A Linear Time Algorithm for High Quality Mesh Simplification

Aiming at the problem that the mesh simplification algorithm loses the geometric features of the model in large-scale simplification, an improved half-edge collapse mesh simplification algorithm is proposed. The concept of approximate measurement of edge curvature is introduced, and the edge curvature is added to the error measure, so that the order of half-edge collapse of the mesh is changed, and the simplified details of the mesh model can be preserved accurately. At the same time, by analyzing the quality of simplified triangular mesh, optimizing triangular mesh locally, reducing the amount of narrow triangles, the quality of the simplified model is improved. The proposed algorithm was tested on Cow model, Car model and Bunny model, and compared with another three algorithms, one of them is a classical mesh simplification algorithm based on edge collapse, the other is an improved algorithm of the classical one. The experimental results show that the improved algorithm can better retain the detail features of the original model at the same reduction ratio, and has reasonable mesh allocation, fast execution speed and small error.

A simple polygon is said to be weakly internally visible from a line segment lying inside it (chord, if the endpoints of the segment lie on the boundary of the polygon), if every point on the boundary of the polygon is visible from some point of this line segment. In this paper we describe a simple linear time algorithm for computing such a chord, when it exists. We also computed all non-redundant components of P which must be intersected by any chord of P from which P is wiv. The most interesting aspects of this algorithm are that we have been able to dispense with the complicated triangulation algorithm of Chazelle [2] as well as the algorithm in [7] for computing all shortest paths from a given vertex in a triangulated polygon. As a consequence of this, we can now use this algorithm to derive an optimal algorithm for triangulating a weakly internally visible polygon. Also these results can now be used in [3] to find a shortest line segment from which a polygon is weakly internally visible in optimal linear time.

This paper discusses the application of another type of subdivision pattern, namely, the subdivision surfaces work as a mesh surface approximation of the original data, thus the grid data is resampled by using sub-sampling points. Specifically, in a number of algorithms based on edge collapse or triangle, the triangle is removed to a new edge collapse or vertex, the algorithm presented with the segmentation method to calculate the new vertex location to replace the traditional quadratic surface fitting algorithm or direct averaging method of calculation. polyhedral mesh simplification, especially triangular mesh simplification has been for nearly a decade, during which a large number of proposed algorithm, also published a number of reviews, here for a comprehensive review will not only simplify the background needed for the grid and make a brief review of related work, the rest of the content of the proposed algorithm is described.Keywordssegmentation modeltriangular meshresolution

Most existing mesh simplification algorithms ignore some important geometric features during the simplification process, especially in the low-level models. A novel edge collapse based mesh simplification algorithm, which utilizes the absolute curvature of the simplified vertex as one factor of the simplification error metrics is presented in this paper. Since the half-edge collapse is adopted in the proposed algorithm, the storage requirements can be efficiently reduced. Experimental results demonstrate that the proposed algorithm can preserve the geometric feature of the original mesh very well as well as maintaining the simplification error.

The simplification of nonconvex tetrahedral meshes with the help of edge collapses (edge contractions) is considerably more complex than the corresponding simplification of convex meshes. In particular, edge collapses in nonconvex meshes may cause intersections of cells that are hard to detect, and therefore hard to avoid. More precisely spoken, the problem is quadratic in the number of cells.However, we show how to reduce the complexity of this problem by employing a pre-processing step, which was originally proposed in order to sort nonconvex meshes. Moreover, our method is able to handle meshes with topologically nontrivial boundaries and to control the modification of the topology of the mesh’s boundary.KeywordsConvex HullTetrahedral MeshVertex NeighborEdge ContractionEdge NeighborThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Large-scale visualization systems are typically designed to efficiently “push” datasets through the graphics hardware. However, exploratory visualization systems are increasingly expected to support scalable data manipulation, restructuring, and querying capabilities in addition to core visualization algorithms. We posit that new emerging abstractions for parallel data processing, in particular computing clouds, can be leveraged to support large-scale data exploration through visualization. In this paper, we take a first step in evaluating the suitability of the MapReduce framework to implement large-scale visualization techniques. MapReduce is a lightweight, scalable, general-purpose parallel data processing framework increasingly popular in the context of cloud computing. Specifically, we implement and evaluate a representative suite of visualization tasks (mesh rendering, isosurface extraction, and mesh simplification) as MapReduce programs, and report quantitative performance results applying these algorithms to realistic datasets. For example, we perform isosurface extraction of up to l6 isovalues for volumes composed of 27 billion voxels, simplification of meshes with 30GBs of data and subsequent rendering with image resolutions up to 80000 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> pixels. Our results indicate that the parallel scalability, ease of use, ease of access to computing resources, and fault-tolerance of MapReduce offer a promising foundation for a combined data manipulation and data visualization system deployed in a public cloud or a local commodity cluster.

With the increasing accessibility of the mobile head-mounted displays (HMDs), mobile virtual reality (VR) systems are finding applications in various areas. However, mobile HMDs are highly constrained with limited graphics processing units (GPUs), low processing power and onboard memory. Hence, VR developers must be cognizant of the number of polygons contained within their virtual environments to avoid rendering at low frame rates and inducing simulator sickness. The most robust and rapid approach to keeping the overall number of polygons low is to use mesh simplification algorithms to create low-poly versions of preexisting, high-poly models. Unfortunately, most existing mesh simplification algorithms cannot adequately handle meshes with lots of boundaries or non-manifold meshes, which are common attributes of 3D models made with computer-aided design tools.; AB@In this paper, we present a high-fidelity mesh simplification algorithm specifically designed for VR. This new algorithm, QEM4VR, addresses the deficiencies of prior quadric error metric (QEM) approaches by leveraging the insight that the most relevant boundary edges lie along curvatures while linear boundary edges can be collapsed. Additionally, our QEM4VR algorithm preserves key surface properties, such as normals, texture coordinates, colors, and materials. It pre-processes the 3D models and generate their low-poly approximations offline. We used six publicly available, high-poly models, with and without textures to compare the accuracy and fidelity of our QEM4VR algorithm to previous QEM variations. We also performed a frame rate analysis with original high-poly models and low-poly models obtained using QEM4VR and previous QEM variations. Our results indicate that QEM4VR creates low-poly, high-fidelity virtual environments for VR applications on devices that are constrained by the low number of polygons they can work with.

Mesh simplification is an efficient method of processing complex models by computer. In this paper, we have emphatically analyzed and researched the problem of the sharp changes of visual features because of not fully considering the visual effect during the process of simplification, and the problem of easily and wrongly choosing the collapse edge caused by the error accumulation of the simplification algorithm, the paper has proposed a mesh simplification algorithm based on the shape transform of triangles. In the algorithm, we make the length of the edge and the difference between the internal angles of the newly created triangle and equilateral triangle's as weight genes when computing the edge collapse cost, and consider the factor of area of triangles around the vertex when computing the quadric error metrics of the vertex, and find the average value of quadric error metrics for each vertex as the vertex's error metrics. The algorithm resolves the above problems effectively, avoiding the problems of causing narrow triangles and producing holes on the surface of the model which are made by premature simplification of too large area of triangles around the vertex. By experiment validating and analyzing contrastively, it proves the validity of the algorithm.

We consider the problem of finding the staircase kernel in orthogonal polygons, with or without holes, in the plane. Orthogonal polygon is a simple polygon in the plane whose sides are either horizontal or vertical. We generalize the notion of visibility in the following way: We say that two points a and b in an orthogonal polygon P are visible to each other via staircase paths if and only if there exist an orthogonal chain connecting a and b and lying entirely in the interior of P. Furthermore, the orthogonal chain should have the property that the angles between the consecutive segments in the chain are either +90 ∘ or −90 ∘ , and these should alternate along the chain. There are two principal types of staircases, NW-SE and NE-SW. The notion of staircase visibility has been studied in the literature for the last three decades. Based on this notion we can generalize the notion of star-shapedness. A polygon P is called star-shaped under staircase visibility, or simply s-star if and only if there is nonempty set of points S in the interior of P, such that any point of S sees any point of P via staircase path. The largest such set of points is called the staircase kernel of P and denoted ker P. Our work is motivated by the work of Breen (1). She proves that the staircase kernel of an orthogonal polygon without holes is the intersection of all maximal orthogonally convex polygons contained in it. We extend Breen's results for the case when the orthogonal polygon has holes. We prove the necessary geometric properties, and use them to derive a quadratic time, O(���� 2 ) algorithm for computing the staircase kernel of an orthogonal polygon with holes, having n vertices in total, including the holes' vertices. The algorithm is based on the plane sweep technique, widely used in Computational Geometry(4). Our result is optimal in the case of orthogonal polygon with holes, since the kernel (as proven) can consist of quadratic number of disjoint regions. In the case of polygon without holes, there is a linear time algorithm by Gewali(3), that is specific to the case of a polygon without holes. We present ex- amples of our algorithm's results.

Let p and q be two points in a simple polygon Π. An open problem in computational geometry asks to devise a simple linear-time algorithm for computing a shortest path between p and q, which is contained in Π, such that the algorithm does not depend on a (complicated) linear-time triangulation algorithm. This report provides a contribution to the solution of this problem by applying the rubberband algorithm. The obtained solution has \({\cal O}\) (nlogn) time complexity (where the super-linear time complexity is only due to preprocessing, i.e. for the calculation of critical edges) and is, altogether, considerably simpler than the triangulation algorithm. It has applications in 2D pattern recognition, picture analysis, robotics, and so forth.Keywordsdigital geometrycomputational geometryrubberband algorithmsimple polygonEuclidean shortest path

The analysis of many processes modeled by directed graphs requires the selection of a subset of vertices which cut all the cycles in the graph. Reducing the size of such a cutset usually leads to a simpler and more efficient analysis, but the problem of finding minimum cutsets in general directed graphs is known to be $NP$-complete. In this paper we show that in reducible graphs (and thus in almost all the “practical” flowcharts of programs), minimum cutsets can be found in linear time. We further show that the linear algorithm can check its own applicability to a given graph, thus eliminating the need of prechecking (in nonlinear time) whether it is reducible or not. An immediate application of this result is in program verification systems based on Floyd’s inductive assertions method.

Let P be a simple planar polygon. We present a linear worst-case time algorithm for constructing the bounded Voronoi diagram of P in the Manhattan metric, where each point z in P belongs to the region of the closest vertex of P that is visible from z. Among other consequences, the minimal spanning tree of the vertices in the Manhattan metric that is contained in P can be computed within optimal linear time.

Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear time recognition algorithms. However, the problems of testing whether a given (di)graph has k vertices whose deletion makes it planar, bipartite or a directed acyclic graph (DAG) are all fundamental NP-complete problems when k is part of the input. As a result, a significant amount of research has been devoted to understanding whether, for every fixed k, these problems admit a polynomial time algorithm (where the exponent in the polynomial is independent of k) and in particular, whether they admit linear time algorithms. While we now know that for any fixed k, we can test in linear time whether a graph is k vertices away from being planar [FOCS 2009, SODA 2014] or bipartite [SODA 2014, SICOMP 2016], the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 78] which runs in time O(nk–1m) and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008] which runs in time O(k!4kk4nm). In other words, it has remained open whether it is possible to recognize in linear time, a graph which is 2 vertices away from being acyclic! In this paper, we settle this question by giving an algorithm that decides whether a given graph is k vertices away from being acyclic, in time O(k!4kk5(n + m)). That is, for every fixed k, our algorithm runs in time O(m + n), thus mirroring the case for planarity and bipartiteness. Our algorithm is designed via a general methodology that shaves off a factor of n from some algorithms that use the powerful technique of iterative compression. The two main features of our methodology are: (i) This is the first generic technique for designing linear time algorithms for directed cut-problems and (ii) it can be used in combination with future improvements in algorithms for the compression version of other well-studied cut-problems such as Multicut and Directed Subset Feedback Vertex Set.

We present the first polynomial time algorithm that finds the shortest route in a simple polygon such that all points of the polygon are visible from the route. This route is called the shortest watchman route, and we do not assume any restrictions on the route or on the simple polygon. Our algorithm runs in worst case O(n 6 ) time, but it is adaptive, making it run faster on polygons with a simple structure.

formula omitted]MST: A robust unified algorithm for quadrilateral mesh adaptation in 2D and 3D