Analysis and processing of 3D digital shapes is a significant research area with
numerous medical, industrial, and entertainment applications which has gained
enormously in importance as optical scanning modalities have started to make
acquired 3D geometry commonplace. The area holds many challenges. One
such challenge, which is addressed in this thesis, is to develop computational
methods for classifying shapes which are in agreement with the human way of
understanding and classifying shapes.
In this dissertation we first present a shape descriptor based on the process
of diffusion on the surface of the shape – the auto diffusion function. When
all heat is inserted at a single point, the function describes how much of that
heat will remain at the same point after a period of time. This method allows
for finding shape features at different scales related to time parameter. For
instance, in conjunction with the method of Reeb graphs for skeletonization, it
is an effective tool for generating scale dependent skeletons of shapes represented
as 3D triangle meshes.
The second part of the thesis aims at capturing the style phenomenon. The style
of an object is easily recognized by humans but a computational method for
finding the style of an object is elusive. Instead of codifying the style explicitly,
which can be only done within a specific context, we develop a general method
for dealing with both style and function which uses the supervision provided
by a set of training examples and can be evaluated using any shape descriptor,
that produces dissimilarity measures between different shapes. Our methods
decouple the effect of style from the effect of function and assess how suitable a
descriptor is to a specific problem.