Ph.D Student | Azencot Omri |
---|---|

Subject | Operator Representations in Geometry Processing |

Department | Department of Computer Science |

Supervisor | Professor Mirela Ben-Chen |

This thesis introduces fundamental equations as well as discrete tools
and numerical methods for carrying out various geometrical tasks on
three-dimensional surfaces via *operators*. An example for an operator is
the Laplacian which maps real-valued functions to their sum of second
derivatives. More generally, many mathematical objects feature an operator
interpretation, and in this work, we consider a few of them in the context of
geometry processing and numerical simulation problems. The operator point of
view is useful in applications since high-level algorithms can be devised for
the problems at hand with operators serving as the main building blocks. While
this approach has received some attention in the past, it has not reached its
full potential, as the following thesis tries to hint.

The contribution of this document is twofold. First, it describes the
analysis and discretization of *derivations* and related operators such as
covariant derivative, Lie bracket, pushforward and flow on triangulated
surfaces. These operators play a fundamental role in numerous computational
science and engineering problems, and thus enriching the readily available differential
tools with these novel components offers multiple new avenues to explore.
Second, these objects are then used to solve certain differential equations on
curved domains such as the advection equation, the Navier-Stokes equations and
the thin films equations. Unlike previous work, our numerical methods are *intrinsic*
to the surface?that is, independent of a particular geometry flattening. In
addition, the suggested machinery *preserves structure*?namely, a central
quantity to the problem, as the total mass, is exactly preserved. These two
properties typically provide a good balance between computation times and
quality of results.

From a broader standpoint, recent years have brought an expected increase in computation power along with extraordinary advances in the theory and methodology of geometry acquisition and processing. Consequently, many approaches which were infeasible before, became viable nowadays. In this view, the operator perspective and its application to differential equations, as depicted in this work, provides an interesting alternative, among the other approaches, for working with complex problems on non-flat geometries. In the following chapters, we study in which cases operators are applicable, while providing a fair comparison to state-of-the-art methods.