On the Algebraic Geometry of Multiview

Abstract

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras.