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**1 - 3**of**3**### Reviewers

, 2012

"... Rendering computer-generated images is both memory and runtime intensive. This is particularly true in realtime computer graphics where large amounts of content have to be produced very quickly and from limited data. Tile-based methods offer a solution to this problem by generating large portions of ..."

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Rendering computer-generated images is both memory and runtime intensive. This is particularly true in realtime computer graphics where large amounts of content have to be produced very quickly and from limited data. Tile-based methods offer a solution to this problem by generating large portions of a specific content out of a much smaller data set of tiles. This dissertation investigates the use of corner tiles for this purpose—unit square tiles with color-coded corners. They tile the plane by placing them without gaps or overlaps such that tiles have matching corner colors. We present efficient algorithms to perform such a tiling that are both more flexible and less prone to artifacts than existing algorithms. We also present solutions to combinatorial problems that arise when using corner tiles, and introduce high-quality methods to perform the tile-based generation of two fundamental components of any rendering system: textures and two-dimensional sample point sets. The results of this dissertation are advantageous for both realtime and offline rendering systems where they improve state-of-the-art results in texture synthesis, image plane sampling, and lighting computations based on numerical integration. iii Zusammenfassung Die Berechnung computergenerierter Bilder ist sehr speicher- und laufzeitintensiv. Besonders kritisch ist dies in der Echtzeit-Computergraphik, in der große Inhaltsmengen sehr schnell und auf einer limitieren Datenbasis generiert werden müssen. Kachelbasierte Methoden bieten für diese Herausforderung eine mögliche Lösung an: Ein bestimmter Inhalt wird dynamisch anhand einer kleinen

### Efficient Computation of Clipped Voronoi Diagram for Mesh Generation

"... The Voronoi diagram is a fundamental geometric structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact domain (i.e. a bounded and closed 2D region or a 3D volume), some Voronoi cells of their Voronoi diagram are infinite or pa ..."

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The Voronoi diagram is a fundamental geometric structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact domain (i.e. a bounded and closed 2D region or a 3D volume), some Voronoi cells of their Voronoi diagram are infinite or partially outside of the domain, but in practice only the parts of the cells inside the domain are needed, as when computing the centroidal Voronoi tessellation. Such a Voronoi diagram confined to a compact domain is called a clipped Voronoi diagram. We present an efficient algorithm to compute the clipped Voronoi diagram for a set of sites with respect to a compact 2D region or a 3D volume. We also apply the proposed method to optimal mesh generation based on the centroidal Voronoi tessellation.

### Non-periodic corner tilings in computer graphics

, 2012

"... Rendering computer-generated images is both memory and runtime intensive. This is particularly true in realtime computer graphics where large amounts of content have to be produced very quickly and from limited data. Tile-based methods offer a solution to this problem by generating large portions of ..."

Abstract
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Rendering computer-generated images is both memory and runtime intensive. This is particularly true in realtime computer graphics where large amounts of content have to be produced very quickly and from limited data. Tile-based methods offer a solution to this problem by generating large portions of a specific content out of a much smaller data set of tiles. This dissertation investigates the use of corner tiles for this purpose—unit square tiles with color-coded corners. They tile the plane by placing them with-out gaps or overlaps such that tiles have matching corner colors. We present efficient algorithms to perform such a tiling that are both more flexible and less prone to artifacts than existing algorithms. We also present solutions to combinatorial problems that arise when using corner tiles, and introduce high-quality methods to perform the tile-based generation of two fundamental components of any rendering system: textures and two-dimensional sample point sets. The results of this dissertation are advantageous for both realtime and offline rendering systems where they improve state-of-the-art results in texture syn-thesis, image plane sampling, and lighting computations based on numerical integration.