Orientation, sphericity and roundness evaluation of particles using alternative 3D representations

Sphericity and roundness indices have been used mainly in geology to analyze the shape of particles. In this paper, geometric methods are proposed as an alternative to evaluate the orientation, sphericity and roundness indices of 3D objects. In contrast to previous works based on digital images, which use the voxel model, we represent the particles with the Extreme Vertices Model, a very concise representation for binary volumes. We define the orientation with three mutually orthogonal unit vectors. Then, some sphericity indices based on length measurement of the three representative axes of the particle can be computed. In addition, we propose a ray-casting-like approach to evaluate a 3D roundness index. This method provides roundness measurements that are highly correlated with those provided by the Krumbein's chart and other previous approach. Finally, as an example we apply the presented methods to analyze the sphericity and roundness of a real silica nano dataset.Postprint (published version

3D SPATIAL OPERATIONS FOR GEO-DBMS: GEOMETRY VS. TOPOLOGY

Geo-DBMS becomes very important medium for GIS as it can handle and manage (e.g. retrieve and update) large volume of spatial data. Providing 3D spatial database with appropriate operation tools such as 3D spatial operations would be very useful for next generation of GIS software (i.e. 3D GIS) since the software would highly depend on the Geo-DBMS in both modeling and analysis. One of the desired components in such future software or system is geometric modeling capability that works with 3D spatial operations. The literature reveals 3D spatial database would be greatly enhanced if analytical operations on the spatial data could be manipulated in real 3D domain. Fundamentally, it can be considered that the aspect of 3D spatial operations within GIS software are still not much been addressed and solved as expected (i.e. up to the level where an operational 3D system could be realized). The main problem from this aspect is the unavailability of 3D spatial data type within geo-DBMS environment. It is the aim of this paper to describe 3D spatial operations for geometrical and topological data types within geo-DBMS environment. In the experiment, we utilize an existing geo-DBMS, PostgreSQL, later known as PostGIS, which complied with the standard specifications from Open Geospatial Consortium (OGC), e.g. abstract and geometry specification. The second factor why we utilise the PostGIS is because its an open source based technology and suitable for academic and research purposes. In this paper, we discuss a suitable way of developing a new 3D data type, polyhedron, for both geometrical and topological data types and spatial operations using C language. 1

3D spatial operations – topology vs. geometry

Geo-DBMS becomes very important medium for GIS as it can handle and manage (e.g. retrieve and update) large volume of spatial data. Providing 3D spatial database with appropriate operation tools such as 3D spatial operations would be very useful for next generation of GIS software (i.e. 3D GIS) since the software would highly depend on the Geo-DBMS in both modeling and analysis. One of the desired components in such future software or system is geometric modeling capability that works with 3D spatial operations. The literature reveals 3D spatial database would be greatly enhanced if analytical operations on the spatial data could be manipulated in real 3D domain. Fundamentally, it can be considered that the aspect of 3D spatial operations within GIS software are still not much been addressed and solved as expected (i.e. up to the level where an operational 3D system could be realized). The main problem from this aspect is the unavailability of 3D spatial data type within geo-DBMS environment. It is the aim of this paper to describe 3D spatial operations for geometrical and topological data types within geo-DBMS environment. In the experiment, we utilize an existing geo-DBMS, PostgreSQL, later known as PostGIS, which complied with the standard specifications from Open Geospatial Consortium (OGC), e.g. abstract and geometry specification. The second factor why we utilise the PostGIS is because its an open source based technology and suitable for academic and research purposes. In this paper, we discuss a suitable way of developing a new 3D data type, polyhedron, for both geometrical and topological data types and spatial operations using C language

Sphericity and roundness computation for particles using the extreme vertices model

Shape is a property studied for many kinds of particles. Among shape parameters, sphericity and roundness indices had been largely studied to understand several processes. Some of these indices are based on length measurements of the particle obtained from its oriented bounding box (OBB). In this paper we follow a discrete approach based on Extreme Vertices Model and devise new methods to compute the OBB and the mentioned indices. We apply these methods to synthetic sedimentary rocks and to a real dataset of silicon nanocrystals (Si NC) to analyze the obtained results and compare them with those obtained with a classical voxel model.Peer ReviewedPostprint (author's final draft

Compact union of disjoint boxes: An efficient decomposition model for binary volumes

This paper presents in detail the CompactUnion of Disjoint Boxes (CUDB), a decomposition modelfor binary volumes that has been recently but brieflyintroduced. This model is an improved version of aprevious model called Ordered Union of Disjoint Boxes(OUDB). We show here, several desirable features thatthis model has versus OUDB, such as less unitary basicelements (boxes) and thus, a better efficiency in someneighborhood operations. We present algorithms forconversion to and from other models, and for basiccomputations as area (2D) or volume (3D). We alsopresent an efficient algorithm for connected-componentlabeling (CCL) that does not follow the classical two-passstrategy. Finally we present an algorithm for collision (oradjacency) detection in static environments. We test theefficiency of CUDB versus existing models with severaldatasets.Peer ReviewedPostprint (published version

Orthogonal polyhedra as geometric bounds in constructive solid geometry

Set membership classification and, specifically, the evaluation of a CSG tree are problems of a certain complexity. Several techniques to speed up these processes have been proposed such as Active Zones, Geometric Bounds and the Extended Convex Differences Tree. Boxes are the most common geometric bounds studied but other bounds such as spheres, convex hulls and prisms have also been proposed. In this work we propose orthogonal polyhedra as geometric bounds in the CSG model. CSG primitives are approximated by orthogonal polyhedra and the orthogonal bound of the object is obtained by applying the corresponding boolean algebra. A specific model for orthogonal polyhedra is presented that allows a simple and robust boolean operations algorithm between orthogonal polyhedra. This algorithm has linear complexity (is based on a merging process) and avoids floating-point computation.Postprint (published version

Orthogonal polyhedra as geometric bounds in constructive solid geometry

Set membership classification and, specifically, the evaluation of a CSG tree are problems of a certain complexity. Several techniques to speed up these processes have been proposed such as Active Zones, Geometric Bounds and the Extended Convex Differences Tree. Boxes are the most common geometric bounds studied but other bounds such as spheres, convex hulls and prisms have also been proposed. In this work we propose orthogonal polyhedra as geometric bounds in the CSG model. CSG primitives are approximated by orthogonal polyhedra and the orthogonal bound of the object is obtained by applying the corresponding boolean algebra. A specific model for orthogonal polyhedra is presented that allows a simple and robust boolean operations algorithm between orthogonal polyhedra. This algorithm has linear complexity (is based on a merging process) and avoids floating-point computation

The three-dimensional art gallery problem and its solutions

This thesis addressed the three-dimensional Art Gallery Problem (3D-AGP), a version of the art gallery problem, which aims to determine the number of guards required to cover the interior of a pseudo-polyhedron as well as the placement of these guards. This study exclusively focused on the version of the 3D-AGP in which the art gallery is modelled by an orthogonal pseudo-polyhedron, instead of a pseudo-polyhedron. An orthogonal pseudopolyhedron provides a simple yet effective model for an art gallery because of the fact that most real-life buildings and art galleries are largely orthogonal in shape. Thus far, the existing solutions to the 3D-AGP employ mobile guards, in which each mobile guard is allowed to roam over an entire interior face or edge of a simple orthogonal polyhedron. In many realword applications including the monitoring an art gallery, mobile guards are not always adequate. For instance, surveillance cameras are usually installed at fixed locations. The guard placement method proposed in this thesis addresses such limitations. It uses fixedpoint guards inside an orthogonal pseudo-polyhedron. This formulation of the art gallery problem is closer to that of the classical art gallery problem. The use of fixed-point guards also makes our method applicable to wider application areas. Furthermore, unlike the existing solutions which are only applicable to simple orthogonal polyhedra, our solution applies to orthogonal pseudo-polyhedra, which is a super-class of simple orthogonal polyhedron. In this thesis, a general solution to the guard placement problem for 3D-AGP on any orthogonal pseudo-polyhedron has been presented. This method is the first solution known so far to fixed-point guard placement for orthogonal pseudo-polyhedron. Furthermore, it has been shown that the upper bound for the number of fixed-point guards required for covering any orthogonal polyhedron having n vertices is (n3/2), which is the lowest upper bound known so far for the number of fixed-point guards for any orthogonal polyhedron. This thesis also provides a new way to characterise the type of a vertex in any orthogonal pseudo-polyhedron and has conjectured a quantitative relationship between the numbers of vertices with different vertex configurations in any orthogonal pseudo-polyhedron. This conjecture, if proved to be true, will be useful for gaining insight into the structure of any orthogonal pseudo-polyhedron involved in many 3-dimensional computational geometrical problems. Finally the thesis has also described a new method for splitting orthogonal polygon iv using a polyline and a new method for splitting an orthogonal polyhedron using a polyplane. These algorithms are useful in applications such as metal fabrication

Reconstruction of Orthogonal Polyhedra

In this thesis I study reconstruction of orthogonal polyhedral surfaces and orthogonal polyhedra from partial information about their boundaries. There are three main questions for which I provide novel results. The first question is "Given the dual graph, facial angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the dihedral angles?" The second question is "Given the dual graph, dihedral angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the facial angles?" The third question is "Given the vertex coordinates of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the edges and faces, possibly after rotating?" For the first two questions, I show that the answer is "yes" for genus-0 orthogonal polyhedra and polyhedral surfaces under some restrictions, and provide linear time algorithms. For the third question, I provide results and algorithms for orthogonally convex polyhedra. Many related problems are studied as well