Abstract
The orthogonal cover of a 3D digital object is a minimum-volume 3D polytope having surfaces parallel to the coordinate planes, and containing the entire object so as to capture its approximate shape information. An efficient algorithm for construction of such an orthogonal cover imposed on a background grid is presented in this paper. A combinatorial technique is used to classify the grid faces constituting the polytope while traversing along the surface of the object in a breadth-first manner. The eligible grid faces are stored in a doubly connected edge list, using which the faces are finally merged to derive the isothetic polygons parallel to the coordinate planes, thereby obtaining the orthogonal cover of the object. The complexity of the cover decreases with increasing grid size. The algorithm requires computations in integer domain only and runs in a time linear in the number of voxels constituting the object surface. Experimental results demonstrate the effectiveness of the algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berg, M.D., Cheong, O., Kreveld, M.V., Overmars, M.: Computational Geometry—Algorithms and Applications. Springer, Heidelberg (1997)
Bhaniramka, P., Wenger, R., Crawfis, R.: Isosurface construction in any dimension using convex hulls. IEEE Trans. on Visualization and Computer Graphics 10, 130–141 (2004)
Bhaniramka, P., Wenger, R., Crawfis, R.: Isosurfacing in higher dimensions. In: Proceedings of Visualization, Salt Lake City, pp. 267–273 (2000)
Biswas, A., Bhowmick, P., Bhattacharya, B.B.: Construction of Isothetic Covers of a Digital Object: A Combinatorial Approach. Journal of Visual Communication and Image Representation 21, 295–310 (2010)
Biswas, A., Bhowmick, P., Bhattacharya, B.B.: TIPS: On Finding a Tight Isothetic Polygonal Shape Covering a 2D Object. In: Kalviainen, H., Parkkinen, J., Kaarna, A. (eds.) SCIA 2005. LNCS, vol. 3540, pp. 930–939. Springer, Heidelberg (2005)
Brimkov, V.: Discrete volume polyhedrization: Complexity and bounds on performance. In: Proc. of the International Symposium on Computational Methodology of Objects Represented in Images: Fundamentals, Methods and Applications, CompIMAGE 2006, pp. 117–122. Taylor and Francis, Coimbra (2006)
Coeurjolly, D., Sivignon, I.: Reversible discrete volume polyhedrization using Marching Cubes simplification. In: SPIE Vision Geometry XII, vol. 5300, pp. 1–11 (2004)
Cohen-Or, D., Shamir, A., Shapira, L.: Consistent Mesh Partitioning and Skeletonization using the Shape Diameter Function. The Visual Computer 24, 249–259 (2008)
Giles, M., Haimes, R.: Advanced interactive visualization for CFD. Computing Systems in Engineering 1, 51–62 (1990)
Golovinskiy, A., Funkhouser, T.: Randomized cuts for 3D mesh analysis. ACM Transactions on Graphics (Proc. SIGGRAPH ASIA) 27, Article 145 (2008)
Hearn, D., Baker, M.P.: Computer Graphics with OpenGL. Pearson Education Inc., London (2004)
Hill, F.S., Kelley, S.M.: Computer Graphics Using OpenGL. Pearson Edcation Inc., London (2007)
Katz, S., Leifman, G., Tal, A.: Mesh segmentation using feature point and core extraction. The Visual Computer 21, 649–658 (2005)
Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)
Livnat, Y., Shen, H.-W., Johnson, C.: A near optimal isosurface extraction algorithm using span space. IEEE Trans. Visualization and Computer Graphics 2, 73–84 (1996)
Lorensen, W.E., Cline, H.E.: Marching Cubes: A high resolution 3D surface construction algorithm. Computer Graphics 21, 163–169 (1987)
Newman, T.S., Yi, H.: A Survey of the Marching Cubes Algorithm. Computers & Graphics 30, 854–879 (2006)
Preparata, F.P., Shamos, M.I.: Computational Geometry—An Introduction. Spinger, New York (1985)
Shapira, L., Shalom, S., Shamir, A., Cohen-Or, D., Zhang, H.: Contextual Part Analogies in 3D Objects. International Journal of Computer Vision 89, 309–326 (2010)
Shlafman, S., Tal, A., Katz, S.: Metamorphosis of Polyhedral Surfaces using Decomposition. In: Eurographics 2002, pp. 219–228 (2002)
Turk, G., Levoy, M.: Zippered polygon meshes from range images. In: SIGGRAPH 1994, pp. 311–318 (1994)
Weber, G., Kreylos, O., Ligocki, T., Shalf, J., Hagen, H., Hamann, B.: Extraction of crack-free isosurfaces from adaptive mesh refinement data. In: Proceedings of VisSym 2001, Ascona, Switzerland, pp. 25–34 (2001)
Wilhelms, J., van Gelder, A.: Topological considerations in isosurface generation extended abstract. Computers Graphics 24, 79–86 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Karmakar, N., Biswas, A., Bhowmick, P., Bhattacharya, B.B. (2011). Construction of 3D Orthogonal Cover of a Digital Object. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds) Combinatorial Image Analysis. IWCIA 2011. Lecture Notes in Computer Science, vol 6636. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21073-0_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-21073-0_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21072-3
Online ISBN: 978-3-642-21073-0
eBook Packages: Computer ScienceComputer Science (R0)