# A Dissimilarity Measure for Comparing Origami Crease Patterns

### Seung Man Oh, Godfried T. Toussaint, Erik D. Demaine, Martin L. Demaine

#### Abstract

A measure of dissimilarity (distance) is proposed for comparing origami crease patterns represented as geometric graphs. The distance measure is determined by minimum-weight matchings calculated between the edges as well as the vertices of the graphs being compared. The distances between pairs of edges and pairs of vertices of the graph are weighted linear combinations of six parameters that constitute geometric features of the edges and vertices. The results of a preliminary study performed with a collection of 45 crease patterns obtained from Mitani’s ORIPA web page, revealed which of these features appear to be more salient for obtaining a clustering of the crease patterns that appears to agree with human intuition.

#### References

- Akitaya, H. A., Mitani, J., Kanamori, and Fukui, Y. (2013). Generating folding sequences from crease patterns of flat-foldable origami. In Proceedings of the Special Interest Group on Graphics, pages 991-1000. ACM.
- Ambauen, R., Fischer, S., and Bunke, H. (2003). Graph edit distance with node splitting and merging, and its application to diatom idenfication. In Proc. 4th IAPR Intl. Conf. Graph Based Representations in Pattern Recognition, pages 95-106.
- Arkin, E. M., Bender, M. A., Demaine, E. D., Demaine, M. L., Mitchell, J. S. B., Sethia, S., and Skiena, S. S. (2004). When can you fold a map? Computational Geometry: Theory and Applications, 29(1):166-195.
- Balkom, D. J., Demaine, E. D., and Demaine, M. L. (2004). Folding paper shopping bags. In Proceedings of the 14th Annual Fall Workshop on Computational Geometry, pages 14-15. MIT, Cambridge.
- Bern, M. and Hayes, B. (1996). The complexity of flat origami. In Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 175-183, Atlanta.
- Berretti, S., Del Bimbo, A., and Pala, P. (2004). A graph edit distance based on node merging. In Proceedings of the ACM International Conference on Image and Video Retrieval (CIVR), pages 464-472, Dublin, Ireland.
- Cheong, O., Gudmundsson, J., Kim, H.-S., Schymura, D., and Stehn, F. (2009). Measuring the similarity of geometric graphs. In Experimental Algorithms, pages 101-112. Springer.
- Clapper, B. (2008). Munkres algorithm for the assignment problem ver.1.0.6.
- Colannino, J., Damian, M., Hurtado, F., Langerman, S., Meijer, H., Ramaswami, S., Souvaine, D., and Toussaint, G. T. (2007). Efficient many-to-many point matching in one dimension. Graphs and Combinatorics, 23:169-178.
- Demaine, E. and O'Rourke, J. (2007). Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, New York.
- Demaine, E. D. and Demaine, M. L. (2001). Recent results in computational origami. In Origami3: Proc. of the 3rd International Meeting of Origami Science, Math, and Education, pages 3-16, Monterey, California.
- Eiter, T. and Mannila, H. (1997). Distance measures for point sets and their computation. Acta Informatica, 34(2):109-133.
- Fei, H. and Huan, J. (2008). Structure feature selection for graph classification. In Proceedings of the 17th ACM conference on Information and knowledge management, pages 991-1000. ACM.
- Gao, X., Xiao, B., and Tao, D. (2010). A survey of graph edit distance. Pattern Analysis and Applications, 13:113-129.
- Gascuel, O. (1997). Bionj: an improved version of the nj algorithm based on a simple model of sequence data. Molecular biology and evolution, 14(7):685-695.
- Graham, R. L. (1987). A similarity measure for graphs. Los Alamos Science, pages 114-121.
- Gu, C. and Guibas, L. (2011). Distance between folded objects. In Proceedings of the European Workshop on Computational Geometry, pages 40-42.
- Hodge, T., Jamie, M., and Cope, T. (2000). A myosin family tree. Journal of Cell Science, 113(19):3353-3354.
- Huson, D. H. and Bryant, D. (2006). Application of phylogenetic networks in evolutionary studies. Molecular biology and evolution, 23(2):254-267.
- Kuhn, H. W. (1955). The Hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2(1-2):83-97.
- Lang, R. J. (1996). A computational algorithm for origami design. In Proceedings of the Twelfth Annual Symposium on Computational Geometry, SCG 7896, pages 98-105, New York, NY, USA. ACM.
- Lang, R. J. (2012). Origami Design Secrets: Mathematical Methods for an Ancient Art. CRC Press.
- Levenshtein, V. I. (1966). Binary codes capable of correcting deletions, insertions, and reversals. Soviet Physics Doklady, 10(8):707-710.
- Liapi, K. A. (2002). Transformable architecture inspired by the origami art: Computer visualization as a tool for form exploration. In Proceedings of the 2002 Annual Conference of the Association for Computer Aided Design In Architecture, pages 381-388.
- McArthur, M. and Lang, R. J. (2013). Folding Paper: The Infinite Possibilities of Origami. Tuttle Publishing, Hong Kong.
- Mitani, J. (2011). Oripa origami pattern editor. http://mitani.cs.tsukuba.ac.jp/oripa.
- Mohamad, M., Rappaport, D., and Toussaint, G. T. (2014). Minimum many-to-many matchings for computing the distance between two sequences. Graphs and Combinatorics.
- Munkres, J. (1957). Algorithms for the assignment and transportation problems. Journal of the Society of Industrial and Applied Mathematics, 5(1):32-38.
- O'Rourke, J. (2011). How to Fold It: The Mathematics of Linkages, Origami and Polyhedra. Cambridge University Press, New York.
- Pach, J. (2004). Towards a Theory of Geometric Graphs. No. 342. American Mathematical Society.
- Post, O. and Toussaint, G. T. (2011). The edit distance as a measure of perceived rhythmic similarity. Empirical Musicology Review, 6(3):164-179.
- Robles-Kelly, A. and Hancock, E. R. (2005). Graph edit distance from spectral seriation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 27(3):365-378.
- Tachi, T. (2010). Rigid-foldable structure using bidirectionally flat-foldable planar quadrilateral mesh. In Advances in Architectural Geometry, pages 87-102.
- Wu, W. and You, Z. (2011). A solution for folding rigid tall shopping bags. Proceedings of the Royal Society - A, pages 1-14.

#### Paper Citation

#### in Harvard Style

Oh S., T. Toussaint G., D. Demaine E. and L. Demaine M. (2015). **A Dissimilarity Measure for Comparing Origami Crease Patterns** . In *Proceedings of the International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,* ISBN 978-989-758-076-5, pages 386-393. DOI: 10.5220/0005291203860393

#### in Bibtex Style

@conference{icpram15,

author={Seung Man Oh and Godfried T. Toussaint and Erik D. Demaine and Martin L. Demaine},

title={A Dissimilarity Measure for Comparing Origami Crease Patterns},

booktitle={Proceedings of the International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,},

year={2015},

pages={386-393},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005291203860393},

isbn={978-989-758-076-5},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,

TI - A Dissimilarity Measure for Comparing Origami Crease Patterns

SN - 978-989-758-076-5

AU - Oh S.

AU - T. Toussaint G.

AU - D. Demaine E.

AU - L. Demaine M.

PY - 2015

SP - 386

EP - 393

DO - 10.5220/0005291203860393