A structure from motion inequality

TL;DR

This paper presents a mathematical inequality that determines the feasibility of 3D reconstruction in structure from motion problems, considering camera and point configurations, and extends the formalism to moving points.

Contribution

It introduces a new inequality based on Frobenius theorem that provides a rigorous criterion for the impossibility of reconstruction in structure from motion.

Findings

Provides a necessary condition for successful reconstruction

Extends the formalism to moving points during imaging

Offers a mathematical framework for analyzing structure from motion

Abstract

We state an elementary inequality for the structure from motion problem for m cameras and n points. This structure from motion inequality relates space dimension, camera parameter dimension, the number of cameras and number points and global symmetry properties and provides a rigorous criterion for which reconstruction is not possible with probability 1. Mathematically the inequality is based on Frobenius theorem which is a geometric incarnation of the fundamental theorem of linear algebra. The paper also provides a general mathematical formalism for the structure from motion problem. It includes the situation the points can move while the camera takes the pictures.