Results 11  20
of
717
Isosurface Reconstruction with Topology Control
, 2002
"... Extracting isosurfaces from volumetric datasets is an essential step for indirect volume rendering algorithms. For physically measured data like it is used, e.g. in medical imaging applications one often introduces topological errors such as small handles that stem from measurement inaccuracy and ca ..."
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Cited by 36 (2 self)
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Extracting isosurfaces from volumetric datasets is an essential step for indirect volume rendering algorithms. For physically measured data like it is used, e.g. in medical imaging applications one often introduces topological errors such as small handles that stem from measurement inaccuracy and cavities that are generated by tight folds of an organ. During isosurface extraction these measurement errors result in a surface whose genus is much higher than that of the actual surface. In many cases however, the topological type of the object under consideration is known beforehand, e.g., the cortex of a human brain is always homeomorphic to a sphere. By using topology preserving morphological operators we can exploit this knowledge to gradually dilate an initial set of voxels with correct topology until it fits the target isosurface. This approach avoids the formation of handles and cavities and guarantees a topologically correct reconstruction of the object's surface.
Homological sensor networks
 Notices of the American Mathematical Society
, 2007
"... A sensor is a device that measures some feature of a domain or environment and returns a signal from which information may be extracted. Sensors vary in scope, resolution, and ability. The information they return can be as simple as a binary flag, as with a metal detector that beeps to indicate a de ..."
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Cited by 31 (0 self)
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A sensor is a device that measures some feature of a domain or environment and returns a signal from which information may be extracted. Sensors vary in scope, resolution, and ability. The information they return can be as simple as a binary flag, as with a metal detector that beeps to indicate a detection threshold being crossed. A more complex sensor, such as a video camera, can return a signal requiring sophisticated analysis to extract relevant data. An increasingly common application for sensors is to scan a region for a particular object or substance. For example, one might wish to determine the existence and location of an outbreak of fire in a national forest. Questions of more interest to national security involve detection of radiological or biological hazards, hidden mines and munitions, or specific individuals in a crowd. All of these scenarios pose difficult and challenging data management problems. Numerous strategies exist, aided by the fact that sensor technology provides an expansive array of available hardware. A fundamental dichotomy exists in the approach to sensing an environment based on the number and complexity of sensors. For a fixed cost (monetary, or perhaps “total complexity”), one can deploy a small number of sophisticated “global ” sensors with high signal complexity and precise readings. In contrast, one can deploy a large number of small, coarse, “local ” devices that may
Topology and Data
, 2008
"... An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that ..."
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Cited by 31 (0 self)
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An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data
Proximity of persistence modules and their diagrams
, 2008
"... Topological persistence has proven to be a key concept for the study of realvalued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case o ..."
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Cited by 31 (7 self)
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Topological persistence has proven to be a key concept for the study of realvalued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable spaces. In this paper, we present new stability results that do not suffer from the above restrictions. Furthermore, by working at an algebraic level directly, we make it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence. Along the way, we extend the definition of persistence diagram to a larger setting, introduce the notions of discretization of a persistence module and associated pixelization map, define a proximity measure between persistence modules, and show how to interpolate between persistence modules, thereby lending a more analytic character to this otherwise algebraic setting. We believe these new theoretical concepts and tools shed new light on the theory of persistence, in addition to simplifying proofs and enabling new applications.
Higher–order polynomial invariants of 3–manifolds giving lower bounds for the Thurston norm
, 2002
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Computing shortest nontrivial cycles on orientable surfaces of bounded genus in almost linear time
 In SOCG ’06: Proc. 22nd Symp. Comput. Geom
, 2006
"... We present an algorithm that computes a shortest noncontractible and a shortest nonseparating cycle on an orientable combinatorial surface of bounded genus in O(n log n) time, where n denotes the complexity of the surface. This solves a central open problem in computational topology, improving upon ..."
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Cited by 29 (0 self)
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We present an algorithm that computes a shortest noncontractible and a shortest nonseparating cycle on an orientable combinatorial surface of bounded genus in O(n log n) time, where n denotes the complexity of the surface. This solves a central open problem in computational topology, improving upon the currentbest O(n 3/2)time algorithm by Cabello and Mohar (ESA 2005). Our algorithm uses universalcover constructions to find short cycles and makes extensive use of existing tools from the field. 1
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 28 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
Graph grammars for self assembling robotic systems
 In Proceedings of the 2004 IEEE International Conference on Robotics and Automation
, 2004
"... Abstract — In this paper we define a class of graph grammars that can be used to model and direct distributed robotic assembly or formation forming processes. We focus on the problem of synthesizing a grammar so that it generates a given, prespecified assembly. In particular, to generate an acyclic ..."
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Cited by 27 (2 self)
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Abstract — In this paper we define a class of graph grammars that can be used to model and direct distributed robotic assembly or formation forming processes. We focus on the problem of synthesizing a grammar so that it generates a given, prespecified assembly. In particular, to generate an acyclic graph we synthesize a binary grammar (rules involve at most two parts), and for a general graph we synthesize a ternary grammar (rules involve at most three parts). We then show a general result that implies that no binary grammar can generate a unique stable assembly. We conclude the paper with a discussion of how graph grammars can be used to direct the synthesis of parts floating in a fluid or for selfmotive robotic parts. I.
Discrete OneForms on Meshes and Applications to 3D Mesh Parameterization
 Journal of CAGD
, 2006
"... We describe how some simple properties of discrete oneforms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "springembedding" theorem for planar graphs, which is widely used for parameterizing mesh ..."
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Cited by 25 (1 self)
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We describe how some simple properties of discrete oneforms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "springembedding" theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be nonconvex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being nonconvex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.