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Axioms
Special Issues
Discrete Geometry and its Applications
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Discrete Geometry and its Applications

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (30 June 2017) | Viewed by 33343

Special Issue Editors

Dr. Emil Saucan

E-Mail Website
Guest Editor
Department of Applied Mathematics, Braude College, Karmiel 2161002, Israel
Interests: discrete differential geometry and its applications; mathematical imaging and vision; complex networks; quasi-conformal mappings and applications; topology and its applications
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. David Gu

E-Mail Website
Guest Editor
Department of Computer Science and Applied Mathematics & Statistics Department, Stony Brook University, Room 2425 Computer Science Building, State University of New York at Stony Brook, Stony Brook, NY 11794-4400, USA
Interests: computer graphics; computer vision; visualization; geometric modelling; networking
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

With the advent of the “digital age”, discrete geometry has been developed and recognized as a self-standing, multifaceted, active and important field of study. A special place is deserved for the sub-field of discrete differential geometry; here “discrete” means that one does not merely restrict oneself to approximations, but rather operates on a deeper level, by considering various possible discretizations of such classical notions as curvature, geodesics and connection, to mention just some of the most basic and essential ones. Papers covering other interesting and important branches of discrete geometry, such as digital geometry and convex geometry, with their multiple and far-reaching applications, are also most welcome.

The ensuing applications are manifold, and range from sampling and reconstruction to segmentation, and from smoothing and denoising to registration and modeling, as well as DNA microarray analyses and neural networks understanding. Moreover, they transcend their specific boundaries (already far from narrow), and have—as already suggested above—applications in medical imaging, complex networks, pattern recognition, manifold learning, mathematical biology and robotics.

It is the goal of this Special Issue to explore, through its constituting papers, the various, dynamic and ever-evolving fields of study, both in its more theoretical facets, as well as its cornucopia of applications.

Prof. Emil Saucan
Prof. David Gu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • discrete (Ricci) curvature
  • geometric flows
  • convex geometry
  • digital geometryimaging
  • graphics
  • complex networks
  • smoothing
  • denoising
  • registration
  • segmentation

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Published Papers (6 papers)

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Editorial

Jump to: Research

166 KiB  
Editorial
Discrete Geometry—From Theory to Applications: A Case Study
by David Gu and Emil Saucan
Axioms 2016, 5(4), 27; https://doi.org/10.3390/axioms5040027 - 9 Dec 2016
Viewed by 4276
Abstract
Science does not necessarily evolve along the lines that are taught to us in High School history classes and in popular films, that is, from simple to complex.[...] Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)

Research

Jump to: Editorial

1508 KiB  
Article
From Normal Surfaces to Normal Curves to Geodesics on Surfaces
by Eli Appleboim
Axioms 2017, 6(3), 26; https://doi.org/10.3390/axioms6030026 - 20 Sep 2017
Viewed by 4281
Abstract
This paper gives a study of a two dimensional version of the theory of normal surfaces; namely, a study o normal curves and their relations with respect to geodesic curves. This study results with a nice discrete approximation of geodesics embedded in a [...] Read more.
This paper gives a study of a two dimensional version of the theory of normal surfaces; namely, a study o normal curves and their relations with respect to geodesic curves. This study results with a nice discrete approximation of geodesics embedded in a triangulated orientable Riemannian surface. Experimental results of the two dimensional case are given as well. Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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3431 KiB  
Article
Topological Signals of Singularities in Ricci Flow
by Paul M. Alsing, Howard A. Blair, Matthew Corne, Gordon Jones, Warner A. Miller, Konstantin Mischaikow and Vidit Nanda
Axioms 2017, 6(3), 24; https://doi.org/10.3390/axioms6030024 - 1 Aug 2017
Cited by 3 | Viewed by 6634
Abstract
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data [...] Read more.
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications. Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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46152 KiB  
Article
Scalable and Fully Distributed Localization in Large-Scale Sensor Networks
by Miao Jin, Su Xia, Hongyi Wu and Xianfeng David Gu
Axioms 2017, 6(2), 15; https://doi.org/10.3390/axioms6020015 - 15 Jun 2017
Cited by 1 | Viewed by 4552
Abstract
This work proposes a novel connectivity-based localization algorithm, well suitable for large-scale sensor networks with complex shapes and a non-uniform nodal distribution. In contrast to current state-of-the-art connectivity-based localization methods, the proposed algorithm is highly scalable with linear computation and communication costs with [...] Read more.
This work proposes a novel connectivity-based localization algorithm, well suitable for large-scale sensor networks with complex shapes and a non-uniform nodal distribution. In contrast to current state-of-the-art connectivity-based localization methods, the proposed algorithm is highly scalable with linear computation and communication costs with respect to the size of the network; and fully distributed where each node only needs the information of its neighbors without cumbersome partitioning and merging process. The algorithm is theoretically guaranteed and numerically stable. Moreover, the algorithm can be readily extended to the localization of networks with a one-hop transmission range distance measurement, and the propagation of the measurement error at one sensor node is limited within a small area of the network around the node. Extensive simulations and comparison with other methods under various representative network settings are carried out, showing the superior performance of the proposed algorithm. Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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422 KiB  
Article
Orientation Asymmetric Surface Model for Membranes: Finsler Geometry Modeling
by Evgenii Proutorov and Hiroshi Koibuchi
Axioms 2017, 6(2), 10; https://doi.org/10.3390/axioms6020010 - 25 Apr 2017
Cited by 2 | Viewed by 4298
Abstract
We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping r from a two-dimensional parameter space M to the three-dimensional Euclidean space R 3 . The metric variable g a b , which is [...] Read more.
We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping r from a two-dimensional parameter space M to the three-dimensional Euclidean space R 3 . The metric variable g a b , which is always fixed to the Euclidean metric δ a b , can be extended to a more general non-Euclidean metric on M in the continuous model. The problem we focus on in this paper is whether such an extension is well defined or not in the discrete model. We find that a discrete surface model with a nontrivial metric becomes well defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in M depends on the direction. It is also shown that the discrete FG model is orientation asymmetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some asymmetries for orientation-changing transformations. Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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2282 KiB  
Article
Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets
by Melanie Weber, Jürgen Jost and Emil Saucan
Axioms 2016, 5(4), 26; https://doi.org/10.3390/axioms5040026 - 10 Nov 2016
Cited by 26 | Viewed by 8095
Abstract
We present a viable geometric solution for the detection of dynamic effects in complex networks. Building on Forman’s discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on [...] Read more.
We present a viable geometric solution for the detection of dynamic effects in complex networks. Building on Forman’s discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. We study the classical Ricci-flow in a network-theoretic setting and introduce an analytic tool for characterizing dynamic effects. The formalism suggests a computational method for change detection and the identification of fast evolving network regions and yields insights into topological properties and the structure of the underlying data. Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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