Abstract
In this paper, we study the problem of approximate topological matching for quadrilateral meshes, that is, the problem of finding as large a set as possible of matching portions of two quadrilateral meshes. This study is motivated by applications in graphics that involve the modeling of different shapes that have results needing to be merged in order to produce a final unified representation of an object. We show that the problem of producing a maximum approximate topological match of two quad meshes is NP-hard and that its decision version is NP-complete. Given these results, which make an exact solution extremely unlikely, we show that the natural greedy algorithm derived from polynomial-time graph isomorphism can produce poor results, even when it is possible to find matches with only a few nonmatching quads. Nevertheless, we provide a “lazy-greedy” algorithm that is guaranteed to find good matches when mismatching portions of mesh are localized. Finally, we provide empirical evidence that this approach produces good matches between similar quad meshes.
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References
Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., Desbrun, M.: Anisotropic polygonal remeshing. In: SIGGRAPH’03: ACM SIGGRAPH 2003 Papers, New York, NY, USA, pp. 485–493. ACM, New York (2003)
Armstrong, C.G., Robinson, D.J., McKeag, R.M., Li, T.S., Bridgett, S.J., Donaghy, R.J.: Applications of the medial axis transform in analysis modelling. In: NAFEMS, Proc. 5th Int. Conf. Reliability of FEM for Engineering Applications, pp. 415–426 (1995)
Battista, G.D., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall PTR, Upper Saddle River (1998)
Baumgart, B.G.: Winged edge polyhedron representation. Technical Report CS-TR-72-320, Stanford University (1972)
Bern, M.W., Eppstein, D.: Quadrilateral meshing by circle packing. Int. J. Comput. Geom. Appl. 10(4), 347–360 (2000)
Berretti, S., Bimbo, A.D., Pala, P.: A graph edit distance based on node merging. In: Image and Video Retrieval. Lecture Notes in Computer Science, vol. 3115, pp. 464–472 (2004)
Blacker, T.D., Stephenson, M.B.: Paving: a new approach to automated quadrilateral mesh generation. Int. J. Numer. Methods Eng. 32, 811–847 (1991)
Bunke, H.: On a relation between graph edit distance and maximum common subgraph. Pattern Recognit. Lett. 18(8), 689–694 (1997)
Chae, S.-W., Jeong, J.-H.: Unstructured surface meshing using operators. In: Proc. 6th Int. Meshing Roundtable, pp. 281–291 (1997)
Cohen-Or, D., Levin, D., Remez, O.: Progressive compression of arbitrary triangular meshes. In: Proc. 10th IEEE Visualization (VIS’99), p. 11 (1999)
Corneil, D.G., Gotlieb, C.C.: An efficient algorithm for graph isomorphism. J. ACM 17(1), 51–64 (1970)
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, Berlin (2000)
de Fraysseix, H., Pach, J., Pollack, R.: Small sets supporting Fáry embeddings of planar graphs. In: STOC’88: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, New York, NY, USA, pp. 426–433. ACM Press, New York (1988)
Dong, S., Bremer, P.-T., Garland, M., Pascucci, V., Hart, J.C.: Spectral surface quadrangulation. In: SIGGRAPH ’06: ACM SIGGRAPH 2006 Papers, New York, NY, USA, pp. 1057–1066. ACM, New York (2006)
Eppstein, D., Goodrich, M.T., Kim, E., Tamstorf, R.: Approximate topological matching of quadrilateral meshes. In: IEEE Int. Conf. on Shape Modeling and Applications (SMI), pp. 83–92 (2008)
Eppstein, D., Goodrich, M.T., Kim, E., Tamstorf, R.: Motorcycle graphs: canonical quad mesh partitioning. Comput. Graph. Forum 27(5), 1477–1486 (2008)
Eshera, M.A., Fu, K.S.: An image understanding system using attributed symbolic representation and inexact graph-matching. IEEE Trans. Pattern Anal. Mach. Intell. 8(5), 604–618 (1986)
Feder, T., Motwani, R.: Clique partitions, graph compression and speeding-up algorithms. In: Proc. 23rd ACM Symp. Theory of Computing, pp. 123–133 (1991)
Ferragina, P., Manzini, G.: Opportunistic data structures with applications. In: Proc. 41st Symp. Foundations of Computer Science, pp. 390–398 (2000)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
Grohe, M.: Isomorphism testing for embeddable graphs through definability. In: STOC’00: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, New York, NY, USA, pp. 63–72. ACM Press, New York (2000)
Gurnhold, S., Strasser, W.: Real time compression of triangle mesh connectivity. In: Proc. 25th Conf. Computer Graphics and Interactive Technology, pp. 133–140 (1998)
Hopcroft, J.E., Wong, J.K.: Linear time algorithm for isomorphism of planar graphs. In: Proc. 6th ACM Symp. Theory of Computing, pp. 172–184 (1974)
Khodakovsky, A., Alliez, P., Desbrun, M., Schröder, P.: Near-optimal connectivity encoding of 2-manifold polygon meshes. Graph. Models 64(3/4), 147–168 (2002)
King, D., Rossignac, J., Szymczak, A.: Connectivity compression for irregular quadrilateral meshes. Technical Report GIT-GVU-99-36, Georgia Institute of Technology (1999)
Marini, S., Spagnuolo, M., Falcidieno, B.: From exact to approximate maximum common subgraph. In: Graph-Based Representations in Pattern Recognition. Lecture Notes in Computer Science, vol. 3434, pp. 263–272. Springer, Berlin (2005)
Miller, G.: Isomorphism testing for graphs of bounded genus. In: Proc. 12th ACM Symp. Theory of Computing, pp. 225–235 (1980)
Miller, G.L.: Graph isomorphism, general remarks. In: STOC’77: Proceedings of the Ninth Annual ACM Symposium on Theory of Computing, New York, NY, USA, pp. 143–150. ACM Press, New York (1977)
Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins University Press, Baltimore (2001)
Müller-Hannemann, M.: High quality quadrilateral surface meshing without template restrictions: a new approach based on network flow techniques. In: Proc. 6th Int. Meshing Roundtable, pp. 293–307 (1997)
Neuhaus, N., Bunke, H.: Automatic learning of cost functions for graph edit distance. Inf. Sci. 177(1), 239–247 (2007)
Nowottny, D.: Quadrilateral mesh generation via geometrically optimized domain decomposition. In: Proc. 6th Int. Meshing Roundtable, pp. 309–320 (1997)
Palacios, J., Zhang, E.: Rotational symmetry field design on surfaces. ACM Trans. Graph. 26(3), 55 (2007)
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1985)
Ray, N., Li, W.C., Lévy, B., Sheffer, A., Alliez, P.: Periodic global parameterization. ACM Trans. Graph. 25(4), 1460–1485 (2006)
Schnyder, W.: Embedding planar graphs on the grid. In: SODA’90: Proceedings of the First Annual ACM–SIAM Symposium on Discrete Algorithms, Philadelphia, PA, USA, pp. 138–148. Society for Industrial and Applied Mathematics, Philadelphia (1990)
Schonfeld, T., Weinerfelt, P.E.R.: The automatic generation of quadrilateral multi-block grids by the advancing front technique. In: Numerical Grid Generation in Computational Fluid Dynamics and Related Fields; Proceedings of the 3rd International Conference, Barcelona, Spain, pp. 743–754 (1991)
Shimada, K., Liao, J.-H., Itoh, T.: Quadrilateral meshing with directionality control through the packing of square cells. In: Proc. 7th Int. Meshing Roundtable, pp. 61–75 (1998)
Smith, B.C., Rowe, L.A.: Algorithms for manipulating compressed images. Comput. Graph. Appl. 13(5), 34–42 (1993)
Talbert, J.A., Parkinson, A.R.: Development of an automatic two-dimensional finite element mesh generator using quadrilateral elements and Bezier curve boundary definition. Int. J. Numer. Methods Eng. 29, 1551–1567 (1991)
Taubin, G., Rossignac, J.: Geometric compression through topological surgery. ACM Trans. Graph. 17(2), 84–115 (1998)
Tong, Y., Alliez, P., Cohen-Steiner, D., Desbrun, M.: Designing quadrangulations with discrete harmonic forms. In: SGP’06: Proceedings of the Fourth Eurographics Symposium on Geometry Processing, Aire-la-Ville, Switzerland, pp. 201–210. Switzerland Eurographics Association (2006)
Touma, C., G.C.: Triangle mesh compression. In: Proc. Graphics Interface, pp. 26–34 (1998)
Wood, D.R., Telle, J.A.: Planar decompositions and the crossing number of graphs with an excluded minor. N. Y. J. Math. 13, 117–146 (2007)
Yeo, B.-L., Liu, B.: Rapid scene analysis on compressed video. IEEE Trans. Circuits Syst. Video Technol. 5(6), 533–544 (1995)
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Eppstein, D., Goodrich, M.T., Kim, E. et al. Approximate topological matching of quad meshes. Vis Comput 25, 771–783 (2009). https://doi.org/10.1007/s00371-009-0363-z
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DOI: https://doi.org/10.1007/s00371-009-0363-z