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17
Computing Persistent Homology
 Discrete Comput. Geom
"... We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enabl ..."
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Cited by 119 (20 self)
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We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This results generalizes and extends the previously known algorithm that was restricted to subcomplexes of S and Z2 coefficients. Finally, our study implies the lack of a simple classification over nonfields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.
Removing excess topology from isosurfaces
 ACM Trans. Graph
"... Many highresolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied ..."
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Cited by 80 (1 self)
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Many highresolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied. However, isosurfaces often contain many topological errors in the form of tiny handles. These nearly invisible artifacts hinder subsequent operations like mesh simplification, remeshing, and parametrization. In this article we present a practical method for removing handles in an isosurface. Our algorithm makes an axisaligned sweep through the volume to locate handles, compute their sizes, and selectively remove them. The algorithm is designed to facilitate outofcore execution. It finds the handles by incrementally constructing and analyzing a Reeb graph. The size of a handle is measured by a short nonseparating cycle. Handles are removed robustly by modifying the volume rather than attempting “mesh surgery. ” Finally, the volumetric modifications are spatially localized to preserve geometrical detail. We demonstrate topology simplification on several complex models, and show its benefits for subsequent surface processing.
Persistence Diagrams of Cortical Surface Data
 Information Processing in Medical Imaging, LNCS
"... Abstract. We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persis ..."
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Cited by 15 (6 self)
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Abstract. We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persistence diagrams. These are scatter plots of paired local critical values of the signal. One of these diagrams visually shows how the number of connected components of the sublevel sets of the signal changes. The use of local critical values of a function differs from the usual statistical parametric mapping framework, which mainly uses the mean signal in quantifying imaging data. Our proposed method uses all the local critical values in characterizing the signal and by doing so offers a completely new data reduction and analysis framework for quantifying the signal. As an illustration, we apply this method to a 1D simulated signal and 2D cortical thickness data. 1
COREDUCTION HOMOLOGY ALGORITHM FOR INCLUSIONS AND PERSISTENT HOMOLOGY
"... Abstract. We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the p ..."
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Abstract. We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the persistence concept to twosided filtrations. In addition to describing the theoretical background, we present results of numerical experiments, as well as several applications to concrete problems in materials science. 1.
Topology based selection and curation of level sets
 In TopoInVis 2007, Accepted
"... Summary. The selection of appropriate level sets for the quantitative visualization of three dimensional imaging or simulation data is a problem that is both fundamental and essential. The selected level set needs to satisfy several topological and geometric constraints to be useful for subsequent ..."
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Cited by 6 (3 self)
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Summary. The selection of appropriate level sets for the quantitative visualization of three dimensional imaging or simulation data is a problem that is both fundamental and essential. The selected level set needs to satisfy several topological and geometric constraints to be useful for subsequent quantitative processing and visualization. For an initial selection of an isosurface, guided by contour tree data structures, we detect the topological features by computing stable and unstable manifolds of the critical points of the distance function induced by the isosurface. We further enhance the description of these features by associating geometric attributes with them. We then rank the attributed features and provide a handle to them for curation of the topological anomalies. 1
Computational topology algorithms for discrete 2manifolds
, 2003
"... 1947–2003 iv ..."
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Efficient computation of a hierarchy of discrete 3d gradient vector fields
 in Proc. TopoInVis
, 2011
"... Abstract This paper introduces a novel combinatorial algorithm to compute a hierarchy of discrete gradient vector fields for threedimensional scalar fields. The hierarchy is defined by an importance measure and represents the combinatorial gradient flow at different levels of detail. The presented ..."
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Abstract This paper introduces a novel combinatorial algorithm to compute a hierarchy of discrete gradient vector fields for threedimensional scalar fields. The hierarchy is defined by an importance measure and represents the combinatorial gradient flow at different levels of detail. The presented algorithm is based on Forman’s discrete Morse theory, which guarantees topological consistency and algorithmic robustness. In contrast to previous work, our algorithm combines memory and runtime efficiency. It thereby lends itself to the analysis of large data sets. A discrete gradient vector field is also a compact representation of the underlying extremal structures – the critical points, separation lines and surfaces. Given a certain level of detail, an explicit geometric representation of these structures can be extracted using simple and fast graph algorithms. 1
Constructive MayerVietoris Algorithm: Computing the Homology of Unions of Simplicial Complexes
, 2010
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Topological Analysis of Scalar Functions for Scientific Data Visualization
, 2004
"... Scientists attempt to understand physical phenomena by studying various quantities measured over the region of interest. A majority of these quantities are scalar (realvalued) functions. These functions are typically studied using traditional visualization techniques like isosurface extraction, ..."
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Scientists attempt to understand physical phenomena by studying various quantities measured over the region of interest. A majority of these quantities are scalar (realvalued) functions. These functions are typically studied using traditional visualization techniques like isosurface extraction, volume rendering etc. As the data grows in size and becomes increasingly complex, these techniques are no longer e#ective. State of the art visualization methods attempt to automatically extract features and annotate a display of the data with a visualization of its features. In this thesis, we study and extract the topological features of the data and use them for visualization. We have three results: .
Abstract ControlledTopology Filtering
"... Many applications require the extraction of isolines and isosurfaces from scalar functions defined on regular grids. These scalar functions may have many different origins: from MRI and CT scan data to terrain data or results of a simulation. As a result of noise and other artifacts, curves and surf ..."
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Many applications require the extraction of isolines and isosurfaces from scalar functions defined on regular grids. These scalar functions may have many different origins: from MRI and CT scan data to terrain data or results of a simulation. As a result of noise and other artifacts, curves and surfaces obtained by standard extraction algorithms often suffer from topological irregularities and geometric noise. While it is possible to remove topological and geometric noise as a postprocessing step, in the case when a large number of isolines are of interest there is a considerable advantage in filtering the scalar function directly. While most smoothing filters result in gradual simplification of the topological structure of contours, new topological features typically emerge and disappear during the smoothing process. In this paper, we describe an algorithm for filtering functions defined on regular 2D grids with controlled topology changes, which ensures that the topological structure of the set of contour lines of the function is progressively simplified.