Rolling-shutter cameras & Kummer's classification of order-one line congruences

Abstract: In this talk, we explain how algebraic geometry can be used to model and understand rolling-shutter cameras. Most consumer cameras today (e.g. in smartphones) use rolling shutters that do not capture an image at the same time but rather scan rapidly across the scene to be captured. When such a camera moves and rotates, the resulting picture can show the same 3D point several times, and straight lines in 3-space become higher-degree curves on the image. The set of light rays through such a camera form an algebraic surface in the Grassmannian of lines in projective 3-space. Kummer classified such surfaces (classically called line congruence) of order-one in 1866. We explain how Kummer's classification essentially characterizes all rolling-shutter cameras that see a generic 3D point exactly once. When such a camera takes a picture of a line in 3-space, the image is a high-degree curve. We compute that degree D in terms of the movement and rotation of the camera, and show that the image curve has multiplicity D-1 at one special point on the image plane. This talk is based on ongoing work with Marvin Hahn, Orlando Marigliano, and Tomas Pajdla.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html