The living in a lattice rule language

This paper presents a rule-based query language for an object-oriented database model. The database model supports complex objects, object identity, classes and types, and a class/type hierarchy. The instances are described by ‘object relations’ which are functions from a set of objects to value sets and other object sets. The rule language is based on object-terms which provide access to objects via the class hierarchy. Rules are divided into two classes: object-preserving rules manipulating existing objects (yielding a new ‘view’ on objects available in the object base) and object-generating rules creating new objects with properties derived from existing objects. The derived object sets are included in a class lattice. We give conditions for whether the instances of the ‘rules’ heads are ‘consistent’, i.e. represent object relations where the properties of the derived objects are functionally determined by the objects.     

Representation of an alpine treeline ecotone in SPOT 5 HRG data

An ecotone is a zone of vegetation transition between two communities, often resulting from a natural or anthropogenic environmental gradient. In remotely sensed imagery, an ecotone may appear as an edge, a boundary of mixed pixels or a zone of continuous variation, depending on the spatial scale of the vegetation communities and their transition zone in relation to the spatial resolution of the imagery. Often in image classification, an ecotone is either ignored if it falls within a width of one or two pixels, or part of it may be mapped as a separate vegetation community if it covers an area of several pixel widths. A soft classification method, such as probability mapping, is inherently appealing for mapping vegetation transition. Ideally, the probability of membership each pixel has to each vegetation class corresponds with the proportional composition of vegetation classes per pixel. In this paper we investigate the use of class probability mapping to produce a softened classification of an alpine treeline ecotone in Austria using a SPOT 5 HRG image. Here the transition with altitude is from dense subalpine forest to treeless alpine meadow and herbaceous vegetation. The posterior probabilities from a Maximum Likelihood algorithm are shown to reflect the land-cover composition of mixed pixels in the ecotone. The relationships between the posterior probability of class membership for the two end-member classes of ‘scrub and forest’ and ‘non-forest vegetation’ and the percentage ground cover of these vegetation classes (enumerated in 15 quadrats from 1:1500 aerial photographs) were highly significant: r² = 0.83 and r² = 0.85 respectively (p < 0.001, n = 15). We identify thresholds (alpha-cuts) in the posterior probabilities of class membership of ‘scrub and forest’ and ‘non-forest vegetation’ to map the alpine treeline ecotone as a transition zone of five intermediate vegetation classes between the end-member communities. In addition, we investigate the representation of the ecotone as a ratio between the posterior probabilities of ‘scrub and forest’ and ‘non-forest vegetation’. This displays the vegetation transition without imposing subjective boundaries, and has greater emphasis on the ecotone transition rather than on the end-member communities. We comment on the fitness for purpose of the different ways investigated for representing the alpine treeline ecotone.     

Computing multidimensional Delaunay tessellations

We recall some properties of Voronoi and Delaunay tessellations in any numbers of dimensions. We then propose a solution to the following problem: Given the Delaunay tessellation of n d-dimensional data points X₁,…, X_n, the proble is to insert a new data point X and to update the tessellation accordingly. The solution proposed achieves minimum space-complexity.     

Integration of polynomials over n-dimensional polyhedra

Explicit formulae and algorithms for computing integrals of polynomials over n-dimensional polyhedra are given. Two different approaches are discussed: one uses a decompositive representation, while the other one uses a boundary representation of the polyhedron. The algorithms are followed by a discussion of the complexity. In the appendix the pseudo-code and some examples of calculation are given.     

Integration of polynomials over n-dimensional linear polyhedra

This paper is concerned with explicit integration formulae for computing integrals of n-variate polynomials over linear polyhedra in n-dimensional space ⁿ. Two different approaches are discussed; the first set of formulae is obtained by mapping the polyhedron in n-dimensional space ⁿ into a standard n-simplex in ⁿ, while the second set of formulae is obtained by reducing the n-dimensional integral to a sum of n − 1 dimensional integrals which are n + 1 in number. These formulae are followed by an application example for which we have explained the detailed computational scheme. The symbolic integration formulae presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, such as, for example, volume, centre of mass, moments of inertia etc., required in engineering design problems.     

Experiments with some cluster analysis algorithms

In this paper, we shall show the following experimental results: (1) the one-dimensional clustering algorithm advocated by Slagle and Lee⁽¹⁾ can be generalized to the n-dimensional case, n > 1: (2) if a set of points in some n-space (n > 1) are linearly ordered through the short spanning path algorithm, then this set of points can be considered as occupying a one-dimensional space and the original n-dimensional clustering problem can now be viewed as a one-dimensional clustering problem; (3) a short spanning path usually contains as much information as a minimal spanning tree; (4) the one-dimensional clustering algorithm can be used to find the long links in a short spanning path or a minimal spanning tree. These long links have to be broken to obtain clusters.     

一种新的自适应多分辨率细分曲面的表示法*

针对已有自适应细分方法中存在的问题,提出了一种新的自适应多分辨率细分曲面的表示方法。该方法结合拓扑细分的特点,运用二维组合映射对半边数据结构进行形式化定义,并引出超映射的概念。在超映射这个通用的理论框架中引出了半边结构在多分辨率上的扩展,并对其在自适应细分方法中的应用进行了讨论。结果表明这种结构具有普适性,支持多种细分方法,允许在网格的任何多分辨率层次上及时有效地导航,并且在自适应细分过程中避免了拓扑裂缝的产生。     

n-Dimensional multiresolution representation of subdivision meshes with arbitrary topology

We present a new model for the representation of n-dimensional multiresolution meshes. It provides a robust topological representation of arbitrary meshes that are combined in closely interlinked levels of resolution. The proposed combinatorial model is formalized through the mathematical model of combinatorial maps allowing us to give a general formulation, in any dimensions, of the topological subdivision process that is a key issue to robustly and soundly define mesh hierarchies. It fully supports multiresolution edition what allows the implementation of most mesh processing algorithms – like filtering or compression – for n-dimensional meshes with arbitrary topologies.We illustrate this model, in dimension 3, with an new truly multiresolution representation of subdivision volumes. It allows us to extend classical subdivision schemes to arbitrary polyhedrons and to handle adaptive subdivision with an elegant solution to compliance issues. We propose an implementation of this model as an effective and relatively inexpensive data structure.     

Addition and removal of information for a knowledge base with incomplete information

A knowledge base containing incomplete information in the form of disjunctions and negative information shows difficulties regarding the update operators. In this paper simple and straightforward definitions are given for an ‘adding’ operator (‘+’) and a ‘removing’ operator (‘−’) using Hebrand models.     

Lower bounds for stability margin of n-dimensional discrete systems

This paper derives lower bounds for the stability margin of n-dimensional discrete systems in the Roesser’s state space setting. The lower bounds for stability margin are derived based on the MacLaurine series expansion. Numerical examples are given to illustrate the results.     

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.     

On describing complex surface shapes

The fractal surface model¹ is extended to provide a formalism that is competent to describe complex, natural three-dimensional surfaces in either a quantitative or qualitative manner and which, in addition, closely mimics human perceptual judgments of surface structure (eg ‘peaks’, ‘ridges’ or ‘valleys’) and threedimensional texture. This representation is then used to develop a statistical version of scale-space filtering that is applicable to one-, two- or three-dimensional data.     

Consumer-oriented versus technology-oriented systems: designing services for people

Current computer systems providing consumer services seem to be designed primarily on ‘technology-oriented’ thinking, based on the ‘efficiency’ and the operating methods of the computer. To be of benefit to a wider variety of consumers, the system design may have to reflect a more ‘consumer-oriented’ approach, based on factors such as the needs, preferences, skills and knowledge of the user. This article demonstrates the differences between user evaluations of two computer systems designed to help in house-hunting, modelled on (a) consumer-oriented, and (b) technology-oriented approaches. Although particularly relevant to consumer systems, the article may have important implications for consumer and public services in general.     

Ray tracing CSG trees using the Sticks representation scheme

A fast ray tracing algorithm of CSG tree is presented. The algorithm uses an adaptive space subdivisions approach, based on the conversion of the objects of the scene into the volumetric Sticks representation scheme. This conversion scheme, which requires O(kn²) memory space to represent data in a n³ grid, makes it possible either to obtain very fast low-quality frontal orthographic projections, or to produce a high-quality rendering of the scene in time less than that needed by classical ray tracing and nearly independent of the number of objects in the scene. Furthermore, the characteristics of the Sticks scheme can be exploited to compute geometric or topological properties of the represented objects. Comparative analyses between the Sticks representation scheme and classical space subdivision schemes and between our Sticksbased ray tracing and classical algorithms are presented.     

A method for synthesis of optimal weight structures

A method to synthesize structures of optimal or least weight is presented and demonstrated for structures employing cylindrical tube members. The method employs a unique reduction in complexity from the general n-dimensional problem to a two dimensional “design plane”. Comparison with previous authors' results are included.     

Blossoms are polar forms

Consider the functions H(t):=t² and h(u,v):=uv. The identity H(t)=h(t,t) shows that h is the restriction of h to the diagonal u=v in the uv-plane. Yet, in many ways, a bilinear function like h is simpler than a homogeneous quadratic function like H. More generally, if F(t) is some n-ic polynomial function, it is often helpful to study the polar form of F, which is the unique symmetric, multiaffine function ƒ(u_1,…u_n) satisfying the identity F(t)=f(t,…,t). The mathematical theory underlying splines is one area where polar forms can be particularly helpful, because two pieces F and G of an n-ic spline meet at a point r with C^k parametric continuity if and only if their polar forms ƒ and g agree on all sequences of n arguments that contain at least n-k copies of r.

The polar approach to the theory of splines emerged in rather different guises in three independent research efforts: Paul de Faget Casteljau called it ‘shapes through poles’; Carl de Boor called it ‘B-splines without divided differences’; and Lyle Ramshaw called it ‘blossoming’. This paper reviews the work of de Casteljau, de Boor, and Ramshaw in an attempt to clarify the basic principles that underly the polar approach. It also proposes a consistent system of nomenclature as a possible standard.     

Boundary matching algorithm for connected component labelling using linear quadtrees

A computationally fast top-down recursive algorithm for connected component labelling using linear quadtrees is presented. The input data structure used is a linear quadtree representing only black leaf nodes. The boundary matching approach used ensures that at most two adjacencies of any black leaf node are considered. Neighbour searching is carried out within restricted subsets of the input quadtree. The time and space complexities of the algorithm are O(Bn) and O(B) respectively for a linear quadtree with B black leaves constructed from a binary array of size 2ⁿ × 2ⁿ. Simulations show the algorithm to be twice as fast as an existing algorithm that uses an identical input data structure. The top-down algorithm presented can also be used to efficiently generate a linear quadtree representing all nodes — ‘grey’, ‘black’ and ‘white’ — in preorder when given an input linear quadtree representing only ‘black’ leaf nodes. The boundary matching algorithm is computationally fast and has low static and dynamic storage costs, making it useful for applications where linear quadtrees are held in main memory.     

Constrained multibody system dynamics an automated approach

The governing equations for constrained multibody systems are formulated in a manner suitable for their automated, numerical development and solution. Specifically, the “closed loop” problem of multibody chain systems is addressed.

The governing equations are developed by modifying dynamical equations obtained from Lagrange's form of d'Alembert's principle. This modification, which is based upon a solution of the constraint equations obtained through a “zero eigenvalues theorem,” is, in effect, a contraction of the dynamical equations.

It is observed that, for a system with n generalized coordinates and m constraint equations, the coefficients in the constraint equations may be viewed as “constraint vectors” in n-dimensional space. Then, in this setting the system itself is free to move in the n−m directions which are “orthogonal” to the constraint vectors.     

An Abstract Böhm-normalization

We study normalization by neededness with respect to ‘infinite results’, such as Böhm-trees, in an abstract framework of Stable Deterministic Residual Structures. We formalize the concept of ‘infinite results’ for finite terms as suitable sets of infinite reductions, and prove an abstract infinitary normalization theorem with respect to such sets. We also give a sufficient and necessary condition for existence of minimal normalizing reductions.     

Autonomous online generation of a motor representation of the workspace for intelligent whole-body reaching

We describe a learning strategy that allows a humanoid robot to autonomously build a representation of its workspace: we call this representation Reachable Space Map. Interestingly, the robot can use this map to: (i) estimate the Reachability of a visually detected object (i.e. judge whether the object can be reached for, and how well, according to some performance metric) and (ii) modify its body posture or its position with respect to the object to achieve better reaching. The robot learns this map incrementally during the execution of goal-directed reaching movements; reaching control employs kinematic models that are updated online as well. Our solution is innovative with respect to previous works in three aspects: the robot workspace is described using a gaze-centered motor representation, the map is built incrementally during the execution of goal-directed actions, learning is autonomous and online. We implement our strategy on the 48-DOFs humanoid robot Kobian and we show how the Reachable Space Map can support intelligent reaching behavior with the whole-body (i.e. head, eyes, arm, waist, legs).