The largest convex patches: A boundary-based method for obtaining object parts

The importance of boundaries for shape decomposition into component parts has been discussed from different points of view by Koenderink and van Doorn (1982), and by Hoffman and Richards (1984). The former define part boundaries as parabolic contours, whereas the latter propose that part boundaries should be defined by contours of negative minima (or maxima) of principal curvature. In this article, building on aspects of both approaches, we develop a new method for shape decomposition. This method relies exclusively on global properties of the surface which are fully characterized by local surface properties. We propose that a useful parcellation of shapes into parts can be obtained by decomposing the shape boundary into the largest convex surface patches and the smallest nonconvex surface patches. The essential computational steps of this method are the following: (i) build initial parts from the largest locally convex patches, (ii) consider an initial part as a constituent part if it is essentially convex, and (iii) obtain the remaining constituent parts by merging adjacent initial parts generated by the largest locally convex and the smallest nonconvex patches of nearly the same sizes. The method is illustrated on both smooth and continuous shapes. We show that the decomposition of shapes into the largest convex patches aims to maximize the “thingness” in an object, and to minimize its “non-thingness”. The method is conducive to a natural parcellation of shapes into constituent parts useful for recognition and for inferring function.