# Modeling a 3D City Model and Its Levels of Detail as a True 4D Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Level of Detail in 3D City Modeling

#### 2.2. Identifying and Linking Corresponding Objects

#### 2.3. LOD as a Dimension

#### 2.4. Storing a 4D Model

#### 2.5. Construction Algorithms for a 4D Model

#### 2.6. Slicing to Extract a 3D Model at a Certain LOD

## 3. Methodology to Model LOD as an Extra Geometric Dimension in a True 4D Model

## 4. Constructing a 4D Model from a 3D City Model and Its LODs

- Identifying corresponding 0D–3D cells;
- Linking them by creating 1D–4D cells connecting them;
- Using an incremental construction algorithm [50] to build a 4D cell complex using all 0D–4D cells.

#### 4.1. Step 1: Identifying Corresponding Cells in 3D Models

#### 4.2. Step 2: Linking Corresponding Cells

#### 4.2.1. Method 1: Simple Linking of Corresponding Cells

**Figure 1.**The four linking schemes for three levels of detail (LODs) of a house, here depicted in 2D: (

**a**) simple linking. (

**b**) unmatched are collapsed. (

**c**) modification of topology and (

**d**) matching all to existing. The objects that would be obtained by slicing between the LODs can be seen in dashed green contours; the red dashed lines reflect the cells that need to be added and split in order to ensure a valid 3D (2D + LOD) cell complex.

#### 4.2.2. Method 2: Unmatched Cells Are Collapsed into Existing Ones

#### 4.2.3. Method 3: Modifying the Topology

#### 4.2.4. Method 4: Matching All Cells to Existing Ones

## 5. Use Cases

#### 5.1. Using Method 1: Simple Linking

#### 5.2. Using Method 2: Collapsing

**Figure 3.**Two LODs of a house with differing geometry and topology are integrated into a 4D model by collapsing cells in the model at the highest LOD.

#### 5.3. Using Method 3: Modifying the Topology

**Figure 4.**Two LODs of two houses being aggregated are integrated into a 4D model by modifying the topology of the model at the lowest LOD so as to match the topology of the model at the highest LOD.

#### 5.4. Combination of Methods 2 and 3

**Figure 5.**Three LODs of a 3D model of a house are integrated into a 4D model by modifying the topology of the model at the lowest LOD and collapsing a part of the model in the highest LOD.

#### 5.5. Using Method 4: Matching to Existing Cells

**Figure 6.**Two LODs of a 3D model of a house (left and right) are linked despite not being isomorphic, with an intermediate LOD that shows the result of slicing the construction at an intermediate LOD (center).

## 6. A Concrete Example: Implementing Cell Matching to Construct a 4D Model

**Figure 7.**Code excerpts that show how (

**a**) a 4D model of the house shown in Figure 6 is constructed. (

**b**) faces are created as cycles of vertices, volumes as sets of faces and four-cells as sets of three-cells. (

**c**) vertices are created based on 4D points. The colors referred to in (b) correspond to the highlighted faces and volumes in (a).

## 7. Discussion and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Biljecki, F.; Ledoux, H.; Stoter, J.; Zhao, J. Formalisation of the level of detail in 3D city modeling. Comput. Environ. Urban Syst.
**2014**, 48, 1–15. [Google Scholar] [CrossRef] - Zhao, J.; Zhu, Q.; Du, Z.; Feng, T.; Zhang, Y. Mathematical morphology-based generalization of complex 3D building models incorporating semantic relationships. ISPRS J. Photogramm. Remote Sens.
**2012**, 68, 95–111. [Google Scholar] [CrossRef] - Zhu, Q.; Zhao, J.; Liu, X.; Zhang, Y. Perceptually guided geometrical primitive location method for 3D complex building simplification. In Proceedings of GeoWeb 2009 Academic Track – Cityscapes, Vancouver, Canada, 27–31 July 2009; pp. 74–79.
- Luebke, D.; Reddy, M.; Cohen, J.D.; Varshney, A.; Watson, B.; Huebner, R. Level of Detail for 3D Graphics; Morgan Kaufmann: San Francisco, CA, USA, 2003. [Google Scholar]
- OGC. OGC City Geography Markup Language (CityGML) Encoding Standard; Open Geospatial Consortium Inc.: Wayland, MA, USA, 2012. [Google Scholar]
- Gröger, G.; Plümer, L. CityGML—Interoperable semantic 3D city models. ISPRS J. Photogramm. Remote Sens.
**2012**, 71, 12–33. [Google Scholar] [CrossRef] - Biljecki, F.; Ledoux, H.; Stoter, J. Improving the consistency of multi-LOD CityGML datasets by removing redundancy. In 3D Geoinformation Science Lecture Notes in Geoinformation and Cartography; Breunig, M., Mulhim, A.D., Butwilowski, E., Kuper, P.V., Benner, J., Häfele, K.H., Eds.; Springer International Publishing: Dubai, UAE, 2015; pp. 1–17. [Google Scholar]
- Zhang, X.; Ai, T.; Stoter, J.; Zhao, X. Data matching of building polygons at multiple map scales improved by contextual information and relaxation. ISPRS J. Photogramm. Remote Sens.
**2014**, 92, 147–163. [Google Scholar] [CrossRef] - van Oosterom, P.; Stoter, J. 5D data modeling: Full integration of 2D/3D space, time and scale dimensions. In Geographic Information Science: 6th International Conference, GIScience 2010, Zurich, Switzerland, September 14-17, 2010. Proceedings; Springer: Berlin/Heidelberg, Germany, 2010; pp. 311–324. [Google Scholar]
- Arroyo Ohori, K.; Ledoux, H.; Stoter, J. An evaluation and classification of nD topological data structures for the representation of objects in a higher-dimensional GIS. Int. J. Geogr. Inf. Sci.
**2015**, 29, 825–849. [Google Scholar] [CrossRef] - Arroyo Ohori, K.; Biljecki, F.; Stoter, J.; Ledoux, H. Manipulating higher dimensional spatial information. In Proceedings of the 16th AGILE International Conference on Geographic Information Science, Leuven, Belgium, 14–17 May 2013.
- Stadler, A.; Kolbe, T.H. Spatio-semantic coherence in the integration of 3D city models. In Proceedings of the WG II/7 5th International Symposium Spatial Data Quality ISSDQ 2007, Enschede, The Netherlands, 13–15 June 2007.
- Meng, L.; Forberg, A. 3D building generalization. In Challenges in the Portrayal of Geographic Information: Issues of Generalisation and Multi Scale Representation; Mackaness, W., Ruas, A., Sarjakoski, T., Eds.; Elsevier Science: Amsterdam, The Netherlands, 2007; pp. 211–232. [Google Scholar]
- Fan, H.; Meng, L. A three-step approach of simplifying 3D buildings modeled by CityGML. Int. J. Geogr. Inf. Sci.
**2012**, 26, 1091–1107. [Google Scholar] [CrossRef] - Döllner, J.; Buchholz, H. Continuous level-of-detail modeling of buildings in 3D city models. In Processings of GIS' 05 Proceedings of the 13th Annual ACM International Workshop on Geographic Information Systems, Bremen, Germany, 31 October–05 November 2005; pp. 173–181.
- Clark, J.H. Hierarchical geometric models for visible surface algorithms. Commun. ACM
**1976**, 19, 547–554. [Google Scholar] [CrossRef] - Coors, V.; Flick, S. Integrating levels of detail in a web-based 3D-GIS. In Proceedings of the 6th ACM International Symposium on Advances in Geographic Information Systems, Washington, DC, USA, 6–7 November 1998; pp. 40–45.
- Guercke, R.; Götzelmann, T.; Brenner, C.; Sester, M. Aggregation of LoD 1 building models as an optimization problem. ISPRS J. Photogramm. Remote Sens.
**2011**, 66, 209–222. [Google Scholar] [CrossRef] - Zhao, J.; Qingg, Z.; Du, Z.; Feng, T.; Zhang, Y. Mathematical morphology-based generalization of complex 3D building models incorporating semantic relationships. ISPRS J. Photogramm. Remote Sens.
**2012**, 68, 95–111. [Google Scholar] [CrossRef] - Semmo, A.; Trapp, M.; Kyprianidis, J.E.; Döllner, J. Interactive visualization of generalized virtual 3D city models using level-of-abstraction transitions. Comput. Graph. Forum
**2012**, 31, 885–894. [Google Scholar] [CrossRef] - Glander, T.; Döllner, J. Abstract representations for interactive visualization of virtual 3D city models. Comput. Environ. Urban Syst.
**2009**, 33, 375–387. [Google Scholar] [CrossRef] - Filho, W.C.; de Figueiredo, L.H.; Gattass, M.; Carvalho, P.C. A Topological Data Structure for Hierarchical Planar Subdivisions; Technical Report CS-95-53; Department of Computer Science, University of Waterloo: Waterloo, Canada, 1995. [Google Scholar]
- Rigaux, P.; Scholl, M. Multi-scale partitions: Application to spatial and statistical databases. In Advances in Spatial Databases; Egenhofer, M.J., Herring, J.R., Eds.; Springer: Berlin/Heidelberg, Germany, 1995; Vol. 951, pp. 170–183. [Google Scholar]
- Plümer, L.; Gröger, G. Achieving integrity in geographic information systems—Maps and nested maps. GeoInformatica
**1997**, 1, 345–367. [Google Scholar] [CrossRef] - Van Oosterom, P. Variable-scale topological data structures suitable for progressive data transfer: The GAP-face tree and GAP-edge forest. Cartogr. Geogr. Inf. Sci.
**2005**, 32, 331–346. [Google Scholar] [CrossRef] - Weibel, R. Generalization of spatial data: Principles and selected algorithms. In Algorithmic Foundations of Geographic Information Systems; Van Kreveld, M., Nievergelt, J., Roos, T., Widmayer, P., Eds.; Springer: Berlin/Heidelberg, Germany, 1997; Vol. 1340, pp. 99–152. [Google Scholar]
- Hampe, M.; Anders, K.H.; Sester, M. MRDB applications for data revision and real-time generalization. In Proceedings of the 21 International Cartographic Conference, Durban, South Africa, 10–16 August 2003; pp. 192–202.
- Veltkamp, R.C.; Hagedoorn, M. State of the art in shape matching. In Principles of Visual Information Retrieval; Springer London: London, UK, 2001; pp. 87–119. [Google Scholar]
- Devogele, T.; Trevisan, J.; Raynal, L. Building a multi-scale database with scale-transition relationships. In Proceedings of the 7th International Symposium on Spatial Data Handling, Delft, The Netherlands, 12–16 August 1996; pp. 337–351.
- Van Oosterom, P.J.M.; Meijers, M. Vario-scale data structures supporting smooth zoom and progressive transfer of 2D and 3D data. Int. J Geogr. Inf. Sci.
**2014**, 28, 455–478. [Google Scholar] [CrossRef] - Mason, N.C.; O'Conaill, M.A.; Bell, S.B.M. Handling four-dimensional geo-referenced data in environmental GIS. Int. J. Geogr. Inf. Syst.
**1994**, 8, 191–215. [Google Scholar] [CrossRef] - Bernard, L.; Schmidt, B.; Streit, U. AtmoGIS—Integration of atmospheric models and GIS. In Proceedings of the 8th International Symposium on Spatial Data Handling, Vancouver, Canada, 11–15 July 1998.
- Samet, H.; Tamminen, M. Bintrees, CSG trees, and time. In Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH '85, San Francisco, CA, USA, 22–26 July 1985; pp. 121–130.
- Varma, H.; Boudreau, H.; Prime, W. A data structure for spatio-temporal databases. Int. Hydrogr. Rev.
**1990**, 67, 71–92. [Google Scholar] - OGC. OpenGIS Implementation Specification for Geographic Information-Simple Feature Access-Part 1: Common Architecture, 1.2.1 ed.; Open Geospatial Consortium: Herndon, VA, USA, 2011. [Google Scholar]
- Rossignac, J.; O'Connor, M. SGC: A dimension-independent model for pointsets with internal structures and incomplete boundaries. In Proceedings of the IFIP Workshop on CAD/CAM, Rensselaerville, NY, USA, 17–21 June 1989; pp. 145–180.
- Masuda, H. Topological operators and boolean operations for complex-based non-manifold geometric models. Comput. Aided Des.
**1993**, 25, 119–129. [Google Scholar] [CrossRef] - Sohanpanah, C. Extension of a boundary representation technique for the description of n dimensional polytopes. Comput. Graph.
**1989**, 13, 17–23. [Google Scholar] [CrossRef] - ESRI. GIS Topology; ESRI: Redlands, CA, USA, 2005. [Google Scholar]
- Mäntylä, M. An Introduction to Solid Modeling; Computer Science Press: New York, NY, USA, 1988. [Google Scholar]
- de Floriani, L.; Hui, A. Data structures for simplicial complexes: an analysis and a comparison. In Eurographics Symposium on Geometry Processing; Desbrunn, M., Pottmann, H., Eds.; The Eurographics Association: Vienna, Austria, 2005. [Google Scholar]
- Shewchuk, J.R. Sweep algorithms for constructing higher-dimensional constrained delaunay triangulations. In Proceedings of the 16th Annual Symposium on Computational Geometry, Hong Kong, China, 12–14 June 2000; pp. 350–359.
- Shewchuk, J.R. General-dimensional constrained delaunay and constrained regular triangulations, I: Combinatorial properties. Discret. Comput. Geom.
**2008**, 39, 580–637. [Google Scholar] [CrossRef] - Lienhardt, P. N-dimensional generalized combinatorial maps and cellular quasi-manifolds. Int. J. Comput. Geom. Appl.
**1994**, 4, 275–324. [Google Scholar] [CrossRef] - Brisson, E. Representing geometric structures in d dimensions: topology and order. Discret. Comput. Geom.
**1993**, 9, 387–426. [Google Scholar] [CrossRef] - Poudret, M.; Arnould, A.; Bertrand, Y.; Lienhardt, P. Cartes Combinatoires Ouvertes; Technical Report 2007-01; Laboratoire SIC, UFR SFA, Université de Poitiers: Poiters, France, 2007. [Google Scholar]
- Kraemer, P.; Untereiner, L.; Jund, T.; Thery, S.; Cazier, D. CGoGN: N-dimensional meshes with combinatorial maps. In Proceedings of the 22nd International Meshing Roundtable, Orlando, FL, USA, 13–16 October 2013; pp. 485–503.
- Arroyo Ohori, K.; Ledoux, H. Using extrusion to generate higher-dimensional GIS datasets. In Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, Orlando, FL, USA, 5–8 November 2013; pp. 398–401.
- Arroyo Ohori, K.; Ledoux, H.; Stoter, J. A dimension-independent extrusion algorithm using generalized maps. Int. J. Geogr. Inf. Sci.
**2015**, 17, 32–46. [Google Scholar] - Arroyo Ohori, K.; Damiand, G.; Ledoux, H. Constructing an n-dimensional cell complex from a soup of (n-1)-dimensional faces. In ICAA 2014; Gupta, P., Zaroliagis, C., Eds.; Springer International Publishing Switzerland: Colkata, India, 2014; Volume 8321, pp. 36–47. [Google Scholar]
- Gosselin, S.; Damiand, G.; Solnon, C. Efficient search of combinatorial maps using signatures. Theor. Comput. Sci.
**2011**, 412, 1392–1405. [Google Scholar] [CrossRef] - Eppstein, D. Subgraph isomorphism in planar graphs and related problems. J. Gr. Algorithms Appl.
**1999**, 3, 1–27. [Google Scholar] [CrossRef] - Rubner, Y.; Tomasi, C.; Guibas, L.J. A metric for distributions with applications to image databases. In Proceedings of the 6th International Conference on Computer Vision, Mumbai, India, 4–7 January 1998; pp. 59–66.
- Granados, M.; Hachenberger, P.; Hert, S.; Kettner, L.; Mehlhorn, K.; Seel, M. Boolean operations on 3D selective Nef complexes: Data structure, algorithms, and implementation. In Proceedings 11th Annual European Symposium on Algorithms (ESA'03), Budapest, Hungary, 15–20 September 2003; pp. 654–666.
- Hachenberger, P. Boolean Operations on 3D Selective Nef Complexes Data Structure, Algorithms, Optimized Implementation, Experiments and Applications. Ph.D. Thesis, Saarland University, Saarbrücken, Germany, 1 December 2006. [Google Scholar]
- Damiand, G.; Lienhardt, P. Removal and contraction for n-dimensional generalized maps. In Proceedings of the 11th Discrete Geometry for Computer Imagery, Naples, Italy, 19–21 November 2003; Volume 2886, pp. 408–419.
- Gröger, G.; Plümer, L. Provably correct and complete transaction rules for updating 3D city models. Geoinformatica
**2011**, 16, 131–164. [Google Scholar] [CrossRef] - Hoppe, H. View-dependent refinement of progressive meshes. In Proceedings of SIGGRAPH'97, Los Angeles, CA, USA, 3–8 August 1997; pp. 189–198.
- De Berg, M.; Dobrindt, K.T.G. On levels of detail in terrains. Gr. Model. Imag. Process.
**1998**, 60, 1–12. [Google Scholar] [CrossRef] - Biljecki, F.; Ledoux, H.; Stoter, J. Redefining the level of detail for 3D models. GIM Int.
**2014**, 28, 21–23. [Google Scholar] - Hornsby, K.; Egenhofer, M.J. Identity-based change: A foundation for spatio-temporal knowledge representation. Int. J. Geogr. Inf. Sci.
**2000**, 14, 207–224. [Google Scholar] [CrossRef] - Peuquet, D.J. Representations of Space and Time; Guilford Press: New York, NY, USA, 2002. [Google Scholar]
- Raper, J.; Livingstone, D. Let's get real: Spatio-temporal identity and geographic entities. Trans. Inst. Br. Geogr.
**2001**, 26, 237–242. [Google Scholar] [CrossRef] - Worboys, M.F. A unified model for spatial and temporal information. Comput. J.
**1994**, 37, 26–34. [Google Scholar] [CrossRef] - de Roo, B.; van de Weghe, N.; Bourgeois, J.; de Maeyer, P. The temporal dimension in a 4D archaeological data model: Applicability of the geoinformation standard. In Proceedings of the 8th 3DGeoInfo Conference & WG II/2 Workshop, Istanbul, Turkey, 27–29 November 2013.
- Huisman, O.; Santiago, I.F.; Kraak, M.J.; Retsios, B. Developing a geovisual analytics environment for investigating archaeological events: Extending the space-time cube. Cartogr. Geogr. Inf. Sci.
**2013**, 36, 225–236. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Ohori, K.A.; Ledoux, H.; Biljecki, F.; Stoter, J.
Modeling a 3D City Model and Its Levels of Detail as a True 4D Model. *ISPRS Int. J. Geo-Inf.* **2015**, *4*, 1055-1075.
https://doi.org/10.3390/ijgi4031055

**AMA Style**

Ohori KA, Ledoux H, Biljecki F, Stoter J.
Modeling a 3D City Model and Its Levels of Detail as a True 4D Model. *ISPRS International Journal of Geo-Information*. 2015; 4(3):1055-1075.
https://doi.org/10.3390/ijgi4031055

**Chicago/Turabian Style**

Ohori, Ken Arroyo, Hugo Ledoux, Filip Biljecki, and Jantien Stoter.
2015. "Modeling a 3D City Model and Its Levels of Detail as a True 4D Model" *ISPRS International Journal of Geo-Information* 4, no. 3: 1055-1075.
https://doi.org/10.3390/ijgi4031055