Efficient ray shooting and hidden surface removal

In this paper we study the ray-shooting problem for three special classes of polyhedral objects in space: axis-parallel polyhedra, curtains (unbounded polygons with three edges, two of which are parallel to thez-axis and extend downward to minus infinity), and fat horizontal triangles (triangles parallel to thexy-plane whose angles are greater than some given constant). For all three problems structures are presented usingO(n ²⁺) preprocessing, for any fixed > 0, withO(logn) query time. We also study the general ray-shooting problem in an arbitrary set of triangles. Here we present a structure that usesOn ⁴⁺) preprocessing and has a query time ofO(logn).We use the ray-shooting structure for curtains to obtain an algorithm for computing the view of a set of nonintersecting prolyhedra. For any > 0, we can obtain an algorithm with running time , wheren is the total number of vertices of the polyhedra andk is the size of the output. This is the first output-sensitive algorithm for this problem that does not need a depth order on the faces of the polyhedra.This research was supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). The first and third authors were also supported by the Dutch Organization for Scientific Research (N.W.O.).     

An optimal algorithm for reporting visible rectangles

We consider the following problem as defined by Grove et al. [Internat. J. Comput. Geom. Appl. 9 (1999) 207-217]: Given a set of n isothetic rectangles in 3D space determine the subset of rectangles, that are not completely hidden. We present an optimal algorithm for this problem that runs in O(nlogn) time and O(n) space. Our result is an improvement over the one of Grove et al. by a logarithmic factor in storage and is achieved by using a different approach. An analogous approach gives non-trivial solutions for other kinds of objects too.     

A review of recent hidden surface removal techniques

Techniques for the removal of hidden surfaces and/or hidden lines from computer generated pictures have continued to be developed and to be applied in other areas. This review covers much recent work in removal methods and some of the newer applications. A broad classification of the many references is given.     

A systolic algorithm for hidden surface removal

With the advent of VLSI it has become possible to map parallel algorithms for compute-bound problems directly on silicon. Systolic architecture is very good candidate for VLSI implementation because of its regular and simple design, and regular communication pattern. In this paper, a systolic algorithm and corresponding systolic architecture, a linear systolic array, for the scanline-based hidden surface removal problem in three-dimensional computer graphics have been proposed. The algorithm is based on the concept of sample spans or intervals. The worst case time taken by the algorithm is O(n), n being the number of segments in a scanline. The time taken by the algorithm for a given scene depends on the scene itself, and on an average considerable improvement over the worst case behaviour is expected. A pipeline scheme for handling the I/O process has also been proposed which is suitable for VLSI implementation of the algorithm.     

Variations on ray shooting

We solve some problems related toray shooting in the plane, such as finding the first object hit by a query ray or counting the number of objects intersected by the query line. Our main results are an algorithm for finding the first hit when the objects are lines, and an algorithm for the case when the objects are segments. If the segments form simple polygons, this information can be used for reducing the complexity of the algorithms. The algorithms are efficient in space and in query time. Moreover, they are simple and therefore of practical use.This research was partially supported by the NY Metropolitan Research Fund. The second author is currently at IBM Haifa Research Group, Haifa 32000, Israel.     

Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons

Given a triangulation of a simple polygonP, we present linear-time algorithms for solving a collection of problems concerning shortest paths and visibility withinP. These problems include calculation of the collection of all shortest paths insideP from a given source vertexS to all the other vertices ofP, calculation of the subpolygon ofP consisting of points that are visible from a given segment withinP, preprocessingP for fast "ray shooting" queries, and several related problems.Work on this paper by this author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, the IBM Corporation, and from the U.S.-Israel Binational Science Foundation.Work on this paper by this author has been supported by National Science Foundation Grant DCR-86-05962.     

Optimum sweeps of simple polygons with two guards

A polygon P admits a sweep if two mobile guards can detect an unpredictable, moving target inside P  , no matter how fast the target moves. Two guards move on the polygon boundary and are required to always be mutually visible. The objective of this study is to find an optimum sweep such that the sum of the distances travelled by the two guards in the sweep is minimized. We present an O(n²)$O(n^2)$ time and O(n)$O(n)$ space algorithm for optimizing this metric, where n   is the number of vertices of the given polygon. Our result is obtained by reducing this problem to finding a shortest path between two nodes in a graph of size O(n)$O(n)$.     

Efficient traversal of well-behaved hierarchical trees of extents for ray-tracing complex scenes

Traversal of hierarchial trees of extents (HTE) requires computation of intersections between rays and bounding volumes whose faces are parallel to the cartesian axes. By redefining the HTE so that nonoverlapping bounding volumes are generated, a well-behaved data structure is obtained in which geometrical coherence is applied to speed up its traversal. We distinguish two types of bounding volumes: internal boxes contain the ray's origin whileexternal bounding volumes do not contain the ray's origin. To traverse the HTE, we look first to polygons in the internal bounding volumes and deal with external boxes only when no ray-polygon intersection is found in internal nodes. As external nodes in the HTE define pruned subtrees of external bounding volumes, geometrical characteristics of the boxes are exploited for HTE traversal. A coding scheme allows a 6-bit code to determine which faces of a bounding volume need to be tested for intersection. Also, our well-behaved HTE allows for reuse of intersection points at lower levels of the tree.     

三维动态几何中直线消隐的实现

由于OpenGL图形系统没有提供现成的直线动态消隐的功能，因此在设计三维动态几何软件时，直线的动态消隐很困难。利用OpenGL图形系统，在研究现有的消隐技术——深度缓存算法的基础上，通过两条直线的叠加和视点移动角度的计算，实现了直线的动态消隐。而且，这种方法也适用于被曲面遮挡的直线的消隐，和被平面或者曲面遮挡的曲线的消隐。     

An efficient scan conversion and hidden surface removal algorithm

A scan conversion and hidden surface algorithm is discussed in detail. The algorithm is of the Watkins class but differs markedly in implementation. It is running on an 8K Varian 620 computer, and is used to produce shaded color renderings of three-dimensional objects. Details of the data structure used and of the internal sortings is given. Together with a video disk subsystem, the minicomputer system is capable of producing studio quality T.V. images in 8–70 s, depending on image complexity.     

An improved noise-robust voice activity detector based on hidden semi-Markov models

To improve the performance of voice activity detector (VAD) in noisy environments, this paper concentrates on three critical aspects related to noise robustness including speech features, feature distributions and temporal dependence. Based on the statistic on TIMIT and NOIZEUS, Mel-frequency cepstrum coefficients (MFCCs) are selected as speech features, Gaussian Mixture distributions (GMD) are applied to associate the observations in MFCC domain with both speech and non-speech states, and Weibull and Gamma distributions are used to explicitly model noise and speech durations, respectively. To integrate these aspects into VAD, the hidden semi-Markov model (HSMM) as a generalized hidden Markov model (HMM) is introduced first. Then the VAD decision is made according to the likelihood ratio test (LRT) incorporating state prior knowledge and modified forward variables of HSMM. We design a recursive way to efficiently calculate modified forward variables. Finally a series of experiments demonstrate: (1) the positive effect of different robustness-related schemes adopted in the proposed VAD; (2) better performance against the standard ITU-T G.729B, Adaptive MultiRate VAD phase 2 (AMR2), Advanced Front-end (AFE), HMM-based VAD and VAD using Laplacian-Gaussian model (LD-GD based VAD).     

Parameterized complexity and improved inapproximability for computing the largest j-simplex in a V-polytope

We consider the problem of computing the squared volume of the largest j-simplex contained in an n-dimensional polytope presented by its vertices (a V-polytope). We show that the related decision problem is W[1]-complete, with respect to the parameter j. We also improve the constant inapproximability factor given in [A. Packer, Polynomial-time approximation of largest simplices in V-polytopes, Discrete Appl. Math. 134 (1-3) (2004) 213-237], by showing that there are constants μ<1,c>1 such that it is NP-hard to approximate within a factor of cμn the volume of the largest ⌊μn⌋-simplex contained in an n-dimensional polytope with O(n) vertices.     

Applying the improved Chen and Han’s algorithm to different versions of shortest path problems on a polyhedral surface

The computation of shortest paths on a polyhedral surface is a common operation in many computer graphics applications. There are two best known exact algorithms for the “single source, any destination” shortest path problem. One is proposed by Mitchell et al. (1987) [1]. The other is by Chen and Han (1990) [11]. Recently, Xin and Wang (2009) [9] improved the CH algorithm by exploiting a filtering theorem and achieved a practical method that outperforms both the CH algorithm and the MMP algorithm whether in time or in space.In this paper, we apply the improved CH algorithm to different versions of shortest path problems. The contributions of this paper include: (1) For a surface point p∈△v₁v₂v₃, we present an unfolding technique for estimating the distance value at p using the distances at v₁,v₂ and v₃. (2) We show that the improved CH algorithm can be naturally extended to the “multiple sources, any destination” version. Also, introducing a well-chosen heuristic factor into the improved CH algorithm will induce an exact solution to the “single source, single destination” version. (3) At the conclusion of multi-source shortest path algorithms, we can use the distance values at vertices to approximately compute the geodesic-distance-based offsets, the Voronoi diagram and the Delaunay triangulation in O(n) time. (4) By importing a precision parameter λ, we obtain a precision controlled approximant which varies from the improved CH algorithm to Dijkstra’s algorithm as λ increases from 0 to 1. Thus, an interesting relationship between them can be naturally established.     

OPTIMAL MESH ALGORITHMS FOR PROXIMITY AND VISIBILITY PROBLEMS IN SIMPLE POLYGONS*

We present optimal parallel algorithms that run in time on mesh-connected computer for a number of fundamental problems concerning proximity and visibility in a simple polygon. These include computing shortest paths, shortest path trees, shortest path partitions, all-farthest neighbors, the visibility polygon of a point, the weak visibility polygon of an edge, and the ray shooting problem.     

Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

We investigate three-dimensional visibility problems for scenes that consist ofn non-intersecting spheres. The viewing point moves on a flightpath that is part of a circle at infinity given by a planeP and a range of angles {(t)¦t[01]} [02]. At timet, the lines of sight are parallel to the ray inP, which starts in the origin ofP and represents the angle(t) (orthographic views of the scene). We give an algorithm that computes the visibility graph at the start of the flight, all time parameters at which the topology of the scene changes, and the corresponding topology changes. The algorithm has running time0(n + k + p) logn), wheren is the number of spheres in the scene;p is the number of transparent topology changes (the number of different scene topologies visible along the flight path, assuming that all spheres are transparent); andk denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.The second author was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).     

An in-place algorithm for Klee's measure problem in two dimensions

We present the first in-place algorithm for solving Klee's measure problem for a set of n axis-parallel rectangles in the plane. Our algorithm runs in O(n3/2logn) time and uses O(1) extra words in addition to the space needed for representing the input. The algorithm is surprisingly simple and thus very likely to yield an implementation that could be of practical interest. As a byproduct, we develop an optimal algorithm for solving Klee's measure problem for a set of n intervals; this algorithm runs in optimal time O(nlogn) and uses O(1) extra space.     

An optimal algorithm for the boundary of a cell in a union of rays

In this paper we study a cell of the subdivision induced by a union ofn half-lines (or rays) in the plane. We present two results. The first one is a novel proof of theO(n) bound on the number of edges of the boundary of such a cell, which is essentially of methodological interest. The second is an algorithm for constructing the boundary of any cell, which runs in optimal (n logn) time. A by-product of our results are the notions of skeleton and of skeletal order, which may be of interest in their own right.This work was partly supported by CEE ESPRIT Project P-940, by the Ecole Normale Supérieure, Paris, and by NSF Grant ECS-84-10902.This work was done in part while this author was visiting the Ecole Normale Supérieure, Paris, France.     

An efficient algorithm for maxdominance, with applications

Given a planar setS ofn points,maxdominance problems consist of computing, for everyp S, some function of the maxima of the subset ofS that is dominated byp. A number of geometric and graph-theoretic problems can be formulated as maxdominance problems, including the problem of computing a minimum independent dominating set in a permutation graph, the related problem of finding the shortest maximal increasing subsequence, the problem of enumerating restricted empty rectangles, and the related problem of computing the largest empty rectangle. We give an algorithm for optimally solving a class of maxdominance problems. A straightforward application of our algorithm yields improved time bounds for the above-mentioned problems. The techniques used in the algorithm are of independent interest, and include a linear-time tree computation that is likely to arise in other contexts.The research of this author was supported by the Office of Naval Research under Grants N00014-84-K-0502 and N00014-86-K-0689, and the National Science Foundation under Grant DCR-8451393, with matching funds from AT&T.This author's research was supported by the National Science Foundation under Grant DCR-8506361.