3D Shape Analysis in a Nutshell - Mathematical Tools for Shape Analysis and Description - page 7

C H A P T E R 2

3D Shape Analysis in a Nutshell

2.1 3D SHAPE ANALYSIS: PROBLEMS AND SOLUTIONS
Computer graphics and geometric modeling emerged as new disciplines focusing on the design
and visualization of digital 3D objects. At the beginning, they mostly focused on solving primary
problems related to representation issues [ 134 , 167 ]; the seminal paper by Requicha [ 167 ] intro-
duced the basic terminology and definitions which shaped the whole field of geometric modeling.
en, computer graphics saw a gradual shift of interest from methods to represent shapes toward
methods to describe shapes. e distinction between representation and description can be expressed
as follows [ 150 ]:
“an object representation contains enough information to reconstruct (an approxima-
tion to) the object, while a description only contains enough information to identify
an object as a member of some class.”
In other words, the representation of an object is more detailed and accurate than a description, but
it does not necessarily contain any high-level information on the object explicitly. e description
is more concise, still it conveys an elaborate and composite view of the object.
Over the years, computer graphics started addressing the same basic issues targeted by
computer vision, from shape recognition to shape segmentation and understanding. e analysis
of 3D models is completely different from the analysis of 2D images. e advantage in 3D is
that we rely on a complete representation of the object shape, while the prospective projection in
2D may cause gaps due to occlusions or to lighting conditions. e disadvantage in 3D is that
geometric models and representations are much more complex to handle than simple grids of
pixels. Nowadays, 3D shape analysis deals with an ever-growing number of challenging issues,
from the analysis of single complex shapes (e.g., deformable models), to the reasoning on the
content of entire collections of shapes. e range of problems pertaining to 3D shape analysis is
reflected by the list of research topics below (see also Figure 2.1 ); throughout the topic we will see
how mathematics can support the solution of these problems.

3D shape analysis problems
• Feature detection: How can we find significant geometric features robustly? Feature de-
tection refers to the identification of significant shape features, where the term “significant”
has a different meaning in different domains. e automatic detection of surface features is
a fundamental task in 3D shape analysis, as it facilitates operations such as editing, simplifi-
cation, morphing, and compression of digital 3D models. Also, feature detection is often the