Geometry of distorted visual space and Cremona transformation

Abstract

An important issue concerning the design of any vision system is the choice of a proper space representation. In order to search for clues to a suitable representation, we look at the distortion of space arising from errors in motion or stereo estimates. Understanding this space distortion has important epistemological implications for the problem of space representation because it tells us what can be and what cannot be computed. This paper is therefore an enquiry into the nature of space representation through the study of the space distortion, though it is not a psychophysical or physiological study but rather a computational one. We show that the distortion transformation is a quadratic Cremona transformation, which is bijective almost everywhere except on the set of fundamental elements. We identify the fundamental elements of both the direct and the inverse transformations, and study the behaviour of the space distortion by analyzing the transformation of space elements (lines, planes) that pass through these fundamental elements.