Regular triangulations of dynamic sets of points

Traditional geometric algorithms are often presented as if input imprecision does not exist, even though it is often unavoidable. We argue that in some cases, it may be desirable for geometric algorithms to treat this imprecision as an explicit component of the input, and to reflect this imprecision in the output. Starting with three problems from computational geometry whose inputs are planar point sets (Voronoi diagrams, convex hulls, and smallest bounding discs), we recast these as problems where each input point’s location is imprecise, but known to lie within a particular region of uncertainty. Where algorithms to solve each of the original problems produce a single geometric object as output, the algorithms that we present typically produce either guaranteed or possible output objects. A guaranteed object represents qualities that can be guaranteed for every point set that is consistent with the uncertain regions, and a possible object represents qualities that exist for at least one such point set. By dealing with input imprecision explicitly, these guaranteed and possible objects can represent a more accurate view of what can be reliably inferred from the input data.

Structuring finite sets of points is at the heart of computational geometry. Such point sets arise naturally in many applications. Examples in R3 are point sets sampled from the surface of a solid or the locations of atoms in a molecule. A first step in processing these point sets is to organize them in some data structure. Structuring a point set into a simplicial complex like the Delaunay triangulation has turned out to be appropriate for many modeling tasks. Here we introduce the flow complex which is another simplicial complex that can be computed efficiently from a finite set of points. The flow complex turned out to be well suited for surface reconstruction from a finite sample and for some tasks in structural biology. Here we study mathematical and algorithmic properties of the flow complex and show how to exploit it in applications.

Chapter 5 - Voronoi Diagrams

A wireless sensor network can be represented by a graph. While the network graph is extremely useful, it often exhibits undesired irregularity. Therefore, special treatment of the graph is required by a variety of network algorithms and protocols. In particular, many geometry-oriented algorithms depend on a type of subgraph called Delaunay triangulation. However, when location information is unavailable, it is nontrivial to achieve Delaunay triangulation by using connectivity information only. The only connectivity-based algorithm available for Delaunay triangulation is built upon the property that the dual graph for a Voronoi diagram is a Delaunay triangulation. This approach, however, often fails in practical wireless sensor networks because the boundaries of Voronoi cells can be arbitrarily short in discrete sensor network settings. In a sensor network with connectivity information only, it is fundamentally unattainable to correctly judge neighboring cells when a Voronoi cell boundary is less than one hop. Consequently, the Voronoi diagram-based Delaunay triangulation fails. The proposed algorithm employs a distributed approach to perform centroidal Voronoi tessellation, and constructs its dual graph to yield Delaunay triangulation. It exhibits several distinctive properties. First, it eliminates the problem due to short cell boundaries and thus effectively avoids crossing edges. Second, the proposed algorithm is proven to converge and succeed in constructing a Delaunay triangulation, if the CVT cell size is greater than a constant threshold. Third, the established Delaunay triangulation consists of close-to-equilateral triangles, benefiting a range of applications such as geometric routing, localization, coverage, segmentation, and data storage and processing. Extensive simulations are carried out under various 2D network models to evaluate the effectiveness and efficiency of the proposed CVT-based triangulation algorithm.

It is difficult to extract the boundary of complex planar points with nonuniform distribution of point density, concave envelopes, and holes. To solve this problem, an algorithm is proposed in this paper. Based on Delaunay triangulation, the maximum boundary angle threshold is introduced as the parameter in the extraction of the rough boundary. Then, the point looseness threshold is introduced, and the fine boundary extraction is conducted for the local areas such as concave envelopes and holes. Finally, the complete boundary result of the whole point set is obtained. The effectiveness of the proposed algorithm is verified by experiments on the simulated point set and practical measured point set. The experimental results indicate that it has wider applicability and more effectiveness in engineering applications than the state‐of‐the‐art boundary construction algorithms based on Delaunay triangulation.

In this paper we present new optimality results for the Delaunay triangulation of a set of points in ℝ d . These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ℝ d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.

The three-dimensional [Formula: see text]-shape is based on a mathematical formalism which determines exact relationships between points and shapes. It reconstructs surface and volume and detects 3D dot patterns for a given point cloud. [Formula: see text]-shape of a set of points is a sub-complex of Delaunay triangulation of this set. It generates a family of shapes according to the selected [Formula: see text] (a set of points). A method to compute the positions of the points of [Formula: see text] is proposed. These points are selected from the vertices of Voronoi diagram by analyzing the form of the polytopes; their elongation. This method allows the [Formula: see text]-shape to reflect different levels of detail in different parts of space. An efficient algorithm computing the three-dimensional [Formula: see text]-shape is presented, the [Formula: see text]-shape of a set of points is derived from the Delaunay triangulation of the same set. The speed of the algorithm is determined by the speed of the algorithm computing the Delaunay triangulation.

Delaunay and Voronoi tessellations and minimal simple cycles in triangular region and regular-3 undirected planar graphs

With the development of machine learning in various fields, it can also be applied to combinatorial optimization problems, automatically discovering generic and fast heuristic algorithms based on training data, and requires fewer theoretical and empirical knowledge. Pointer network improves the attention mechanism, instead of allocating different attention to hidden states of encoder to generate context vectors, using attention as a pointer to select an element of the input sequence at every step of decoding, which solves the problem of variable dictionary size of the output sequence. Pointer net (Ptr-Net) applied to three combinatorial optimization problems, convex hull, Delaunay triangulation, and traveling salesman problem (TSP), obtains good approximate solutions. Point matching is also a special kind of combinatorial optimization problems that is to obtain the optimal corresponding references, which can be modeled by Ptr-Net. However, Ptr-Net can’t be used to solve point matching problem because it doesn’t take full advantage of the correspondences between the two point sets. We propose multi-pointer network, which draws the idea from multi-label classification, to address this limitation by pointing out a set of input elements. These applications are all based on supervised learning to approximate expected known solutions. However, high-quality labeled data is often expensive, unreliable, or simply unavailable and may be infeasible for new problem statements, making supervised learning being unpractical. Reinforcement learning, as another research hotspot in the field of machine learning, does not require labeled sample data. It interacts with the environment through trial-and-error mechanism and focuses more on learning problem-solving strategies. We introduce a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning, focusing on the traveling salesman problem. We also introduce a framework, a unique combination of reinforcement learning and graph embedding network, to solve graph optimization problems, focusing on maximum cut (MAXCUT) and minimum vertex cover (MVC) problems.

The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a well-shaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a by-product, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other by convex duality or embedding. Moreover, we can always compute them indirectly as power diagrams in primal or dual spaces, or directly after linearization in an extra-dimensional space as the projection of a Euclidean polytope. Finally, our paper proposes to generalize Bregman divergences to higher-order terms, called κ-jet Bregman divergences, and touch upon their Voronoi diagrams.

Digital elevation model (DEM) based on Delaunay triangulation can better express the feature of the terrain surface and avoid significantly data redundancy. There is, however, litter research on the surface model of DEM based on Delaunay triangulation. The main surface model of DEM is linear interpolation function based on triangulated irregular network (TIN). Many researches show that the effect of smoothness and continuity is not ideal. Aiming at the problem of linear interpolation based on TIN of DEM that local part is substituted by plane, the whole surface is not smooth and there is great difference in the actual terrain, this paper put forward a new surface interpolation model based on Convex Hull (CH) in Delaunay triangulation. This method is that the known reference points can be well-distributed around the inserting point. The interpolation model is based on the effective dynamic subdivision of CH area of the triangle influenced set as the weight. This paper combines with the actual alpine terrain data and uses Root Mean Square Error (RMSE) and Maximum Error (ME) to evaluate and analysis the experimental results. The experiment shows that our method has more effective in structure and precision.

Evaluation methods for the spatial distribution uniformity of a large number of particles have important applications in a range of scientific and industrial problems. We report a quantitative image-processing method to evaluate the spatial distribution uniformity of particles using the Voronoi tessellation and Delaunay triangulation. Given the geometric particle centroids, we constructed the Voronoi tessellation and Delaunay triangulation. For each of the Voronoi cell, the volume, facet area, and number of neighbors (facets) were calculated. For the Delaunay simplices, the volume and the distance between neighbors were calculated and the distributive characteristics were compared between different particle patterns. Simulation results demonstrated that distributions of the above metrics were numerically related to the uniformity of the particle spatial displacement. More studies are needed to further validate the results and refine the method.

Stationary point processes in ?2 with two different types of points, say H and L, are considered where the points are located on the edge set G of a random geometric graph, which is assumed to be stationary and connected. Examples include the classical Poisson---Voronoi tessellation with bounded and convex cells, aggregate Voronoi tessellations induced by two (or more) independent Poisson processes whose cells can be nonconvex, and so-called β-skeletons being subgraphs of Poisson---Delaunay triangulations. The length of the shortest path along G from a point of type H to its closest neighbor of type L is investigated. Two different meanings of "closeness" are considered: either with respect to the Euclidean distance (e-closeness) or in a graph-theoretic sense, i.e., along the edges of G (g-closeness). For both scenarios, comparability and monotonicity properties of the corresponding typical shortest-path lengths C e? and C g? are analyzed. Furthermore, extending the results which have recently been derived for C e?, we show that the distribution of C g? converges to simple parametric limit distributions if the edge set G becomes unboundedly sparse or dense, i.e., a scaling factor ? converges to zero and infinity, respectively.

Computing the shape of a planar points set

With a given set of points, Voronoi diagram is a technique of partitioning a plane into set of points having the same closest sites. A number of studies are made on the generalization of Voronoi diagram. These studies are mainly conducted on different kind of distance metric. Some studies are also made upon Voronoi diagram of sites those are not necessarily points. The current researchers present a new generalization of the construction of Voronoi diagram in the plane with axes parallel rectangles. Here, the interest has been shifted from points to simple convex orthogonal blocks i.e. rectangles. The current researchers have computed the partition of the set of the rectangles in the plane into sets of lines having the same closest sites (where sites are rectangles in the plane. This paper represents a method to find the Voronoi diagram, which considered rectangles as sites. Prototype implementation has been done and the results are quite encouraging. The algorithm is easy to implement and finds application in VLSI layouts.