Exploring Material Representations for Sparse Voxel DAGs

Ray tracing is a popular technique used in movies and video games to create compelling visuals. Ray traced computer images are increasingly becoming more realistic and almost indistinguishable from real-word images. Due to the complexity of scenes and the desire for high resolution images, ray tracing can become very expensive in terms of computation and memory. To address these concerns, researchers have examined data structures to efficiently store geometric and material information. Sparse voxel octrees (SVOs) and directed acyclic graphs (DAGs) have proven to be successful geometric data structures for reducing memory requirements. Moxel DAGs connect material properties to these geometric data structures, but experience limitations related to memory, build times, and render times. This thesis examines the efficacy of connecting an alternative material data structure to existing geometric representations. The contributions of this thesis include the creation of a new material representation using hashing to accompany DAGs, a method to calculate surface normals using neighboring voxel data, and a demonstration and validation that DAGs can be used to super sample based on proximity. This thesis also validates the visual acuity from these methods via a user survey comparing different output images. In comparison to the Moxel DAG implementation, this work increases render time, but reduces build times and memory, and improves the visual quality of output images

Spanners for Geometric Intersection Graphs

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.Comment: 16 pages, 5 figures, Late

Recurrence-based time series analysis by means of complex network methods

Complex networks are an important paradigm of modern complex systems sciences which allows quantitatively assessing the structural properties of systems composed of different interacting entities. During the last years, intensive efforts have been spent on applying network-based concepts also for the analysis of dynamically relevant higher-order statistical properties of time series. Notably, many corresponding approaches are closely related with the concept of recurrence in phase space. In this paper, we review recent methodological advances in time series analysis based on complex networks, with a special emphasis on methods founded on recurrence plots. The potentials and limitations of the individual methods are discussed and illustrated for paradigmatic examples of dynamical systems as well as for real-world time series. Complex network measures are shown to provide information about structural features of dynamical systems that are complementary to those characterized by other methods of time series analysis and, hence, substantially enrich the knowledge gathered from other existing (linear as well as nonlinear) approaches.Comment: To be published in International Journal of Bifurcation and Chaos (2011

The Geometric Block Model

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model generalizes the random geometric graphs in the same way that the well-studied stochastic block model generalizes the Erdos-Renyi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancement in community detection. While being a topic of fundamental theoretical interest, our main contribution is to show that many practical community structures are better explained by the geometric block model. We also show that a simple triangle-counting algorithm to detect communities in the geometric block model is near-optimal. Indeed, even in the regime where the average degree of the graph grows only logarithmically with the number of vertices (sparse-graph), we show that this algorithm performs extremely well, both theoretically and practically. In contrast, the triangle-counting algorithm is far from being optimum for the stochastic block model. We simulate our results on both real and synthetic datasets to show superior performance of both the new model as well as our algorithm.Comment: A shorter version of this paper has appeared in 32nd AAAI Conference on Artificial Intelligence. The AAAI proceedings version as well as the previous version in arxiv contained some errors that have been corrected in this versio