Bruce F. Naylor

Set operations on polyhedra using binary space partitioning trees

We introduce a new representation for polyhedra by showing how Binary Space Partitioning Trees (BSP trees) can be used to represent regular sets. We then show how they may be used in evaluating set operations on polyhedra. The BSP tree is a binary tree representing a recursive partitioning of d-space by (sub-)hyperplanes, for any dimension d. Their previous application to computer graphics has been to organize an arbitrary set of polygons so that a fast solution to the visible surface problem could be obtained. We retain this property (in 3D) and show how BSP trees can also provide an exact representation of arbitrary polyhedra of any dimension. Conversion from a boundary representation (B-reps) of polyhedra to a BSP tree representation is described. This technique leads to a new method for evaluating arbitrary set theoretic (boolean) expressions on B-reps, represented as a CSG tree, producing a BSP tree as the result. Results from our language-driven implmentation of this CSG evaluator are discussed. Finally, we show how to modify a BSP tree to represent the result of a set operation between the BSP tree and a B-rep. We describe the embodiment of this approach in an interactive 3D object design program that allows incremental modification of an object with a tool. Placement of the tool, selection of views, and performance of the set operation are all performed at interactive speeds for modestly complex objects.

Predicting RF coverage in large environments using ray-beam tracing and partitioning tree represented geometry

We present a system for efficient prediction of RF power distribution in site specific environments using a variation of ray tracing, which we calledray-beam tracing. The simulation results were validated against measured data for a number of large environments with good statistical correlation between the two. We represent geometric environments in full 3D which facilitates rooftop deployment along with any other 3D locations. We use broadcast mode of propagation, whose cost increases more slowly with an increase in the number of receiving bins. The scheme works well both for indoor and outdoor environments. Simple ray tracing has a major disadvantage in that adjacent rays from a transmitter diverge greatly after large path lengths due to multiple reflections, such that arbitrarily large geometric entities could fall in between these rays. This results in asampling error problem. The error increases arbitrarily as the incident angle approaches 90°. The problem is addressed by introducing the notion of beams while retaining the simplicity of rays for intersection calculations. A beam is adaptively split into child beams to limit the error. A major challenge for computational efficiency is to quickly determine the closest ray-surface intersection. We achieve this by using partitioning trees which allows representation of arbitrarily oriented polygonal environments. We also use partitioning trees for our full 3D interactive visualization along with interactive placement of transmitters, receiving bins, and querying of power.

Merging BSP trees yields polyhedral set operations

BSP trees have been shown to provide an effective representation of polyhedra through the use of spatial subdivision, and are an alternative to the topologically based b-reps. While bsp tree algorithms are known for a number of important operations, such as rendering, no previous work on bsp trees has provided the capability of performing boolean set operations between two objects represented by bsp trees, i.e. there has been no closed boolean algebra when using bsp trees. This paper presents the algorithms required to perform such operations. In doing so, a distinction is made between the semantics of polyhedra and the more fundamental mechanism of spatial partitioning. Given a partitioning of a space, a particular semantics is induced on the space by associating attributes required by the desired semantics with the cells of the partitioning. So, for example, polyhedra are obtained simply by associating a boolean attribute with each cell. Set operations on polyhedra are then constructed on top of the operation of merging spatial partitionings. We present then the algorithm for merging two bsp trees independent of any attributes/semantics, and then follow this by the additional algorithmic considerations needed to provide set operations on polyhedra. The result is a simple and numerically robust algorithm for set operations.

Piecewise linear approximations of digitized space curves with applications

Generating piecewise linear approximations of digitized or “densely sampled” curves is an important problem in many areas. Here, we consider how to approximate an arbitrary digitized 3-D space curve, made of n+1 points, with m line segments. We present an O(n 3 log m) time, O(n 2 log m) space, dynamic programming algorithm which finds an optimal approximation. We then introduce an iterative heuristic algorithm, based upon the notions of curve length and spherical image, which quickly computes a good approximation of a space curve in O(N iter n) time and O(n) space. We apply this fast heuristic algorithm to display space curve segments and implicit surface patches, and to linearly approximate curved 3D objects, made by rotational sweeping, by binary space partitioning trees that are well-balanced.

Predetermining visibility priority in 3-D scenes (Preliminary Report)

The principal calculation performed by all visible surface algorithms is the determination of the visible polygon at each pixel in the image. Of the many possible speedups and efficiencies found for this problem, only one published algorithm (developed almost a decade ago by a group at General Electric) took advantage of an observation that many visibility calculations could be performed without knowledge of the eventual viewing position and orientation—once for all possible images. The method is based on a “potential obscuration” relation between polygons in the simulated environment. Unfortunately, the method worked only for certain objects; unmanagable objects had to be manually (and expertly!) subdivided into managable pieces. Described in this paper is a solution to this problem which allows substantial a priori visibility determination for all possible objects without any manual intervention. The method also identifies the ( hopefully, few) visibility calculations which remain to be performed after the viewing position is specified. Also diescussed is the development of still stronger solutions which could further reduce the number of these visibility calculations remaining at image generation time. The reduction in overall processing and memory requirements enabled by this approach may be quite significant, especially for those applications (e.g., 3-D simulation, animation, interactive design) in which numerous visible surface images are generated from a relatively stable data base.

Converting discrete images to partitioning trees

The discrete space representation of most scientific datasets, generated through instruments or by sampling continuously defined fields, while being simple, is also verbose and structureless. We propose the use of a particular spatial structure, the binary space partitioning tree as a new representation to perform efficient geometric computation in discretely defined domains. The ease of performing affine transformations, set operations between objects, and correct implementation of transparency makes the partitioning tree a good candidate for probing and analyzing medical reconstructions, in such applications as surgery planning and prostheses design. The multiresolution characteristics of the representation can be exploited to perform such operations at interactive rates by smooth variation of the amount of geometry. Application to ultrasound data segmentation and visualization is proposed. The paper describes methods for constructing partitioning trees from a discrete image/volume data set. Discrete space operators developed for edge detection are used to locate discontinuities in the image from which lines/planes containing the discontinuities are fitted by using either the Hough transform or a hyperplane sort. A multiresolution representation can be generated by ordering the choice of hyperplanes by the magnitude of the discontinuities. Various approximations can be obtained by pruning the tree according to an error metric. The segmentation of the image into edgeless regions can yield significant data compression. A hierarchical encoding schema for both lossless and lossy encodings is described.

Binary space partitioning tree representation of images

Keywords: LTS1 Reference LCAV-ARTICLE-1991-004doi:10.1016/1047-3203(91)90023-9 Record created on 2005-04-18, modified on 2017-05-12

Interactive playing with large synthetic environments

Until recently, opportunities to experience large synthetic rnvironmcnts have been limited primarily to cxpcnsive training simulators. However, with the advent of “location based entertainment” at theme parks and cvcn CD-ROM based games for PCs. these kinds of experiences are beginning to be made available to the gcncral public as well. The constraints on the possibilities for appealing “content” arises from the technological capabilities that are possible for a given pcrformancc level on a given platform. Currently, for 3D graphics, performance is closely tied to the number of tcxturc mapped polygons that can be rendered for each frame as well as the rntc at which collisions of various kinds can be computed. Large synthetic environments require at least tens of thousands of polygons, and could easily entail millions. However. for each image, only a small subset of these polygons are typically required to synthesize the image. Similarly. collisions bctwcen two objects, or between a viewer and the environment, involve an cvcn smaller subset. The task then for efficient geometric computations is to. if possible. quickly identify the relevant subset. The principal methodology for finding the minimal subset of polygons is to use spatial search structures. such as regular grids, octrees, or binary space partitioning trees. In this paper, we describe briefly the current status of our efforts at using binary space partitioning trees for navigating through and playing with large environments, including rendering and collision dctcction. as well as permitting intcractivc modifications of the environment using set operations that should prove appealing for entertainment applications.

Representing medical images with partitioning trees

The binary space partitioning tree is a method of converting a discrete space representation to a particular continuous space representation. The conversion is accomplished using standard discrete space operators developed for edge detection, followed by a Hough transform to generate candidate hyperplanes that are used to construct the partitioning tree. The result is a segmented and compressed image represented in continuous space suitable for elementary computer vision operations and improved image transmission/storage. Examples of 256*256 medical images for which the compression is estimated to range between 1 and 0.5 b/pixel are given.<<ETX>>

Image representation using binary space partitioning trees

Representation of two and three-dimensional objects by tree structures has been used extensively in solid modeling, computer graphics, computer vision and image processing. (See for example [Mantyla] [Chen] [Hunter] [Rosenfeld] [Leonardi].) Quadtrees, which are used to represent objects in 2-D space, and octrees, which are the extension of quadtrees in 3-D space, have been studied thoroughly for applications in graphics and image processing.