A POLYHEDRAL APPROACH TO COMPUTE THE GENUS OF A VOLUME DATASET

D. Ayala, E. Vergés, I. Cruz

Abstract

Topological characteristics are fundamental in many areas. In this paper we present a method that computes the Euler characteristic and the genus of a volume dataset. The followed approach, based on the analogy between binary volume datasets and orthogonal pseudo-polyhedra (OPP), computes the mentioned values using two models that are suitable for representing OPP: the Extreme Vertices Model (EVM) and the Ordered Union of Disjoint Boxes (OUDB). We show the results of these methods for phantom models as well as real datasets, and compare the efficiency of the presented methods with those based on the classic voxel model.

References

  1. Aguilera, A. (1998). Orthogonal Polyhedra: Study and Application. PhD thesis, LSI-UPC.
  2. Aguilera, A. and Ayala, D. (2001). Geometric Modeling, volume 14 of Computing Supplement, chapter Converting Orthogonal Polyhedra from Extreme Vertices Model to B-Rep and to Alternating Sum of Volumes, pages 1 - 28. Springer.
  3. Andújar, C., Brunet, P., and Ayala, D. (2002). Topologyreducing surface simplification using a discrete solid rep. ACM Trans. on Graphics, 21(2):88 - 105.
  4. Attene, M., Giorgi, D., Ferri, M., and Falcidieno, B. (2009). On converting sets of tetrahedra to combinatorial and pl manifolds. Computer Aided Geometric Design, 26:850-864.
  5. Bertrand, G. and Malandain, G. (1994). A new characterization of three-dimensional simple points. Pattern Recog. Lett., 2:169 - 175.
  6. Borgefors, G. B., Nystrom, I., and Baja, G. S. D. (1999). Computing skeletons in three dimensions. Pattern Recognition, 32:1225-1236.
  7. Bourke, P. (1994). Polygonising a scalar field. http://paulbourke.net/geometry/polygonise/.
  8. Delfinado, C. J. A. and Edelsbrunner, H. (1993). An incremental algorithm for betti numbers of simplicial complexes. In SCG'93 Proceedings of the 9th annual symposium on Computational geometry, pages 232-239.
  9. Dillencourt, M. B., Samet, H., and Tamminen, M. (1992). A general approach to ccl for arbitrary image representations. Journal of the ACM, 39(2):253 - 280.
  10. Floriani, L., Mesmoudi, M., Morando, F., and Puppo, E. (2003). Decomposing non-manifold objects in arbitrary dimensions. Graphical Models, 65:2 - 22.
  11. Ghosh, P. K. and Haralick, R. M. (1996). Mathematical morphological operations of boundary-represented geometric objects. Journal of Mathematical Imaging and Vision, 6:199-222. morphological.
  12. Gonzalez, R. C. and Woods, R. E. (1992). Digital Image Processing. Addison-Wesley.
  13. Greb , A. and Klein, R. (2004). Efficient representation and extraction of 2-manifold isosurfaces using kd-trees. Graphical Models, 66:370 - 397.
  14. Grevera, G. J., Udupa, J. K., and Odhner, D. (2000). An Order of Magnitude Faster Isosurface Rendering in Software on a PC than Using Dedicated, General Purpose Rendering Hardware. IEEE Transactions Visualization and Computer Graphics, 6(4):335-345.
  15. Guèziec, A., Taubin, G., Lazarus, F., and Horn, B. (2001). Cutting and stitching: converting sets of polygons to manifold surfaces. IEEE Transactions on Visualization and Computer Graphics, 7(2):136 - 151.
  16. Ioannidis, M. A. and Chatzis, I. (2000). On the Geometry and Topology of 3D Stochastic Porous Media. Journal of Colloid and Interface Science, 229:323 - 334.
  17. Juan-Arinyo, R. (1995). Domain extension of isothetic polyhedra with minimal CSG representation. Computer Graphics Forum, 5:281 - 293.
  18. Khachan, M., Chenin, P., and Deddi, H. (2000). Polyhedral representation and adjacency graph in n-dimensional digital images. Computer Vision and Image understanding, 79:428 - 441.
  19. Kim, B. H., Seo, J., and Shin, Y. G. (2001). Binary volume rendering using Slice-based Binary Shell. The Visual Computer, 17:243 - 257.
  20. Kong, T. and Rosenfeld, A. (1989). Digital topology: Introduction and survey. Computer Vision, Graphics and Image Processing, 48:357-393.
  21. Konkle, S. F., Moran, P. J., Hamann, B., and Joy, K. I. (2003). Fast methods for computing isosurface topology with Betti numbers. In Data Visualization: the sate of the art proceedings Dagstuhl Seminar on Scientific Visualization, pages 363 - 376.
  22. Lachaud, J. and Montanvert, A. (2000). Continuous analogs of digital boundaries: A topological approach to isosurfaces. Graphical Models, 62:129 - 164.
  23. Latecki, L. (1997). 3D Well-Composed Pictures. Graphical Models and Image Processing, 59(3):164-172.
  24. Lorensen, W. E. and Cline, H. E. (1987). Marching cubes: A high resolution 3D surface construction algorithm. ACM Computer Graphics, 21(4):163-169.
  25. Mantyla, M. (1988). An Introduction to Solid Modeling. Computer Science Press.
  26. Martín-Badosa, E., Elmoutaouakkil, A., Nuzzo, S., Amblard, D., Vico, L., and Peyrin, F. (2003). A method for the automatic characterization of bone architecture in 3D mice microtomographic images. Computerized Medical Imaging and Graphics, 27:447-458.
  27. Massey, W. S. (1991). A Basic Course in Algebraic Topology. Springer-Verlag.
  28. Odgaard, A. and Gundersen, H. J. (1993). Quantification of Connectivity in Cancellous Bone, with Special Emphasis on 3-D Reconstructions. Bone, 14:173 - 182.
  29. Peyrin, F., Peter, Z., Larrue, A., Bonnassie, A., and Attali, D. (2007). Local geometrical analysis of 3D porous network based on medial axis: Application to bone micro-architecture microtomography images. Image Analysis and Stereology, 26(3):179 - 185.
  30. Pothuaud, L., Newitt, D. C., Lu, Y., MacDonald, B., and Majumdar, S. (2004). In vivo application of 3d-line skeleton graph analysis (lsga) technique with highresolution magnetic resonance imaging of trabecular bone structure. Osteoporos Int., 15:411 - 419.
  31. Pothuaud, L., Rietbargen, B. V., Mosekilde, L., Beuf, O., Levitz, P., Benhamou, C. L., and Majumdar, S. (2002). Combination of topological param. and bone volume fraction better predicts the mechanical properties of trabecular bone. J. of Biomechanics, 35:1091 - 1099.
  32. Quadros, W. R., Shimada, K., and Owen, S. J. (2004). 3d discrete skeleton generation by wave propagation on pr-octree for finite element mesh sizing. In Proc. ACM Symposium on Solid Modeling and Applications, pages 327 - 332.
  33. Requicha, A. (1980). Representations for rigid solids: Theory, methods and systems. ACM Computing Surveys, 12(4):73-82.
  34. Rodríguez, J. and Ayala, D. (2003). Fast neighborhood operations for images and volume data sets. Computers & Graphics, 27:931-942.
  35. Rodríguez, J., Ayala, D., and Aguilera, A. (2004). Geometric Modeling for Scientific Visualization, chapter EVM: A Complete Solid Model for Surface Rendering, pages 259-274. Springer Verlag.
  36. Rosenfeld, A., Kong, T. Y., and Nakamura, A. (1998). Topology-preserving deformations of two valued digital pictures. Graphical Models and Image Processing, 60(1):24 - 34.
  37. Rossignac, J. and Cardoze, D. (1999). Matchmaker: manifold BReps for non-manifold r-sets. In Proc. Fifth Symposium on Solid Modeling, pages 31 - 40.
  38. Rossignac, J. R. and Requicha, A. A. G. (1991). Contructive Non-Regularized Geometry. Computer-aided design, 23(1):21 - 32.
  39. Rossignac, J. R. and Requicha, A. G. (1986). Offsetting operations in solid modeling. Computer Aided Geometric Design, 3:129 - 148.
  40. Samet, H. (1990). Applications of spatial data structures. Computer Graphics, Image Processing and GIS.
  41. Sarioz, D., Herman, G., and Kong, T. Y. (2004). A technology for retrieval of volume images from biomedical databases*. In Proc. 30th IEEE/EMB Annual Northeast Bioengineering Conference, pages 67-68.
  42. Schaefer, S., Ju, T., and Warren, J. (2007). Manifold dual contouring. IEEE Transactions on Visualization and Computer Graphics, 13(3):610 - 619.
  43. Stauber, M. and Müller, R. (2006). Volumetric spatial decomposition of trabecular bone into rods and plates: A new method for local bone morphometry. Bone, 38(4):475-484.
  44. Tang, K. and Woo, T. (1991). Algorithmic aspects of alternating sum of volumes. Part 1: Data structure and difference operation. CAD, 23(5):357 - 366.
  45. Thurfjell, L., Bengtsson, E., and Nordin, B. (1995). A boundary approach to fast neighborhood operations on three-dimensional binary data. CVGIP: Graphical Models and Image Processing, 57(1):13 - 19.
  46. Toriwaki, J. and Yonekura, T. (2002). Euler number and connectivity indexes of a three dimensional digital picture. Forma, 17:183-209.
  47. Vanderhyde, J. and Szymczak, A. (2008). Topological simplification of isosurfaces in volume data using octrees. Graphical Models, 70:16 - 31.
  48. Vergés, E., Ayala, D., Grau, S., and Tost, D. (2008). Virtual porosimeter. Computer-Aided Design and Applications, 5(1-4):557-564.
  49. Vogel, H. J., T ölke, J., Schulz, V. P., Krafczyk, M., and Roth, K. (2005). Comparison of a lattice-boltzmann model, a full-morphology model, and a pore network model for determining capillary pressure-saturation relationships. Vadose Zone Journal, 4:380 -388.
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Paper Citation


in Harvard Style

Ayala D., Vergés E. and Cruz I. (2012). A POLYHEDRAL APPROACH TO COMPUTE THE GENUS OF A VOLUME DATASET . In Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2012) ISBN 978-989-8565-02-0, pages 38-47. DOI: 10.5220/0003821400380047


in Bibtex Style

@conference{grapp12,
author={D. Ayala and E. Vergés and I. Cruz},
title={A POLYHEDRAL APPROACH TO COMPUTE THE GENUS OF A VOLUME DATASET},
booktitle={Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2012)},
year={2012},
pages={38-47},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0003821400380047},
isbn={978-989-8565-02-0},
}


in EndNote Style

TY - CONF
JO - Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2012)
TI - A POLYHEDRAL APPROACH TO COMPUTE THE GENUS OF A VOLUME DATASET
SN - 978-989-8565-02-0
AU - Ayala D.
AU - Vergés E.
AU - Cruz I.
PY - 2012
SP - 38
EP - 47
DO - 10.5220/0003821400380047