Metric Invariants for Camera Calibration: Designing Algorithms from Algebraic Rank Analysis

Reconstructing a metric structure of a scene from images has been one of the important topics in computer vision. In this book, we focus on a simple diagonal rank-deficient form of a 2D metric invariant, a conic dual to the circular points. By manipulating image features to constrain the simple form algebraically, the metric reconstruction can be achieved. We start from second order curves such as concentric circles or confocal conics to be used as basic features. By simply subtracting them, affine and metric properties of a plane are recovered. The geometric meanings of the resulting subtraction matrices are also investigated. The idea of algebraically manipulating features extend to an ``addition method'' using human recognizable features such as a rectangle. Its parallelism and orthogonality enables us to obtain information of the scene structure. As a generalization, we propose a framework to unify the geometric constraints used in camera calibration and in metric reconstruction. We show that scene constraints can be converted into constraints of cameras, and that a flexible algorithm to metric-reconstruct scenes from images can be developed in the proposed unified framework.