A 3D Topology Model for Multiple Topological Representations based on Combinatorial Maps

Mingguang WU1*, Bai Guimei2, Yongning WEN1, Jingwei SHEN1, TaiSheng CHEN1

1 Key Laboratory of Virtual Geographic Environment (Ministry of Education), Nanjing Normal University, Nanjing, Jiangsu Province, 210046, P.R.China

2 Dept. of Computer Science and Information Engineering Luoyang Institute of Science and Technology, Luoyang, China，4710232

Corresponding Author: blumanw@163.com

Abstract—The formal representation of topological relations plays a very important role in process of spatial query, spatial analysis and spatial reasoning. Most current models support a single representation of geometry and topology at a fixed resolution. While it often happens that different applications take into consideration spatial data/topological relationship, with the same 3D scene date, under different viewpoints, in different geometric forms and/or with different resolutions. In this case, a topology model for multiple topological representations is needed. In this paper, we propose a new approach based on combinatorial maps, which focus on both multiple topological representations under the situation of LOD and Feature semantic/spatial aggregation and the consistency of topology.

Keywords-Topology; Data model; Multiple topological representation; 3D Combinatorial Maps

I. INTRODUCTION Most current models support a single representation of

geometry and topology at a fixed resolution [1]. While it often happens that different applications take into consideration spatial data /topological relationship, with the same 3D scene date, under different viewpoints, in different geometric forms and/or with different resolutions [2]. In this case, a topology model for multiple topological representations is needed. But, it is still not straightforward for three main reasons: (i) the memory/CPU resources needed for processing the massive 3D dataset, (ii) the high number of geometric primitives as well as the large variety of topological information (incidence, adjacency, etc.) necessary to describe, and (iii) the lack of methods for automated complex 3D geometry/topology generalization.

For these reasons, existing 3D spatial data models provide no support for multiple topological representations and the control of consistency constraints on multiple topological representations.

In this paper, we propose a new multiple topological representations model, supporting multiple representations and topology management during vector data multiple representations, which relies on these five topological operators. In this new model we investigate the issues of how to representation topology at different levels of detail, and how to control consistency.

The rest of the paper is organized as follows: In Section Ⅱ, we review some related work. In Section Ⅲ, we describe the integrated data structure, in terms of the types of 3D spatial entities required. Section Ⅳ presents experimental results of studies aimed at integration of aboveground, ground and underground. Finally, in Section Ⅴ, we draw some conclusions and discuss future work.

II. RELATED WORK The problem of supporting multiple topological

representations of the same object/complex, possibly at different resolutions, has received some attention in the GIS and image segmentation communities.

In the field of remote sensing image analyzing, pyramidal models have been used for long to handle several subdivisions or different scales of a given remote sensing image [3]. Bertolotto [4] showed that seven minimal atomic generalization operators maintain topological consistency in a pure topological model. Based on the Point Sets theory, Bruegger and Werner proposed [1] a hierarchical relations approach to multiple topological representations, and introduced largest homogeneous cells (LHC's) to guarantee the hierarchical structure of these links. In [5] a conceptual data model with integrated approach and inter-relationship approach was proposed which providing full support for multiple representations of the same real world data. A joint topology-geometry model was presented for dealing with multiple representations and topology management to support map generalization [6]. Trott and Greasley [7] proposed a level of topology to supporting three-dimensional representations of natural and man-made environments that include either partial or full 3D topology. Lévy [8] introduced hierarchical G-Maps for the representation of multi-scale topology.

Several different issues have been investigated with the aim of providing partial solutions to this problem (e.g. [9], [10], [11] and [12]).

These models on multi-topological representations are often limited to a conceptual description, without (besides some small experiments) addressing the problems concerning the embedding of the 3D geometry primitives. In addition, they

This research is supported by the National Science Foundation of China under Grant No. 40901186, and China Postdoctoral Science Foundation funded project No. 200904501109.

provide no consistency constraints on multiple 3D representations.

III. DESIGN OF THE TOPOLOGY MODEL In this section a topology structure for modeling different

levels of detail is described. It is composed of three basic components: topology graph based on 3D Combinatorial Maps, topological operators and Euler characteristics. topology graph is used to describe the relationships of the cells under the dynamic, view point related and LOD situations. Topological operators are used to build the topology graph. The Euler characteristic is used to control the consistency of topology.

A. 3D Multiple Topological Representations Based on Combinatorial Maps Based on the theory of cellular complex, a 3D object is an

abstract cellular complex consisting of dimensions 0, 1, 2, 3 ... (i.e. 0-cells are vertices, 1-cells are edges, 2-cells are faces, 3-cells are volumes, ...) and the boundary and coboundary relationships. The structure, or the topology, of the object is related to the decomposition of the object into sub-objects, and to the relations between these sub-objects [13].

Most current models support a single representation of cell relationships at a fixed resolution. While it often happens that different applications take into consideration spatial data/topological relationship, with the same 3D scene date, under different viewpoints, in different geometric forms and/or with different resolutions. In this case, a topology model for multiple topological representations is needed.

Combinatorial maps define a general framework which allows us to encode any subdivision of nD topological spaces orientable or non-orientable with or without boundaries [15]. A 3D combinatorial map is formally defined by G =(D;a1; a2; a3). The permutation a1 connects darts belonging to the same face and the same volume, preserving their ordering on the boundary of the face, and the involution a2 connects 2 darts, part of the same edge and the same volume. The involution a3 , links 2 darts that belong to the same face and same edge (and the 2 volumes separated by the respective face) [15]. More detail about 3D combinatorial maps could be fond in [15]. The topological relations in the models are represented by a geometric representation of the cells and their neighborhoods, which are in turn defined in terms of their boundary and co-boundary cells based upon boundary representation. ai could be regarded as a sewing tool, which brings together neighboring elements(Node, Edge, Face and Volume).

Considering the multiple topological representations requirements, we present a sub graph and hierarchical graph approach to catch the focused topological relations.

Cell relationships for different theme or spatial area can be represented by sub Graph. Spatial object may be grouped according to attributes or location. The layer approach (feature grouped by attributes) and coverage approach (feature grouped by location) have increasingly gained acceptance as two alternative approaches for collecting and managing geographical data in GIS. In this topological model, cell also could by grouped according to attributes or location.

Cell relationships at different resolution levels can be represented by hierarchical Graph. The first level corresponds to the highest resolution, and other levels are obtained by cell generalizations. Each level is an adjacency graph, which describes adjacent relations of cells at a certain granularity level.

By implicitly encoding the adjacency relationships of cell ， sub graph and hierarchical graph can catch the topological information at different area, levels and for different themes.

B. Topological Operators Hierarchical graph and sub graph are two key stones to

construct multiple topological representations. In order to maintain multiple topological representations that may differ in resolution, geometry form, or view point, we apply five operations to a k-cell(0≤k≤3): delete, merge, split, group and link(show in Fig.1 ).

We represent the topology of vector maps by following the

ISO 19107 Spatial schema [16] topological data structure. The primitive topological elements are nodes, edges, faces and volumes. A node is a zero dimensional object. An edge is bounded by two nodes: a start node and an end node. A face is an atomic two-dimensional element. Its associated geometric object is a simply connected polygon. A volume is a three dimensional object which is bounded by a list of faces.

We are interested here in representing multi-level topological relationships. So, we also propose the mechanisms for building Hierarchical graph and sub graph. In this model, sub graph and hierarchical graph are built in a particular way: they use only five topological operators. The following gives their definitions:

• Split: this decomposes a k-cell into two k-cells. i.e. split an edge into two edges 1-cells by inserting a node, split a face into two faces by inserting an edge, ....

• Merge: this composes two k-cells into one k-cell. i.e. composes two edges into one edges, composes two faces into one face,…

• Delete: this removes a k-cell from the complex.

• Link: this links the cells at different topological levels.

Figure 1. Five topological operators and its relationships with multiple topological representations

Multiple topological Representations

Hierarchical Graph

Sub Graph create split merge

delete link

group

• Group: this brings together several cells into a collection.

C. Modeling Topological Relations Using Euler Characteristics Another problem arising from topological operators is how

the control the consistency of topological during the processing of topological operators. In this model, Euler characteristics based topological modeling is proposed as one solution for dealing with this problem.

Based on the 4-intersection model, DENG Min [17] presents a new model to modeling topological relations using Euler characteristics, which can distinguish more details in topological aspect. In this model, Topology-preserving constraints based on Euler characteristics are introduced for these five operators to prevent topological inconsistencies from arising multiple representations. Euler characteristics can be used for the measure for the separation is the topological invariant of a complex [18].

where Ni denotes the number of cells of dimension i. Egenhofer [18] proposed a model of binary topological relationships, representing the topological relationship between two spatial objects, and , as the intersections of the boundary (denoted by ∂ )，interior( denoted by o) and exterior( denoted by -)of with the boundary, interior and exterior of . Boundary, interior and exterior can be combined to form the nine fundamental criteria of the topological relationships between two spatial objects (A, B), called the 9-intersection. Based on the 9-intersection model (Fig. 2(2)), topology relations can by calculated as show in Fig. 2(3).

If any topological violations occur during topology generalization based on the five topological operators, the operation is halted. By using this mechanism, the topology relationships can be made consistency.

IV. EXPERIMENT At the present stage, we concentrate on the meet

relationships of vector dataset. In the prototype, geometry primitives are defined. These primitives include:

• Point: A point is defined by its unique spatial location (longitude, latitude).

• lineString: A lineString is an ordered sequence of vertices that are connected by straight-line segments.

Firstly, we implement a 3D global viewer based on the OpenGL library and C++ program language for the browsing of vector datasets (boundary of research area). In this global viewer, users can browser the dataset at any altitudes and any view angles (show in Fig. 3(a)). Then, in order to test usability of the topological operators, a lineString simplification Algorithm is developed. In the context of simplification, the Square Root Model of Topfer [19] is used as a vertices selection criterion according to the distance between the lineString and the center of the view port.

Following the definition of Hierarchical graph and topological operators described in section Ⅲ, we define a graph stack by a list of Hierarchical graph. For our experiments we used the Merge and split operator to define a Hierarchical graph for the topology graph of the vector dataset, used the link operation to define the consistency on multiple sub graphs, used the delete and the merge operations to define a hierarchical graph. If any topological violations occur during lineString simplification, the operation is halted and an exception is returned. By using the merge operator base on Euler characteristics, the meet relationships can be make consistency show in Fig. 3(b).

V. CONCLUSION In this paper we have discussed multiple 3D topological

representations and the consistence of topology. A general

Figure 2. Modeling topological relations using Euler characteristics

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approach to the handling of level of topology based on Combinatorial Maps has been proposed. Our contributions concern the following points:

•Hierarchical graph and sub graph are introduced to catch the topological relationships at a dynamic, viewpoint-related, and LOD scenario;

• Five topological operators applied to cell based on Euler characteristics are introduced to creating Hierarchical graph and sub graph.

Future work will involve extending our model to support consistent topological generalization of the complex 3D model. The integration with 2D/3D geometric generalization algorithms is also one of our future perspectives.

ACKNOWLEDGEMENT The authors would like to thank ShanShan Tan and Qi Luo

of Nanjing Normal University for the suggestion of English translation and paper organization.

REFERENCES [1] Bruegger and W. Kuhn, “Multiple Topological Representations,”

NCGIA Technical Report 91-17, Department of surveying Engineering University of Maine, 1993.

[2] M. Breunig, A. Thomsen, B. Broscheit, E. Butwilowski, and U. Sander. (2007), “Representation and analysis of topology in multi-representation databases”. In: PIA 2007, Photogrammetric Image Analysis, pp.167-172, Munich, 2007

[3] G. Damiand and P. Resch. “Topological Map Based Algorithms for 3D Image Segmentation,” DGCI 2002, pp.220-231.

[4] M. Bertolotto. “Geometric Modelling of Spatial Entities at Multiple Levels of Resolution,” Ph.D. Thesis, Department of Computer and Information Sciences, University of Genova, Italy,1998

[5] S. Spaccaietra, C. Parent, and C Vangenot. “GIS databases: from multi-scale to multi-representation,”In Proceedings of the International Workshop on Emerging Technologies for Geo-based Applications,LNAll864 ， Ascona Switzerland ， EPFL-DI-LBD ，Springer,May,2000.

[6] H. Qiang , M. Bertolotto, and J. Weakliam, “A Conceptual Model for Supporting Multiple Representations and Topology Management,” ER (Workshops) 2004, pp.5-16.

[7] K. Trott and I. Greasley, “A 3D spatial data model for terrain reasoning,” In: 4th International Conference on GeoComputation, Mary Washington College in Fredericksburg, VA-USA,1999.

[8] B. Lévy, “Topologie Algorithmique - Combinatoire etPlongement,” PhD Thesis, INPL Nancy, 1999: 202p.

[9] E. Danovaro, L. D. Floriani, E. Puppo, and H. Samet, “Out-of-core Multi-resolution Modeling,” In: A. Belussi, B. Catania, E. Clementini, E. Ferrari, editors. Spatial Data on the Web - Modeling and Management, Springer Germany, pp.51-72, 2007.

[10] J. Stell and M. Worboys, “Stratified Map spaces: a formal basis for multi-resolution spatial database,” In T K Poiker and N Chrisman, editors,SDH’98 Proceedings 8th International Symposium on Spatial Data Handing, pp.180-189, 1998.

[11] C. Vangenot, “Multi-representation in spatial databases using the MADS conceptual model,” ICA Workshop on Generalization and multiple representation -20-21 august 2004.

[12] D. Gubiani and A. Montanari, “A conceptual spatial model supporting topologically-consistent multiple representations,” ACM GIS ’08. Irvine, CA, US, pp.325-329, 2008.

[13] S. Pigot, “A topological model for 3D spatial information system,” in Proceedings of the 5th Int. Symp. on Spatial Data Handling, Charleston, USA, 1(1), pp.344-360, 1992.

[14] C. Ellul and M. Haklay, “Requirements for Topology in 3D GIS,” Transactions in GIS, Vol. 10(2), 157-175, 2006.

[15] T. Illetschko, A. Ion, Y. Haxhimusa, and W. G. Kropatsch, “Collapsing 3D Combinatorial Maps.” In F. Lenzen, O. Scherzer and Vincze, M., editors, Proceedings of 30th Austrian Association for Pattern Recognition Workshop, Oesterreichische Computer Gesellschaft, pp.85-93, 2006.

[16] The OpenGIS Abstract Specification, Topic 1: Feature Geometry (ISO 19107 Spatial Schema), Version 5, 2001

[17] M. Deng, X. S. Zhang, and Z. J. Lin. “Modeling Topological Relations Based on Euler Characteristics,” Geomatics and Information Science of Wuhan University, Vol. 29(10), pp.872-876, 2004.

[18] M. Egenhofer, A Model for Detailed Binary Topological Relationships. Geomatica , Vol. 47(3/ 4) , 261-273, 1993.

[19] F. Töpfer, and W. Pillewizer. “The Principles of Selection,” The Cartographic Journal, Vol. 3, No. 1, pp.10-16, 1966.