Linear Size Binary Space Partitions for Uncluttered Scenes

We describe a new and simple method for constructing binary space partitions (BSPs) in arbitrary dimensions. We also introduce the concept of uncluttered scenes, which are scenes with a certain property that we suspect many realistic scenes exhibit, and we show that our method constructs a BSP of size O(n) for an uncluttered scene consisting of n objects. The construction time is O(n log n) . Because any set of disjoint fat objects is uncluttered, our result implies an efficient method to construct a linear size BSP for fat objects. We use our BSP to develop a data structure for point location in uncluttered scenes. The query time of our structure is O( log n) , and the amount of storage is O(n) . This result can in turn be used to perform range queries with not-too-small ranges in scenes consisting of disjoint fat objects or, more generally, in so-called low-density scenes. Received December 11, 1997; revised May 9, 1998.     

Alternating paths through disjoint line segments

We show that every segment endpoint visibility graph on n disjoint line segments in the plane admits an alternating path of length , and this bound is optimal apart from a constant factor.     

The number of guillotine partitions in d dimensions

Guillotine partitions play an important role in many research areas and application domains, e.g., computational geometry, computer graphics, integrated circuit layout, and solid modeling, to mention just a few. In this paper we present an exact summation formula for the number of structurally-different guillotine partitions in d dimensions by n hyperplanes, and then show that it is .     

Weaving patterns of lines and line segments in space

Aweaving W is a simple arrangement of lines (or line segments) in the plane together with a binary relation specifying which line is above the other. A system of lines (or line segments) in 3-space is called arealization ofW, if its projection into the plane isW and the above-below relations between the lines respect the specifications. Two weavings are equivalent if the underlying arrangements of lines are combinatorially equivalent and the above-below relations are the same. An equivalence class of weavings is said to be aweaving pattern. A weaving pattern isrealizable if at least one element of the equivalence class has a three-dimensional realization. A weaving (pattern)W is calledperfect if, along each line (line segment) ofW, the lines intersecting it are alternately above and below. We prove that (i) a perfect weaving pattern ofn lines is realizable if and only ifn 3, (ii) a perfect m byn weaving pattern of line segments (in a grid-like fashion) is realizable if and only if min(m, n) 3, (iii) ifn is sufficiently large, then almost all weaving patterns ofn lines are nonrealizable.Jànos Pach has been supported in part by Hungarian NFSR Grant 1812, NSF Grant CCR-8901484, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center, under NSF Grant STC88-09648. Richard Pollack has been supported in part by NSA Grant MDA904-89-H-2030, NSF Grants DMS-85-01947 and CCR-8901484, and DIMACS. Emo Welzl has been supported in part by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM) and DIMACS.     

Clustering of collinear line segments

A number of methods are presented for finding clusters in collinear collections of line segments. The methods are of two kinds — merging methods and splitting methods. Both make use of an evaluation function, and several alternative functions are illustrated. The methods are evaluated using randomly generated clusters on backgrounds containing varying amounts of noise.     

Cluster detection in a collection of collinear line segments

Two novel methods of clustering of a collection of collinear line segments are presented. Both the methods are of merging type, global in nature and take into account both gap and line segment lengths. The methods are also useful for clustering collinear dot patterns. The efficacy of the algorithms is brought out by successfully applying them on different types of data sets. A comparative study is made by comparing the results of the proposed procedures with that of another procedure already existing.     

Improved Binary Space Partition merging

This paper presents a new method for evaluating boolean set operations between Binary Space Partition (BSP) trees. Our algorithm has many desirable features, including both numerical robustness and O(n) output sensitive time complexity, while simultaneously admitting a straightforward implementation. To achieve these properties, we present two key algorithmic improvements. The first is a method for eliminating null regions within a BSP tree using linear programming. This replaces previous techniques based on polygon cutting and tree splitting. The second is an improved method for compressing BSP trees based on a similar approach within binary decision diagrams. The performance of the new method is analyzed both theoretically and experimentally. Given the importance of boolean set operations, our algorithms can be directly applied to many problems in graphics, CAD and computational geometry.     

A Hough transform based line recognition method utilizing both parameter space and image space

Hough Transform (HT) is recognized as a powerful tool for graphic element extraction from images due to its global vision and robustness in noisy or degraded environment. However, the application of HT has been limited to small-size images for a long time. Besides the well-known heavy computation in the accumulation, the peak detection and the line verification become much more time-consuming for large-size images. Another limitation is that most existing HT-based line recognition methods are not able to detect line thickness, which is essential to large-size images, usually engineering drawings. We believe these limitations arise from that these methods only work on the HT parameter space. This paper therefore proposes a new HT-based line recognition method, which utilizes both the HT parameter space and the image space. The proposed method devises an image-based gradient prediction to accelerate the accumulation, introduces a boundary recorder to eliminate redundant analyses in the line verification, and develops an image-based line verification algorithm to detect line thickness and reduce false detections as well. It also proposes to use pixel removal to avoid overlapping lines instead of rigidly suppressing the N×N neighborhood. We perform experiments on real images with different sizes in terms of speed and detection accuracy. The experimental results demonstrate the significant performance improvement, especially for large-size images.     

An efficient algorithm for line and polygon clipping

We present an algorithm for clipping a polygon or a line against a convex polygonal window. The algorithm demonstrates the practicality of various ideas from computational geometry. It spendsO(logp) time on each edge of the clipped polygon, wherep is the number of window edges, while the Sutherland-Hodgman algorithm spendsO(p) time per edge. Theoretical and experimental analyses show that the constants involved are small enough to make the algorithm competitive even for windows with four edges. The algorithm enables image-space clipping against windows whose boundaries are convex spline curves. The paper contains detailed pseudo-code implementation of the algorithm and an adaptation of the simulation of simplicity method for handling degenerate cases.     

Fast line segment grouping method for finding globally more favorable line segments

In this paper, we propose a new method for extracting line segments from edge images. Our method basically follows a line segment grouping approach. This approach has many advantages over a Hough transform based approach in practical situations. However, since the process of the conventional line segment grouping approach is purely local, it does not provide a mechanism for finding more favorable line segments from a global point of view. Our method overcomes the local nature of the conventional line segment grouping approach, while retaining most of its advantages, by incorporating the useful concept of the Hough transform based approach into the line segment grouping approach. Our method is fast and allows elementary line segments to be shared simultaneously by several line segments, and the degree of sharing is determined by a user-specified threshold. We performed a series of tests to compare the performance of our method with that of six other methods. Throughout the tests our method ranked in the top two of the tested methods both in detection rate and computation time.     

A new efficient motion-planning algorithm for a rod in two-dimensional polygonal space

We present here a new and efficient algorithm for planning collision-free motion of a line segment (a rod or a ladder) in two-dimensional space amidst polygonal obstacles. The algorithm uses a different approach than those used in previous motion-planning techniques, namely, it calculates the boundary of the (three-dimensional) space of free positions of the ladder, and then uses this boundary for determining the existence of required motions, and plans such motions whenever possible. The algorithm runs in timeO(K logn) =O(n ² logn) wheren is the number of obstacle corners and whereK is the total number of pairs of obstacle walls or corners of distance less than or equal to the length of the ladder. The algorithm has thus the same complexity as the best previously known algorithm of Leven and Sharir [5], but if the obstacles are not too cluttered together it will run much more efficiently. The algorithm also serves as an initial demonstration of the viability of the technique it uses, which we expect to be useful in obtaining efficient motion-planning algorithms for other more complex robot systems.Work on this paper has been supported in part by a grant from the Joint Ramot-Israeli Ministry of Industry Foundation.     

隐私保护的四方判定线段是否相交协议

鉴于目前已有的隐私保护的线段相交判定协议的参与方都是两方,不能解决多方之间判定线段是否相交的问题,因此提出了一个隐私保护的四方相互合作判定线段是否相交的协议.4个参与方各自拥有线段的一个端点,采用安全多方计算中的一些基础运算协议,通过两两计算,判定由这4个端点构成的两条线段是否相交,同时确保不会向其它的参与方泄漏线段以及端点的信息.最后给出了该协议在现实中的一个应用.     

A note on searching line arrangements and applications

We study the problem of searching for a vertex with a desired property in the arrangement of a set of lines in the plane. We show that this problem can be solved efficiently by modifying (and simplifying) two slope selection algorithms without using parametric search. We apply this result to a points approximation problem and obtain an optimal solution for it without using parametric search. Since this line arrangement searching problem is quite natural, our result may find other applications as well.     

Fast dynamic intersection searching in a set of isothetic line segments

We reconsider the (isothetic) line segment intersection searching problem: Given a set S of n horizontal and vertical line segments and a query segment q, find all t segments in S intersecting q. We describe the first dynamic solution for this problem which achieves O(log n + t) query time. This, however, has to be paid by O(n log² n) space requirements and O(log³ n) update time. If segments are defined over a grid of size N × N (the semidynamic case), then the problem can be solved in O(logN + t) query time, O(n log² N) space and O(log² N) update time. The solution is based on the use of segment tree and range tree and the halfobject technique.     

Pattern Matching for Sets of Segments

In this paper we present algorithms for a number of problems in geometric pattern matching where the input consists of a collection of orthogonal segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new criterion for segment similarity called the coverage measure, and present efficient algorithms for maximizing it between sets of axis-parallel segments under translations. In the general case, we present a procedure running in time O(n³ log² n), and for the case when all segments are horizontal, we give a procedure that runs in time O(n² log² n). Here n is the number of segments. In addition, we show that an -approximation to the Hausdorff distance between two sets of horizontal segments under vertical translations can be computed in time O(n^3/2 max(poly(log M, log n, 1/))). Here, M denotes the ratio of the diameter to the closest pair of points in the sets of segments (where pairs of points lie on different segments). These algorithms are significant improvements over the general algorithm of Agarwal et al. that required time O(n⁴ log²     

Robust and efficient map-to-image registration with line segments

This work investigates map-to-image registration for planar scenes in the context of robust parameter estimation. Registration is posed as the problem of estimating a projective transformation which optimally aligns transformed model line segments from a map with data line segments extracted from an image. Matching and parameter estimation is solved simultaneously by optimizing an objective function which is based on M-estimators, and depends on overlap and the weighted orthogonal distance between transformed model segments and data segments. An extensive series of registration experiments was conducted to test the performance of the proposed parameter estimation algorithm. More than 200 000 registration experiments were run with different objective functions for 12 aerial images and randomly corrupted maps distorted by randomly selected projective transformations. Received: 10 August 2000 / Accepted: 29 January 2001