Birational geometry of critical loci in Algebraic Vision
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866909279679676416 |
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| author | Bertolini, Marina Notari, Roberto Turrini, Cristina |
| author_facet | Bertolini, Marina Notari, Roberto Turrini, Cristina |
| contents | In Algebraic Vision, the projective reconstruction of the position of each camera and scene point from the knowledge of many enough corresponding points in the views is called the structure from motion problem. It is known that the reconstruction is ambiguous if the scene points are contained in particular algebraic varieties, called critical loci. To be more precise, from the definition of criticality, for the same reconstruction problem, two critical loci arise in a natural way. In the present paper, we investigate the relations between these two critical loci, and we prove that, under some mild smoothness hypotheses, (some of) their irreducible components are birational. To this end, we introduce a unified critical locus that restores the symmetry between the two critical loci, and a natural commutative diagram relating the unified critical locus and the two single critical loci. For technical reasons, but of interest in its own, we also consider how a critical locus change when one increases the number of views. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02067 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Birational geometry of critical loci in Algebraic Vision Bertolini, Marina Notari, Roberto Turrini, Cristina Algebraic Geometry Primary: 14J25, Secondary: 14M12, 14M15, 14N05 In Algebraic Vision, the projective reconstruction of the position of each camera and scene point from the knowledge of many enough corresponding points in the views is called the structure from motion problem. It is known that the reconstruction is ambiguous if the scene points are contained in particular algebraic varieties, called critical loci. To be more precise, from the definition of criticality, for the same reconstruction problem, two critical loci arise in a natural way. In the present paper, we investigate the relations between these two critical loci, and we prove that, under some mild smoothness hypotheses, (some of) their irreducible components are birational. To this end, we introduce a unified critical locus that restores the symmetry between the two critical loci, and a natural commutative diagram relating the unified critical locus and the two single critical loci. For technical reasons, but of interest in its own, we also consider how a critical locus change when one increases the number of views. |
| title | Birational geometry of critical loci in Algebraic Vision |
| topic | Algebraic Geometry Primary: 14J25, Secondary: 14M12, 14M15, 14N05 |
| url | https://arxiv.org/abs/2408.02067 |