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.

Article Free Access Share on Interactive playing with large synthetic environments Author: Bruce F. Naylor AT&T Bell Laboratories, Murray Hill, NJ AT&T Bell Laboratories, Murray Hill, NJView Profile Authors Info & Claims I3D '95: Proceedings of the 1995 symposium on Interactive 3D graphicsApril 1995 Pages 107–ff.https://doi.org/10.1145/199404.199423Online:15 April 1995Publication History 1citation292DownloadsMetricsTotal Citations1Total Downloads292Last 12 Months2Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my Alerts New Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

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.>

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.

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.

In computer graphics, geometric modeling of complex objects is a difficult process. An important class of complex objects arise from natural phenomena: trees, plants, clouds, mountains, etc. Researchers are at present investigating a variety of techniques for extending modeling capabilities to include these as well as other classes. One mathematical concept that appears to have significant potential for this is fractals. Much interest currently exists in the general scientific community in using fractals as a model of complex natural phenomena. However, only a few methods for generating fractal sets are known. We have been involved in the development of a new approach to computing fractals. Any set of linear maps (affine transformations) and an associated set of probabilities determines an Iterated Function System (IFS). Each IFS has a unique "attractor" which is typically a fractal set (object). Specification of only a few maps can produce very complicated objects. Design of fractal objects is made relatively simple and intuitive by the discovery of an important mathematical property relating the fractal sets to the IFS. The method also provides the possibility of solving the inverse problem. given the geometry of an object, determine an IFS that will (approximately) generate that geometry. This paper presents the application of the theory of IFS to geometric modeling.

This paper describes a new algorithm for solving the hidden surface (or line) problem, to more rapidly generate realistic images of 3-D scenes composed of polygons, and presents the development of theoretical foundations in the area as well as additional related algorithms. As in many applications the environment to be displayed consists of polygons many of whose relative geometric relations are static, we attempt to capitalize on this by preprocessing the environment's database so as to decrease the run-time computations required to generate a scene. This preprocessing is based on generating a “binary space partitioning” tree whose in order traversal of visibility priority at run-time will produce a linear order, dependent upon the viewing position, on (parts of) the polygons, which can then be used to easily solve the hidden surface problem. In the application where the entire environment is static with only the viewing-position changing, as is common in simulation, the results presented will be sufficient to solve completely the hidden surface problem.

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.