A Hilbert scheme in computer vision (Q2871237)

Motivated by applications in computer vision, the authors study multiview varieties. These are subvarieties of \((\mathbb P^2)^n\) that contain the images of \(n\) linear projections from \(\mathbb P^3\) to \(\mathbb P^2\) (``cameras''). Knowledge about multiview varieties and their ideals is important for the numeric reconstruction of points from their images. The authors' findings go well-beyond these immediate applications and reveal a fascinating algebraic and combinatorial geometry of the varieties under scrutiny.NEWLINENEWLINEA first important result is that a known generating system of generic multiview ideals is in fact a universal Gröbner basis. This observation is followed by a thorough investigation of multiview ideals and, in particular, a distinguished monomial subideal. An explicit description is given and interesting results on its Hilbert function in the \(\mathbb Z^n\)-grading are provided. The article's main result states that the Hilbert scheme that parametrizes the \(\mathbb Z^n\)-homogeneous ideals with this Hilbert function contains the space of camera positions as a special component.NEWLINENEWLINEThe above statements are true in generic cases. Other results pertain to collinear and infinitesimally neighbouring cameras or less than five cameras with a toric multiview variety.