Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ℝn to the processing of motion o...

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Published in:	Computer graphics forum Vol. 35; no. 2; pp. 295 - 305
Main Authors:	Brandt, Christopher, von Tycowicz, Christoph, Hildebrandt, Klaus
Format:	Journal Article
Language:	English
Published:	Oxford Blackwell Publishing Ltd 01.05.2016
Subjects:	Algorithms Analysis Animation Categories and Subject Descriptors (according to ACM CCS) Computation Computer animation Curves (geometry) Deformation Formability Geodesy I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Physically based modeling Image processing systems Smoothing Studies
ISSN:	0167-7055, 1467-8659
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Abstract	We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ℝn to the processing of motion of non‐rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced‐order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non‐linear “Bézier curves” by executing de Casteljau's algorithm in shape space.
AbstractList	We introduce techniques for the processing of motion and animations of non-rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in n to the processing of motion of non-rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced-order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non-linear "Bézier curves" by executing de Casteljau's algorithm in shape space. We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ℝ n to the processing of motion of non‐rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced‐order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non‐linear “Bézier curves” by executing de Casteljau's algorithm in shape space. We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ℝn to the processing of motion of non‐rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced‐order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non‐linear “Bézier curves” by executing de Casteljau's algorithm in shape space. We introduce techniques for the processing of motion and animations of non-rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in super(n) to the processing of motion of non-rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced-order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non-linear "Bezier curves" by executing de Casteljau's algorithm in shape space.
Author	von Tycowicz, Christoph Brandt, Christopher Hildebrandt, Klaus
Author_xml	– sequence: 1 givenname: Christopher surname: Brandt fullname: Brandt, Christopher organization: Delft University of Technology – sequence: 2 givenname: Christoph surname: von Tycowicz fullname: von Tycowicz, Christoph organization: Zuse Institute Berlin – sequence: 3 givenname: Klaus surname: Hildebrandt fullname: Hildebrandt, Klaus organization: Delft University of Technology
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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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W., Popović J.: Deformation transfer for triangle meshes. ACM Transactions on Graphics 23, 3 (2004), 399-405. 2 Heeren B., Rumpf M., Wardetzky M., Wirth B.: Time-discrete geodesics in the space of shells. Computer Graphics Forum 31, 5 (2012), 1755-1764. 1, 2, 4, 5, 7, 9 Kurtek S., Klassen E., Gore J., Ding Z., Srivastava A.: Elastic geodesic paths in shape space of parameterized surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 34, 9 (2012), 1717-1730. 2 Marsden J. E., Hughes T. J. R.: Mathematical Foundations of Elasticity. Dover Publications, 1994. 3 An S. S., Kim T., James D. L.: Optimizing cubature for efficient integration of subspace deformations. ACM Transactions on Graphics 27, 5 (2008), 1-10. 6 Wirth B., Bar L., Rumpf M., Sapiro G.: A continuum mechanical approach to geodesics in shape space. Int. J. Comput. Vision 93, 3 (2011), 293-318. 2 Chen R., Weber O., Keren D., Ben-Chen M.: Planar shape interpolation with bounded distortion. 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ACM Transactions on Graphics 34, 3 (2015), 34:1-34:10. 2, 3, 5, 6 LI S., Huang J., de Goes F., Jin X., Bao H., Desbrun M.: Space-time editing of elastic motion through material optimization and reduction. ACM Transactions on Graphics 33, 4 (2014), 108:1-108:10. 3 Winkler T., Drieseberg J., Alexa M., Hormann K.: Multi-scale geometry interpolation. Computer Graphics Forum 29, 2 (2010), 309-318. 2 LI H., Luo L., Vlasic D., Peers P., Popović J., Pauly M., Rusinkiewicz S.: Temporally coherent completion of dynamic shapes. ACM Transactions on Graphics 31, 1 (2012), 2. 2 Botsch M., Sorkine O.: On linear variational surface deformation methods. IEEE Transactions on Visualization and Computer Graphics 14, 1 (2008), 213-230. 2 Lipman Y., Sorkine O., Levin D., Cohen-Or D.: Linear rotation-invariant coordinates for meshes. 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References_xml	– reference: Barbič J., James D. L.: Real-time subspace integration for St. Venant-Kirchhoff deformable models. ACM Transactions on Graphics 24, 3 (2005), 982-990. 6 – reference: Kurtek S., Klassen E., Gore J., Ding Z., Srivastava A.: Elastic geodesic paths in shape space of parameterized surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 34, 9 (2012), 1717-1730. 2 – reference: Kilian M., Mitra N. J., Pottmann H.: Geometric modeling in shape space. ACM Transactions on Graphics 26, 3 (2007), 64:1-64:8. 1, 2 – reference: Hildebrandt K., Schulz C., von Tycowicz C., Polthier K.: Interactive spacetime control of deformable objects. ACM Transactions on Graphics 31, 4 (2012), 71:1-71:8. 3 – reference: Sumner R. W., Popović J.: Deformation transfer for triangle meshes. ACM Transactions on Graphics 23, 3 (2004), 399-405. 2 – reference: von Tycowicz C., Schulz C., Seidel H.-P., Hildebrandt K.: An efficient construction of reduced deformable objects. ACM Transactions on Graphics 32, 6 (2013), 213:1-10. 6 – reference: Kurtek S., Srivastava A., Klassen E., Laga H.: Landmark-guided elastic shape analysis of spherically-parameterized surfaces. Computer Graphics Forum 32, 2 (2013), 429-438. – reference: Heeren B., Rumpf M., Schröder P., Wardetzky M., Wirth B.: Exploring the geometry of the space of shells. Computer Graphics Forum 33, 5 (2014), 247-256. 2 – reference: Witkin A., Kass M.: Spacetime constraints. ACM SIGGRAPH 22 (1988), 159-168. 3 – reference: Rumpf M., Wirth B.: A nonlinear elastic shape averaging approach. SIAM Journal on Imaging Science 2, 3 (2009), 800-833. 3, 5 – reference: LI S., Huang J., de Goes F., Jin X., Bao H., Desbrun M.: Space-time editing of elastic motion through material optimization and reduction. ACM Transactions on Graphics 33, 4 (2014), 108:1-108:10. 3 – reference: von Tycowicz C., Schulz C., Seidel H.-P., Hildebrandt K.: Real-time nonlinear shape interpolation. ACM Transactions on Graphics 34, 3 (2015), 34:1-34:10. 2, 3, 5, 6 – reference: Younes L.: Shapes and Diffeomorphisms. Springer, 2010. 2 – reference: Chao I., Pinkall U., Sanan P., Schröder P.: A simple geometric model for elastic deformations. ACM Transactions on Graphics 29 (2010), 38:1-38:6. 3 – reference: LI H., Luo L., Vlasic D., Peers P., Popović J., Pauly M., Rusinkiewicz S.: Temporally coherent completion of dynamic shapes. ACM Transactions on Graphics 31, 1 (2012), 2. 2 – reference: Wirth B., Bar L., Rumpf M., Sapiro G.: A continuum mechanical approach to geodesics in shape space. Int. J. Comput. Vision 93, 3 (2011), 293-318. 2 – reference: An S. S., Kim T., James D. L.: Optimizing cubature for efficient integration of subspace deformations. ACM Transactions on Graphics 27, 5 (2008), 1-10. 6 – reference: Fröhlich S., Botsch M.: Example-driven deformations based on discrete shells. Computer Graphics Forum 30, 8 (2011), 2246-2257. 2 – reference: Barbič J., Sin F., Grinspun E.: Interactive editing of deformable simulations. ACM Transactions on Graphics 31, 4 (2012). 3 – reference: Heeren B., Rumpf M., Wardetzky M., Wirth B.: Time-discrete geodesics in the space of shells. Computer Graphics Forum 31, 5 (2012), 1755-1764. 1, 2, 4, 5, 7, 9 – reference: Barbič J., da Silva M., Popović J.: Deformable object animation using reduced optimal control. ACM Transactions on Graphics 28, 3 (2009), 53:1-53:9. 3 – reference: Botsch M., Sorkine O.: On linear variational surface deformation methods. IEEE Transactions on Visualization and Computer Graphics 14, 1 (2008), 213-230. 2 – reference: Lipman Y., Sorkine O., Levin D., Cohen-Or D.: Linear rotation-invariant coordinates for meshes. ACM Transactions on Graphics 24, 3 (2005), 479-487. 2 – reference: Marsden J. E., Hughes T. J. R.: Mathematical Foundations of Elasticity. 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Transactions on Graphics doi: 10.1145/2185520.2185566
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Snippet	We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in... We introduce techniques for the processing of motion and animations of non-rigid shapes. The idea is to regard animations of deformable objects as curves in...
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SubjectTerms	Algorithms Analysis Animation Categories and Subject Descriptors (according to ACM CCS) Computation Computer animation Curves (geometry) Deformation Formability Geodesy I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Physically based modeling Image processing systems Smoothing Studies
Title	Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects
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