition system with 20 complex objects, an illumination planning technique for robust object recognition, and a real-time visual positioning and tracking system are described. The simplicity and generality of the pr...