The Chern-Mather class of the multiview variety

Abstract

The multiview variety associated to a collection of $N$ cameras records which sequences of image points in ${\mathbb{P}}^{2N}$ can be obtained by taking pictures of a given world point $x\in {\mathbb{P}}^3$ with the cameras. In order to reconstruct a scene from its picture under the different cameras it is important to be able to find the critical points of the function which measures the distance between a general point $u\in {\mathbb{P}}^{2N}$ and the multiview variety. In this paper we calculate a specific degree $3$ polynomial that computes the number of critical points as a function of $N$. In order to do this, we construct a resolution of the multiview variety, and use it to compute its Chern-Mather class.

12 pages, 1 figure