On Ullman's theorem in computer vision

TL;DR

This paper investigates the Ullman theorem in computer vision, providing explicit formulas for reconstructing camera and point configurations from image data, and establishing local uniqueness results for specific cases.

Contribution

It extends Ullman's theorem by deriving explicit reconstruction formulas and demonstrating local uniqueness for three cameras and three points.

Findings

Explicit reconstruction formulas derived

Local uniqueness established for 3 cameras and 3 points

Conditions for realizability of picture data identified

Abstract

Both in the plane and in space, we invert the nonlinear Ullman transformation for 3 points and 3 orthographic cameras. While Ullman's theorem assures a unique reconstruction modulo a reflection for 3 cameras and 4 points, we find a locally unique reconstruction for 3 cameras and 3 points. Explicit reconstruction formulas allow to decide whether picture data of three cameras seeing three points can be realized as a point-camera configuration.