Erin Connelly: Classical and Computational Algebraic Geometry in Computer Vision

Abstract

A camera is a linear projective map \(\mathbb{P}^3 \to \mathbb{P}^2\), represented by a full rank matrix in \(\mathbb{P}(\mathbb{R}^{3\times 4})\), which sends world points \(q \in \mathbb{P}^3\) to image points \(p \in \mathbb{P}^2\). A multi-view arrangement is a collection of cameras \(A_i\), world points \(q_j\) and image points \(p_{ij}\) satisfying \(A_iq_j=p_{ij}\). We study the problem of reconstructing such systems from partial data. For the problem of reconstructing from the image data alone (3D Image Reconstruction) we find answers by studying the conditions under which \(k\) rank one tensors \(x_i\otimes y_i\) are linearly dependent for \(2\leq k\leq 9\). We also consider reconstruction from both the camera and image data (Triangulation) and from both the world point and image data (Resectioning). We utilize Carlson-Weinshall Duality and existing results for Triangulation to produce dual results for Resectioning.