**FULL TUTORIAL T1:
**

**FREEFORM
SHAPE REPRESENTATIONS FOR EFFICIENT GEOMETRY
PROCESSING**

**Stephan Bischoff
Mario Botsch
Leif Kobbelt **

The most important data structures for handling and storage of free

form shapes in geometry processing applications and computer graphics

can be classified according to the dimensionality of their basic

structural elements: point-sets, polygon meshes, as well as volumetric

representations are well established concepts to describe the shape of

arbitrarily complex objects. On the other hand, algorithms to process

a given geometric object can be classified according to the dominant

access operation which can be of the type evaluation (sampling), query

(e.g. inside/outside tests), or modification (of the geometry or the

topology). On a more abstract level, we can in principle distinguish

between unstructured geometry representations which do not imply any

global regularity and uniformly structured representations or even

hierarchically structured representations.

All these different types of shape representations will be presented

in this tutorial and the respective strengths and disadvantages will

be discussed in detail. For a given application we can analyze which

types of access operations are the most critial ones and then design

a custom data structure accoring to the usage profile. In some cases

it turns out that a proper combination of two different representations

provides the optimal performance or robustness. We will present a

couple of important geometry processing problems such as topology

control for deforming implicit surfaces, mesh decimation and smoothing

with global error control, and mesh restoration where this combination

of geometry data representations leads to superior algorithms compared

to the standard approaches.

Short (incomplete) list
of topics:

* introduction

* point-based representations

(generation, storage, high-quality rendering, deformation)

* polygon meshes

(data structures, hierarchies)

* volumetric representations

(data structures, level sets)

* geometry processing tasks

remeshing
: (points and polygons)

topology control
: (points and volumes)

smoothing/decimation : (polygons and volumes)

multiresolution
: (polygons and volumes)

... and others