An important goal has been reached regarding shape representation with
reduced dimensionality for three-dimensional objects (refer to "Curve
skeletonization of surface-like objects in 3D images guided by voxel
classification", S. Svensson, I. Nyström, G. Sanniti di Baja, Pattern
Recognition Letters, 23/12 1419-1426, 2002). It is well-known that
skeletonization is a process to extract from a two-dimensional object a
set of arcs and curves, called skeleton, that is a sketched and faithful
representation of the object. In fact, the object can be reconstructed
starting from its skeleton. Similarly, a three-dimensional object can be
faithfully represented by a two-dimensional skeleton, called surface
skeleton, consisting of surfaces and curves centrally placed within the
object. In turn, further dimensionality reduction of the skeleton of three-dimensional objects, so as to obtain a representation in terms of arcs and
curves, is a rather complex problem. In fact, it is not possible to
guarantee that the curve skeleton of a three-dimensional object includes
the information necessary to reconstruct the object itself. Thus, the one-dimensional representation is likely to lose faithfulness and, as such,
its use for applications (e.g., biomedical applications like angiography
or colonoscopy) is questionable. The proposed method to extract the curve
skeleton of three-dimensional objects overcomes this problem, since the
obtained skeleton provides an adequate representation of shape, though
object reconstruction is still not possible. The aim is pursued by first
computing the two-dimensional surface skeleton and, then, by identifying
on the surface skeleton the elements carrying shape information, in order
to guarantee their inclusion in the curve skeleton. The elements carrying
shape information have been identified as the elements placed in junctions
among surfaces in the surface skeleton. These elements are actually placed
in the most internal regions of the object, so that they play a role
analogous to the role of the elements that, in the case of skeletonization
of two-dimensional objects, are assigned to the skeleton. Therefore, to
implement the algorithm for the computation of the curve skeleton a deep
investigation of the structure of the surface skeleton has been necessary,
in order to classify its elements (refer to "A new shape descriptor for
surfaces in 3D images", G. Sanniti di Baja, S. Svensson, Pattern
Recognition Letters, 23/6, 703-711, 2002). It should be pointed out that
the proposed classification, besides its undoubted usefulness for the
innovative curve skeletonization algorithm, is an important result in the
field of digital geometry. In fact, classification methods available in
the current literature provide correct classification only if dealing
with "ideal" surfaces, i.e., surfaces characterised everywhere by unit
thickness. As it is well-known, unit thickness cannot be guaranteed in
junctions among surfaces, due to the discrete nature of the digital space.
Focus