Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.

Implicit reconstruction of solids from cloud point sets

Abstract: Chek T. Lirnl George M. Turkiyyah2 hark A. G’ante# Duane W. storti~ University of Washington Seattle, WA 98195 {ctlim@u, george@ce,ganter@u, storti@u}.Washington.edu This paper describes a new technique that combines numerical optimization methods with triangulation methods for generating mathematical representations of solids from 3D point data. The solid representation obtained takes the form of an algebraic function whose level surface closely approximates the surface described by the data, The algebraic function is obtained via Implicit Solid Modeling, a constructive scheme for approximating Boolean volume set operations on implicitly defined primitive volumes, and is comprised of a blended union of spherical primitives. The parameters of the algebraic function are the spatial locations and radii of the spheres as well as the parameters that describe the blending of these primitives, Fitting an implicit solid model to a data set is formulated as a sequence of non-linear optimization problems of an increasing number of variables. The cost function we employ in these optimizations is a weighted combination of discrepancies in location (distance from points to boundary of reconstructed object), discrepancies in surface normals, and desired curvature characteristics of the reconstructed solid. Since a set of trivariate data points without any connectivity information is ambiguous, an infinite number of solids, in principle, can be constructed to fit them. Different characteristics of the solid can be specified through the cost function to create the most desirable interpretation of the data. The starting point of the optimization—corresponding to the starting configuration of the primitives—is determined by performing a 3D Delaunay triangulation on the data set, and is based on the locations and sizes of the resulting tetrahedral. The effectiveness of the algorithm is demonstrated through the reconstruction of several sample data sets, including a molar and a femur. Tradeoffs between accuracy and compactness of the representations are also examined. 1Department of Mechanical Engineering, FU-10 ‘Department of Civil Engineering, FX-10. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association of Computing Machinery.To copy otherwise, or to republish, requires a fee andlor specific permission. Solid Modeling ’95, Salt Lake City, Utah USA

Joint triangulations and triangulation maps

Abstract: In rubber-sheeting applications in cartography, it is useful to seek piecewise-linear homeomorphisms (PLH maps) between rectangular regions which map an arbitrary sequence of n points {p1, p2, …,pn} from the interior of one rectangle to a corresponding sequence {q1, q2, …, qn} of n points in the interior of the second region. This paper proves that it is always possible to find such PLH maps and describes then in terms of a joint triangulation of the domain and the range rectangular regions.One naive approach to finding a PLH map is to triangulate (in any fashion) the domain rectangle on its n points and four corners and to define a piecewise affine map on each triangle up11p12p13 to be the unique affine map that sends the three vertices p11, p12, p13 of the triangle to the three corresponding vertices q11, q12, q13 of the image triangle uq11q12q13. Such piecewise affine maps send triangles to triangles, agree on shared edges, and thus extend globally, and will be called triangulation maps. The shortcoming of building transformations in this fashion is that the resulting triangulation map need not be one-to-one, although there is a simple test to determine if such a map is one-to-one (see Theorem 2 below). If the map is one-to-one, then the image triangles will form a triangulation of the range space; and we will have a joint triangulation. If the map is not one-to-one, then there will be folding over of triangles. It may be possible to alleviate this folding by choosing a different triangulation of the n domain points, or it may be the case that no triangulation of the n domain points will work. (See figures 5 and 6 below). We show that it will be possible, in all cases, to rectify the folding by adding appropriate additional triangulation vertex pairs {pn+1, pn+2, …, pn+m} and {qn+1, qn+2, …, qn+m} and retriangulating (see Theorem 1 below). This paper examines conditions for triangulation maps to be homeomorphisms and explores different ways of modifying triangulations and triangulation maps to make them joint triangulations and homeomorphisms.The paper concludes with a section on alternative constructive approaches to the open problem of finding joint triangulations on the original sequences of vertex pairs without augmenting those sequences of pairs.The existence proofs in this paper do not solve computational geometry problems per se; instead they permit us to formulate new computational geometry problems. The problems we pose are of interest to us because of a particular application in automated cartography.

Incremental Surface Extraction from Sparse Structure-from-Motion Point Clouds

Abstract: In this paper we propose a new method to incrementally extract a surface from a consecutively growing Structure-from-Motion (SfM) point cloud in real-time. Our method is based on a Delaunay triangulation (DT) on the 3D points. The core idea is to robustly label all tetrahedra into freeand occupied space using a random field formulation and to extract the surface as the interface between differently labeled tetrahedra. For this reason, we propose a new energy function that achieves the same accuracy as state-of-the-art methods but reduces the computational effort significantly. Furthermore, our new formulation allows us to extract the surface in an incremental manner, i. e. whenever the point cloud is updated we adapt our energy function. Instead of minimizing the updated energy with a standard graph cut, we employ the dynamic graph cut of Kohli et al. [1] which enables efficient minimization of a series of similar random fields by re-using the previous solution. In such a way we are able to extract the surface from an increasingly growing point cloud nearly independent of the overall scene size. Energy Function for Surface Extraction Our method formulates surface extraction as a binary labeling problem, with the goal of assigning each tetrahedron either a free or occupied label. For this reason, we model the probabilities that a tetrahedron is free- or occupied space analyzing the set of rays that connect all 3D points to image features. Following the idea of the truncated signed distance function (TSDF), which is known from voxel-based surface reconstructions, a tetrahedron in front of a 3D point X has a high probability to be free space, whereas a tetrahedron behind X is presumably occupied space. We further assume that it is very unlikely that neighboring tetrahedra obtain different labels, except for pairs of tetrahedra that have a ray through the face connecting both. Such a labeling problem can be elegantly formulated as a pairwise random field and since our priors are submodular, we can efficiently find a global optimal labeling solution e. g. using graph cuts. In contrast to existing methods like [2], our energy depends only on the visibility information that is directly connected to the four 3D points that span the tetrahedraVi. Hence a modification of the tetrahedral structure by inserting new points has only limited effect on the energy function. This property enables us to easily adopt the energy function to a modified tetrahedral structure. Incremental Surface Extraction To enable efficient incremental surface reconstruction, our method has to consecutively integrate new scene information (3D points as well as visibility information) in the energy function and to minimize the modified energy efficiently. Integrating new visibility information, i. e. adding rays for newly available 3D points, affects only those terms of the energy function that relate

Fast Triangulation of Simple Polygons

Abstract: We present a new algorithm for triangulating simple polygons that has four advantages over previous solutions [GJPT, Ch].