Shape Matching via Quotient Spaces

We introduce a novel method for non‐rigid shape matching, designed to address the symmetric ambiguity problem present when matching shapes with intrinsic symmetries. Unlike the majority of existing methods which try to overcome this ambiguity by sampling a set of landmark correspondences, we address...

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Veröffentlicht in:	Computer graphics forum Jg. 32; H. 5; S. 1 - 11
Hauptverfasser:	Ovsjanikov, Maks, Mérigot, Quentin, Pătrăucean, Viorica, Guibas, Leonidas
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Oxford, UK Blackwell Publishing Ltd 01.08.2013 Wiley
Schlagworte:	Ambiguity Combinatorial analysis Computational Geometry Computer graphics Computer Science I.3.3 [Computer Graphics]:-Shape Analysis Information management Landmarks Matching Quotients Sampling Studies Symmetry Symmetry detection United States > US shape maps symmetry Shape matching
ISSN:	0167-7055, 1467-8659
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Abstract	We introduce a novel method for non‐rigid shape matching, designed to address the symmetric ambiguity problem present when matching shapes with intrinsic symmetries. Unlike the majority of existing methods which try to overcome this ambiguity by sampling a set of landmark correspondences, we address this problem directly by performing shape matching in an appropriate quotient space, where the symmetry has been identified and factored out. This allows us to both simplify the shape matching problem by matching between subspaces, and to return multiple solutions with equally good dense correspondences. Remarkably, both symmetry detection and shape matching are done without establishing any landmark correspondences between either points or parts of the shapes. This allows us to avoid an expensive combinatorial search present in most intrinsic symmetry detection and shape matching methods. We compare our technique with state‐of‐the‐art methods and show that superior performance can be achieved both when the symmetry on each shape is known and when it needs to be estimated.
AbstractList	We introduce a novel method for non‐rigid shape matching, designed to address the symmetric ambiguity problem present when matching shapes with intrinsic symmetries. Unlike the majority of existing methods which try to overcome this ambiguity by sampling a set of landmark correspondences, we address this problem directly by performing shape matching in an appropriate quotient space, where the symmetry has been identified and factored out. This allows us to both simplify the shape matching problem by matching between subspaces, and to return multiple solutions with equally good dense correspondences. Remarkably, both symmetry detection and shape matching are done without establishing any landmark correspondences between either points or parts of the shapes. This allows us to avoid an expensive combinatorial search present in most intrinsic symmetry detection and shape matching methods. We compare our technique with state‐of‐the‐art methods and show that superior performance can be achieved both when the symmetry on each shape is known and when it needs to be estimated. We introduce a novel method for non-rigid shape matching, designed to address the symmetric ambiguity problem present when matching shapes with intrinsic symmetries. Unlike the majority of existing methods which try to overcome this ambiguity by sampling a set of landmark correspondences, we address this problem directly by performing shape matching in an appropriate quotient space, where the symmetry has been identified and factored out. This allows us to both simplify the shape matching problem by matching between subspaces, and to return multiple solutions with equally good dense correspondences. Remarkably, both symmetry detection and shape matching are done without establishing any landmark correspondences between either points or parts of the shapes. This allows us to avoid an expensive combinatorial search present in most intrinsic symmetry detection and shape matching methods. We compare our technique with state-of-the-art methods and show that superior performance can be achieved both when the symmetry on each shape is known and when it needs to be estimated. [PUBLICATION ABSTRACT]
Author	Ovsjanikov, Maks Mérigot, Quentin Pătrăucean, Viorica Guibas, Leonidas
Author_xml	– sequence: 1 givenname: Maks surname: Ovsjanikov fullname: Ovsjanikov, Maks organization: LIX, École Polytechnique – sequence: 2 givenname: Quentin surname: Mérigot fullname: Mérigot, Quentin organization: LJK, Université Joseph Fourier / CNRS – sequence: 3 givenname: Viorica surname: Pătrăucean fullname: Pătrăucean, Viorica organization: LIX, École Polytechnique – sequence: 4 givenname: Leonidas surname: Guibas fullname: Guibas, Leonidas organization: Stanford University
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Cites_doi	10.1137/0206024 10.1145/1833349.1778840 10.1109/ICCVW.2011.6130444 10.1145/1531326.1531378 10.1145/359842.359859 10.1111/j.1467-8659.2008.01285.x 10.1111/j.1467-8659.2011.02024.x 10.1145/2185520.2185526 10.1073/pnas.0508601103 10.1109/CVPR.2009.5206775 10.1111/j.1467-8659.2011.01879.x 10.1111/j.1467-8659.2010.01764.x 10.1007/s11263-010-0320-3 10.1111/cgf.12007
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References	Lipman Y., Funkhouser T.: Möbius voting for surface correspondence. In Proc. of SIGGRAPH (2009), vol. 28:3, pp. 72:1-72:12. 1, 2. Huang Q.-X., Adams B., Wicke M., Guibas L. J.: Non-rigid registration under isometric deformations. CGF (Proc. SGP) 27, 5 (2008), 1449-1457. 1. Ovsjanikov M., Huang Q.-X., Guibas L. J.: A condition number for non-rigid shape matching. Comput. Graph. Forum (Proc. SGP) 30, 5 (2011), 1503-1512. 2. Sahillioglu Y., Yemez Y.: Coarse-to-fine isometric shape correspondence by tracking symmetric flips. Comput. Graph. Forum 32, 1 (2013), 177-189. 1, 2. Raviv D., Bronstein A. M., Bronstein M. M., Kimmel R.: Full and partial symmetries of non-rigid shapes. Int. J. Comput. Vision 89 (2010), 18-39. 2, 3. Knuth D. E., Morris J., Pratt V. R.: Fast pattern matching in strings. SIAM Journal of Computing 6, 2 (1977), 323-350. 2. Boyer R. S., Moore J. S.: A fast string searching algorithm. Commun. ACM 20 (October 1977). 2. Kim V. G., Lipman Y., Funkhouser T.: Blended intrinsic maps. ACM TOG (Proc. SIGGRAPH) 30, 4 (2011). 1, 2, 8, 9, 10, 11. Sun J., Ovsjanikov M., Guibas L.: A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion. CGF (Proc. SGP) 28, 5 (2009). 3. Ovsjanikov M., Merigot Q., Memoli F., Guibas L.: One point isometric matching with the heat kernel. CGF 29, 5 (2010), 1555-1564. 1, 2. Bronstein A., Bronstein M., Kimmel R.: Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. PNAS 103, 5 (2006). 1, 11. Bronstein A., Bronstein M., Kimmel R.: Numerical Geometry of Non-Rigid Shapes. Springer, 2008. 8. Ovsjanikov M., Ben-Chen M., Solomon J., Butscher A., Guibas L.: Functional maps: a flexible representation of maps between shapes. ACM Trans. Graph. 31, 4 (July 2012), 30:1-30:11. 1, 2, 3, 5, 6, 8, 9, 10, 11. 2010; 89 July 2012; 31 2012 2011 2013; 32 2010 2010; 29 2008; 27 October 1977; 20 2009 2008 2011; 30 2003 2006; 103 1977; 6 2009; 28 e_1_2_10_11_2 e_1_2_10_22_2 e_1_2_10_8_2 e_1_2_10_12_2 e_1_2_10_20_2 e_1_2_10_10_2 Sun J. (e_1_2_10_19_2) 2009; 28 e_1_2_10_21_2 Chazal F. (e_1_2_10_6_2) 2011 Bronstein A. (e_1_2_10_4_2) 2008 Boyer R. S. (e_1_2_10_5_2) 1977; 20 e_1_2_10_3_2 e_1_2_10_17_2 e_1_2_10_2_2 e_1_2_10_18_2 e_1_2_10_15_2 Kim V. G. (e_1_2_10_9_2) 2011; 30 e_1_2_10_16_2 e_1_2_10_7_2 e_1_2_10_13_2 e_1_2_10_14_2
References_xml	– reference: Boyer R. S., Moore J. S.: A fast string searching algorithm. Commun. ACM 20 (October 1977). 2. – reference: Ovsjanikov M., Ben-Chen M., Solomon J., Butscher A., Guibas L.: Functional maps: a flexible representation of maps between shapes. ACM Trans. Graph. 31, 4 (July 2012), 30:1-30:11. 1, 2, 3, 5, 6, 8, 9, 10, 11. – reference: Sun J., Ovsjanikov M., Guibas L.: A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion. CGF (Proc. SGP) 28, 5 (2009). 3. – reference: Bronstein A., Bronstein M., Kimmel R.: Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. PNAS 103, 5 (2006). 1, 11. – reference: Sahillioglu Y., Yemez Y.: Coarse-to-fine isometric shape correspondence by tracking symmetric flips. Comput. Graph. Forum 32, 1 (2013), 177-189. 1, 2. – reference: Kim V. G., Lipman Y., Funkhouser T.: Blended intrinsic maps. ACM TOG (Proc. SIGGRAPH) 30, 4 (2011). 1, 2, 8, 9, 10, 11. – reference: Ovsjanikov M., Merigot Q., Memoli F., Guibas L.: One point isometric matching with the heat kernel. CGF 29, 5 (2010), 1555-1564. 1, 2. – reference: Knuth D. E., Morris J., Pratt V. R.: Fast pattern matching in strings. SIAM Journal of Computing 6, 2 (1977), 323-350. 2. – reference: Ovsjanikov M., Huang Q.-X., Guibas L. J.: A condition number for non-rigid shape matching. Comput. Graph. Forum (Proc. SGP) 30, 5 (2011), 1503-1512. 2. – reference: Bronstein A., Bronstein M., Kimmel R.: Numerical Geometry of Non-Rigid Shapes. Springer, 2008. 8. – reference: Huang Q.-X., Adams B., Wicke M., Guibas L. J.: Non-rigid registration under isometric deformations. CGF (Proc. SGP) 27, 5 (2008), 1449-1457. 1. – reference: Lipman Y., Funkhouser T.: Möbius voting for surface correspondence. In Proc. of SIGGRAPH (2009), vol. 28:3, pp. 72:1-72:12. 1, 2. – reference: Raviv D., Bronstein A. M., Bronstein M. M., Kimmel R.: Full and partial symmetries of non-rigid shapes. Int. J. Comput. Vision 89 (2010), 18-39. 2, 3. – year: 2011 – start-page: 97 year: 2011 end-page: 106 – volume: 30 issue: 4 year: 2011 publication-title: Blended intrinsic maps. ACM TOG (Proc. SIGGRAPH) – volume: 28 start-page: 72:1 issue: 3 year: 2009 end-page: 72:12 article-title: Möbius voting for surface correspondence publication-title: Proc. of SIGGRAPH – start-page: 8 year: 2008 – year: 2008 – year: 2003 – volume: 27 start-page: 1449 issue: 5 year: 2008 end-page: 1457 article-title: Non‐rigid registration under isometric deformations publication-title: CGF (Proc. SGP) – volume: 20 year: October 1977 article-title: A fast string searching algorithm publication-title: Commun. ACM – volume: 31 start-page: 30:1 issue: 4 year: July 2012 end-page: 30:11 article-title: Functional maps: a flexible representation of maps between shapes publication-title: ACM Trans. Graph. – volume: 30 start-page: 1503 issue: 5 year: 2011 end-page: 1512 article-title: A condition number for non‐rigid shape matching publication-title: Comput. Graph. Forum (Proc. SGP) – volume: 28 issue: 5 year: 2009 article-title: A Concise and Provably Informative Multi‐Scale Signature Based on Heat Diffusion publication-title: CGF (Proc. SGP) – volume: 6 start-page: 323 issue: 2 year: 1977 end-page: 350 article-title: Fast pattern matching in strings publication-title: SIAM Journal of Computing – volume: 103 issue: 5 year: 2006 article-title: Generalized multidimensional scaling: a framework for isometry‐invariant partial surface matching publication-title: PNAS – volume: 32 start-page: 177 issue: 1 year: 2013 end-page: 189 article-title: Coarse‐to‐fine isometric shape correspondence by tracking symmetric flips publication-title: Comput. Graph. Forum – volume: 89 start-page: 18 year: 2010 end-page: 39 article-title: Full and partial symmetries of non‐rigid shapes publication-title: Int. J. Comput. Vision – start-page: 1185 year: 2009 end-page: 1192 – year: 2010 – year: 2012 – volume: 29 start-page: 1555 issue: 5 year: 2010 end-page: 1564 article-title: One point isometric matching with the heat kernel publication-title: CGF – ident: e_1_2_10_10_2 doi: 10.1137/0206024 – ident: e_1_2_10_13_2 – start-page: 8 volume-title: Numerical Geometry of Non‐Rigid Shapes year: 2008 ident: e_1_2_10_4_2 – ident: e_1_2_10_11_2 doi: 10.1145/1833349.1778840 – ident: e_1_2_10_2_2 doi: 10.1109/ICCVW.2011.6130444 – ident: e_1_2_10_12_2 doi: 10.1145/1531326.1531378 – volume: 28 issue: 5 year: 2009 ident: e_1_2_10_19_2 article-title: A Concise and Provably Informative Multi‐Scale Signature Based on Heat Diffusion publication-title: CGF (Proc. SGP) – volume: 20 year: 1977 ident: e_1_2_10_5_2 article-title: A fast string searching algorithm publication-title: Commun. ACM doi: 10.1145/359842.359859 – ident: e_1_2_10_8_2 doi: 10.1111/j.1467-8659.2008.01285.x – ident: e_1_2_10_15_2 doi: 10.1111/j.1467-8659.2011.02024.x – ident: e_1_2_10_14_2 doi: 10.1145/2185520.2185526 – ident: e_1_2_10_3_2 doi: 10.1073/pnas.0508601103 – ident: e_1_2_10_7_2 – ident: e_1_2_10_21_2 doi: 10.1109/CVPR.2009.5206775 – ident: e_1_2_10_22_2 doi: 10.1111/j.1467-8659.2011.01879.x – start-page: 97 volume-title: Proc. SOCG year: 2011 ident: e_1_2_10_6_2 – ident: e_1_2_10_16_2 doi: 10.1111/j.1467-8659.2010.01764.x – ident: e_1_2_10_17_2 – ident: e_1_2_10_18_2 doi: 10.1007/s11263-010-0320-3 – volume: 30 issue: 4 year: 2011 ident: e_1_2_10_9_2 publication-title: Blended intrinsic maps. ACM TOG (Proc. SIGGRAPH) – ident: e_1_2_10_20_2 doi: 10.1111/cgf.12007
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SubjectTerms	Ambiguity Combinatorial analysis Computational Geometry Computer graphics Computer Science I.3.3 [Computer Graphics]:-Shape Analysis Information management Landmarks Matching Quotients Sampling Studies Symmetry Symmetry detection
Title	Shape Matching via Quotient Spaces
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