We present a new particle-based approach to sampling and controlling implicit surfaces. A simple constraint locks a set of particles onto a surface while the particles and the surface move. We use the constraint to make surfaces follow particles, and to make particles follow surfaces. We implement control points for direct manipulation by specifying particle motions, then solving for surface motion that maintains the constraint. For sampling and rendering, we run the constraint in the other direction, creating floater particles that roam freely over the surface. Local repulsion is used to make floaters spread evenly across the surface. By varying the radius of repulsion adaptively, and fissioning or killing particles based on the local density, we can achieve good sampling distributions very rapidly, and maintain them even in the face of rapid and extreme deformations and changes in surface topology. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling:...

This paper presents a state of the art report of our project, the main objectives of which are: -- Categorization and summary of the geometric concepts required in a functionally based modeling environment; -- Elaboration of a rich system of geometric operations closed on functionally represented objects; -- Treatment of multidimensional and particularly space-time objects in a uniform manner; 6

We present an approach for the reconstruction and approximation of 3D CAD models from an unorganized collection of points. Applications include rapid reverse engineering of existing objects for use in a synthetic computer environment, including computer aided design and manufacturing. Our reconstruction approach is flexible enough to permit interpolation of both smooth surfaces and sharp features, while placing few restrictions on the geometry or topology of the object. Our algorithm is based on alpha-shapes to compute an initial triangle mesh approximating the object's surface. A mesh reduction technique is applied to the dense triangle mesh to build a simplified approximation, while retaining important topological and geometric characteristics of the model. The reduced mesh is interpolated with piecewise algebraic surface patches which approximate the original points. The process is fully automatic, and the reconstruction is guaranteed to be homeomorphic and error bounded with respec...

We discuss methods for fitting implicitly defined (e.g. piecewise algebraic) curves to scattered data, which may contain problematic regions, such as edges, cusps or vertices. As the main idea, we construct a bivariate function, whose zero contour approximates a given set of points, and whose gradient field simultaneously approximates an estimated normal field. The coefficients of the implicit representation are found by solving a system of linear equations. In order to allow for problematic input data, we introduce a criterion for detecting points close to possible singularities. Using this criterion we split the data into segments and develop methods for propagating the orientation of the normals globally. Furthermore we present a simple fallback strategy, that can be used when the process of orientation propagation fails. The method has been shown to work successfully

A general paradigm for recognizing 3D objects is offered, and applied to some geometric primitives (spheres, cylinders, cones, and tori). The assumption is that a curve on the surface, or a pair of intersecting curves, was measured with high accuracy (for instance, by a sensory robot). Differential invariants of the curve(s) are then used to recognize the surface. The motivation is twofold: the output of some devices is not surface range data, but such curves. Also, a considerable speedup is obtained by using curve data, as opposed to surface data which usually contains a much higher number of points. We survey global, algebraic methods for recognizing surfaces, and point out their limitations. After introducing some notions from differential geometry and elimination theory, the differential and "semi-differential" approach to the problem is described, and novel invariants which are based on the curve's curvature and torsion are derived. 1 Introduction and Previous Work One task an ...

Implicit representations of curves have certain advantages over explicit representation, one of them being the ability to determine with ease whether a point is inside or outside the curve (inside-outside functions). However, save for some special cases, it is not known how to construct implicit representations which are guaranteed to preserve the curve's topology. As a result, points may be erroneously classified with respect to the curve. The paper offers to overcome this problem by using a representation which is guaranteed to yield the correct topology of a simple closed curve by using homeomorphic mappings of the plane to itself. If such a map carries the curve onto the unit circle, then a point is inside the curve if and only if its image is inside the unit circle.

The implementation of morphological connected set operators for image filtering and pattern recognition is discussed. Two earlier algorithms based on priority queues and hierarchical queues, respectively, are compared to a more recent union-find approach.

We present a technique for fitting a smooth, locally parameterized surface model (called the manifold surface model) to unevenly scattered data describing an anatomical structure. This data is acquired from medical imaging modalities such as CT scans or MRI. The manifold surface is useful for problems which require analyzable or parametric surfaces fitted to data acquired from surfaces of arbitrary topology (e.g., entire bones). This surface modeling work is part of a larger project to model and analyze skeletal joints, in particular the complex of small bones within the wrist and hand. To demonstrate the suitability of this model we fit to several di#erent bones in the hand, and to the same bone from multiple people.

We introduce a new variational formulation for the problem of reconstructing a watertight surface defined by an implicit equation, from a finite set of oriented points; a problem which has attracted a lot of attention for more than two decades. As in the Poisson Surface Reconstruction approach, discretizations of the continuous formulation reduce to the solution of sparse linear systems of equations. But rather than forcing the implicit function to approximate the indicator function of the volume bounded by the implicit surface, in our formulation the implicit function is forced to be a smooth approximation of the signed distance function to the surface. Since an indicator function is discontinuous, its gradient does not exist exactly where it needs to be compared with the normal vector data. The smooth signed distance has approximate unit slope in the neighborhood of the data points. As a result, the normal vector data can be incorporated directly into the energy function without implicit function smoothing. In addition, rather than first extending the oriented points to a vector field within the bounding volume, and then approximating the vector field by a gradient field in the least squares sense, here the vector field is constrained to be the gradient of the implicit function, and a single variational problem is solved directly in one step. The formulation allows for a number of different efficient discretizations, reduces to a finite least squares problem for all linearly parameterized families of functions, and does not require boundary conditions. The resulting algorithms are significantly simpler and easier to implement, and produce results of quality comparable with state-of-the-art algorithms. An efficient implementation based on a primal-graph octree-based hybrid finite element-finite difference discretization, and the Dual Marching Cubes isosurface extraction algorithm, is shown to produce high quality crack-free adaptive manifold polygon meshes.