The DOI matches its Crossref record; 2 of the 2 references it cites were found in a registry or an index.
Geometric modeling in shape space
Abstract
We present a novel framework to treat shapes in the setting of Riemannian geometry. Shapes -- triangular meshes or more generally straight line graphs in Euclidean space -- are treated as points in a shape space. We introduce useful Riemannian metrics in this space to aid the user in design and modeling tasks, especially to explore the space of (approximately) isometric deformations of a given shape. Much of the work relies on an efficient algorithm to compute geodesics in shape spaces; to this end, we present a multi-resolution framework to solve the interpolation problem -- which amounts to solving a boundary value problem -- as well as the extrapolation problem -- an initial value problem -- in shape space. Based on these two operations, several classical concepts like parallel transport and the exponential map can be used in shape space to solve various geometric modeling and geometry processing tasks. Applications include shape morphing, shape deformation, deformation transfer, and intuitive shape exploration.
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Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“A deformation of a shape represented by a triangular mesh is isometric if and only if all edge lengths are preserved during the deformation [9].”
From MORPHING OF TRIANGULAR MESHES IN SHAPE SPACE (2010) · cites this paper as [9]
In the citing paper
“All of the newly presented approaches solve the morphing problem without the need to solve a minimization problem.”
From MORPHING OF TRIANGULAR MESHES IN SHAPE SPACE (2010) · cites this paper as [9]
In the citing paper
“Recently, Kilian et al. [9] used shape space representations to guide morphs and other more general deformations between shapes represented as triangular meshes. Each shape is represented by a point in a high-dimensional shape space and deformations are modeled as geodesics in shape space. The geodesic paths in shape space are found using an energy-minimization approach.”
From MORPHING OF TRIANGULAR MESHES IN SHAPE SPACE (2010) · cites this paper as [9]
In the citing paper
“Just as in the algorithm by Kilian et al. [9], the results depend heavily on the initial rigid alignment of the shapes.”
From MORPHING OF TRIANGULAR MESHES IN SHAPE SPACE (2010) · cites this paper as [9]
Citations
Cites
SCAPE: shape completion and animation of people
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The space of human body shapes: reconstruction and parameterization from range scans
Allen, B., Curless, B., & Popović, Z. (2003) · ACM SIGGRAPH 2003 Papers · doi:10.1145/1201775.882311 · Cited by 913
Flagged first, then by citations
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Cited by
Sorted by citations
Shape Analysis of Elastic Curves in Euclidean Spaces
Srivastava, A., Klassen, E., Joshi, S. H., & Jermyn, I. H. (2011) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2010.184 · Cited by 683
Optimization Methods on Riemannian Manifolds and Their Application to Shape Space
Ring, W., & Wirth, B. (2012) · SIAM Journal on Optimization · doi:10.1137/11082885x · Cited by 342
“introduced a Riemannian shape space in [13], in which shapes are represented by meshes with fixed connectivity.”
BackgroundA simple geometric model for elastic deformations
Chao, I., Pinkall, U., Sanan, P., & Schröder, P. (2010) · ACM Transactions on Graphics · doi:10.1145/1833349.1778775 · Cited by 307
One Point Isometric Matching with the Heat Kernel
Ovsjanikov, M., Mérigot, Q., Mémoli, F., & Guibas, L. (2010) · Computer Graphics Forum · doi:10.1111/j.1467-8659.2010.01764.x · Cited by 301
“High quality correspondences are also essential in morphing, and shape interpolation [KMP07].”
Non‐Rigid Registration Under Isometric Deformations
Huang, Q., Adams, B., Wicke, M., & Guibas, L. J. (2008) · Computer Graphics Forum · doi:10.1111/j.1467-8659.2008.01285.x · Cited by 279
Metadata
- Authors
- Kilian, M., Mitra, N. J., & Pottmann, H.
- Affiliations
- TU Wien (AT)
- Venue
- ACM SIGGRAPH 2007 papers
- Publisher
- ACM
- Published
- 2007-07-29
- Pages
- 64
- OpenAlex
- W2154084374
- Cited by
- 224
- References
- 0
- Citations by year
- 2026: 5 · 2025: 3 · 2024: 4 · 2023: 6 · 2022: 6 · 2021: 11 · 2020: 12 · 2019: 12 · 2018: 10 · 2017: 14 · 2016: 11 · 2015: 19 · 2014: 16 · 2013: 12 · 2012: 18
- FWCI
- 37.03
- Type
- conference paper
- Language
- en
- Open access
- closed
- Locations
- ACM SIGGRAPH Conference (SIGGRAPH) (publishedVersion)
- UCL Discovery (University College London) (submittedVersion)
- Topics
- Physical Sciences › Computer Science › Computer Graphics and Computer-Aided Design › 3D Shape Modeling and Analysis
- Physical Sciences › Computer Science › Computer Graphics and Computer-Aided Design › Computational Geometry and Mesh Generation
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Keywords
- shape space, Riemannian metrics, triangular meshes, straight line graphs, geodesics, boundary value problems, initial value problems, parallel transport, exponential map, shape morphing, shape deformation, deformation transfer, shape exploration, isometric deformations
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