Lifting Directional Fields to Minimal Sections
Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role in downstream applications such as meshing, existing metho...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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ACM
2024
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| Online Access: | https://hdl.handle.net/1721.1/155933 |
| _version_ | 1811093848267423744 |
|---|---|
| author | Palmer, David Chern, Albert Solomon, Justin |
| author_facet | Palmer, David Chern, Albert Solomon, Justin |
| author_sort | Palmer, David |
| collection | MIT |
| description | Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role in downstream applications such as meshing, existing methods for computing directional fields either require them to be specified in advance, ignore them altogether, or treat them as zeros of a relaxed field. While fields are ill-defined at their singularities, the graphs of directional fields with singularities are well-defined surfaces in a circle bundle. By lifting optimization of fields to optimization over their graphs, we can exploit a natural convex relaxation to a minimal section problem over the space of currents in the bundle. This relaxation treats singularities as first-class citizens, expressing the relationship between fields and singularities as an explicit boundary condition. As curvature frustrates finite element discretization of the bundle, we devise a hybrid spectral method for representing and optimizing minimal sections. Our method supports field optimization on both flat and curved domains and enables more precise control over singularity placement. |
| first_indexed | 2024-09-23T15:51:39Z |
| format | Article |
| id | mit-1721.1/155933 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:51:39Z |
| publishDate | 2024 |
| publisher | ACM |
| record_format | dspace |
| spelling | mit-1721.1/1559332024-09-20T04:13:07Z Lifting Directional Fields to Minimal Sections Palmer, David Chern, Albert Solomon, Justin Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role in downstream applications such as meshing, existing methods for computing directional fields either require them to be specified in advance, ignore them altogether, or treat them as zeros of a relaxed field. While fields are ill-defined at their singularities, the graphs of directional fields with singularities are well-defined surfaces in a circle bundle. By lifting optimization of fields to optimization over their graphs, we can exploit a natural convex relaxation to a minimal section problem over the space of currents in the bundle. This relaxation treats singularities as first-class citizens, expressing the relationship between fields and singularities as an explicit boundary condition. As curvature frustrates finite element discretization of the bundle, we devise a hybrid spectral method for representing and optimizing minimal sections. Our method supports field optimization on both flat and curved domains and enables more precise control over singularity placement. 2024-08-05T16:56:17Z 2024-08-05T16:56:17Z 2024-07-19 2024-08-01T07:49:41Z Article http://purl.org/eprint/type/JournalArticle 0730-0301 https://hdl.handle.net/1721.1/155933 Palmer, David, Chern, Albert and Solomon, Justin. 2024. "Lifting Directional Fields to Minimal Sections." ACM Transactions on Graphics, 43 (4). PUBLISHER_CC en 10.1145/3658198 ACM Transactions on Graphics Creative Commons Attribution-NoDerivs License https://creativecommons.org/licenses/by-nd/4.0/ The author(s) application/pdf ACM Association for Computing Machinery |
| spellingShingle | Palmer, David Chern, Albert Solomon, Justin Lifting Directional Fields to Minimal Sections |
| title | Lifting Directional Fields to Minimal Sections |
| title_full | Lifting Directional Fields to Minimal Sections |
| title_fullStr | Lifting Directional Fields to Minimal Sections |
| title_full_unstemmed | Lifting Directional Fields to Minimal Sections |
| title_short | Lifting Directional Fields to Minimal Sections |
| title_sort | lifting directional fields to minimal sections |
| url | https://hdl.handle.net/1721.1/155933 |
| work_keys_str_mv | AT palmerdavid liftingdirectionalfieldstominimalsections AT chernalbert liftingdirectionalfieldstominimalsections AT solomonjustin liftingdirectionalfieldstominimalsections |