Chirality

Abstract

It is known that a set of points in three-dimensions is determined up to projectivity from two views with uncalibrated cameras. It is shown in this paper that this result may be improved by distinguishing between points in front of and behind the camera. Any point that lies in an image must lie in front of the camera producing that image. Using this idea, it is shown that the scene is determined from two views up to a more restricted class of mappings known as quasi-affine transformations, which are precisely those projectivities that preserve the convex hull of an object of interest. An invariant of quasi-affine transformation known as the chiral sequence of a set of points is defined and it is shown how the chiral sequence may be computed using two uncalibrated views. As demonstrated theoretically and by experiment the chiral sequence may distinguish between sets of points that are projectively equivalent. These results lead to necessary and sufficient conditions for a set of corresponding pixels in two images to be realizable as the images of a set of points in three dimensions.Using similar methods, a necessary and sufficient condition is given for the orientation of a set of points to be determined by two views. If the perspective centres are not separated from the point set by a plane, then the orientation of the set of points is determined from two views.