為了構建3D 虛擬環境，我們需要許多模型資料；例如，地形、建築物、車輛等。如果此虛擬環境中含有大量的模型資料，繪圖的效率必然降低。因此我們需要一些技巧來改善繪圖的效率而又不失視覺品質。多重解析度模塑技術就是用來達成此目標的一種技術。
我們的研究著重在多重解析度3D 建物的建構上。建築物模型與地形模型有著相當不同的特徵，例如建築物外觀通常比較方正 (rectangular) 或是擁有較多銳利的外觀平面。即使在建築物模型並非相當方正的情形下，我們仍然希望在模型簡化的過程中能保留建築物的特徵。因此，我們提出了以半空間 (half-space) 法來尋找建築物的半空間基準面 (half-space primitive plane)，然後用這些基準面加上一些判定的限制，並訂出一些規則來判定建築物模型中哪些三角平面需要被保留、那些需要被簡化以維持建築物的外型特徵。
此外，我們還以螢幕空間誤差 (screen-space error) 表示觀察者與建築物模型的距離，以定義不同門檻值來決定並產生離散的多重解析度建築物模型，並可以利用此一門檻值在虛擬環境中快速的決定並瀏覽多重解析度建築物模型。

To construct a 3D virtual environment, we need many models such as, terrain, buildings, vehicles, and others. If there are huge amount of models in the virtual environment, the rendering efficiency of the visualization system will degrade; thus we need advanced techniques to improve the visualization performance without reducing the visual quality. The multiresolution modeling technique is one way to improve the visualization performance without degrading the visual quality.
In this study, we focus on the multiresolution of 3D building models. The building models have different characteristics to the terrain models. We always need to keep the sharp outlines of the building models. If the building models are non-rectangular, we need to create level-of-detailed (LOD) 3D building models without squaring them before. We adopt the half-space method with some constraints to distinguish the triangles which should be simplified in a building model and the other triangles should be preserved in order to maintain the special characteristics of the building models. Then we define some regularization rules for the simplification of building models. Moreover, we use the idea similar to screen-space error to obtain the relationship between the threshold of half-space method and the distance between viewer and the LOD building models. Afterward we can use the relationship to quickly browse the LOD building models in a 3D virtual environment.

[1] Anders, K. H., “Level of detail generation of 3D building groups by aggregation and typification,” in Proc. 22nd Int. Cartographic Conf., La Coruna, Spain, July 11-16, 2005, in CD.
[2] Chang, J.-H., Adaptive Multiresolution Terrain Modeling and Dynamic-loading for Flight Simulation, Master Thesis, Inst. of Computer Science and Information Engineering, National Central University, Chungli, Taiwan, June 2002.
[3] D?llner, J. and H. Buchholz, “Continuous level-of-detail modeling of buildings in 3D city models,” in Proc. of the 13th ACM Int. of Geographic Information Systems, Bremen, Germany, Nov.4-5, 2005, pp.173-181.
[4] Erikson, C. and D. Manocha, Simplification Culling of Static and Dynamic Scene Graphs, Tech. Report 98-009, Computer Science Dept., Univ. of North Carolina, Chapel Hill, 1998.
[5] Forberg, A. and H. Mayer, “Generalization of 3D building data based on scale-spaces,” in Proc. of the Joint Int. Symp. on Geospatial Theory, Processing and Applications, Ottawa, Canada, July 9-12, 2002, pp.225-230.
[6] Forberg, A., “Generalization of 3D building data based on a scale-space approach,” ISPRS Journal of Photogrammetry and Remote Sensing, vol.62, no.2, pp.104-111, 2007.
[7] Garland, M. and P. S. Heckbert, Fast Polygonal Approximation of Terrain and Height Fields, Tech. Report CMU-CS-95-181, Carnegie Mellon Univ., School of Computer Science, Sep. 1995.
[8] Garland, M. and P. S. Heckbert, “Surface simplification using quadric error metrics,” in Proc. SIGGRAPH`97, Los Angeles, CA, Aug.3-8, 1997, pp.209-216.
[9] Garland, M., A. Willmott, and P. Heckbert. “Hierarchical face clustering on polygonal surfaces,” in Proc. of ACM Sym. on Interactive 3D Graphics, Chapel Hill, NC, Mar.26-29, 2001, pp.49-58.
[10] Glander, T., Automated Generalization of 3D Building Models, Master Thesis, Computer Graphics Systems, Hasso Plattner Institute, Univ. of Potsdam, Potsdam, Germany, 2007.
[11] Grabner, M., “Feature preservation in view-dependent multiresolution meshes,” in Proc. of Spring Conf. on Computer Graphics, Budmerice, Slovakia, Apr.24-27, 2002, pp.153-162.
[12] Hoppe, H., “Progressive meshes,” in Proc. SIGGRAPH’96, New Orleans, LA, Aug.4-9, 1996, pp.99-108.
[13] Hoppe, H., “View-dependent refinement of progressive meshes,” in Proc. SIGGRAPH’97, Los Angeles, CA, Aug.3-8, 1997, pp.189-198.
[14] Hoppe, H., “Progressive simplicial complexes,” in Proc. SIGGRAPH’97, Los Angeles, CA, Aug.3-8, 1997, pp.217-224.
[15] Hoppe, H., “Smooth view-dependent level-of-detail control and its application to terrain rendering,” in Proc. IEEE Visualization’98, Research Triangle Park, NC, Oct.18-23, 1998, pp.35-42.
[16] Hsieh, Y.-W., Multiresolution Terrain Visualization with Multiresolution Building Modeling, Master Thesis, Inst. of Computer Science and Information Engineering, National Central University, Chungli, Taiwan, June 2007.
[17] Huang, C.-C., View-dependent Progressive-mesh Terrain Browsing with Dynamic Loading, Master Thesis, Inst. of Computer Science and Information Engineering, National Central University, Chungli, Taiwan, June 1999.
[18] Huang, W.-K., A Tactical Simulation System with Dynamic-loading Multiresolution Terrain Modeling, Master Thesis, Inst. of Computer Science and Information Engineering, National Central University, Chungli, Taiwan, June 2001.
[19] Jang, J., P. Wonka, W. Ribarsky, and C. D. Shaw, “Punctuated simplification of man-made objects,” The Visual Computer, vol.22, no.2, pp.136-145, 2006.
[20] Kada, M., “Automatic generalisation of 3D building models,” in Proc. of the Joint Int. Symp. on Geospatial Theory, Processing and Applications, Ottawa, Canada, July 9-12, 2002, pp.243-248.
[21] Kada, M., “3D Building generalization based on half-space modeling,” in Proc. of the ISPRS Workshop on Multiple Representation and Interoperability of Spatial Data, Hanover, Germany, Feb.22-24, 2006, pp.58-64.
[22] Kada, M., “3D building generalisation by roof simplification and typification,” in Proc. 23rd Int. Cartographic Conf., Moscow, Russia, Aug.4-10, 2007, in CD.
[23] Kada, M., “Scale-dependent simplification of 3D building models based on cell decomposition and primitive instancing,” Lecture Notes in Computer Science, Springer-Verlag, Heidelberg, vol.4736, pp.222-237, 2007.
[24] Lee, Y.-L., Adaptive Dynamic Loading with Multiresolution Terrain Visualization, Master Thesis, Inst. of Computer Science and Information Engineering, National Central University, Chungli, Taiwan, June, 2005.
[25] Liu, S.-C., View-dependent Multiresolution Modeling with Appearance Attributes Using Quadric Error Metrics, Master Thesis, Inst. of Computer Science and Information Engineering, National Central University, Chungli, Taiwan, June 2001.
[26] Mayer, H., Three Dimensional Generalization of Buildings Based on Scale-spaces, Tech. Report, Chair for Photogrammetry and Remote Sensing, Technische Universit?t M?nchen, Munich, Germany, 1998.
[27] Mayer, H., “Scale-spaces for generalization of 3D buildings,” Int. Journal of Geographical Information Science, vol.19, no.8-9, pp.975-997, 2005.
[28] Rau, J.-Y., L.-C. Chen, F. Tsai, K.-H. Hsiao, and W.-C. Hsu, “LOD generation for 3D polyhedral building model,” Lecture Notes in Computer Science, Springer-Verlag, Heidelberg, vol.4319, pp.44-53, 2006.
[29] Ribelles, J., P. Heckbert, M. Garland, T. Stahovich, and V. Srivastava, “Finding and removing features from polyhedra,” in Proc. ASME Design Engineering Technical Conf., Pittsburgh, Pennsylvania, Sep.9-12, 2001.
[30] Thiemann F., “Generalization of 3D building data,” in Proc. of the Joint Int. Symp. on Geospatial Theory, Processing and Applications, Ottawa, Canada, July 9-12, 2002, in CD.
[31] Thiemann F. and M. Sester, “Segmentation of buildings for 3D generalization,” in Proc. of the ICA workshop on generalisation and multiple representation, Leicester, UK, Aug.20-21, 2004, in CD.
[32] Thiemann F. and M. Sester, “3D-symbolization using adaptive templates,” in Proc. of the ISPRS Technical Commission II Symp., Vienna, Austria, July 12-16, 2006, pp.109-113.
[33] Wang, Y.-X., B.-Y. Li, W.-B. Liu, and X.-X. Shi, “3D urban visualization with LOD techniques,” Journal of China University of Mining and Technology, vol.16, no.1, pp.64-67, 2006.
[34] Yang, S., C.-S. Kim, and C.-C. J. Kuo, “A progressive view-dependent technique for interactive 3-D mesh transmission,” IEEE Trans. on Circuits and Systems for Video Technology, vol.14, no.11, pp.1249-1264, 2004.
[35] Yang, T.-S., Dynamic-loading Multiresolution Terrain Modeling in A Flight Simulation System, Master Thesis, Inst. of Computer Science and Information Engineering, National Central University, Chungli, Taiwan, June 2000.
[36] Yang, Y.-S., Flight Simulation with Adaptive Dynamic Loading Multiresolution Terrain Visualization, Master Thesis, Inst. of Computer Science and Information Engineering, National Central University, Chungli, Taiwan, June 2006.