Computing Minimum Area Homologies
Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a simila...
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| Vydáno v: | Computer graphics forum Ročník 34; číslo 6; s. 13 - 21 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Oxford
Blackwell Publishing Ltd
01.09.2015
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| ISSN: | 0167-7055, 1467-8659, 1467-8659 |
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| Abstract | Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure.
Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure. |
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| AbstractList | Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure. Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure. Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure. Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure. Calculating and categorizing the similarity of curves is a fundamental problem which has generated much recent interest. However, to date there are no implementations of these algorithms for curves on surfaces with provable guarantees on the quality of the measure. In this paper, we present a similarity measure for any two cycles that are homologous, where we calculate the minimum area of any homology (or connected bounding chain) between the two cycles. The minimum area homology exists for broader classes of cycles than previous measures which are based on homotopy. It is also much easier to compute than previously defined measures, yielding an efficient implementation that is based on linear algebra tools. We demonstrate our algorithm on a range of inputs, showing examples which highlight the feasibility of this similarity measure. |
| Author | Vejdemo-Johansson, Mikael Chambers, Erin Wolf |
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| Cites_doi | 10.1006/jsco.1996.0125 10.4310/jdg/1214439286 10.1007/b97315 10.1145/2462356.2462375 10.1016/S0007-4497(02)01119-3 10.1007/978-3-319-04099-8_7 10.1002/cpa.20132 10.1137/100800245 10.1109/78.258082 10.1109/CVPR.1996.517122 10.1109/ACSSC.1993.342465 10.1109/SMI.2004.1314502 10.1142/S0218195995000064 10.1016/j.comgeo.2009.02.008 10.1145/1868237.1868247 10.1090/conm/470/09186 10.1007/978-3-662-30697-0 10.1007/978-3-642-03456-5_16 |
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| References | [BCP97] Bosma W., Cannon J., Playoust C.: The magma algebra system I: The user language. Journal of Symbolic Computation 24, 3 (1997), 235-265. [AS71] Atiyah M., Singer I.: The index of elliptic operators on compact. Bulletin of the American Mathematical Society 93 (1971), 119-138. [HSB66] Hirzebruch F., Schwarzenberger R. L., Borel A.: Topological Methods in Algebraic Geometry, vol. 232. Springer, New York, NY, 1966. [DHK11] Dey T., Hirani A., Krishnamoorthy B.: Optimal homologous cycles, total unimodularity, and linear programming. SIAM Journal on Computing 40, 4 (2011), 1026-1044. http://epubs.siam.org/doi/abs/10.1137/100800245, http://arxiv.org/abs/epubs.siam.org/doi/pdf/10.1137/100800245 arXiv:http://epubs.siam.org/doi/pdf/10.1137/100800245, http://dx.doi.org/10.1137/100800245 doi:10.1137/100800245. [CCE*10] Chambers E. W., Colin de Verdière É., Erickson J., Lazard S., Lazarus F., Thite S.: Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time. Computational Geometry 43, 3 (2010), 295-311. Special Issue on 24th Annual Symposium on Computational Geometry (SoCG' 08). http://www.sciencedirect.com/science/article/pii/S0925772109000637, http://dx.doi.org/10.1016/j.comgeo.2009.02.008 doi:10.1016/j.comgeo.2009.02.008. [PVG*11] Pedregosa F., Varoquaux G., Gramfort A., Michel V., Thirion B., Grisel O., Blondel M., Prettenhofer P., Weiss R., Dubourg V., Vanderplas J., Passos A., Cournapeau D., Brucher M., Perrot M., Duchesnay E.: Scikit-learn: Machine learning in Python. Journal of Machine Learning Research 12 (2011), 2825-2830. [RS02] Rubio J., Sergeraert F.: Constructive algebraic topology. Bulletin des Sciences Mathématiques 126, 5 (2002), 389-412. [KMM04] Kaczynski T., Mischaikow K., Mrozek M.: Computational Homology, vol. 157. Springer, 2004. [Mun00] Munkres J. R.: Topology (2nd edition). Pearson Education, Upper Saddle River, NJ. 2000. [Don06] Donoho D. L.: For most large underdetermined systems of linear equations the minimal ð1-norm solution is also the sparsest solution. Communications on Pure and Applied Mathematics 59, 6 (2006), 797-829. [Hat02] Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge, UK, 2002. http://www.math.cornell.edu/hatcher/AT/ATpage.html. [Law80] Lawson H.: Lectures on Minimal Submanifolds, vol. 1 of Mathematics Lecture Series. Publish or Perish, Houston, TX, 1980. http://books.google.com/books?id=vlbvAAAAMAAJ. [SJ05] Stein W., Joyner D.: Sage: System for algebra and geometry experimentation. Communications in Computer Algebra (SIGSAM Bulletin) (2005). http://sage.sourceforge.net. Accessed 1 October 2012. [TV04] Tangelder J., Veltkamp R.: A survey of content based 3D shape retrieval methods. In Proceedings of Shape Modeling Applications (June 2004), IEEE, pp. 145-156. http://dx.doi.org/10.1109/SMI.2004.1314502 doi:10.1109/SMI.2004.1314502. [VJ12] Vejdemo-Johansson M.: Gap persistence-a computational topology package for gap. Book of Abstracts Minisymposium on Publicly Available Geometric/Topological Software (2012), 43. [AG95] Alt H., Godau M.: Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry and its Applications 5 (1995), 75-91. [CW10] Cook A. F., Wenk C.: Geodesic Fréchet distance inside a simple polygon. ACM Transactions on Algorithms 7 (2010). 1-19. [EG08] Ellis G.: Homological algebra programming. In Computational Group Theory and the Theory of Groups 470; Americal Mathematical Society, Providence,RI. (2008), pp. 63-74. [RZE08] Rubinstein R., Zibulevsky M., Elad M.: Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit. CS Technion 40, 8 (2008), 1-15. [Whi84] White B.: Mappings that minimize area in their homotopy classes. Journal of Differential Geometry 20, 2 (1984), 433-446. http://projecteuclid.org/euclid.jdg/1214439286. [Mil07] Milnor J.: A survey of cobordism theory. Collected Papers of John Milnor: Differential topology 3 (2007), 291. 1984; 20 1966; 232 2012 1997; 24 2011; 40 2006; 59 1996 2005 2011; 12 2004 2002 2009; 5760 1993; 1 1995; 5 2010; 43 1971; 93 2004; 157 2001 2000 1980; 1 2002; 126 2014 2013 2007; 3 2008; 40 2010; 7 2008; 470 e_1_2_7_6_1 e_1_2_7_4_1 e_1_2_7_3_1 e_1_2_7_9_1 Hatcher A. (e_1_2_7_18_1) 2002 Pedregosa F. (e_1_2_7_29_1) 2011; 12 e_1_2_7_8_1 e_1_2_7_7_1 e_1_2_7_19_1 e_1_2_7_17_1 Lawson H. (e_1_2_7_22_1) 1980 Rubinstein R. (e_1_2_7_31_1) 2008; 40 e_1_2_7_16_1 Stein W. (e_1_2_7_33_1) 2005 e_1_2_7_2_1 e_1_2_7_15_1 e_1_2_7_14_1 e_1_2_7_13_1 e_1_2_7_12_1 e_1_2_7_11_1 e_1_2_7_10_1 e_1_2_7_26_1 e_1_2_7_27_1 e_1_2_7_28_1 Atiyah M. (e_1_2_7_5_1) 1971; 93 Milnor J. (e_1_2_7_23_1) 2007; 3 White B. (e_1_2_7_39_1) 1984; 20 Munkres J. R. (e_1_2_7_25_1) 2000 e_1_2_7_30_1 e_1_2_7_24_1 e_1_2_7_32_1 e_1_2_7_34_1 Tausz A. (e_1_2_7_37_1) 2012 e_1_2_7_21_1 e_1_2_7_35_1 e_1_2_7_20_1 e_1_2_7_36_1 Vejdemo‐Johansson M. (e_1_2_7_38_1) 2012 |
| References_xml | – reference: [DHK11] Dey T., Hirani A., Krishnamoorthy B.: Optimal homologous cycles, total unimodularity, and linear programming. SIAM Journal on Computing 40, 4 (2011), 1026-1044. http://epubs.siam.org/doi/abs/10.1137/100800245, http://arxiv.org/abs/epubs.siam.org/doi/pdf/10.1137/100800245 arXiv:http://epubs.siam.org/doi/pdf/10.1137/100800245, http://dx.doi.org/10.1137/100800245 doi:10.1137/100800245. – reference: [Mun00] Munkres J. R.: Topology (2nd edition). Pearson Education, Upper Saddle River, NJ. 2000. – reference: [BCP97] Bosma W., Cannon J., Playoust C.: The magma algebra system I: The user language. Journal of Symbolic Computation 24, 3 (1997), 235-265. – reference: [Don06] Donoho D. L.: For most large underdetermined systems of linear equations the minimal ð1-norm solution is also the sparsest solution. Communications on Pure and Applied Mathematics 59, 6 (2006), 797-829. – reference: [PVG*11] Pedregosa F., Varoquaux G., Gramfort A., Michel V., Thirion B., Grisel O., Blondel M., Prettenhofer P., Weiss R., Dubourg V., Vanderplas J., Passos A., Cournapeau D., Brucher M., Perrot M., Duchesnay E.: Scikit-learn: Machine learning in Python. Journal of Machine Learning Research 12 (2011), 2825-2830. – reference: [AS71] Atiyah M., Singer I.: The index of elliptic operators on compact. Bulletin of the American Mathematical Society 93 (1971), 119-138. – reference: [TV04] Tangelder J., Veltkamp R.: A survey of content based 3D shape retrieval methods. In Proceedings of Shape Modeling Applications (June 2004), IEEE, pp. 145-156. http://dx.doi.org/10.1109/SMI.2004.1314502 doi:10.1109/SMI.2004.1314502. – reference: [VJ12] Vejdemo-Johansson M.: Gap persistence-a computational topology package for gap. Book of Abstracts Minisymposium on Publicly Available Geometric/Topological Software (2012), 43. – reference: [KMM04] Kaczynski T., Mischaikow K., Mrozek M.: Computational Homology, vol. 157. Springer, 2004. – reference: [CCE*10] Chambers E. W., Colin de Verdière É., Erickson J., Lazard S., Lazarus F., Thite S.: Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time. Computational Geometry 43, 3 (2010), 295-311. Special Issue on 24th Annual Symposium on Computational Geometry (SoCG' 08). http://www.sciencedirect.com/science/article/pii/S0925772109000637, http://dx.doi.org/10.1016/j.comgeo.2009.02.008 doi:10.1016/j.comgeo.2009.02.008. – reference: [Hat02] Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge, UK, 2002. http://www.math.cornell.edu/hatcher/AT/ATpage.html. – reference: [RZE08] Rubinstein R., Zibulevsky M., Elad M.: Efficient implementation of the K-SVD algorithm using batch orthogonal matching pursuit. CS Technion 40, 8 (2008), 1-15. – reference: [AG95] Alt H., Godau M.: Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry and its Applications 5 (1995), 75-91. – reference: [EG08] Ellis G.: Homological algebra programming. In Computational Group Theory and the Theory of Groups 470; Americal Mathematical Society, Providence,RI. (2008), pp. 63-74. – reference: [Mil07] Milnor J.: A survey of cobordism theory. Collected Papers of John Milnor: Differential topology 3 (2007), 291. – reference: [CW10] Cook A. F., Wenk C.: Geodesic Fréchet distance inside a simple polygon. ACM Transactions on Algorithms 7 (2010). 1-19. – reference: [Law80] Lawson H.: Lectures on Minimal Submanifolds, vol. 1 of Mathematics Lecture Series. Publish or Perish, Houston, TX, 1980. http://books.google.com/books?id=vlbvAAAAMAAJ. – reference: [RS02] Rubio J., Sergeraert F.: Constructive algebraic topology. Bulletin des Sciences Mathématiques 126, 5 (2002), 389-412. – reference: [SJ05] Stein W., Joyner D.: Sage: System for algebra and geometry experimentation. Communications in Computer Algebra (SIGSAM Bulletin) (2005). http://sage.sourceforge.net. Accessed 1 October 2012. – reference: [HSB66] Hirzebruch F., Schwarzenberger R. L., Borel A.: Topological Methods in Algebraic Geometry, vol. 232. Springer, New York, NY, 1966. – reference: [Whi84] White B.: Mappings that minimize area in their homotopy classes. Journal of Differential Geometry 20, 2 (1984), 433-446. http://projecteuclid.org/euclid.jdg/1214439286. – volume: 470 start-page: 63 year: 2008 end-page: 74 article-title: Homological algebra programming publication-title: Computational Group Theory and the Theory of Groups – volume: 126 start-page: 389 issue: 5 year: 2002 end-page: 412 article-title: Constructive algebraic topology publication-title: Bulletin des Sciences Mathématiques – volume: 7 start-page: 1 year: 2010 end-page: 19 article-title: Geodesic Fréchet distance inside a simple polygon publication-title: ACM Transactions on Algorithms – volume: 1 year: 1980 – volume: 40 start-page: 1026 issue: 4 year: 2011 end-page: 1044 article-title: Optimal homologous cycles, total unimodularity, and linear programming publication-title: SIAM Journal on Computing – volume: 12 start-page: 2825 year: 2011 end-page: 2830 article-title: Scikit‐learn: Machine learning in Python publication-title: Journal of Machine Learning Research – 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| SubjectTerms | Algorithms Analysis Chains Clustering algorithms Computation computational geometry Computer graphics curves and surfaces Efficient implementation Feasibility Geometry Homology Homotopies I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling-Geometric algorithms I.5.3 [Pattern Recognition]: Clustering-Similarity measures Image processing systems languages and systems languages and systems; I.5.3 [Pattern Recognition]: Clustering–Similarity measures Linear algebra Linear algebra tools Mathematical analysis Object modelling Pattern recognition Pattern recognition systems Similarity Similarity measure Similarity measures Studies Topological manifolds weird math |
| Title | Computing Minimum Area Homologies |
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