Differential Geometry and Shape Analysis - Mathematical Tools for Shape Analysis and Description - page 32

Figure 4.5: Representation of the angle excess around a vertex.
compression and simplification of 3D shapes. Several features, such as flats, tips, pits, mounts and
blends, are characterized nicely in terms of curvature: for instance, pits are generally meant to be
the extrema of an overall concave area, mounts have an overall convex shape without being sharp
protrusions.
Going beyond the curvature alone, Mortara et al. [ 145 ] proposed a multi-scale analysis of
the shape, exploiting the idea of neighborhoods of variable size, though the so-called blowing
bubbles : a set of spheres of increasing radius, centered at each vertex of the mesh, are used to
analyze the shape from a local to a more global view. e number of intersections between spheres
and the shape tells how the object shape is embedded in the 3D space, while angle excess quantifies
the curvature for points on protrusions of the object (see Fig. 4.6 ).
Figure 4.6: Examples of spheres intersecting shapes: the number of intersections (1, 2, 3) is related
to the embedding of the shape in the 3D space.
e result is a morphological decomposition of the object (Figure 4.7 ), which stores features
together with their morphological type, persistence at scale variation, amplitude and/or size, so
that it can be used to automate processes such as shape matching and comparison.