Designing and Implementing a General Purpose Halfedge Data Structure

This paper describes an algorithm for morphing polyhedral surfaces based on their decompositions into patches. The given surfaces need neither be genus-zero nor two-manifolds. We present a new algorithm for decomposing surfaces into patches. We also present a new projection scheme that handles topologically cylinder-like polyhedral surfaces. We show how these two new techniques can be used within a general framework and result with morph sequences that maintain the distinctive features of the input models. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computational Geometry and Object Modeling]: Boundary representations I.3.7 [Three-Dimensional Graphics and Realism]: Animation

Chapter 5 - Voronoi Diagrams

Chapter 20 - Voronoi Diagram Distances

Smooth surfaces are approximated by polyhedral surfaces both for display, and for other computational purposes. This is probably the most common surface representation in computer graphics, because polygons are what the current generation of graphics hardware can display efficiently. In computer vision this representation has not been very popular for a long time, probably because it was not seen as very appropriate for object recognition and positioning applications. But it is now being used in more applications, particularly in the medical domain, where polyhedral surface approximations of unknown surfaces are routinely generated with iso-surface construction algorithms. We look at polyhedral surfaces as a generalization of digital images, where the polygon takes the place of the pixel. The polygon is the fundamental surface element. Digital images are functions defined at the nodes of a regular rectangular grid. Discrete surface signals are functions defined at the vertices of a polyhedral surface of arbitrary topology. Signal processing operations, and Fourier analysis in particular, are the fundamental tools of low level computer vision. The lack of regularity and self-similarity of arbitrary polyhedral surface meshes, as opposed to the regular rectangular grids of digital images, complicates the analysis and processing of digital surface signals. Nevertheless, in this paper we generalize Fourier analysis to discrete surface signals. As a first application of this theory we consider the problem of surface smoothing, which corresponds to low-pass filtering within this framework. As in the classical cases of one-dimensional signals, and of digital images, the analysis of discrete surface signals is reduced to matrix analysis, and matrix computation techniques are used to achieve fast discrete surface signal processing operations. We intend to continue along this line of research in the near future, extending higher level computer vision operations to arbitrary polyhedral surfaces.

In stark contrast to the case of images, finding a concise, learnable discrete representation of 3D surfaces remains a challenge. In particular, while polygon meshes are arguably the most common surface representation used in geometry processing, their irregular and combinatorial structure often make them unsuitable for learning-based applications. In this work, we present VoroMesh, a novel and differentiable Voronoi-based representation of watertight 3D shape surfaces. From a set of 3D points (called generators) and their associated occupancy, we define our boundary representation through the Voronoi diagram of the generators as the subset of Voronoi faces whose two associated (equidistant) generators are of opposite occupancy: the resulting polygon mesh forms a watertight approximation of the target shape’s boundary. To learn the position of the generators, we propose a novel loss function, dubbed VoroLoss, that minimizes the distance from ground truth surface samples to the closest faces of the Voronoi diagram which does not require an explicit construction of the entire Voronoi diagram. A direct optimization of the Voroloss to obtain generators on the Thingi32 dataset demonstrates the geometric efficiency of our representation compared to axiomatic meshing algorithms and recent learning-based mesh representations. We further use VoroMesh in a learning-based mesh prediction task from input SDF grids on the ABC dataset, and show comparable performance to state-of-the-art methods while guaranteeing closed output surfaces free of self-intersections.

Nonorientable manifold surfaces not only play an important role in topology, but also have numerous applications in many topics such as visualization and computation of minimal surfaces. From the topological point of view, a 2-manifold surface is locally homeomorphic to an open disk. This property is independent of the global orientability. However, as far as the discrete representation is concerned, orientable manifold surfaces are usually discretized with halfedge data structure, while nonorientable surfaces are discretized into polygon soups, which is inconvenient for digital geometry processing that often takes orientable meshes as input. In this paper, we propose a uni ed framework for transforming geodesic distance problems de ned on nonorientable 2-manifold meshes to the counter-parts on orientable surfaces, and thereby bridging up nonorientable 2-manifold meshes and conventional geometric algorithms. In order to illustrate the universal adaptability, we apply this new approach to study three problems on nonorientable meshes, including computing exact geodesic paths, discrete exponential mapping and farthest point sampling.

After analyzing the characteristics of rock salt and the feasibility and importance of using underground rock salt cavern as energy stockpiles, this paper explains in detail about the principle of sonar detection technology in cavern measuring and its data organization. This paper studies the technology of three-dimensional surface topological reconstruction of underground rock salt cavern based on sonar detection data with half-edge data structure and boundary representation models. Meanwhile, by conducting validity check for constructed surface model and repair of possible defects, the paper finds correct and effective three-dimensional surface models of underground rock salt cavern, and provides effective data base for the stability analysis, creep analysis and other numerical simulations of underground rock salt cavern. This method, with important economic and social significance, can provide a scientific basis and technological support for the construction of caverns, especially for the safe and rational use of underground rock salt cavern.

In order to meet the requirements of terahertz radar for space target detection, the radar cross sections (RCS) of typical satellite targets in terahertz low frequency region is studied. A typical satellite target with rough body is established by using half edge data structure. The RCS distributions at different heights, root-mean-square, correlation lengths and different frequencies are calculated by using the hybrid method of Physical Optics method (PO), Shooting and Bouncing Ray method (SBR) and Truncated-wedge Incremental-length Diffraction Coefficients (TWILDCs). Based on the principle of turntable imaging, the scattering center distribution of satellite target with different roughness at 220GHz and 0.05m imaging resolution is simulated and analyzed. The results show that RCS and image are more sensitive to the fluctuation of the target surface, the frequency and incident angle in terahertz band.

In this paper, we deal with the enumeration of spanning trees in planar graphs using some combinatorial structures such as duality, bipartition and reduction. It is well known that the number of spanning trees in a planar graph is equal to the one in its dual. Based on this result, our contribution is to propose a formula that state the relation between the number of spanning trees in a given graph and in its bipartite. Then, similarly we establish a formula for the reduced graph. Finally, we extend the results of the deletion, the contraction and the splitting techniques to investigate the complexity in these combinatorial structures. The main advantage of our proposal is to derive spanning trees recursions in order to over- come the hardiness of computing the complexity of a given planar graph using the existing methods.

Taking hundreds of pieces of hazardous geological maps (1: 10 000) of Three Gorges reservoir area (3GR) as background, we establish regional three-dimensional (3D) geo-hazard model using DEM (digital elevation model) superposed surface images and geo-hazards elements. Based on landslides and other geo-hazard survey data, using improved B-REP (boundary representation) entity data structure (two-body 3D data structure), we set up 3D solid models for each hazardous bodies in each hazardous geological maps. Then we integrate the two types of 3D models with different scales from area to point, which are the regional geo-hazard 3D model and the solid models of each disaster body, in order to provide a visual processing and analysis platform for danger partition, stability evaluation, disaster prevention and control, early warning and command.

We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line l in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line l. We present two data structures for this problem. The first data structure uses O(n1+e) space and preprocessing time, and answers queries in O(21/e log n) time. The second data structure uses O(n log3n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n2) space. In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position. This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make Θ(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) pointer changes per operation while keeping O(log n)-time point-location queries.

We propose data structures for maintaining shortest path in planar graphs in which the weight of an edge is modified. Our data structures allow us to compute after an update the shortest-path tree rooted at an arbitrary query node in time O(n√log log n) and to perform an update in O((log n)3). Our data structure can be also applied to the problem of maintaining the maximum flow problem in an s − t planar network.

This paper revisits design and analysis techniques for fixed parameter algorithms for Planar Dominating Set and other problems on planar structures. As our main result, we use new geometric arguments concerning treewidth-based algorithms to show that determining whether a planar graph G has a dominating set of size k can be solved in \(O(2^{16.4715 \sqrt{k}}+ n^3)\) steps. This result improves on the best known treewidth-based algorithm by Kanj and Perkovič that runs in time \(O(2^{27\sqrt{k}}n)\). Our main result nearly matches the new branchwidth-based algorithm for Planar Dominating Set by Fomin and Thilikos that runs in time \(O(2^{15.13 \sqrt{k}}k +n^3)\). Algorithms for other problems on planar structures are explored. In particular, we show that Planar Red/Blue Dominating Set can be solved in time \(O(2^{24.551 \sqrt{k}}n)\). This leads to the main results, namely, that faster parameterized algorithms can be obtained for a variety of problems that can be described by planar boolean formulae. This gives the best-known parameterized algorithms for Planar Vertex Cover, Planar Edge Dominating Set, and Face Cover.

To leverage modern hardware platforms to their fullest, more and more database systems embrace compilation of query plans to native code. In the research community, there is an ongoing debate about the best way to architect such query compilers. This is perceived to be a difficult task, requiring techniques fundamentally different from traditional interpreted query execution. In this dissertation, we contribute to this discussion by drawing attention to an old but underappreciated idea known as Futamura projections, which fundamentally link interpreters and compilers. Guided by this idea, we demonstrate that efficient query compilation can actually be very simple, using techniques that are no more difficult than writing a query interpreter in a high-level language. We first develop LB2: a high-level query compiler implemented in this style that is competitive with the best compiled query engines both in sequential and parallel execution on the standard TPC-H benchmark. Query engines process a variety of data types and structures including text, spatial, graphs, etc. Several spatial and graph engines are implemented as extensions to relational query engines to leverage optimized memory, storage, and evaluation. Still, the performance of these extensions is often stymied by the interpretive nature of the underlying data management, generic data structures, and the need to execute domain-specific external libraries. On that basis, compiling spatial and graph queries to native code is a desirable avenue to mitigate existing limitations and improve performance. To support compiling spatial queries, we extend the LB2 main-memory query compiler with spatial predicates, indexing structures, and spatial operators. To support compiling graph queries, we extend LB2 with graph data structures and operators. The spatial extension matches the performance of hand-written code and outperforms relational query engines and map-reduce extensions. Similarly, the graph extension matches, and sometimes outperforms, low-level graph engines.

Practical modeling technique for large-scale 3D building models from ground images