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On the role of non-local Menger curvature in image processing

Guy Gilboa, Eli Appleboim, Emil Saucan, Yehoshua Y. Zeevi
2015 2015 IEEE International Conference on Image Processing (ICIP)  
Examples on natural and textural images highlight potential applications of these new concepts. Index Terms-Menger curvature, non-local curvature, non-local diffusion, texture features.  ...  In this paper we extend the notion of curvature in two ways.  ...  proposed versions of Algorithm 1, building the weights for the non-local process based on the input image f , W (f ), or on the pointwise Menger, W (κ).  ... 
doi:10.1109/icip.2015.7351625 dblp:conf/icip/GilboaASZ15 fatcat:4kmr3bbq2jgsxbb5hp2xpmbawu

Introducing total curvature for image processing

Bastian Goldluecke, Daniel Cremers
2011 2011 International Conference on Computer Vision  
as a regularizer in image processing problems.  ...  It is defined as the Menger-Melnikov curvature of the Radon measure |Du|, which can be understood as a measure theoretic formulation of curvature mathematically related to mean curvature.  ...  In this paper, we introduce the concept of total curvature of a function, which is based on the Menger-Melnikov curvature of a measure.  ... 
doi:10.1109/iccv.2011.6126378 dblp:conf/iccv/GoldluckeC11 fatcat:zvmlxm6wgfam7koyexuzxg7bpy

Geometric curvature energies: facts, trends, and open problems [chapter]

Paweł Strzelecki, Heiko von der Mosel, Philipp Reiter, Simon Blatt, Armin Schikorra
2017 New Directions in Geometric and Applied Knot Theory  
We briefly describe the impact of geometric curvature energies on geometric knot theory.  ...  Geometric curvature functionals are characterised as geometrically defined energies on a priorily non-smooth k-dimensional subsets Σ k of R d , and these functionals are designed to penalise self-intersections  ...  one covers the set (up to a set of measure zero) by Lipschitz images, and one might think about improving the regularity of the covering images.  ... 
doi:10.1515/9783110571493-002 fatcat:suxf4lbq6rembov3en7wp4vxlq

Stretching and folding in finite time

Tian Ma, Nicholas T. Ouellette, Erik M. Bollt
2016 Chaos  
Note that the image of the line segment l e;v ðzÞ is generally a curve / t 0 þs t 0 ðl e;v ðzÞÞ, and the role of the limit is to isolate the curvature to the locality of the image of the center point,  ...  estimating curvature of the image by the Menger curvature by Eq. (3.6).  ...  (A4) map to and the angle at the image is (A9) from which the Menger curvature of the image points becomes Finally, we compute the ratio of the growth of the Menger curvature using Eqs.  ... 
doi:10.1063/1.4941256 pmid:26931593 fatcat:k7zjcnzarfdk5jxyedgs7uqwgy

Adaptive region-growing with maximum curvature strategy for tumor segmentation in 18F-FDG PET

Shan Tan, Laquan Li, Wookjin Choi, Min Kyu Kang, Warren D D'Souza, Wei Lu
2017 Physics in Medicine and Biology  
factor; segmentation; f-volume curve; local maximum curvature spectrum of classical automated thresholding algorithms for tumor segmentation in PET [33] .  ...  The ARG_MC was robust to parameter settings and region of interest selection, and it did not depend on scanners, imaging protocols, or tumor types.  ...  The interrun variation in DSI is defined as: where Q is the number of ROIs (Q=8), (k=p or q) is the value of DSI of one tested method on the ith image segmented with the kth ROI, N is the number of images  ... 
doi:10.1088/1361-6560/aa6e20 pmid:28604372 pmcid:PMC5497763 fatcat:3qocjaopy5gzfox6btzwu3casy

A Simple Differential Geometry for Networks and its Generalizations [article]

Emil Saucan, Areejit Samal, Jürgen Jost
2019 arXiv   pre-print
Based on two classical notions of curvature for curves in general metric spaces, namely the Menger and Haantjes curvatures, we introduce new definitions of sectional, Ricci and scalar curvature for networks  ...  In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature, and derived via a fitting discrete Gauss-Bonnet Theorem, are quite flexible.  ...  A Local Gauss-Bonnet Theorem and the Curvature of 2-Cells. Because of its advantages over Menger curvature, we shall now use Haantjes curvature to define scalar and Ricci curvatures of networks.  ... 
arXiv:1910.06258v1 fatcat:pepaqifgjjcelhffvx7dcxv75e

A Simple Differential Geometry for Complex Networks [article]

Emil Saucan, Areejit Samal, Jürgen Jost
2020 arXiv   pre-print
In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss-Bonnet theorem.  ...  spaces, namely, the Menger curvature and the Haantjes curvature.  ...  AS would like to thank Max Planck Society, Germany, for the award of a Max Planck Partner Group in Mathematical Biology.  ... 
arXiv:2004.11112v2 fatcat:cw4tyzt2ubdp5ilckyhosqlwoi

Machine learning paradigm for dynamic contrast-enhanced MRI evaluation of expanding bladder

Dee H Wu
2020 Frontiers in Bioscience  
Rheal Towner for bringing the ICBPS clinical problem to our attention. We thank Phillip Rhoton and Saurabh Kumar Srivastava for their organizational and technical assistance.  ...  For example, we selected the maximum first-order difference of the curvature (MDC) between local Menger curvature (eq. 5) at the current point (at x0) with the curvature of its neighbors.  ...  Incorporated features that capture shape characteristics include local Menger curvature (MDC) and changes in angle values (SOA).  ... 
doi:10.2741/4876 fatcat:6yjt6azpozbcrc2p6k2sfkwkpu

Metric Curvatures Revisited: A Brief Overview [chapter]

Emil Saucan
2017 Lecture notes in mathematics  
In addition, a simple, metric way of defining curvature for metric measure spaces is proposed.  ...  We survey metric curvatures, special accent being placed upon the Wald curvature, its relationship with Alexandrov curvature, as well as its application in defining a metric Ricci curvature for P L cell  ...  Acknowledgement The author would like to thank to organizers of 2013 CIRM Meeting on Discrete Curvature for the opportunity they gave him to write this bookchapter, and especially Pascal Romon for his  ... 
doi:10.1007/978-3-319-58002-9_2 fatcat:ecmemhxh2fcb7o3hvtvvstposy

Fat Triangulations and Differential Geometry [article]

Emil Saucan
2011 arXiv   pre-print
We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, Müller and Schrader, regarding the convergence of  ...  In addition, the notion of fatness of triangulations and its relation to metric curvature and to excess is explored.  ...  Curvature analysis of this type of data sets plays a major role in a variety of applications, such as reconstruction, segmentation and recognition and non photorealistic rendering (see the bibliography  ... 
arXiv:1108.3529v1 fatcat:trk3iwsu7vdpvcryjluh3svfta

Geometric Knot Theory

Dorothy Buck, Jason Cantarella, John Sullivan, Heiko von der Mosel
2013 Oberwolfach Reports  
Increasing interest in this area over the past decade is partly due to various applications, for instance to random knots and polymers, to topological fluid dynamics and to the molecular biology of DNA  ...  This workshop focused on the mathematics behind these applications, drawing on techniques from algebraic topology, differential geometry, integral geometry, geometric measure theory, calculus of variations  ...  Yuanan Diao is currently supported by grants DMS-0920880 and DMS-1016460 from National Science Foundation of USA. Acknowledgments. The author was partially supported by NSF grant DMS-06-04164.  ... 
doi:10.4171/owr/2013/22 fatcat:kfayvl3kmneehfwi2o2yo65mo4

Metric Curvatures and their Applications 2: Metric Ricci Curvature and Flow [article]

Emil Saucan
2019 arXiv   pre-print
In addition to our previous metric approaches to the discretization of Ricci curvature, we consider yet another one, based on the Haantjes curvature, interpreted as a geodesic curvature.  ...  In this second part of our overview of the different metric curvatures and their various applications, we concentrate on the Ricci curvature and flow for polyhedral surfaces and higher dimensional manifolds  ...  Acknowledgement The author would like to thank his former and present students who helped produce the figures in the text, to Jürgen Jost for his warm hosting  ... 
arXiv:1902.03438v2 fatcat:ci5ypmjyszd7rlvew76a4qamvq

On the analyticity of critical points of the generalized integral Menger curvature in the Hilbert case

Daniel Steenebrügge, Nicole Vorderobermeier
2022 Nonlinear analysis / Theory  
The main new idea is an additional iteration in the recursive estimate of the derivatives to obtain a sufficient difference in the order of regularity.  ...  We prove the analyticity of smooth critical points for generalized integral Menger curvature energies intM (p,2) , with p ∈ ( 7 3 , 8 3 ), subject to a fixed length constraint.  ...  In addition, the authors would like to sincerely thank Simon Blatt for his constant support and indispensable discussions throughout the development of this research paper, Heiko von der Mosel for his  ... 
doi:10.18154/rwth-2022-03411 fatcat:ny34ffqo2jgqtbfzk3clfs6ofm

Phospholipase A2 promotes raft budding and fission from giant liposomes

Galya Staneva, Miglena I. Angelova, Kamen Koumanov
2004 Chemistry and Physics of Lipids  
This leads to the hypothesis that the PLA 2 , apart from its role in second messenger generation, might play a direct and general role in the vesiculation processes underlying the intermembrane transport  ...  In addition, the participation of phospholipases in the vesiculation of Golgi and other membranes has been already established [Traffic 1 (2000) 504] essentially in their role in the production of second  ...  Seigneuret for critical correction of the text and W. Sacks for English improvements.  ... 
doi:10.1016/j.chemphyslip.2003.11.005 pmid:14998727 fatcat:thkw2f36w5h25b44yvcvz4zsge

Epithelial organ shape is generated by patterned actomyosin contractility and maintained by the extracellular matrix

Ali Nematbakhsh, Megan Levis, Nilay Kumar, Weitao Chen, Jeremiah J. Zartman, Mark Alber, Philip K. Maini
2020 PLoS Computational Biology  
These results provide general insight into how the subcellular forces are generated and maintained within individual cells to induce tissue curvature.  ...  Basally generated actomyosin forces generate epithelial bending of the wing disc pouch.  ...  Acknowledgments The authors acknowledge support from the Notre Dame Integrated Imaging Facility and CRC Computing center. We would like to thank members of Dr. Zartman's and Dr. Alber's groups  ... 
doi:10.1371/journal.pcbi.1008105 pmid:32817654 fatcat:kmgusm4h7veelhwpxmz3d5y5jy
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