On minimizing errors in 3D reconstruction for stereo camera systems

Abstract

Active reconstruction of 3D surfaces deals with the control of camera viewpoints to minimize error and uncertainty in the reconstructed shape of an object. In this paper we develop a mathematical relationship between the setup and focal lengths of a stereo camera system and the corresponding error in 3D reconstruction of a given surface. We explicitly model the noise in the image plane, which can be interpreted as pixel noise or as uncertainty in the localization of corresponding point features. The results can be used to plan sensor posi- tioning, e.g., using information theoretic concepts for optimal sensor data selection. The paper is structured as follows: first, we describe the setup for 3D reconstruction using triangulation in a normalized stereo camera system. Then we present a mathematical development of the reconstruction error in a simple 2D model, taking explicitly into account noise in a one dimensional image plane. We map the problem of optimal stereo positioning to an optimiza- tion problem. This will be analyzed first, to get the opti- mal focal length and the optimal baseline (translation in x direction) in a normalized 2D stereo system. Further we gradually generalize this model, firstly by rotations, and secondly by translation in x and z directions. In this simple case we can perform a partial analytical analy- sis, but there are visibility assumptions which cannot be fulfilled in real stereo systems. Therefore, we further generalize to a 3D model with visibility constraints. In this model, we cannot perform an analytical analysis; therefore, we optimize the modifiable parameters by a Monte Carlo simulation. The results will be compared with the analytical results of the simple case. We present experimental results and compare them with the theoretical predictions, and conclude this paper with prospects for future study.