Beyond Geometry and Topology - Mathematical Tools for Shape Analysis and Description - page 94

12. BEYOND GEOMETRY AND TOPOLOGY
12.3 RECOGNITION OF FUNCTIONAL PARTS OF
MAN-MADE OBJECTS
Laga et al. in [ 122 ] address the problem of automatic recognition of functional parts of man-made
3D shapes, starting from classes of objects that present significant geometric and topological vari-
ations. e idea is to infer the context of a part within a 3D shape and to use it to learn the se-
mantics of shapes. In practice, the context is modeled in terms of structural relationships between
shape parts that are used, in addition to part geometry, as cues for functionality recognition.
Each shape is represented as a graph that connects the parts that share some spatial rela-
tionships and the context of a shape part is modeled as walks in the graph, adopting a graph kernel
strategy. Each node of the graph corresponds to a shape part and two nodes are connected with an
edge if they share some contextual relationship. During the initial graph construction, two types
of relationships are considered: (i) inter-part symmetries if there exists a rotational, translational,
or reflectional symmetry transform that aligns two parts; (ii) adjacency if two parts share some ver-
tices or if their bounding boxes overlap. en, a hierarchical graph is created, by contracting the
cliques of the original graph on the basis of their geometry and adjacency. Nodes within the same
level are connected with either an adjacency or a symmetry relationship and a node is connected
to its parent with an edge.
Geometric attributes are assigned to each node to encode the geometric properties of its
shape and provide a mean for estimating the similarity between the geometry of two nodes. In
this method the geometric properties chosen are the distance-based shape distribution proposed
in [ 152 ] (shape descriptor), the radius of the sphere that bounds the component (size) and the
three eigenvalues of the Principal Component Analysis (PCA) of each shape part (component
aspect). en to compare the geometry of two nodes, a Gaussian kernel for each of the three
geometric properties is built and the geometric kernel is obtained as the uniformly weighted sum
of the three components.
e contextual relationships are the core component to the proposed shape correspondence
and functionality learning algorithms. e basic assumption is that components sharing common
contextual relations are more likely to have the same functionality. e relations of interest include
enclosure, side contact, symmetry, co-centricity, and adjacency (see Figure 12.5 left and middle).
To define a similarity metric between two edges labeled with the relationship types it is assumed
that the two edges are similar if they are of the same type. As the final assumption, two parts
are similar if their geometry and context are similar. While geometry is informative, under large
geometric and topological variations, the context, i.e., the way shape parts are interconnected,
provides rich information about the semantics of shapes. As context-aware similarity measure
between two nodes, all walks of length p starting from the two nodes are considered; in the
practice, the parameter p is set with the value 3 .
Finally, the functional recognition is seen as a multi-label supervised classification problem
that assigns a shape part to one of the possible functional categories and it is solved with a Support