Apparent contours: an outline
Peter Giblin · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · 1998
A point in space viewed from a camera centre c yields a point in the image. A curve in space generally yields a curve in the image. A surface M in space, the boundary of a three–dimensional region, yields a region in the image whose boundary edge is the apparent countour of M from the given viewpoint c. More precisely supposing M has a well defined tangent plane at every point, those points r of M where the tangent plane passes through c form a curve on M called the contour generator. This projects in the image into the apparent contour, also called the profile or outline, of M. When the camera centre moves, c = c(t), the contour generator moves over M and the changing apparent contours carry a great deal of information about the surface. In fact, provided cameras are calibrated and camera motion is known, there is in theory enough information to reconstruct a region of M that is swept out by the moving contour generators. In this paper I outline this now ‘classical’ (1987) result, and some of its more recent variants. It can happen, however, that the contour generators do not sweep out a region but create a boundary, or frontier on M, with the contour generators all on one side of this frontier. This situation can be used in principle to recover motion as well as the structure of M. I shall describe recent work which seeks to say exactly what is the difference between projections of a space curve, apparent contours of a surface, and apparent contours which yield a frontier. That is, I shall try to describe in a sense all the available information.