# The Chern-Mather class of the multiview variety

## TL;DR

This paper computes the Chern-Mather class of the multiview variety, enabling the calculation of the number of critical points for scene reconstruction in multiview geometry, by constructing a resolution and a degree polynomial.

## Contribution

It introduces a method to compute the Chern-Mather class of the multiview variety, leading to a polynomial for counting critical points in scene reconstruction.

## Key findings

- Derived a degree 3 polynomial for critical point count as a function of N.
- Constructed a resolution of the multiview variety.
- Computed 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.