Geodesic Voronoi Diagram Research Articles

Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

ЧИСЛЕННЫЙ АЛГОРИТМ ПОСТРОЕНИЯ ПОКРЫТИЯ ПОВЕРХНОСТЕЙ ВРАЩЕНИЯ ШАРАМИ С РАВНЫМИ РАДИУСАМИ

The paper focuses on the problem of constructing the thinnest covering for surfaces of revolution by equal balls whose radii are unknown in advance. A heuristic algorithm based on the joint applying the optical-geometric approach and the geodesic Voronoi diagram is proposed. Calculations for some surfaces of revolution, including a sphere, are carried out

О покрытии поверхности цилиндра и конуса равными шарами

The article concerns the problem of covering the lateral surface of a right circular cylinder or a cone with equal balls. The surface is required to belong to their union, and the balls’ radius is minimal. The centers of the balls must lie on the covered surface. The problem is relevant for mathematics and for applications since it arises in security and communications. We develop heuristic algorithms for covering construction based on a geodesic Voronoi diagram. The construction of a covering is a non-trivial task since the line of intersection of a cylinder or a cone with a sphere is a closed curve of the fourth order. To compare the numerical results with the known ones, we unroll the surface of revolution onto a plane. Another feature is that, we use both Euclidean distance and a special non-Euclidean metric, which can describe the speed of signal propagation in a heterogeneous medium. We also perform a numerical experiment and discuss its results. Meanwhile, it is shown that with a small number of circles covering a planification of the cylindrical surface, their radius is significantly less than for a similar rectangle.

GBGVD: Growth-based geodesic Voronoi diagrams

Given a set of generators, the geodesic Voronoi diagram (GVD) defines how the base surface is decomposed into separate regions such that each generator dominates a region in terms of geodesic distance to the generators. Generally speaking, each ordinary bisector point of the GVD is determined by two adjacent generators while each branching point of the GVD is given by at least three generators. When there are sufficiently many generators, straight-line distance serves as an effective alternative of geodesic distance for computing GVDs. However, for a set of sparse generators, one has to use exact or approximate geodesic distance instead, which requires a high computational cost to trace the bisectors and the branching points. We observe that it is easier to infer the branching points by stretching the ordinary segments than competing between wavefronts from different directions. Based on the observation, we develop an unfolding technique to compute the ordinary points of the GVD, as well as a growth-based technique to stretch the traced bisector segments such that they finally grow into a complete GVD. Experimental results show that our algorithm runs 3 times as fast as the state-of-the-art method at the same accuracy level.

Surface Simplification using Intrinsic Error Metrics

This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM) , we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature "drifts" during simplification. This process also yields a bijective map between the fine and coarse mesh, and prolongation operators for both scalar- and vector-valued data. Moreover, we obtain hard guarantees on element quality via intrinsic retriangulation---a feature unique to the intrinsic setting. The overall payoff is a "black box" approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations. We show how our method benefits several fundamental tasks, including geometric multigrid, all-pairs geodesic distance, mean curvature flow, geodesic Voronoi diagrams, and the discrete exponential map.

Parallel Post-processing of Restricted Voronoi Diagram on Thin Sheet Models

Given a thin-sheet surface S in 3D, as well as a collection of sites located on S, it is a fundamental operation in digital geometry processing to partition S into a set of constituent surface patches in terms of proximity. Geodesic Voronoi diagram (GVD) provides a theoretically sound solution to this problem, but has limited use due to its high computational cost. Restricted Voronoi Diagram (RVD) considers the partitioning of S based on the straight-line distance. Under general conditions, RVD can yield a practicable solution with much less running time. However, when the input model contains thin sheets but the sites are sparse, RVD may present some issues due to the significant difference between the straight-line distance and the geodesic distance. The resulting flawed RVD cells limit the use of RVDs on further geometry processing occasions.In this paper, we address two kinds of commonly encountered issues: (1) a site dominates multiple disconnected regions, and (2) a site dominates one single region but the region is kidney-shaped. Both of the issues occur frequently, on a thin-sheet model with a relatively small number of sites. The first kind of flawed RVD cells occurs when the dominance of a site penetrates the other side of the surface. For such flawed RVD cells, it is easy to separate the remote parts from the principal region, based on connectivity. The second kind of flawed RVD cells occurs when a site is nearby a geometry edge and its dominating area crosses the geometry edge, leading to a kidney-shaped region. We observe that for a roughly flat RVD cell, the sum of its turning angles is close to 2π. For a kidney-shaped region, however, the total angle becomes much larger than 2π, which inspires us to develop an angle-based checking rule. We further propose a concept of “virtual site” to cut the region at the narrowest part and re-partition the ownerless region by the surrounding sites (may be virtual). All of the above-mentioned operations can be parallelized and generally only require lightweight computational relative to the RVD. After post-processing, the improved RVD has inconspicuous difference from the GVD, and is of much higher quality than LRVD (Yan et al., 2014) and EDBVD (Xin et al., 2022). We present its usage in intrinsic Delaunay triangulation (IDT) and centroidal Voronoi tessellation (CVT) based meshing.

An efficient algorithm for approximate Voronoi diagram construction on triangulated surfaces

Voronoi diagrams on triangulated surfaces based on the geodesic metric play a key role in many applications of computer graphics. Previous methods of constructing such Voronoi diagrams generally depended on having an exact geodesic metric. However, exact geodesic computation is time-consuming and has high memory usage, limiting wider application of geodesic Voronoi diagrams (GVDs). In order to overcome this issue, instead of using exact methods, we reformulate a graph method based on Steiner point insertion, as an effective way to obtain geodesic distances. Further, since a bisector comprises hyperbolic and line segments, we utilize Apollonius diagrams to encode complicated structures, enabling Voronoi diagrams to encode a medial-axis surface for a dense set of boundary samples. Based on these strategies, we present an approximation algorithm for efficient Voronoi diagram construction on triangulated surfaces. We also suggest a measure for evaluating similarity of our results to the exact GVD. Although our GVD results are constructed using approximate geodesic distances, we can get GVD results similar to exact results by inserting Steiner points on triangle edges. Experimental results on many 3D models indicate the improved speed and memory requirements compared to previous leading methods.

Exact Voronoi diagram for topographic spatial analysis

ABSTRACT The Voronoi diagram (VD) is a fundamental geo-computing structure that has crucial applications. Computing this structure on a topographic surface requires having every point clustered on the geodesic distances, and thus the same challenging task as the geodesic distance mapping. This article proposes a new algorithm for the geodesic VD (GVD) by breaking up the highly complicated task into regular routines on the exact computation. The exact approach is due to the irregular rough nature of the Earth surface, where the discrete computation is more appropriate. The key operation involves a direct window growth devised to avoid the overloaded facet splitting and realized in a conic arrangement. Conventional clustering and GVD structure post-extraction are built on top of the window growth. The fundamental role of GVD in geo-computing is then demonstrated. The experimental results showed that the dual structure of GVD is useful in justifying the potential wrong triangulations from the popularly used 2D Delaunay, and the geometric exactness of GVD is more reliable in guaranteeing the surface process model efficiency and convergence, under the harsh checking of centroidal Voronoi tessellation optimization.

Extracting Skeletons of Two-Manifold Triangular Mesh Surface for Planning Skeleton-Guided Five-Axis Surface Inspection Path

Abstract Skeleton curve-guided five-axis sweep scanning is an emerging surface inspection technique with marvelous inspection efficiency. The precondition of applying such a technique to a complex surface is to extract the skeleton curves of a two-manifold surface associated with partitioning the surface into compact surfaces. However, the work on extracting skeletons of two-manifold triangle mesh surfaces is scarce, as existing skeletonization methods mostly focus on either 2D planar shapes or 3D solid shapes. In this paper, we present a new approach to extract the skeletons on a two-manifold triangle mesh with boundaries. The skeletons are formed in the most intuitive manner of wave front propagation which is based on computing the initial value problem of geodesics on a triangle mesh. The step-length of wave front propagation is adaptively controlled to guarantee the appropriate density of skeleton points for having good connectivity. Experiments show that, as a direct application of the proposed skeleton generation and its associated surface partitioning result, the five-axis sweep scanning path of complex free-form surface can be generated conveniently. Experimental results also validate that the computation of our proposed approach is simpler and faster than the state-of-the-art geodesic Voronoi diagram (GVD) method when most of the two-manifold triangle mesh is nearly planar. Additionally, the augmented parameter used for tracing the wave front’s geodesic propagation information is helpful for the skeleton-based surface partition, which is necessary for skeleton curve-guided five-axis sweep scanning.

A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon

The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is $$\varOmega (n+m\log m)$$, and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve $$O( (n+m) \log (n+m) )$$ and $$O(n+m\log m\log ^2n)$$ time, which are optimal for $$m=\varOmega (n)$$ and $$m=O(\frac{n}{\log ^3n})$$, respectively. In this paper, we give a construction algorithm with $$O( n + m ( \log m+ \log ^2 n ) )$$ time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in $$O(\log n)$$ time, then the construction time will become the optimal $$O(n+m\log m)$$. In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in $$O(\log n)$$ time.

Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon

Given a set of sites in a simple polygon, a geodesic Voronoi diagram of the sites partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for \(m \le n/{\text {polylog}}n\). Moreover, the algorithms for the geodesic nearest-point and farthest-point Voronoi diagrams are optimal for \(m \le n/{\text {polylog}}n\). This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.

Computing $$L_1$$ Shortest Paths Among Polygonal Obstacles in the Plane

Given a point s and a set of h pairwise disjoint polygonal obstacles with a total of n vertices in the plane, suppose a triangulation of the space outside the obstacles is given; we present an $$O(n+h\log h)$$ time and O(n) space algorithm for building a data structure (called shortest path map) of size O(n) such that for any query point t, the length of an $$L_1$$ shortest obstacle-avoiding path from s to t can be computed in $$O(\log n)$$ time and the actual path can be reported in additional time proportional to the number of edges of the path. The previously best algorithm computes such a shortest path map in $$O(n\log n)$$ time and O(n) space. So our algorithm is faster when h is relatively small. Further, our techniques can be extended to obtain improved results for other related problems, e.g., computing the $$L_1$$ geodesic Voronoi diagram for a set of point sites among the obstacles.

Decorating 3D models with Poisson vector graphics

This paper proposes a novel method for decorating 3D surfaces using a new type of vector graphics, called Poisson Vector Graphics (PVG). Unlike other existing techniques that frequently require local/global parameterization, our approach advocates a parameterization-free paradigm, affording decoration of geometric models with any topological type while minimizing the overall computational expenses. Since PVG supports a set of simple discrete curves, it is straightforward for users to edit colors and synthesize geometry details. Meanwhile, the details could be organized by Poisson Region (PR), leading to much smoother decoration than those of Diffusion Curve (DC). Consequently, it is an ideal tool to create smooth relief. It may be noted that, DC is adequate to create sharp or discontinuous results. But PR is superior to DC, supporting level-of-details editing on meshes thanks to its smoothness. To render PVG on meshes efficiently, we develop a Poisson solver based on harmonic B-splines, which could be constructed using geodesic Voronoi diagram. Our Poisson solver is a local solver for rendering with more flexibility and versatility. We demonstrate the efficacy of our approach on synthetic and real-world 3D models.

Constructing Intrinsic Delaunay Triangulations from the Dual of Geodesic Voronoi Diagrams

Intrinsic Delaunay triangulation (IDT) naturally generalizes Delaunay triangulation from R 2 to curved surfaces. Due to many favorable properties, the IDT whose vertex set includes all mesh vertices is of particular interest in polygonal mesh processing. To date, the only way for constructing such IDT is the edge-flipping algorithm, which iteratively flips non-Delaunay edges to become locally Delaunay. Although this algorithm is conceptually simple and guarantees to terminate in finite steps, it has no known time complexity and may also produce triangulations containing faces with only two edges. This article develops a new method to obtain proper IDTs on manifold triangle meshes. We first compute a geodesic Voronoi diagram (GVD) by taking all mesh vertices as generators and then find its dual graph. The sufficient condition for the dual graph to be a proper triangulation is that all Voronoi cells satisfy the so-called closed ball property. To guarantee the closed ball property everywhere, a certain sampling criterion is required. For Voronoi cells that violate the closed ball property, we fix them by computing topologically safe regions, in which auxiliary sites can be added without changing the topology of the Voronoi diagram beyond them. Given a mesh with n vertices, we prove that by adding at most O ( n ) auxiliary sites, the computed GVD satisfies the closed ball property, and hence its dual graph is a proper IDT. Our method has a theoretical worst-case time complexity O ( n 2 + tn log n ), where t is the number of obtuse angles in the mesh. Computational results show that it empirically runs in linear time on real-world models.

On the Expected Complexity of Voronoi Diagrams on Terrains

We investigate the combinatorial complexity of geodesic Voronoi diagrams on polyhedral terrains using a probabilistic analysis. Aronov et al. [2008] prove that, if one makes certain realistic input assumptions on the terrain, this complexity is Θ( n + m √ n ) in the worst case, where n denotes the number of triangles that define the terrain and m denotes the number of Voronoi sites. We prove that, under a relaxed set of assumptions, the Voronoi diagram has expected complexity O ( n + m ), given that the sites are sampled uniformly at random from the domain of the terrain (or the surface of the terrain). Furthermore, we present a construction of a terrain that implies a lower bound of Ω( nm 2/3 ) on the expected worst-case complexity if these assumptions on the terrain are dropped. As an additional result, we show that the expected fatness of a cell in a random planar Voronoi diagram is bounded by a constant.

Polyline‐sourced Geodesic Voronoi Diagrams on Triangle Meshes

AbstractThis paper studies the Voronoi diagrams on 2‐manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point‐source based GVDs, since a typical bisector contains line segments, hyperbolic segments and parabolic segments. To tackle this challenge, we introduce a new concept, called local Voronoi diagram (LVD), which is a combination of additively weighted Voronoi diagram and line‐segment Voronoi diagram on a mesh triangle. We show that when restricting on a single mesh triangle, the GVD is a subset of the LVD and only two types of mesh triangles can contain GVD edges. Based on these results, we propose an efficient algorithm for constructing the GVD with polyline generators. Our algorithm runs in O(nNlogN) time and takes O(nN) space on an n‐face mesh with m generators, where N = max{m, n}. Computational results on real‐world models demonstrate the efficiency of our algorithm.

Intrinsic computation of centroidal Voronoi tessellation (CVT) on meshes

Centroidal Voronoi tessellation (CVT) is a special type of Voronoi diagram such that the generating point of each Voronoi cell is also its center of mass. The CVT has broad applications in computer graphics, such as meshing, stippling, sampling, etc. The existing methods for computing CVTs on meshes either require a global parameterization or compute it in the restricted sense (that is, intersecting a 3D CVT with the surface). Therefore, these approaches often fail on models with complicated geometry and/or topology. This paper presents two intrinsic algorithms for computing CVT on triangle meshes. The first algorithm adopts the Lloyd framework, which iteratively moves the generator of each geodesic Voronoi diagram to its mass center. Based on the discrete exponential map, our method can efficiently compute the Riemannian center and the center of mass for any geodesic Voronoi diagram. The second algorithm uses the L-BFGS method to accelerate the intrinsic CVT computation. Thanks to the intrinsic feature, our methods are independent of the embedding space, and work well for models with arbitrary topology and complicated geometry, where the existing extrinsic approaches often fail. The promising experimental results show the advantages of our method.

The complexity of geodesic Voronoi diagrams on triangulated 2-manifold surfaces

We study the combinatorial complexity of Voronoi diagram of point sites on a general triangulated 2-manifold surface, based on the geodesic metric. Given a triangulated 2-manifold T of n faces and a set of m point sites S={s1,s2,…,sm}∈T, we prove that the complexity of Voronoi diagram VT(S) of S on T is O(mn) if the genus of T is zero. For a genus-g manifold T in which the samples in S are dense enough and the resulting Voronoi diagram satisfies the closed ball property, we prove that the complexity of Voronoi diagram VT(S) is O((m+g)n).

An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions

We present an optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source s on the surface of a convex polytope P in three dimensions. Our algorithm runs in O(nlog n) time and requires O(nlog n) space, where n is the number of edges of P. The algorithm is based on the O(nlog n) algorithm of Hershberger and Suri for shortest paths in the plane (Hershberger, J., Suri, S. in SIAM J. Comput. 28(6):2215–2256, 1999), and similarly follows the continuous Dijkstra paradigm, which propagates a “wavefront” from s along ∂ P. This is effected by generalizing the concept of conforming subdivision of the free space introduced by Hershberger and Suri and by adapting it for the case of a convex polytope in ℝ3, allowing the algorithm to accomplish the propagation in discrete steps, between the “transparent” edges of the subdivision. The algorithm constructs a dynamic version of Mount’s data structure (Mount, D.M. in Discrete Comput. Geom. 2:153–174, 1987) that implicitly encodes the shortest paths from s to all other points of the surface. This structure allows us to answer single-source shortest-path queries, where the length of the path, as well as its combinatorial type, can be reported in O(log n) time; the actual path can be reported in additional O(k) time, where k is the number of polytope edges crossed by the path. The algorithm generalizes to the case of m source points to yield an implicit representation of the geodesic Voronoi diagram of m sites on the surface of P, in time O((n+m)log (n+m)), so that the site closest to a query point can be reported in time O(log (n+m)).

Crest lines for surface segmentation and flattening

We present a method for extracting feature curves called crest lines from a triangulated surface. Then, we calculate the geodesic Voronoi diagram of crest lines to segment the surface into several regions. Afterward, barycentric surface flattening using theory from graph embeddings is implemented and, using the Geodesic Voronoi diagram, we develop a faster surface flattening algorithm.