Grassmann tensors arise from classical problems of scene reconstruction in computer vision. In particular, bifocal Grassmann tensors, related to a pair of projections from a projective space onto view spaces of varying dimensions, generalize the classical notion of fundamental matrices. In this paper, we study in full generality the variety of bifocal Grassmann tensors focusing on its birational geometry. To carry out this analysis, every object of multi-view geometry is described both from an algebraic and geometric point of view, e.g., the duality between the view spaces, and the space of rays is explicitly described via polarity. Next, we deal with the moduli of bifocal Grassmann tensors, thus showing that this variety is both birational to a suitable homogeneous space and endowed with a dominant rational map to a Grassmannian.

The varieties of bifocal Grassmann tensors / M. Bertolini, G. Bini, C. Turrini. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - (2023), pp. 1-23. [Epub ahead of print] [10.1007/s10231-023-01317-y]

The varieties of bifocal Grassmann tensors

M. Bertolini
Primo
;
G. Bini
Penultimo
;
C. Turrini
Ultimo
2023

Abstract

Grassmann tensors arise from classical problems of scene reconstruction in computer vision. In particular, bifocal Grassmann tensors, related to a pair of projections from a projective space onto view spaces of varying dimensions, generalize the classical notion of fundamental matrices. In this paper, we study in full generality the variety of bifocal Grassmann tensors focusing on its birational geometry. To carry out this analysis, every object of multi-view geometry is described both from an algebraic and geometric point of view, e.g., the duality between the view spaces, and the space of rays is explicitly described via polarity. Next, we deal with the moduli of bifocal Grassmann tensors, thus showing that this variety is both birational to a suitable homogeneous space and endowed with a dominant rational map to a Grassmannian.
Fundamental Matrices; Grassmann Tensors; Group actions; Multi-view Geometry;
Settore MAT/03 - Geometria
2023
7-apr-2023
https://link.springer.com/article/10.1007/s10231-023-01317-y
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/968718
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