This work introduces a tool for interactive exploration and visualization using MetaTracts. MetaTracts is a novel method for extraction and visualization of individual fiber bundles and weaving patterns from X-ray computed tomography (XCT) scans of endless carbon fiber reinforced polymers (CFRPs). It is designed specifically to handle XCT scans of low resolutions where the individual fibers are barely visible, which makes extraction of fiber bundles a challenging problem. The proposed workflow is used to analyze unit cells of CFRP materials integrating a recurring weaving pattern. First, a coarse version of integral curves is used to trace sections of the individual fiber bundles in the woven CFRP materials. We call these sections MetaTracts. In the second step, these extracted fiber bundle sections are clustered using a two-step approach: first by orientation, then by proximity. The tool can generate volumetric representations as well as surface models of the extracted fiber bundles to be exported for further analysis. In addition a custom interactive tool for exploration and visual analysis of MetaTracts is designed. We evaluate the proposed workflow on a number of real world datasets and demonstrate that MetaTracts effectively and robustly identifies and extracts fiber bundles.

This work introduces MetaTracts, a novel method for extracting and visualizing individual fiber bundles and weaving patterns from X-ray computed tomography (XCT) scans of endless carbon fiber reinforced polymers (CFRP). The proposed work flow is designed to analyze unit cells of CFRP materials integrating the recurring weaving pattern. It is designed to handle XCT scans of low resolution, in which individual fibers are not visible or are barely visible. First, a coarse version of integral curves is used to trace subsections of the individual fiber bundles in the woven CFRP materials. We call these sections MetaTracts. In the second step, these extracted fiber bundle sections (MetaTracts) are clustered using a two-step approach: first by orientation, then by proximity. The tool can generate volumetric representations as well as surface models of the extracted fiber bundles to be exported for further analysis. We evaluate the proposed work flow on a number of real world datasets and demonstrate that MetaTracts effectively and robustly identifies and separates different fiber bundles.

Let f be a continuous function defined on a topological space. As we increase the function value, connected components in the level set of f appear, disappear, merge, or split, and the Reeb graph tracks these changes. The join and split trees of f track the changes of connected components in the sublevel and superlevel set respectively. If the Reeb graph is loop-free, then it is a tree called the contour tree. Given a piecewise linear function f defined on a simplicial complex K, Carr, Snoeyink and Axen proposed a simple and elegant algorithm to compute the contour tree of f by "merging" its join and split trees in near-linear time.

Large scale scientific simulations frequently use streamline based techniques to visualize flow fields. As the shape of a streamline is often related to some underlying property of the field, it is important to identify streamlines (or their parts) with unique geometric features. In this paper, we introduce a metric, called the box counting ratio, which measures the geometric complexity of streamlines by measuring their space-filling capacity at different scales. We propose a novel interactive visualization framework which utilizes this metric to extract, organize and visualize features of varying density and complexity hidden in large numbers of streamlines. The proposed framework extracts complex regions of varying density from the streamlines, and organizes and presents them on an interactive 2D information space, allowing user selection and visualization of streamlines. We also extend this framework to support exploration using an ensemble of measures including box counting ratio. Our framework allows the user to easily visualize and interact with features otherwise hidden in large vector field data. We strengthen our claims with case studies using combustion and climate simulation data sets.

Abstract A number of papers present algorithms to construct isosurfaces with sharp edges and corners from hermite data, i.e. the exact surface normals at the exact intersection of the surface and grid edges. We discuss some fundamental problems with the previous algorithms and describe a new approach, based on merging grid cubes near sharp edges, that produces significantly better results. Our algorithm requires only gradients at the grid vertices, not at each surface‐edge intersection point. We also give a method for measuring the correctness of the resulting sharp edges and corners in the isosurface.

Given a continuous scalar field ƒ: X → where X is a topological space, a level set of ƒ is a set {x ∈ X : ƒ (x) = α} for some value α ∈ IR. The level sets of ƒ can be subdivided into connected components. As α changes continuously, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of ƒ encodes these changes in connected components of level sets. It provides a simple yet meaningful abstraction of the input domain. As such, it has been used in a range of applications in fields such as graphics and scientific visualization.

A (3D) scalar grid is a regular n1 x n2 x n3 grid of vertices where each vertex v is associated with some scalar value sv. Applying trilinear interpolation, the scalar grid determines a scalar function g where g(v) = sv for each grid vertex v. An isosurface with isovalue σ is a triangular mesh which approximates the level set g(-1)(σ). The fractal dimension of an isosurface represents the growth ;in the isosurface as the number of grid cubes increases. We define and discuss the fractal isosurface dimension. Plotting the fractal ;dimension as a function of the isovalues in a data set provides information about the isosurfaces determined by the data set. We present statistics on the average fractal dimension of 60 publicly available benchmark data sets. We also show the fractal dimension is highly correlated with topological noise in the benchmark data sets, measuring the topological noise by the number of connected components in the isosurface. Lastly, we present a formula predicting the fractal dimension as a function of noise and validate the formula with experimental results.

Abstract We present an algorithm for the reconstruction of a surface with boundaries (including a non‐orientable one) in three dimensions from a sufficiently dense sample. It is guaranteed that the output is isotopic to the unknown sampled surface. No previously known algorithm guarantees isotopic or homeomorphic reconstruction of surfaces with boundaries. Our algorithm is surprisingly simple. It ‘peels’ slivers greedily from an α‐complex of a sample of the surface. No other post‐processing is necessary. We provide several experimental results from an implementation of our basic algorithm and also a modified version of it.

Abstract The Marching Cubes Algorithm may return degenerate, zero area isosurface triangles, and often returns isosurface triangles with small areas, edges or angles. We show how to avoid both problems using an extended Marching Cubes lookup table. As opposed to the conventional Marching Cubes lookup table, the extended lookup table differentiates scalar values equal to the isovalue from scalar values greater than the isovalue. The lookup table has 38= 6561 entries, based on three possible labels, ‘−’ or ‘=’ or ‘+’, of each cube vertex. We present an algorithm based on this lookup table which returns an isosurface close to the Marching Cubes isosurface, but without any degenerate triangles or any small areas, edges or angles.

We present a Delaunay based algorithm for simplifying vector field datasets. Our aim is to reduce the size of the mesh on which the vector field is defined while preserving topological features of the original vector field. We leverage a simple paradigm, vertex deletion in Delaunay triangulations, to achieve this goal. This technique is effective for two reasons. First, we guide deletions by a local error metric that bounds the change of the vectors at the affected simplices and maintains regions near critical points to prevent topological changes. Second, piecewise-linear interpolation over Delaunay triangulations is known to give good approximations of scalar fields. Since a vector field can be regarded as a collection of component scalar fields, a Delaunay triangulation can preserve each component and thus the structure of the vector field as a whole. We provide experimental evidence showing the effectiveness of our technique and its ability to preserve features of both two and three dimensional vector fields.

Let P be a simple polygon and let {( u 1 , u ′ 1 ), ( u 2 , u ′ 2 ),…,( u m , u ′ m )} be a set of m pairs of distinct vertices of P , where for every distinct i , j ≤ m , there exist pairwise disjoint (nonintersecting) paths connecting u i to u ′ i and u j to u ′ j . We wish to construct m pairwise disjoint paths in the interior of P connecting u i to u ′ i for i = 1, …, m , with a minimal total number of line segments. We give an approximation algorithm that constructs such a set of paths using O ( M ) line segments in O ( n log m + M log m ) time, where M is the number of line segments in the optimal solution and n is the size of the polygon.

We present an algorithm for constructing isosurfaces in any dimension. The input to the algorithm is a set of scalar values in a d-dimensional regular grid of (topological) hypercubes. The output is a set of (d-1)-dimensional simplices forming a piecewise linear approximation to the isosurface. The algorithm constructs the isosurface piecewise within each hypercube in the grid using the convex hull of an appropriate set of points. We prove that our algorithm correctly produces a triangulation of a (d-1)-manifold with boundary. In dimensions three and four, lookup tables with 2(8) and 2(16) entries, respectively, can be used to speed the algorithm's running time. In three dimensions, this gives the popular Marching Cubes algorithm. We discuss applications of four-dimensional isosurface construction to time varying isosurfaces, interval volumes, and morphing.

Tracking and visualizing local features from a time-varying volumetric data allows the user to focus on selected regions of interest, both in space and time, which can lead to a better understanding of the underlying dynamics. In this paper, we present an efficient algorithm to track time-varying isosurfaces and interval volumes using isosurfacing in higher dimensions. Instead of extracting the data features such as isosurfaces or interval volumes separately from multiple time steps and computing the spatial correspondence between those features, our algorithm extracts the correspondence directly from the higher dimensional geometry and thus can more efficiently follow the user selected local features in time. In addition, by analyzing the resulting higher dimensional geometry, it becomes easier to detect important topological events and the corresponding critical time steps for the selected features. With our algorithm, the user can interact with the underlying time-varying data more easily. The computation cost for performing time-varying volume tracking is also minimized.

Shape modeling is an integral part of many visualization problems. Recent advances in scanning technology and a number of surface reconstruction algorithms have opened up a new paradigm for modeling shapes from samples. Many of the problems currently faced in this modeling paradigm can be traced back to two anomalies in sampling, namely undersampling and oversampling. Boundaries, non-smoothness and small features create undersampling problems, whereas oversampling leads to too many triangles. We use Voronoi cell geometry as a unified guide to detect undersampling and oversampling. We apply these detections in surface reconstruction and model simplification. Guarantees of the algorithms can be proved. The authors show the success of the algorithms empirically on a number of interesting data sets.

Article A Helly-type theorem for hyperplane transversals to well-separated convex sets Share on Authors: Boris Aronov Polytechnic University, Brooklyn, NY Polytechnic University, Brooklyn, NYView Profile , Jacob E. Goodman City College, City University of New York, New York, NY City College, City University of New York, New York, NYView Profile , Richard Pollack Courant Institute of Mathematical Sciences, New York University, New York, NY Courant Institute of Mathematical Sciences, New York University, New York, NYView Profile , Rephael Wenger The Ohio State University, Columbus, OH The Ohio State University, Columbus, OHView Profile Authors Info & Claims SCG '00: Proceedings of the sixteenth annual symposium on Computational geometryMay 2000 Pages 57–63https://doi.org/10.1145/336154.336178Online:01 May 2000Publication History 0citation233DownloadsMetricsTotal Citations0Total Downloads233Last 12 Months4Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Article Reconstruction curves with sharp corners Share on Authors: Tamal K. Dey View Profile , Rephael Wenger Dept. of CIS, Ohio State University, Columbus, Ohio Dept. of CIS, Ohio State University, Columbus, OhioView Profile Authors Info & Claims SCG '00: Proceedings of the sixteenth annual symposium on Computational geometryMay 2000 Pages 233–241https://doi.org/10.1145/336154.336209Online:01 May 2000Publication History 13citation470DownloadsMetricsTotal Citations13Total Downloads470Last 12 Months7Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Visualization algorithms have seen substantial improvements in the past several years. However, very few algorithms have been developed for directly studying data in dimensions higher than three. Most algorithms require a sampling in three-dimensions before applying any visualization algorithms. This sampling typically ignores vital features that may be present when examined in oblique cross-sections, and places an undo burden on system resources when animation through additional dimensions is desired. For time-varying data of large data sets, smooth animation is desired at interactive rates. We provide a fast Marching Cubes like algorithm for hypercubes of any dimension. To support this, we have developed a new algorithm to automatically generate the isosurface and triangulation tables for any dimension. This allows the efficient calculation of 4D isosurfaces, which can be interactively sliced to provide smooth animation or slicing through oblique hyperplanes. The former allows for smooth animation in a very compressed format. The latter provide better tools to study time-evolving features as they move downstream. We also provide examples in using this technique to show interval volumes or the sensitivity of a particular isovalue threshold.

We prove that a (k-1)-separated family of n compact convex sets in Rd can be met byk-transversals in at most O(d)d2((2k+1-2 / k) (n / k+1))k(d-k) or, for fixed k and d, O(nk(k+1)(d-k)) different order types. This is the first non-trivial bound for 1

Article Free Access Share on There is a universal topological plane Authors: Jacob E. Goodman View Profile , Richard Pollack View Profile , Rephael Wenger View Profile , Tudor Zamfirescu View Profile Authors Info & Claims SCG '92: Proceedings of the eighth annual symposium on Computational geometryJuly 1992Pages 171–176https://doi.org/10.1145/142675.142714Published:01 July 1992Publication History 0citation272DownloadsMetricsTotal Citations0Total Downloads272Last 12 Months25Last 6 weeks3 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Publisher SiteeReaderPDF

We prove that for any set S of n points in the plane and n3-α triangles spanned by the points of S there exists a point (not necessarily of S) contained in at least n3-3α/(512 log5 n) of the triangles. This implies that any set of n points in three-dimensional space defines at most 6.4n8/3 log5/3 n halving planes.

We show that the maximum combinatorial complexity of the space of hyperplane transversals to a family of n separated and strictly convex sets in Rd is Θ(n⌊d/2⌋), which generalizes results of Edelsbrunner and Sharir in the plane. As a key step in the argument, we show that the space of hyperplanes tangent to κ ≤ d separated and strictly convex sets in Rd is a topological (d - κ)-sphere.

In general, finding a line stabber for a family of n objects in the plane takes ω(n log n) time. However, we show how to find a line stabber for a family of n pairwise disjoint convex translates in the plane in linear time. Our algorithm still runs in optimal Ο (n log n) time when the translates are not pairwise disjoint.

We will prove that a finite family B = {B1, B2, …, Bn} of connected compact sets in Rd has a hyperplane transversal if and only if for some k there exists a set of points P = {P1, P2, …, Pn} (i.e. a k-dimensional labeling of the family) which spans Rk and every k + 2 sets of B are met by a k-flat consistent with the order type of P. This is a common generalization of theorems of Hadwiger, Katchalski, Goodman-Pollack and Wenger.

Algorithms are developed for determining if a set of polyhedral objects in R3 can be intersected by a common transversal (stabbing) line. It can be determined in Ο(n) time if a set of n lines in space has a line transversal, and such a transversal can be found in the same time bound. For a set of n line segments, the complexity of finding such a transversal becomes Ο(nlogn). Finally, for a set of polyhedra with a total of n vertices, we give a Ο(n5) algorithm for determining the existence of, and computing, a line transversal. Helly-type theorems for lines and segments are also given. In particular, it is shown that if every six of a set of lines in space are intersected by a common transversal, then the entire set has a common transversal.