Beyond Geometry and Topology - Mathematical Tools for Shape Analysis and Description - page 90

12. BEYOND GEOMETRY AND TOPOLOGY
ject (or part of an object) to sets of unknown objects [ 120 , 127 ]. Another example is viewpoint
selection , referring to the process of finding the most informative view of an object in a given
context [ 124 , 146 ].
A recent path of research aims to derive high-level information about objects by analyzing
single objects in the context of larger collections of objects: the idea is to derive information not
only from the object itself, but also from its relation with the other objects in the collection. is
is the case for example of the co-segmentation of a set of 3D objects [ 105 , 108 , 208 ], i.e., the
segmentation of the objects as a whole into consistent semantic parts with part correspondences.
Another example is the structuring of 3D large datasets to enable navigation and retrieval; this is
often achieved by exploiting not only the similarity between a query and the dataset, but also the
pairwise similarities between the rest of dataset models.
In this chapter, we show how mathematical concepts can support 3D shape analysis even
beyond the analysis of geometric aspects, by taking into account texture and semantic aspects.
We will start with examples of 3D textured shape retrieval (section 12.1 ). en, we present a
technique for the qualitative organization of datasets of 3D models (section 12.2 ). Finally, we
illustrate a method to infer the functionality of an object through co-segmentation and co-analysis
(section 12.3 ). Despite the focus on the technical aspects of the methods, we hope the choice of
these techniques will stimulate further thoughts and investigations in the field.
12.1 3D TEXTURED SHAPE RETRIEVAL
e authors of [ 121 ] generalized heat kernels (section 5.2 ) to textured shapes. e main idea
is to define the diffusion process, hence the heat kernel, on a manifold embedded into a
high-dimensional space, instead of the usual 3-dimensional Euclidean space R 3 : namely, a 6-
dimensional space R 3 C , where C is some color space, for example the RGB or the CIELab
color spaces. In this embedding, three coordinates represent geometric information, in terms
of the usual Cartesian coordinates, whereas the other three represent the photometric informa-
tion, in terms of color components. e definition of the Laplace-Beltrami operator, hence the
diffusion process and the heat kernel, are straightforward. ere it follows the definition of a
photometric Heat Kernel Signature (pHKS), which is used for textured shape retrieval. Similar
ideas have been used in [ 107 ], who defined a family of diffusion distances based on Schrödinger
operators incorporating photometric data.
e authors of [ 23 ] proposed PHOG, a signature for 3D textured shape retrieval which is
able to analyze both colorimetric and geometric properties. e main idea is to look at colorimetric
properties as at multi-value attributes associated with the vertices of a model, and taking advantage
of the persistence homology settings (chapter 11 ). e PHOG signature consists of three parts:
• a colorimetric descriptor: coordinates in the CIELab color space are seen as either scalar
( L channel, i.e., luminosity) or bivariate ( a and b channels, i.e., color hues) real functions
defined over the shape, and used to get persistence diagrams and (approximated) persistence
spaces;