Turning shape decision problems into measures

This paper considers the problem of constructing shape measures; we start by giving a short overview of areas of practical application of such measures. Shapes can be characterised in terms of a set of properties, some of which are Boolean in nature. E.g. is this shape convex? We show how it is possible in many cases to turn such Boolean properties into continuous measures of that property e.g. convexity, in the range [0–1]. We give two general principles for constructing measures in this way, and show how they can be applied to construct various shape measures, including ones for convexity, circularity, ellipticity, triangularity, rectilinearity, rectangularity and symmetry in two dimensions, and 2.5D-ness, stability, and imperforateness in three dimensions. Some of these measures are new; others are well known and we show how they fit into this general framework. We also show how such measures for a single shape can be generalised to multiple shapes, and briefly consider as particular examples measures for containment, resem-blance, congruence, and similarity