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Featurepreserving reconstruction of singular surfaces
 EUROGRAPHICS SYMPOSIUM ON GEOMETRY PROCESSING 2012, EITAN GRINSPUN , NILOY MITRA (EDS.)
, 2012
"... Reconstructing a surface mesh from a set of discrete point samples is a fundamental problem in geometric modeling. It becomes challenging in presence of ‘singularities ’ such as boundaries, sharp features, and nonmanifolds. A few of the current research in reconstruction have addressed handling som ..."
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Reconstructing a surface mesh from a set of discrete point samples is a fundamental problem in geometric modeling. It becomes challenging in presence of ‘singularities ’ such as boundaries, sharp features, and nonmanifolds. A few of the current research in reconstruction have addressed handling some of these singularities, but a unified approach to handle them all is missing. In this paper we allow the presence of various singularities by requiring that the sampled object is a collection of smooth surface patches with boundaries that can meet or intersect. Our algorithm first identifies and reconstructs the features where singularities occur. Next, it reconstructs the surface patches containing these feature curves. The identification and reconstruction of feature curves are achieved by a novel combination of the Gaussian weighted graph Laplacian and the Reeb graphs. The global reconstruction is achieved by a method akin to the well known Cocone reconstruction, but with weighted Delaunay triangulation that allows protecting the feature samples with balls. We provide various experimental results to demonstrate the effectiveness of our featurepreserving singular surface reconstruction algorithm.
TRAJECTORY GROUPING STRUCTURE
 JOURNAL OF COMPUTATIONAL GEOMETRY
, 2015
"... The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept fro ..."
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The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters that allow more global views of the data in group size, group duration, and entity interdistance. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. We also study how the trajectory grouping structure can be made robust, that is, how brief interruptions of groups can be disregarded in the global structure, adding a notion of persistence to the structure. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. The Starkey data describe the movement of elk, deer, and cattle. Although there is no ground truth for the grouping structure in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results.
Probabilistic StreetIntersection Reconstruction from GPS Trajectories: Approaches and Challenges∗
"... Analyzing and mining georeferenced trajectory data has different aspects to researchers from different communities. For example, animal location data provides ecologists live points of contact between ecologies and the species. And by studying movements of individual animals, they have gained insig ..."
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Analyzing and mining georeferenced trajectory data has different aspects to researchers from different communities. For example, animal location data provides ecologists live points of contact between ecologies and the species. And by studying movements of individual animals, they have gained insight into population distributions, important resources, dispersal settings, social interaction or general patterns of how the space was used in an ecological system. Similarly, geologists and environmentalists use earthquake positional data for predicting the location of the next earthquake. Intelligent Transportation Systems and GIS communities use heuristic algorithms on vehicle trajectory data sets to construct or update digital streetmaps that represent the data set. Recently, the Computational Geometry community started to give attention to the streetmap construction problems as well, applying different approaches and providing quality guarantees. Although different communities use different types or aspects of the GPS data, they face one challenge in common: how to model or incorporate the impreciseness of the input data in their output. In this paper we discuss specifically the impact of spatial inaccuracy of GPS trajectory data on streetmap reconstruction algorithms. In particular, we discuss approaches and challenges to associate that impreciseness with the reconstructed streetintersections.
Statistical Analysis of Metric Graph Reconstruction
"... A metric graph is a 1dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. We co ..."
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A metric graph is a 1dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as street maps, neurons, networks of rivers and galaxies. We consider the statistical problem of reconstructing the topology of a metric graph from a random sample. We derive a lower bound on the minimax risk for the noiseless case and an upper bound for the special case of metric graphs embedded in R2. The upper bound is based on the reconstruction algorithm given in Aanjaneya et al. (2012).
Noname manuscript No. (will be inserted by the editor) Can a Black Hole Collapse to a Spacetime Singularity?
, 704
"... Abstract A critique of the singularity theorems of Penrose, Hawking, and Geroch is given. It is pointed out that a gravitationally collapsing black hole acts as an ultrahigh energy particle accelerator that can accelerate particles to energies inconceivable in any terrestrial particle accelerator, a ..."
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Abstract A critique of the singularity theorems of Penrose, Hawking, and Geroch is given. It is pointed out that a gravitationally collapsing black hole acts as an ultrahigh energy particle accelerator that can accelerate particles to energies inconceivable in any terrestrial particle accelerator, and that when the energy E of the particles comprising matter in a black hole is ∼ 10 2 GeV or more, or equivalently, the temperature T is ∼ 10 15 K or more, the entire matter in the black hole is converted into quarkgluon plasma permeated by leptons. As quarks and leptons are fermions, it is emphasized that the collapse of a blackhole to a spacetime singularity is inhibited by Pauli’s exclusion principle. It is also suggested that ultimately a black hole may end up either as a stable quark star, or as a pulsating quark star which may be a source of gravitational radiation, or it may simply explode with a mini bang of a sort. Keywords black hole · gravitational collapse · spacetime singularity · quark star 1
Local, Smooth, and Consistent Jacobi Set Simplification
"... The relation between two Morse functions defined on a smooth, compact, and orientable 2manifold can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the two functions are aligned. Both the Jacobi set itself as well as the segmentation of t ..."
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The relation between two Morse functions defined on a smooth, compact, and orientable 2manifold can be studied in terms of their Jacobi set. The Jacobi set contains points in the domain where the gradients of the two functions are aligned. Both the Jacobi set itself as well as the segmentation of the domain it induces, have shown to be useful in various applications. In practice, unfortunately, functions often contain noise and discretization artifacts, causing their Jacobi set to become unmanageably large and complex. Although there exist techniques to simplify Jacobi sets, they are unsuitable for most applications as they lack finegrained control over the process, and heavily restrict the type of simplifications possible. This paper introduces the theoretical foundations of a new simplification framework for Jacobi sets. We present a new interpretation of Jacobi set simplification based on the perspective of domain segmentation. Generalizing the cancellation of critical points from scalar functions to Jacobi sets, we focus on simplifications that can be realized by smooth approximations of the corresponding functions, and show how these cancellations imply simultaneous simplification of contiguous subsets of the Jacobi set. Using these extended cancellations as atomic operations, we introduce an algorithm to successively cancel subsets of the Jacobi set with minimal modifications to some userdefined metric. We show that for simply connected domains, our algorithm reduces a given Jacobi set to its minimal
A fast and robust algorithm to count topologically persistent holes in noisy clouds
"... Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in noisy clouds in the plane. The holes in a given cloud are quantified by the topological persistence of their boundary contours when the cloud is analyzed at all possible scales. We des ..."
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Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in noisy clouds in the plane. The holes in a given cloud are quantified by the topological persistence of their boundary contours when the cloud is analyzed at all possible scales. We design the algorithm to count holes that are most persistent in the filtration of offsets (neighborhoods) around given points. The input is a cloud of n points in the plane without any userdefined parameters. The algorithm has O(n log n) time and O(n) space. The output is the array (number of holes, relative persistence in the filtration). We prove theoretical guarantees when the algorithm finds the correct number of holes (components in the complement) of an unknown shape approximated by a cloud.