scholarly journals Quantitative estimates for bending energies and applications to non-local variational problems

2019 ◽  
Vol 150 (1) ◽  
pp. 131-169
Author(s):  
Michael Goldman ◽  
Matteo Novaga ◽  
Matthias Röger
Keyword(s):  
Variational Problems ◽  
Three Dimensions ◽  
Variational Model ◽  
Connected Sets ◽  
Isoperimetric Deficit ◽  
Quantitative Estimates ◽  
Non Local ◽  
Simply Connected ◽  
Small Charge ◽  
Large Charge

AbstractWe discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the ‘charge’, that is, the weight of the Riesz interaction energy.In the two-dimensional case, we first prove that for simply connected sets of small elastica energy, the elastica deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centred annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge.

scholarly journals Using the Split Bregman Algorithm to Solve the Self-repelling Snakes Model

2022 ◽  
Author(s):  
Huizhu Pan ◽  
Jintao Song ◽  
Wanquan Liu ◽  
Ling Li ◽  
Guanglu Zhou ◽  
...  
Keyword(s):  
Evolution Equations ◽  
Active Contour Model ◽  
Operator Splitting ◽  
The Self ◽  
Convergence Time ◽  
Variational Model ◽  
Split Bregman ◽  
Geodesic Active Contour ◽  
Non Local ◽  
Split Bregman Algorithm

AbstractPreserving contour topology during image segmentation is useful in many practical scenarios. By keeping the contours isomorphic, it is possible to prevent over-segmentation and under-segmentation, as well as to adhere to given topologies. The Self-repelling Snakes model (SR) is a variational model that preserves contour topology by combining a non-local repulsion term with the geodesic active contour model. The SR is traditionally solved using the additive operator splitting (AOS) scheme. In our paper, we propose an alternative solution to the SR using the Split Bregman method. Our algorithm breaks the problem down into simpler sub-problems to use lower-order evolution equations and a simple projection scheme rather than re-initialization. The sub-problems can be solved via fast Fourier transform or an approximate soft thresholding formula which maintains stability, shortening the convergence time, and reduces the memory requirement. The Split Bregman and AOS algorithms are compared theoretically and experimentally.

On The Kakeya Constant

1965 ◽  
Vol 17 ◽  
pp. 946-956 ◽  
Author(s):  
F. Cunningham ◽  
I. J. Schoenberg
Keyword(s):  
Lower Bound ◽  
Small Area ◽  
Original Position ◽  
Connected Sets ◽  
Unit Segment ◽  
Simply Connected

We shall say that a plane set D has the Kakeya property if a unit segment can be turned continuously in D through 360° back to its original position. The famous solution of this problem by A. S. Besicovitch (1; 2; 4; 5; 6), to the effect that there are sets of arbitrarily small area having the Kayeka property, leaves open the problem obtained by adding the new condition that the set D be also simply connected. Since we do not know whether there is an attainable minimum, we define the Kakeya constant K to be the greatest lower bound of areas of simply connected sets having the Kakeya property. We shall refer to such sets as Kakeya sets.

The bond-based peridynamic system with Dirichlet-type volume constraint

2014 ◽  
Vol 144 (1) ◽  
pp. 161-186 ◽  
Author(s):  
Tadele Mengesha ◽  
Qiang Du
Keyword(s):  
Poisson Ratio ◽  
Variational Problems ◽  
Compact Embedding ◽  
Volume Constraint ◽  
Positive Definite Kernels ◽  
Local Boundary ◽  
Local Operators ◽  
Non Local ◽  
Poincaré Type Inequalities ◽  
Non Locality

In this paper, the bond-based peridynamic system is analysed as a non-local boundary-value problem with volume constraint. The study extends earlier works in the literature on non-local diffusion and non-local peridynamic models, to include non-positive definite kernels. We prove the well-posedness of both linear and nonlinear variational problems with volume constraints. The analysis is based on some non-local Poincaré-type inequalities and the compactness of the associated non-local operators. It also offers careful characterizations of the associated solution spaces, such as compact embedding, separability and completeness. In the limit of vanishing non-locality, the convergence of the peridynamic system to the classical Navier equations of elasticity with Poisson ratio ¼ is demonstrated.

scholarly journals A non-local vectorial total variational model for multichannel SAR image speckle suppression

2015 ◽  
Vol 28 (3) ◽  
pp. 770-779 ◽  
Author(s):  
Rubing Xi ◽  
Zhengming Wang ◽  
Meihua Xie ◽  
Xia Zhao ◽  
Weiwei Wang
Keyword(s):  
Variational Model ◽  
Sar Image ◽  
Non Local ◽  
Total Variational Model ◽  
Speckle Suppression

scholarly journals Spatial Mechanism Design in Virtual Reality With Networking

2001 ◽  
Author(s):  
John N. Kihonge ◽  
Judy M. Vance ◽  
Pierre M. Larochelle
Keyword(s):  
Virtual Reality ◽  
Finite Sequence ◽  
Degree Of Freedom ◽  
Three Dimensions ◽  
Synthesis Process ◽  
Spatial Mechanisms ◽  
Cylindrical Joint ◽  
In Series ◽  
Simply Connected ◽  
Two Degree Of Freedom

Abstract Mechanisms are used in many devices to move a rigid body through a finite sequence of prescribed locations in space. The most commonly used mechanisms are four-bar planar mechanisms that move an object in one plane in space. Spatial mechanisms allow motion in three-dimensions (3D). Spatial 4C mechanisms are two degree of freedom kinematic closed-chains consisting of four rigid links simply connected in series by cylindrical (C) joints. A cylindrical joint is a two degree of freedom joint which allows translation along and rotation about a line in space. This paper describes a synthesis process for the design of 4C spatial mechanisms in a virtual environment. Virtual reality allows the user to view and interact with digital models in a more intuitive way than using the traditional human-computer interface (HCI). The software developed as part of this research also allows multiple users to network and share the designed mechanism. Networking tools have the potential to greatly enhance communication between members of the design team at different industrial sites and therefore reduce design costs.

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