Functional Characterization of Intrinsic and Extrinsic Geometry
We propose a novel way to capture and characterize distortion between pairs of shapes by extending the recently proposed framework of shape differences built on functional maps. We modify the original definition of shape differences slightly and prove that after this change, the discrete metric is f...
| Main Authors: | , , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Association for Computing Machinery (ACM)
2017
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| Online Access: | http://hdl.handle.net/1721.1/111016 |
| _version_ | 1811097988236312576 |
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| author | Corman, Etienne Ben-Chen, Mirela Guibas, Leonidas Ovsjanikov, Maks Solomon, Justin |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Corman, Etienne Ben-Chen, Mirela Guibas, Leonidas Ovsjanikov, Maks Solomon, Justin |
| author_sort | Corman, Etienne |
| collection | MIT |
| description | We propose a novel way to capture and characterize distortion between pairs of shapes by extending the recently proposed framework of shape differences built on functional maps. We modify the original definition of shape differences slightly and prove that after this change, the discrete metric is fully encoded in two shape difference operators and can be recovered by solving two linear systems of equations. Then we introduce an extension of the shape difference operators using offset surfaces to capture extrinsic or embedding-dependent distortion, complementing the purely intrinsic nature of the original shape differences. Finally, we demonstrate that a set of four operators is complete, capturing intrinsic and extrinsic structure and fully encoding a shape up to rigid motion in both discrete and continuous settings. We highlight the usefulness of our constructions by showing the complementary nature of our extrinsic shape differences in capturing distortion ignored by previous approaches. We additionally provide examples where we recover local shape structure from the shape difference operators, suggesting shape editing and analysis tools based on manipulating shape differences. |
| first_indexed | 2024-09-23T17:08:11Z |
| format | Article |
| id | mit-1721.1/111016 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T17:08:11Z |
| publishDate | 2017 |
| publisher | Association for Computing Machinery (ACM) |
| record_format | dspace |
| spelling | mit-1721.1/1110162022-09-29T23:55:58Z Functional Characterization of Intrinsic and Extrinsic Geometry Corman, Etienne Ben-Chen, Mirela Guibas, Leonidas Ovsjanikov, Maks Solomon, Justin Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Research Laboratory of Electronics Solomon, Justin We propose a novel way to capture and characterize distortion between pairs of shapes by extending the recently proposed framework of shape differences built on functional maps. We modify the original definition of shape differences slightly and prove that after this change, the discrete metric is fully encoded in two shape difference operators and can be recovered by solving two linear systems of equations. Then we introduce an extension of the shape difference operators using offset surfaces to capture extrinsic or embedding-dependent distortion, complementing the purely intrinsic nature of the original shape differences. Finally, we demonstrate that a set of four operators is complete, capturing intrinsic and extrinsic structure and fully encoding a shape up to rigid motion in both discrete and continuous settings. We highlight the usefulness of our constructions by showing the complementary nature of our extrinsic shape differences in capturing distortion ignored by previous approaches. We additionally provide examples where we recover local shape structure from the shape difference operators, suggesting shape editing and analysis tools based on manipulating shape differences. National Science Foundation (U.S.) (Award 1502435) National Science Foundation (U.S.) (Grant IIS 1528025) National Science Foundation (U.S.) (Grant IIS 1546206) 2017-08-24T19:47:00Z 2017-08-24T19:47:00Z 2017-03 Article http://purl.org/eprint/type/JournalArticle 0730-0301 http://hdl.handle.net/1721.1/111016 Corman, Etienne, et al. “Functional Characterization of Intrinsic and Extrinsic Geometry.” ACM Transactions on Graphics 36, 2 (March 2017): 1–17 © Association for Computing Machinery (ACM) en_US http://dx.doi.org/10.1145/2999535 ACM Transactions on Graphics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Association for Computing Machinery (ACM) MIT Web Domain |
| spellingShingle | Corman, Etienne Ben-Chen, Mirela Guibas, Leonidas Ovsjanikov, Maks Solomon, Justin Functional Characterization of Intrinsic and Extrinsic Geometry |
| title | Functional Characterization of Intrinsic and Extrinsic Geometry |
| title_full | Functional Characterization of Intrinsic and Extrinsic Geometry |
| title_fullStr | Functional Characterization of Intrinsic and Extrinsic Geometry |
| title_full_unstemmed | Functional Characterization of Intrinsic and Extrinsic Geometry |
| title_short | Functional Characterization of Intrinsic and Extrinsic Geometry |
| title_sort | functional characterization of intrinsic and extrinsic geometry |
| url | http://hdl.handle.net/1721.1/111016 |
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