Multiscale shape equivalence

In this paper we define a property applied to contours and 2D shapes we call `shape equivalence', or more strictly, `virtual shape equivalence'. The intuitive idea is that two contours or 2D shapes are `virtually equivalent' (at a given scale of resolution) if they can possibly give rise to identical area sampled images (at the given scale) with respect to a given sampling regime. The word `virtual' is used because the relationship is not a true equivalence relation--in particular it is not strictly transitive. The idea is similar to the psychological notion of `just noticeable difference' (JND). Two stimuli are within a JND threshold if a subject cannot perceptually distinguish them, even though they may in fact be different. Similarly our notion of virtual equivalence of contours corresponds to there being no noticeable difference between them with respect to a certain class of sampling regimes at a particular scale of resolution. The usefulness of the concept is that it can be used to built a formal theory of shape and contour simplification (at various scales) to assist object recognition.