Non-manifold Objects Research Articles

Autonomous void detection and characterisation in point clouds and triangular meshes

In this paper we propose and demonstrate a novel void characterisation algorithm which is able to distinguish between internal and external voids that are present in point clouds of both manifold and non-manifold objects and 3D scenes. We demonstrate the capabilities of our algorithm using several point clouds representing both scenes and objects. Our algorithm is shown in both a descriptive overview format as well as pseudocode. We also compare a variety of different void detection algorithms and then present a novel refinement to the best performing of these algorithms. Our refinement allows for voids in point clouds to be detected more efficiently, with fewer false positives and with over an order of magnitude improvement in terms of run time. We show our run time performance and compare it to results obtained using alternative algorithms, when tested using popular single board computers. This comparison is important as our work is intended for online robotics applications, where hardware is typically of low computational power. The target application for this work is 3D scene reconstruction to aid in the decommissioning of nuclear facilities.

Autonomous void detection and characterisation in point clouds and triangular meshes

In this paper we propose and demonstrate a novel void characterisation algorithm which is able to distinguish between internal and external voids that are present in point clouds of both manifold and non-manifold objects and 3D scenes. We demonstrate the capabilities of our algorithm using several point clouds representing both scenes and objects. Our algorithm is shown in both a descriptive overview format as well as pseudocode. We also compare a variety of different void detection algorithms and then present a novel refinement to the best performing of these algorithms. Our refinement allows for voids in point clouds to be detected more efficiently, with fewer false positives and with over an order of magnitude improvement in terms of run time. We show our run time performance and compare it to results obtained using alternative algorithms, when tested using popular single board computers. This comparison is important as our work is intended for online robotics applications, where hardware is typically of low computational power. The target application for this work is 3D scene reconstruction to aid in the decommissioning of nuclear facilities.

A Search-and-Validate Method for Face Identification from Single Line Drawings

Several studies have been made in finding the faces of an object depicted in a line drawing, but the problem has not been completely solved. Although existing methods can find the correct faces in most cases, there is no mechanism to ascertain that they are indeed correct, leaving the human user to do so. This paper uses a two-stage approach--find potential faces, then validate their correctness--to ensure that only correct faces are delivered ultimately. The face finding itself uses a double breadth-first search algorithm, which yields the shortest path, to find the potential faces. The basic premise is that the smallest faces found are more likely the correct ones. They serve as the "seed" potential faces, from which the algorithm proceeds to search for more faces. If the potential faces found satisfy the validation rules, then they are accepted as correct. Otherwise, the wrong potential faces are identified and removed, and new ones found in their place. The validation process is then repeated. The algorithm is fast and reliable, can deal with planar-faced manifold and nonmanifold objects, and can deliver the different results when a drawing has multiple interpretations. Our extensive tests show that the method can deal with most cases efficiently, including those that previous methods cannot solve.

On the Validation of Solids Represented with the International Standards for Geographic Information

AbstractThe international standards for geographic information provide unambiguous definitions of geometric primitives, with the aim of fostering exchange and interoperability in the geographical information system (GIS) community. In two dimensions, the standards are well accepted and there are algorithms (and implementations of these) to validate primitives, that is given a polygon, they ensure that it respects the standardized definition (and if it does not a reason is given to the user). However, although there exists an equivalent definition in three dimensions (for solids), it is ignored by most researchers and by software vendors. Several different definitions are indeed used, and none is compliant with the standards: for example solids are often defined as two‐manifold objects only, although in fact they can be nonmanifold objects. Exchanging and converting data sets from one format/platform to another is thus highly problematic. I present in this article a methodology to validate solids according to the international standards. It is hierarchical and permits us to validate the primitives of all dimensionalities. To understand and study the topological relationships between the different parts of a solid (the shells) the concept of Nef polyhedron is used. The methodology has been implemented in a prototype, and I report on the main engineering decisions that were made and on its use for the validation of real‐world three‐dimensional data sets.

CAD CONSTRUCTION METHOD OF 3D BUILDING MODELS FOR GIS ANALYSIS

Abstract. Buildings are often modelled as two-dimensional (2D) footprints which are extruded to simple cubes. Buildings are also represented as more complex objects with roofs, facades, etc. – in this case they are polyhedra, sometimes of a complex shape. These allow for visualisation and analysis of a wide area like a city, but micro-scale analysis of interiors is not possible. An example can be rescue operation simulation where information about the internal structure of a building and the external terrain is crucial to improve the response time. It demands a three-dimensional (3D) model where each room is represented as a separate element; there are also doors, windows, walls and other objects that have to be included. Even complex geometrical models can be easily constructed using Computer-Aided Design (CAD) systems. However, lack of semantic information and topological relations makes such models poor choices for GIS analysis. With the new dual half-edge (DHE) data structure and a set of Euler operators a 3D model can be built as in CAD systems, and represented as a cell complex. Construction of non-manifold objects is also possible. An advantage of the DHE is simplicity – only edges and nodes are used. Because of the 3D duality implemented in the structure volumes (cells) and faces are also present in the model. The geometry of a model is constructed explicitly by using Euler operators: connections between elements are created automatically, and semantic information is represented with attributes which can be assigned to any element of the model.

A Dimension-Independent Representation for Multiresolution Nonmanifold Meshes

Abstract We consider the problem of representing and manipulating nonmanifold objects of any dimension and at multiple resolutions. We present a modeling scheme based on (1) a multiresolution representation, called the vertex-based nonmanifold multitessellation, (2) a compact and dimension-independent data structure, called the Simplified Incidence Graph (SIG), and (3) an atomic mesh update operator, called vertex-pair contraction/vertex expansion. We propose efficient algorithms for performing the vertex-pair contraction on a simplicial mesh encoded as a SIG, and an effective representation for encoding this multiresolution model based on a compact encoding of vertex-pair contractions and vertex expansions.

Decomposing non-manifold objects in arbitrary dimensions

We address the problem of building valid representations of non-manifold d-dimensional objects through an approach based on decomposing a non-manifold d-dimensional object into an assembly of more regular components. We first define a standard decomposition of d-dimensional non-manifold objects described by abstract simplicial complexes. This decomposition splits a non-manifold object into components that belong to a well-understood class of objects, that we call initial quasi-manifold. Initial quasi-manifolds cannot be decomposed without cutting them along manifold faces. They form a decidable superset of d-manifolds for d⩾3, and coincide with manifolds for d⩽2. We then present an algorithm that computes the standard decomposition of a general non-manifold complex. This decomposition is unique, and removes all singularities which can be removed without cutting the complex along its manifold faces.

POINT-SETS AND CELL STRUCTURES RELEVANT TO COMPUTER AIDED DESIGN

Several fields of mathematics are relevant to computer aided design and other software systems involving solid object geometry, topology, differential and algebraic geometry being particularly important. This paper discusses some of this mathematics in order to provide a theoretical foundation for geometric modelling kernels that support non-manifold objects with an internal cellular structure and subsets of different dimensions. The paper shows relationships between relevant concepts from topology, differential geometry and computer aided geometric design that are not widely known in the CAD community. It also discusses semialgebraic, semianalytic and subanalytic sets as candidates for object representation. Stratifications of such sets are proposed for an object's cellular structure and new stratification concepts are introduced to support candidate applications.

On the Topology of Sheet Metal Parts

This paper analyzes the topological properties of sheet metal parts represented schematically (zero thickness, zero bend radii). Although such parts are usually non-manifold objects, the paper establishes a general topological invariant f = s + b + e + w − v − gnm + m regarding the number of facets, components, bends, free edges, welds, vertices holes and volumes, respectively. Corresponding Euler operators are derived, providing a basis for a modeling system for sheet metal parts. With this invariant, it is possible to reason about manufacturing processes, such as number of components and arrangement of bend lines and weld lines, using only a single qualitative model of the product. This capability is particularly useful in the preliminary stage of conceptual design. A corresponding topological invariant v − e + f = s + m − gnm is also proposed for general sheet models and thin walled objects.

Robust interval solid modelling Part I: representations

This paper introduces and develops an interval geometric representation for curved objects. The representation is based on rounded interval arithmetic and addresses robustness problems in current boundary representation solid modellers operating in floating point arithmetic. These problems include topology violation (such as gaps and inappropriate intersections), incidence asymmetry, and incidence intransitivity. The concepts of interval polynomial spline curves and surface patches are employed in this context. A graph-based data structure incorporating the special features of interval geometries is developed and extended to represent non-manifold objects.

An algorithm reducing 3D Boolean operations to a 2D problem: concepts and results

This paper describes a new method for Boolean operations on solids with boundary representation. Its interest is to reduce a basically 3D problem to a 2D one. After a brief description of existing algorithms we present the idea for computing Boolean operations on faces. An important part of the paper is the formalization of the reconstruction which permits the algorithm to be robust. Presently, the method only processes objects with planar surfaces but will be extended to non-manifold objects and non-planar surfaces.

Identification of faces in a 2D line drawing projection of a wireframe object

An important key to reconstructing a three-dimensional object depicted by a two-dimensional line drawing projection is face identification. Identification of edge circuits in a 2D projection corresponding to actual faces of a 3D object becomes complex when the projected object is in wireframe representation. This representation is commonly encountered in drawings made during the conceptual design stage of mechanical parts. When nonmanifold objects are considered, the situation becomes even more complex. This paper discusses the principles underlying face identification and presents an algorithm capable of performing this identification. Face-edge-vertex relationships applicable to nonmanifold objects are also proposed. Examples from a working implementation are given.

Performance evaluation of boundary data structures

Boundary representations, or B-reps, are a solid modeling technique with widespread applications in computer-aided design, computer-aided manufacturing, and robotics. B-reps have evolved considerably in recent years, and advances in theoretic studies-particularly in topological models and associated operators-make it possible now to model nonorientable, nonmanifold objects, as well as orientable, manifold objects. Boundary data structures are normally built on the adjacency relations between faces, edges, and vertices. The analytic method presented here identifies optimal combinations of relations for building these structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

DOMAIN DELAUNAY TETRAHEDRIZATION OF SOLID MODELS

An algorithm is presented for constructing a topologically and geometrically valid Domain Delaunay Tetrahedrization (DDT) of an arbitrarily shaped solid model with quadric curved faces (including objects with holes and nonmanifold objects). The algorithm operates on the boundary representation (B-rep) of the solid, and makes extensive use of properties of the Delaunay triangulation. This algorithm also includes a mechanism for transferring neighborhood information from the solid model to the elements of the tetrahedral model. Neighborhood information is used for identifying tetrahedra to be included in the DDT, and — in combination with geometric criteria — for ensuring that the DDT approximates satisfactorily the curved faces of the solid.

Solid representation and operation using extended octrees

Solid modelers must be based on reliable and fast algorithms for Boolean operations. The octree model, as well as several generalizations (polytrees, integrated polytrees, extended octrees), is specially well suited for these algorithms and can be used either as a primary or as a secondary model in solid modeling systems. This paper is concerned with a precise definition of the extended octree model that allows the representation of nonmanifold objects with planar faces and, consequently, is closed under Boolean operations on polyhedrons. Boolean nodes and nearly vertex nodes are introduced, and the model is discussed in comparison with related representations. A fast algorithm for the direct generation of the extended octree from the geometry of the base polygon in extrusion solids is presented, and its complexity is studied. Boolean operation algorithms are introduced.

An Algorithm for Geometric Set Operations Using Cellular Subdivision Techniques

Geometric set operations play an integral role in systems for CAD/CAM, for robot planning, and for modeling objects such as underground formations from empirical data. Two major issues in the implementation of geometric set operations are efficiency in the search for geometric intersections and effectiveness in the treatment of singular intersection cases. This article presents an algorithm for geometric set operations on planar polyhedral nonmanifold objects that addresses both these issues. First, an efficient search for geometric intersections is obtained by localizing the search to small regions of object space through a cellular subdivision scheme using the polytree data structure. Second, an effective treatment of singular intersection cases is obtained by mapping each singular intersection occurring in a region into one of a small set of cases.