Surface parameterizations have been widely applied to digital geometry processing. In this paper, we propose an efficient conformal energy minimization (CEM) algorithm for computing conformal parameterizations of simply-connected open surfaces with a very small angular distortion and a highly improved computational efficiency. In addition, we generalize the proposed CEM algorithm to computing conformal parameterizations of multiply-connected surfaces. Furthermore, we prove the existence of a nontrivial accumulation point of the proposed CEM algorithm under some mild conditions. Several numerical results show the efficiency and robustness of the CEM algorithm comparing to the existing state-of-the-art algorithms. An application of the CEM on the surface morphing between simply-connected open surfaces is demonstrated thereafter. Thanks to the CEM algorithm, the whole computations for the surface morphing can be performed efficiently and robustly.

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In this paper, we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon and a set  of possibly overlapping polygons fixed in the plane. More specifically, a subset   is called an intersection relation if there exists a placement of that intersects, and only intersects, . The objective of this paper is to design an efficient algorithm that finds a finite and discrete representation of all of the intersection relations between and . Past related research only focuses on the complexity of the free space of the configuration space between and  and how to move or place an object in this free space. However, there are many applications that require the knowledge of not only the free space, but also the intersection relations. Examples are presented to demonstrate the rich applications of the formulated problem on intersection relations.

In this paper, we consider the problem of approximating the longest path in a grid graph that possesses a square-free 2-factor. By (1) analyzing several characteristics of four types of cross cells and (2) presenting three cycle-merging operations and one extension operation, we propose an algorithm that finds in an n-vertex grid graph G, a long path of length at least 5/6 n + 2. Our approximation algorithm runs in quadratic time. In addition to its theoretical value, our work has also an interesting application in image-guided maze generation.

In the research of computer vision and machine perception, 3D objects are usually represented by 2-manifold triangular meshes M. In this paper, we present practical and efficient algorithms to construct iso-contours, bisectors, and Voronoi diagrams of point sites onM, based on an exact geodesic metric. Compared to euclidean metric spaces, the Voronoi diagrams onMexhibit many special properties that fail all of the existing euclidean Voronoi algorithms. To provide practical algorithms for constructing geodesicmetric- based Voronoi diagrams onM, this paper studies the analytic structure of iso-contours, bisectors, and Voronoi diagrams onM. After a necessary preprocessing of model M, practical algorithms are proposed for quickly obtaining full information about iso-contours, bisectors, and Voronoi diagrams on M. The complexity of the construction algorithms is also analyzed. Finally, three interesting applications—surface sampling and reconstruction, 3D skeleton extraction, and point pattern analysis—are presented that show the potential power of the proposed algorithms in pattern analysis.

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).

A natural metric in 2-manifold surfaces is to use geodesic distance. If a 2-manifold surface is represented by a triangle mesh T , the geodesic metric on T can be computed exactly using computational geometry methods. Previous work for establishing the geodesic metric on T only supports using half-edge data structures; i.e., each edge e in T is split into two halves (he1, he2) and each half-edge corresponds to one of two faces incident to e. In this paper, we prove that the exact-geodesic structures on two halfedges of e can be merged into one structure associated with e. Four merits are achieved based on the properties which are studied in this paper: (1) Existing CAD systems that use edge-based data structures can directly add the geodesic distance function without changing the kernel to a half-edge data structure; (2) To find the geodesic path from inquiry points to the source, the MMP algorithm can be run in an onthe- fly fashion such that the inquiry points are covered by correct wedges; (3) The MMP algorithm is sped up by pruning unnecessary wedges during the wedge propagation process; (4) The storage of the MMP algorithm is reduced since fewer wedges need to be stored in an edge-based data structure. Experimental results show that when compared to the classic half-edge data structure, the edge-based implementation of the MMP algorithm reduces running time by 44% and storage by 29% on average.

This 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.

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.

Computing geodesic distances on triangle meshes is a fundamental problem in computational geometry and computer graphics. To date, two notable classes of algorithms, the Mitchell-Mount-Papadimitriou (MMP) algorithm and the Chen-Han (CH) algorithm, have been proposed. Although these algorithms can compute exact geodesic distances if numerical computation is exact, they are computationally expensive, which diminishes their usefulness for large-scale models and/or time-critical applications. In this paper, we propose the fast wavefront propagation (FWP) framework for improving the performance of both the MMP and CH algorithms. Unlike the original algorithms that propagate only a single window (a data structure locally encodes geodesic information) at each iteration, our method organizes windows with a bucket data structure so that it can process a large number of windows simultaneously without compromising wavefront quality. Thanks to its macro nature, the FWP method is less sensitive to mesh triangulation than the MMP and CH algorithms. We evaluate our FWP-based MMP and CH algorithms on a wide range of large-scale real-world models. Computational results show that our method can improve the speed by a factor of 3-10.