Critical configurations for three projective views (Q6539992)

In this paper, the author studies critical configurations for three projective views. This topic is motivated by questions in computer vision, where one of the main problems is that of \textit{structure from motion}, where, given a set of \(2\)-dimensional images, the task is to reconstruct a scene in \(3\)-space and to find the camera positions in the scene. In general, given enough images and enough points in each image, one can uniquely reconstruct the original scene. However, there are some \(3D\)-configurations of points and cameras for which a unique reconstruction from the images is never possible. These are called critical configurations. The main goal of this paper is to study critical configurations for three projective cameras using the approach of algebraic geometry. A set of \(n\) cameras \(\mathbb{P}\) defines a rational map \(\phi_{\mathbb{P}} : \mathbb{P}^{3} \dashrightarrow (\mathbb{P}^{2})^{n}\), which extends to a morphism under the blowing up of the camera centers \(\mathrm{BL}_{\mathbb{P}}(\mathbb{P}^{3})\). A set of \(n\) cameras \(\mathbb{P}\) and a set of points \(X \subset \mathrm{BL}_{\mathbb{P}}(\mathbb{P}^{3})\) is called a configuration. A configuration is critical if there exists another set of cameras \(\mathbb{Q}\) and a set of points \(Y\) such that \(\widetilde{\phi_{\mathbb{P}}} = \widetilde{\phi_{\mathbb{Q}}}\), where \(\widetilde{\phi_{\bullet}} : \mathrm{BL}_{\bullet}(\mathbb{P}^{3}) \rightarrow (\mathbb{P}^{2})^{n}\). The map \(\widetilde{\phi_{\mathbb{P}}}\) takes \(\mathrm{BL}_{\mathbb{P}}(\mathbb{P}^{3})\) to a variety \((\mathbb{P}^{2})^{n}\) called the multi-view variety. Since \(X\) and \(Y\) both map into the intersection of the multi-view varieties, one can try to classify critical configurations by classifying all possible intersections of multi-view varieties. The main result of the paper tells us that all critical configurations lie on the intersection of quadric surfaces, and we can find a classification of intersections that form critical configurations. It is worth noting that a configuration of three projective cameras \(\mathbb{P}\) and \(6\) or less points \(X \subset\mathrm{BL}_{\mathbb{P}}(\mathbb{P}^{3})\) is always a critical configuration.