Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908855234985984 |
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| author | Duff, Timothy Korotynskiy, Viktor Leykin, Anton Pajdla, Tomas |
| author_facet | Duff, Timothy Korotynskiy, Viktor Leykin, Anton Pajdla, Tomas |
| contents | We characterize the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal. This answers the first interesting case of a question recently posed by Bråtelund and Rydell. Our result improves upon previously discovered sets of algebraic constraints in the geometric computer vision literature, which are all incomplete (as they do \emph{not} generate the vanishing ideal) and sometimes make restrictive assumptions about how a matrix triple should be scaled. Our discussion touches more broadly on generalized compatibility varieties, whose multihomogeneous vanishing ideals are much less well understood. One of our key new discoveries is a simple set of quartic constraints vanishing on compatible fundamental matrix triples. These quartics are also significant in the setting of essential matrices: together with some previously known constraints, we show that they locally cut out the variety of compatible essential matrix triples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23450 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices Duff, Timothy Korotynskiy, Viktor Leykin, Anton Pajdla, Tomas Commutative Algebra Computer Vision and Pattern Recognition Algebraic Geometry We characterize the variety of compatible fundamental matrix triples by computing its multidegree and multihomogeneous vanishing ideal. This answers the first interesting case of a question recently posed by Bråtelund and Rydell. Our result improves upon previously discovered sets of algebraic constraints in the geometric computer vision literature, which are all incomplete (as they do \emph{not} generate the vanishing ideal) and sometimes make restrictive assumptions about how a matrix triple should be scaled. Our discussion touches more broadly on generalized compatibility varieties, whose multihomogeneous vanishing ideals are much less well understood. One of our key new discoveries is a simple set of quartic constraints vanishing on compatible fundamental matrix triples. These quartics are also significant in the setting of essential matrices: together with some previously known constraints, we show that they locally cut out the variety of compatible essential matrix triples. |
| title | Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices |
| topic | Commutative Algebra Computer Vision and Pattern Recognition Algebraic Geometry |
| url | https://arxiv.org/abs/2602.23450 |