Marek Teichmann

Surface reconstruction with anisotropic density-scaled alpha shapes

Generation of a three-dimensional model from an unorganized set of points is an active area of research in computer graphics. Alpha shapes can be employed to construct a surface which most closely reflects the object described by the points. However, no /spl alpha/-shape, for any value of /spl alpha/, can properly detail discontinuous regions of a model. We introduce herein two methods of improving the results of reconstruction using /spl alpha/-shapes: density-scaling, which modulates the value of a depending on the density of points in a region; and anisotropic shaping, which modulates the form of the /spl alpha/-ball based on point normals. We give experimental results that show the successes and limitations of our method.

Assisted articulation of closed polygonal models

Creating articulated geometric models is a common task in animation systems. In some instances, models are procedurally instanced, and articulated degrees of freedom are designed into the model. In other instances, the model is some geometric assemblage, and an articulated skeleton (sometimes called an “I-K skeleton”) is bound to the model by the user, typically by manual indication of a correspondence between elements of each structure. In either case, some binding must be made to couple boundary motions to those of the skeleton; this can be done for example by generating spring networks or spatial deformation fields. Both processes can be tedious in the ordinary case, especially when the model to be articulated is given only as a boundary representation, for example a polygonal mesh representing a character’s skin or clothing, or an object’s surface.

Visibility Queries in Simple Polygons and Applications

In this paper we explore some novel aspects of visibility for stationary and moving points inside a simple polygon P. We provide a mechanism for expressing the visibility polygon from a point as the disjoint union of logarithmically many canonical pieces using a quadraticspace data structure. This allows us to report visibility polygons in time proportional to their size, but without the cubic space overhead of earlier methods. The same canonical decomposition can be used to determine visibility within a frustum, or to compute various attributes of the visibility polygon efficiently. By exploring the connection between visibility polygons and shortest path trees, we obtain a kinetic algorithm that can track the visibility polygon as the viewpoint moves along polygonal paths inside P, at a polylogarithmic cost per combinatorial change in the visibility. The combination of the static and kinetic algorithms leads to a space query-time tradeoff for the visibility from a point problem and an output-sensitive algorithm for the weak visibility from a segment problem.

Reactive robotics. I: Reactive grasping with a modified gripper and multifingered hands

We study the problem of grasping an unknown object with constant cross section using various “reactive” robot hands. In the simplest example, we equip a standard parallel-jaw gripper with several light-beam sensors (close to each jaw) and implement a reactive algorithm for grasping polygonal objects with this architecture. Extending these ideas further, we also devise two-and three-fingered reactive hands for objects with smooth boundary and equip these with distance and angle sensors that are located at the finger tips. The sensors used are simple and provide only limited and immediate information, but they allow us to reactively find a good grasp on an object of unknown geometry and dynamics. When grasped, the forces applied will be normal to the object boundary. Furthermore, in all cases, the object is not disturbed as the grasping points are being sought.

Smallest enclosing cylinders

This paper addresses the complexity of computing the smallest-radius infinite cylinder that encloses an input set of n points in 3-space. We show that the problem can be solved in time O(n 4 log O(1) n) in an algebraic complexity model. We also achieve a time of O(n 4 L⋅μ(L)) in a bit complexity model where L is the maximum bit size of input numbers and μ(L) is the complexity of multiplying two L bit integers.

Non convex mesh penetration distance for rigid body dynamics

We show that the framework of [Ehmann and Lin 2001] for finding the minimum distance and detecting collision between meshes generalizes to a large class of distance-like functions. As an application, we describe a fast new algorithms for computing the penetration distance along a direction between two arbitrary meshes. Penetration distance is important for computing collision response for rigid body dynamics solvers. Finally, we describe a simple way to find a good direction of penetration: the gradient of the penetration volume between the meshes. We give a simple algorithm to obtain this direction which requires only the intersection curve between the meshes.

Geometric reconstruction with anisotropic alpha-shapes

of a three-dimensional model from an unorganized set of points is an active area of research in computer graphics. Such point sets come from a number of common sources, such as range data from three-dimensional scanning hardware, implicit surface, and medical imaging. The notion of ␣-shapes provides an elegant mathematical framework for extracting the geometric structure of a set of points in three dimensions. 2 Briefly, the ␣-shape is a set of triangles and tetrahedra that is a subset of the Delaunay tri-angulation of the input point set. The theory of ␣-shapes provides a method for obtaining a surface by selecting a subset of the triangles in the triangulation. A triangle is in the ␣-shape if one of the radii of the following spheres is at most ␣: the spheres circumscribing the trian-gles vertices and the two nearest neighboring points, and the sphere circumscribing the triangle. While this definition gives good results for point sets of roughly uniform density with large separation between surfaces, this definition is clearly not optimal for non-uniform point sets, or for surfaces that are separated by a distance less than their sampling density. For these, there exists no value of ␣ that includes all desired triangles and deletes all undesired triangles. See the first image in the figure for an example of the best possible surface obtained using ␣-shapes. We propose two extensions to alleviate these problems in the case where normal information is available (or estimated as in Hoppe et al. 3) at each point. These extensions allow reconstruction from a larger class of point sets: 1 Anisotropic scaling: we allow the spheres to vary in shape, and change the triangulation accordingly. A fundamental contribution of this work is incremental retriangulation based on a user-specified factor ␶, for the influence of the anisotropy. The spherical forbidden region, which depends on three or four points, is locally deformed along the local average normal direction d. It is compressed along d if the point normal and the normals of the triangle align well. It is stretched otherwise, decreasing the likelihood that this triangle will be selected for the ␣-shape. This in effect varies the local metric tensor. 1 The amount of deformation depends on the normal correlation and is multiplied by an interactively adjustable parameter ␶. The local normal direction d is multiplied by ␶, so the user has direct control of the anisotropy; ␶ = …