|Title:||Operator Representations in Geometry Processing
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|Abstract:||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.
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