Changes in surface convexity and topology caused by distortions of stereoscopic visual space

Abstract

We introduce the notion of a distorted reconstruction from two views, which need not satisfy the epipolar constraint. It can be computed from point correspondences and from a possibly inexact estimate of the stereo configuration. Thus, this scheme avoids the often costly and unstable minimization procedure for establishing the epipolar constraint, at the cost of introducing non-linear distortions. As a consequence, the convexity and topology of curves and surfaces can be changed.The distorted reconstruction is related to the original scene structure by a quadratic Cremona transformation of space. By analyzing the distortion of curves and surfaces geometrically in terms of the singular elements of the associated Cremona transformation, we show that severe distortions are present particularly in the vicinity of the camera centers, thereby indicating that their consideration is of particularly high relevance for near regions of the stereo rig. Our main technical contribution is the derivation of the exact criteria governing changes in surface convexity and topology.