Metric reconstruction from images using rank-deficient relations

Abstract

Inferring geometric information from images has been dealt with very importantly in computer vision area. In this work, we aim to reconstruct a metric structure of a scene from images using the fact that the conic dual to the circular points has a simple diagonal rank-deficient form. By manipulating image``s features to constrain the simple diagonal form algebraically, the metric invariants of an observed scene can be recovered. In the first part of the work, we use "concentric circles" as basic features, and we propose simple "subtraction methods" to find affine and metric properties of a plane with concentric circles. The geometric meanings of the resulting subtraction matrices are revealed. Some experiments are conducted to show the possibilities to use the proposed algorithm, including a calibration of a multi-camera system. As a direct extension, the concentric circle cases are generalized to deal with some general conics whose foci are known and confocal conics whose foci are unknown. In the second part of the work, we propose an "addition method" using one-dimensional basic features such as points and lines. To analyze the geometric information efficiently, we build a new space called "semi-metric space." The parameterization of metric invariants in the semi-metric space is made, and using that, the physical meanings of the parameters of the invariants are derived. Although we cannot measure the scene metrically, some knowledge about the structure of the scene can be retrieved from images, by using only easily obtainable features such as parallelism and orthogonality. Under static camera assumption and with more images, the metric of the planes can be determined. In the third part of the work, we propose a framework to unify the geometric constraints used in camera calibration and in metric reconstruction. The previously used constraints are revisited and reinterpreted in the proposed framework. We show that all kinds of scene constraints can be converte...