General linear cameras : theory and applications

Abstract

I present a General Linear Camera (GLC) model that unifies many previous camera models into a single representation. The GLC model describes all perspective (pinhole), orthographic, and many multiperspective (including pushbroom and two-slit) cameras, as well as epipolar plane images. It also includes three new and previously unexplored multiperspective linear cameras. The GLC model is general and linear in the sense that, given any vector space where rays are represented as points, it describes all 2D affine subspaces (planes) formed by the affine combination of 3 rays. I also present theories of projection and collineation for GLCs and use these theories to explain various multiperspective distortions. Given an arbitrary multiperspective imaging system that captures smoothly varying set of rays, I show how to map the rays onto a 2D ray manifold embedded into a 4D linear vector space. The GLC model can then be use to analyze the tangent planes on this manifold. Geometric structures associated with the local GLC model of each tangent plane provide an intuitive physical interpretation of the imaging system, and they are closely related to the caustics of reflected rays. These geometric structures are characteristic of only 4 of the 8 GLC types. I also prove that the local GLC type at each tangent plane is invariant to the choice of parametrization, and, thus, an intrinsic property of the reflecting surface. Using GLCs to analyze the caustics of reflection extends the previous Jacobian-based approaches, which consider only a pinhole model at each infinitesimal region about each surface point. Finally, I demonstrate how to use the GLC model in computer vision, computer graphics, and optical design applications. In particular, I show how to use GLCs for modelling and rendering multiperspective images and characterizing real multiperspective imaging systems such as catadioptric mirrors.