Three-dimensional alpha shapes
TL;DR: This article introduces the formal notion of the family of α-shapes of a finite point set in R 3 .
Abstract: Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that purpose, this article introduces the formal notion of the family of a-shapes of a finite point set in R3. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter a e R controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in time 0(n2), worst case. A robust implementation of the algorithm is discussed, and several applications in the area of scientific computing are mentioned.
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26 Jun 2006
TL;DR: A spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model, and which reduces to a well conditioned sparse linear system.
Abstract: We show that surface reconstruction from oriented points can be cast as a spatial Poisson problem. This Poisson formulation considers all the points at once, without resorting to heuristic spatial partitioning or blending, and is therefore highly resilient to data noise. Unlike radial basis function schemes, our Poisson approach allows a hierarchy of locally supported basis functions, and therefore the solution reduces to a well conditioned sparse linear system. We describe a spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model. Experimenting with publicly available scan data, we demonstrate reconstruction of surfaces with greater detail than previously achievable.
2,712 citations
Cites background from "Three-dimensional alpha shapes"
...[Boi84,KSO04]), alpha shapes [ EM94 , BBX95, BMR ∗ 99]), or Voronoi diagrams [ABK98,ACK01]....
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TL;DR: It is argued that a topological interaction is indispensable to maintain a flock's cohesion against the large density changes caused by external perturbations, typically predation, and supported by numerical simulations.
Abstract: Numerical models indicate that collective animal behavior may emerge from simple local rules of interaction among the individuals. However, very little is known about the nature of such interaction, so that models and theories mostly rely on aprioristic assumptions. By reconstructing the three-dimensional positions of individual birds in airborne flocks of a few thousand members, we show that the interaction does not depend on the metric distance, as most current models and theories assume, but rather on the topological distance. In fact, we discovered that each bird interacts on average with a fixed number of neighbors (six to seven), rather than with all neighbors within a fixed metric distance. We argue that a topological interaction is indispensable to maintain a flock's cohesion against the large density changes caused by external perturbations, typically predation. We support this hypothesis by numerical simulations, showing that a topological interaction grants significantly higher cohesion of the aggregation compared with a standard metric one.
1,814 citations
TL;DR: Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
Abstract: We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility.
1,671 citations
TL;DR: The new features and major improvements in Membrane Builder that allow users to robustly build realistic biological membrane systems are described, including addition of new lipid types, including phosphoinositides, cardiolipin (CL), sphingolipids, bacterial lipids, and ergosterol.
Abstract: CHARMM-GUI Membrane Builder, http://www.charmm-gui.org/input/membrane, is a web-based user interface designed to interactively build all-atom protein/membrane or membrane-only systems for molecular dynamics simulations through an automated optimized process. In this work, we describe the new features and major improvements in Membrane Builder that allow users to robustly build realistic biological membrane systems, including (1) addition of new lipid types, such as phosphoinositides, cardiolipin (CL), sphingolipids, bacterial lipids, and ergosterol, yielding more than 180 lipid types, (2) enhanced building procedure for lipid packing around protein, (3) reliable algorithm to detect lipid tail penetration to ring structures and protein surface, (4) distance-based algorithm for faster initial ion displacement, (5) CHARMM inputs for P21 image transformation, and (6) NAMD equilibration and production inputs. The robustness of these new features is illustrated by building and simulating a membrane model of the polar and septal regions of E. coli membrane, which contains five lipid types: CL lipids with two types of acyl chains and phosphatidylethanolamine lipids with three types of acyl chains. It is our hope that CHARMM-GUI Membrane Builder becomes a useful tool for simulation studies to better understand the structure and dynamics of proteins and lipids in realistic biological membrane environments. © 2014 Wiley Periodicals, Inc.
1,604 citations
TL;DR: Computed Atlas of Surface Topography of proteins (CASTp) provides an online resource for locating, delineating and measuring concave surface regions on three-dimensional structures of proteins, including pockets located on protein surfaces and voids buried in the interior of proteins.
Abstract: Computed Atlas of Surface Topography of proteins (CASTp) provides an online resource for locating, delineating and measuring concave surface regions on three-dimensional structures of proteins. These include pockets located on protein surfaces and voids buried in the interior of proteins. The measurement includes the area and volume of pocket or void by solvent accessible surface model (Richards' surface) and by molecular surface model (Connolly's surface), all calculated analytically. CASTp can be used to study surface features and functional regions of proteins. CASTp includes a graphical user interface, flexible interactive visualization, as well as on-the-fly calculation for user uploaded structures. CASTp is updated daily and can be accessed at http://cast.engr.uic.edu.
1,312 citations
References
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Book•
01 Jan 1990TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.
Abstract: From the Publisher:
The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Like the first edition,this text can also be used for self-study by technical professionals since it discusses engineering issues in algorithm design as well as the mathematical aspects.
In its new edition,Introduction to Algorithms continues to provide a comprehensive introduction to the modern study of algorithms. The revision has been updated to reflect changes in the years since the book's original publication. New chapters on the role of algorithms in computing and on probabilistic analysis and randomized algorithms have been included. Sections throughout the book have been rewritten for increased clarity,and material has been added wherever a fuller explanation has seemed useful or new information warrants expanded coverage.
As in the classic first edition,this new edition of Introduction to Algorithms presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers. Further,the algorithms are presented in pseudocode to make the book easily accessible to students from all programming language backgrounds.
Each chapter presents an algorithm,a design technique,an application area,or a related topic. The chapters are not dependent on one another,so the instructor can organize his or her use of the book in the way that best suits the course's needs. Additionally,the new edition offers a 25% increase over the first edition in the number of problems,giving the book 155 problems and over 900 exercises thatreinforcethe concepts the students are learning.
21,651 citations
01 Aug 1987
TL;DR: In this paper, a divide-and-conquer approach is used to generate inter-slice connectivity, and then a case table is created to define triangle topology using linear interpolation.
Abstract: We present a new algorithm, called marching cubes, that creates triangle models of constant density surfaces from 3D medical data. Using a divide-and-conquer approach to generate inter-slice connectivity, we create a case table that defines triangle topology. The algorithm processes the 3D medical data in scan-line order and calculates triangle vertices using linear interpolation. We find the gradient of the original data, normalize it, and use it as a basis for shading the models. The detail in images produced from the generated surface models is the result of maintaining the inter-slice connectivity, surface data, and gradient information present in the original 3D data. Results from computed tomography (CT), magnetic resonance (MR), and single-photon emission computed tomography (SPECT) illustrate the quality and functionality of marching cubes. We also discuss improvements that decrease processing time and add solid modeling capabilities.
13,231 citations
01 Jan 1985
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
6,525 citations
Book•
01 Jan 1978TL;DR: In this article, the authors present a coherent treatment of computational geometry in the plane, at the graduate textbook level, and point out the way to the solution of the more challenging problems in dimensions higher than two.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
3,419 citations
Book•
01 Jan 1987TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.
2,284 citations