Image-Swept Volumes

Many graphical objects can be represented by swept volumes (including its subset — generalised cylinders) by sweeping 2D or 3D templates along 3D trajectories. In this paper, we present a new approach for constructing swept volumes using image templates. We utilise scalar fields as our underlying data type, and employ volume ray casting techniques for rendering swept volumes in their original sweeping specifications as well as in their voxelised approximations. In addition to some simple image‐swept volumes, we also treat multi‐channel image templates, video templates, generalised sweeps, and self‐intersecting trajectories. This approach enables us to model swept volumes with heterogeneous interiors and amorphous effects. It also facilitates the use of constructive volume geometry for creating complex scenes in both modelling and rendering space.     

Constructive non-regularized geometry

Solid modelling is concerned with the construction and manipulation of unambiguous computer representations of solid objects. These representations permit us to distinguish the interior, the boundary and the complement of a solid. They are conveniently specified in Constructive Solid Geometry (CSG) by a construction tree that has solid primitives as leaves and rigid body motions or regularized Boolean operations as internal nodes. Algoriths for classifying sets with respect to CSG trees and for evaluating the boundaries of the corresponding solids are known, at least for simple geometric domains. Emerging CAD applications require that we extend the domain of solid modellers to support more general and more structured geometric objects. The focus is on dimensionally non-homogeneous, non-closed pointsets with internal structures. These entities are well suited for dealing with mixed-dimensional (‘non-manifold’) objects in ⁿ that have dangling or missing boundary elements, and that may be composed of several regions. A boundary representation for such objects has been described elsewhere. We propose to specify and represent inhomogeneous objects in terms of Constructive Non-Regularized Geometry (CNRG) trees that extend the domain of CSG by supporting non-regularized primitive shapes as leaves, and by providing more general set-theoretic and topological operators at interior nodes. Filtering operations are also provided that construct CNRG objects from aggregates of selected regions of other CNRG objects. A syntax and semantics of the operators in CNRG are presented, and some basic algorithms for classifying pointsets with respect to the regions of objects represented by CNRG trees are outlined.     

Planar piecewise algebraic curves

An approach is described for piecing together segments of planar algebraic curves with derivative continuity. The application of piecewise algebraic curves to area modelling (the two-dimensional analogue of solid modelling) is discussed. A technique is presented for expressing a planar rational parametric curve as an algebraic curve segment. An upper bound is derived for the farthest distance between two algebraic curves (one of which may also be a parametric curve) within a specified region.     

Generation of configuration space obstacles: The case of moving algebraic curves

We present algebraic algorithms to generate the boundary of planar configuration space obstacles arising from the translatory motion of objects among obstacles. Both the boundaries of the objects and obstacles are given by segments of algebraic plane curves.Research supported in part by NSF Grant MIP-85-21356 and a David Ross Fellowship. An earlier version of this paper appeared in theProceedings of the 1987 IEEE International Conference on Robotics and Automation, pp. 979–984.     

Modelling and Rendering Graphics Scenes Composed of Multiple Volumetric Datasets

This paper presents a method for modelling graphics scenes consisting of multiple volumetric objects. A two-level hierarchical representation is employed, which enables the reduction of the overall storage consumption as well as rendering time. With this approach, different objects can be derived from the same volumetric dataset, and 2D images can be trivially integrated into a scene. The paper also describes an efficient algorithm for rendering such scenes on ordinary workstations, and addresses issues concerning memory requirements and disk swapping.     

Construction and optimization of CSG representations

Boundary representations (B-reps) and constructive solid geometry (CSG) are widely used representation schemes for solids. While the problem of computing a B-rep from a CSG representation is relatively well understood, the inverse problem of B-rep to CSG conversion has not been addressed in general. The ability to perform B-rep to CSG conversion has important implications for the architecture of solid modelling systems and, in addition, is of considerable theoretical interest.

The paper presents a general approach to B-rep to CSG conversion based on a partition of Euclidean space by surfaces induced from a B-rep, and on the well known fact that closed regular sets and regularized set operations form a Boolean algebra. It is shown that the conversion problem is well defined, and that the solution results in a CSG representation that is unique for a fixed set of halfspaces that serve as a ‘basis’ for the representation. The ‘basis’ set contains halfspaces induced from a B-rep plus additional non-unique separating halfspaces.

An important characteristic of B-rep to CSG conversion is the size of a resulting CSG representation. We consider minimization of CSG representations in some detail and suggest new minimization techniques.

While many important geometric and combinatorial issues remain open, a companion paper shows that the proposed approach to B-rep to CSG conversion and minimization is effective in E², In E³, an experimental system currently converts natural-quadric B-reps in PARASOLID to efficient CSG representations in PADL-2.     

Constructive Backpropagation for Recurrent Networks

Choosing a network size is a difficult problem in neural network modelling. In many recent studies, constructive or destructive methods that add or delete connections, neurons or layers have been studied in order to solve this problem. In this work we consider the constructive approach, which is in many cases a very computationally efficient approach. In particular, we address the construction of recurrent networks by the use of constructive backpropagation. The benefits of the proposed scheme are firstly that fully recurrent networks with an arbitrary number of layers can be constructed efficiently. Secondly, after the network has been constructed we can continue the adaptation of the network weights as well as we can of its structure. This includes both addition and deletion of neurons/layers in a computationally efficient manner. Thus, the investigated method is very flexible compared to many previous methods. In addition, according to our time series prediction experiments, the proposed method is competitive in terms of modelling performance and training time compared to the well-known recurrent cascade-correlation method.     

Graphic Objects: A Mathematical Abstract Model for Computer Graphics

This paper defines an abstract mathematical model for graphic systems which incorporates modelling operations. The model is based on the “graphic object” concept. A graphic object is an abstraction of a graphic entity. Graphic objects are a generalization of the Fiume graphic object concept, supporting modelling operations, which give the set of graphic objects an algebraic structure of vector space and booelan algebra. Fiume visualization formalism can be applied to graphic objects. Further, product operations of graphic objects are defined, which are a generalization of solid sweep operations. Graphic object products may be used to build common 2D and 3D objects such as triangles, squares, cubes, circles and spheres from lines and arcs.