Lines, Quadrics, and Cremona Transformations in Two-View Geometry
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914719221153792 |
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| author | Connelly, Erin Thomas, Rekha R. Vinzant, Cynthia |
| author_facet | Connelly, Erin Thomas, Rekha R. Vinzant, Cynthia |
| contents | Given $7 \leq k \leq 9$ points $(x_i,y_i) \in \mathbb{P}^2 \times \mathbb{P}^2$, we characterize rank deficiency of the $k \times 9$ matrix $Z_k$ with rows $x_i^\top \otimes y_i^\top$, in terms of the geometry of the point sets $\{x_i\}$ and $\{y_i\}$. This problem arises in the conditioning of certain well-known reconstruction algorithms in computer vision, but has surprising connections to classical algebraic geometry via the interplay of quadric surfaces, cubic curves and Cremona transformations. The characterization of rank deficiency of $Z_k$, when $k \leq 6$, was completed in arXiv:2301.09826. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02757 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lines, Quadrics, and Cremona Transformations in Two-View Geometry Connelly, Erin Thomas, Rekha R. Vinzant, Cynthia Algebraic Geometry Commutative Algebra Combinatorics Given $7 \leq k \leq 9$ points $(x_i,y_i) \in \mathbb{P}^2 \times \mathbb{P}^2$, we characterize rank deficiency of the $k \times 9$ matrix $Z_k$ with rows $x_i^\top \otimes y_i^\top$, in terms of the geometry of the point sets $\{x_i\}$ and $\{y_i\}$. This problem arises in the conditioning of certain well-known reconstruction algorithms in computer vision, but has surprising connections to classical algebraic geometry via the interplay of quadric surfaces, cubic curves and Cremona transformations. The characterization of rank deficiency of $Z_k$, when $k \leq 6$, was completed in arXiv:2301.09826. |
| title | Lines, Quadrics, and Cremona Transformations in Two-View Geometry |
| topic | Algebraic Geometry Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2308.02757 |