Near-Isometric Level Set Tracking

Implicit representations of geometry have found applications in shape modeling, simulation, and other graphics pipelines. These representations, however, do not provide information about the paths of individual points as shapes move and undergo deformation. For this reason, we reconsider the problem...

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Published in:	Computer graphics forum Vol. 35; no. 5; pp. 65 - 77
Main Authors:	Tao, Michael, Solomon, Justin, Butscher, Adrian
Format:	Journal Article
Language:	English
Published:	Oxford Blackwell Publishing Ltd 01.08.2016
Subjects:	Algorithms Analysis and systems Categories and Subject Descriptors (according to ACM CCS) Computer graphics Computer simulation Deformation Geometry Graphical representations I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems Image processing systems Information dissemination languages Mathematical analysis Representations Simulation Studies Tracking
ISSN:	0167-7055, 1467-8659
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Abstract	Implicit representations of geometry have found applications in shape modeling, simulation, and other graphics pipelines. These representations, however, do not provide information about the paths of individual points as shapes move and undergo deformation. For this reason, we reconsider the problem of tracking points on level set surfaces, with the goal of designing an algorithm that — unlike previous work — can recover rotational motion and nearly isometric deformation. We track points on level sets of a time‐varying function using approximate Killing vector fields (AKVFs), the velocity fields of near‐isometric motions. To this end, we provide suitable theoretical and discrete constructions for computing AKVFs in a narrow band surrounding an animated level set surface. Furthermore, we propose time integrators well‐suited to integrating AKVFs in time to track points. We demonstrate the theoretical and practical advantages of our proposed algorithms on synthetic and practical tasks.
AbstractList	Implicit representations of geometry have found applications in shape modeling, simulation, and other graphics pipelines. These representations, however, do not provide information about the paths of individual points as shapes move and undergo deformation. For this reason, we reconsider the problem of tracking points on level set surfaces, with the goal of designing an algorithm that — unlike previous work — can recover rotational motion and nearly isometric deformation. We track points on level sets of a time‐varying function using approximate Killing vector fields (AKVFs), the velocity fields of near‐isometric motions. To this end, we provide suitable theoretical and discrete constructions for computing AKVFs in a narrow band surrounding an animated level set surface. Furthermore, we propose time integrators well‐suited to integrating AKVFs in time to track points. We demonstrate the theoretical and practical advantages of our proposed algorithms on synthetic and practical tasks.
Author	Solomon, Justin Tao, Michael Butscher, Adrian
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Cites_doi	10.1006/jcph.2002.7166 10.2197/ipsjjip.21.674 10.1109/38.135885 10.1109/ICCV.2003.1238443 10.1145/2487228.2487235 10.1016/0021-9991(88)90002-2 10.1145/1966394.1966400 10.1201/b10635 10.1145/1061347.1061353 10.1145/2185520.2185526 10.1145/2723158 10.1016/B978-044451104-1/50021-6 10.1109/MCG.1983.263183 10.1006/jcph.1998.6090 10.1145/2661229.2661264 10.1090/mmono/174 10.1145/357306.357310 10.2307/1968813 10.1016/S0045-7825(02)00559-5 10.1145/2766963 10.1023/A:1025324613450 10.1016/j.jcp.2003.09.032 10.1299/jamdsm.2014jamdsm0012
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Copyright	2016 The Author(s) Computer Graphics Forum © 2016 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2016 The Eurographics Association and John Wiley & Sons Ltd.
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Cambridge university press, 1999. 5 – reference: Ovsjanikov M., Ben-Chen M., Solomon J., Butscher A., Guibas L.: Functional maps: a flexible representation of maps between shapes. ACM Transactions on Graphics (TOG) 31, 4 (2012), 30. 2 – reference: Wang M.Y., Wang X., Guo D.: A level set method for structural topology optimization. Computer methods in applied mechanics and engineering 192, 1 (2003), 227-246. 3 – reference: Fujisawa M., Mandachi Y., Miura K.T.: Calculation of velocity on an implicit surface by curvature invariance. Journal of Information Processing 21, 4 (2013), 674-680. 2, 7 – reference: Freudenthal H.: Simplizialzerlegung von beschränkter flachheit. Ann. of Math. 43 (1942), 580-582. 5 – reference: Hairer E., Lubich C., Wanner G.: Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, vol. 31. Springer Science & Business Media, 2006. 6 – reference: Enright D., Fedkiw R., Ferziger J., Mitchell I.: A hybrid particle level set method for improved interface capturing. Journal of Computational physics 183, 1 (2002), 83-116. 2 – reference: Blinn J.F.: A generalization of algebraic surface drawing. ACM Trans. Graph. 1, 3 (July 1982), 235-256. 1 – reference: Do Carmo M.: Differential Geometry. Prentice-Hall, 1992. 3 – reference: Dinh H.Q., Yezzi A., Turk G.: Texture transfer during shape transformation. ACM Transactions on Graphics (TOG) 24, 2 (2005), 289-310. 2 – reference: Munkres J.: Analysis on manifolds, ser. Advanced Book Classics. Addison-Wesley Pub. Co., Advanced Book Program, 1991. 3 – reference: Vaillant R., Guennebaud G., Barthe L., Wyvill B., Cani M.-P.: Robust iso-surface tracking for interactive character skinning. ACM Transactions on Graphics (TOG) 33, 6 (2014), 189. 2 – reference: Osher S., Sethian J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of computational physics 79, 1 (1988), 12-49. 2 – reference: Bridson R.: Fluid Simulation for Computer Graphics. Taylor & Francis, 2008. 1 – reference: Max N.: Computer representation of molecular surfaces. IEEE Comput. Graph. Appl. 3, 5 (1983), 21-29. 1 – reference: Stam J., Schmidt R.: On the velocity of an implicit surface. ACM Transactions on Graphics (TOG) 30, 3 (2011), 21. 1, 2, 7, 8, 11 – reference: Solomon J., De Goes F., Peyré G., Cuturi M., Butscher A., Nguyen A., Du T., Guibas L.: Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains. ACM Transactions on Graphics (TOG) 34, 4 (2015), 66. 10, 12 – reference: Azencot O., Ovsjanikov M., Chazal F., Ben-Chen M.: Discrete derivatives of vector fields on surfaces-an operator approach. ACM Transactions on Graphics (TOG) 34, 3 (2015), 29. 2 – reference: Payne B.A., Toga A.W.: Distance field manipulation of surface models. IEEE Comput. Graph. Appl. 12, 1 (Jan. 1992), 65-71. 1 – reference: Museth K.: Vdb: High-resolution sparse volumes with dynamic topology. ACM Transactions on Graphics (TOG) 32, 3 (2013), 27. 5 – reference: Mandachi Y., Usuki S., Miura K.T.: Velocity calculation of 2d geometric objects by use of surface interpolation in 3d. Journal of Advanced Mechanical Design, Systems, and Manufacturing 8, 2 (2014), JAMDSM0012-JAMDSM0012. 2 – volume: 194 start-page: 363 issue: 1 year: 2004 end-page: 393 article-title: Structural optimization using sensitivity analysis and a level‐set method publication-title: Journal of computational physics – volume: 32 start-page: 27 issue: 3 year: 2013 article-title: Vdb: High‐resolution sparse volumes with dynamic topology publication-title: ACM Transactions on Graphics (TOG) – volume: 8 start-page: JAMDSM0012 issue: 2 year: 2014 end-page: JAMDSM0012 article-title: Velocity calculation of 2d geometric objects by use of surface interpolation in 3d publication-title: Journal of Advanced Mechanical Design, Systems, and Manufacturing – volume: 5 article-title: Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science publication-title: vol. 3. 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Graph. Appl. – volume: 34 start-page: 66 issue: 4 year: 2015 article-title: Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains publication-title: ACM Transactions on Graphics (TOG) – volume: 1 start-page: 235 issue: 3 year: 1982 end-page: 256 article-title: A generalization of algebraic surface drawing publication-title: ACM Trans. Graph. – volume: 2007 start-page: 3 year: 2007 end-page: 9 – volume: 24 start-page: 289 issue: 2 year: 2005 end-page: 310 article-title: Texture transfer during shape transformation publication-title: ACM Transactions on Graphics (TOG) – year: 2002 – year: 2008 – volume: 30 start-page: 21 issue: 3 year: 2011 article-title: On the velocity of an implicit surface publication-title: ACM Transactions on Graphics (TOG) – volume: 43 start-page: 580 year: 1942 end-page: 582 article-title: Simplizialzerlegung von beschränkter flachheit publication-title: Ann. of Math. – volume: 19 start-page: 439 issue: 1 year: 2003 end-page: 456 article-title: Semi‐implicit level set methods for curvature and surface diffusion motion publication-title: Journal of Scientific Computing – volume: 148 start-page: 2 issue: 1 year: 1999 end-page: 22 article-title: The fast construction of extension velocities in level set methods publication-title: Journal of Computational Physics – volume: 3 start-page: 21 issue: 5 year: 1983 end-page: 29 article-title: Computer representation of molecular surfaces publication-title: IEEE Comput. Graph. Appl. – volume: 79 start-page: 12 issue: 1 year: 1988 end-page: 49 article-title: Fronts propagating with curvature‐dependent speed: algorithms based on Hamilton‐Jacobi formulations publication-title: Journal of computational physics – start-page: 269 year: 1994 end-page: 277 – start-page: 894 year: 2003 end-page: 899 – year: 1991 – volume: 31 start-page: 30 issue: 4 year: 2012 article-title: Functional maps: a flexible representation of maps between shapes publication-title: ACM Transactions on Graphics (TOG) – volume: 192 start-page: 227 issue: 1 year: 2003 end-page: 246 article-title: A level set method for structural topology optimization publication-title: Computer methods in applied mechanics and engineering – volume: 32 start-page: 73 year: 2013 end-page: 82 – volume: 29 start-page: 1701 year: 2010 end-page: 1711 – volume: 6 article-title: Geometric numerical integration: structure‐preserving algorithms for ordinary differential equations publication-title: vol. 31. 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Snippet	Implicit representations of geometry have found applications in shape modeling, simulation, and other graphics pipelines. These representations, however, do...
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SubjectTerms	Algorithms Analysis and systems Categories and Subject Descriptors (according to ACM CCS) Computer graphics Computer simulation Deformation Geometry Graphical representations I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems Image processing systems Information dissemination languages Mathematical analysis Representations Simulation Studies Tracking
Title	Near-Isometric Level Set Tracking
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