About this Topic - Mathematical Tools for Shape Analysis and Description - page 2

1. ABOUT THIS TOPIC
nologies are causing a fast process of dematerialization in our lives, with a reduction of the material
essence of our reality: material objects are gradually substituted by processes and services which
are more and more immaterial [ 132 ]. While attaining visual realism is within the grasp of current
research and development, further research has to be done to ensure that digital environments
will be accommodating to the needs of human communication and interaction. Shape analysis
methods are defining the building blocks of these future interfaces, laying the basis for associating
meaning to digital assets and for interacting with the digital world in a semantically rich modality.
1.2 WHY MATH FOR 3D SHAPE ANALYSIS?
e focus of this topic is on mathematics for shape analysis: the motivation for this choice is
captured very well by the following citation from the famous topic OnGrowthandForm by D'Arcy
ompson [ 195 ].
“We must learn from the mathematician to eliminate and discard; to keep in mind the
type and leave the single case, with all its accidents, alone; and to find in this sacrifice
of what matters little and conservation of what matters much one of the peculiar
excellences of the method of mathematics.”
In the context of biological taxonomy, D'Arcy ompson clearly states that mathematical
formalisms and tools are needed in shape analysis and classification for their abstraction and syn-
thesis power. He clarifies further: “the study of form may be descriptive merely, or it may become
analytical. We begin by describing the shape of an object in the simple words of common speech:
we end by defining it in the precise language of mathematics: and the one method tends to follow
the other in strict scientific order and historical continuity.” Based on this approach, he classi-
fies animals, bones or plants by using geometric transformations (combinations of dilations and
contractions). A famous example taken from his topic shows two different fish made equivalent
through such a transformation (Figure 1.1 ).
e notion of shape and invariance occurs frequently in many attempts to clarify what a
shape is and what are the important properties to characterize it. In the field of statistical shape
analysis, Kendall [ 111 ] defined a shape as “all the geometrical information that remains when
location, scale, and rotational effects (Euclidean transformations) are filtered out from an object”;
yet, if we look at the examples in Figure 1.2 , we will probably think there should be something
more beside Euclidean transformations. e cups' shapes are indeed similar, even if not all of
them may be obtained by Euclidean transformations applied to the same geometry.
e description of shapes relies on the use of similes, as it emerges nicely in the following
linguistic definition: “shape: the outer form of something by which it can be seen (or felt) to be
different from something else” [ 3 ]. is definition establishes a link between the concept of shape
and the concept of similarity , as used for distinguishing among objects: shape emerges thanks
to invariance, or equivalence, under similarity transformations. ough convincing, this idea can
be puzzling at the same time, as long as we do not define what similar, or different, means. In