Distortions of stereoscopic visual space and quadratic Cremona transformations

Abstract

When incorrect, values for the extrinsic and intrinsic parameters of the two cameras of a stereo rig are used in the reconstruction of a three-dimensional scene from image correspondences, the resulting reconstruction is distorted in a systematic way. We show here that the transformation between the true scene structure and its distorted reconstruction is a quadratic Cremona transformation- a rational transformation which is one-to-one almost everywhere, but which does not in general preserve collinearity. We study the distortion of points in the fixation plane on a global, qualitative, level, and on a local, quantitative level. Both global and local viewpoints provide evidence of severe non-linear distortions in tile vicinity of the camera centers, thereby indicating that their consideration is of particularly high relevance for near regions of the stereo rig. This is consistent with experimental evidence from psychophysics, which shows that distortions in the near range cannot be adequately described by linear transformations. Our distortion framework describes and enables the analysis of situations where insufficient information is available to compute even a weak calibration of a stereo rig. It also makes possible a thorough error analysis of systematic errors in computing structure from motion.