scholarly journals Approximation of the Mumford–Shah functional by phase fields of bounded variation

2020 ◽  
pp. 1-35
Author(s):  
Sandro Belz ◽  
Kristian Bredies
Keyword(s):  
Image Processing ◽  
Image Segmentation ◽  
Bounded Variation ◽  
Phase Field ◽  
Field Approximation ◽  
Field Variable ◽  
Function Of Bounded Variation ◽  
Total Variation Minimization ◽  
Phase Fields ◽  
New Phase

In this paper, we introduce a new phase field approximation of the Mumford–Shah functional similar to the well-known one from Ambrosio and Tortorelli. However, in our setting the phase field is allowed to be a function of bounded variation, instead of an [Formula: see text]-function. In the context of image segmentation, we also show how this new approximation can be used for numerical computations, which contains a total variation minimization of the phase field variable, as it appears in many problems of image processing. A comparison to the classical Ambrosio–Tortorelli approximation, where the phase field is an [Formula: see text]-function, shows that the new model leads to sharper phase fields.

scholarly journals Phase-field approximation for a class of cohesive fracture energies with an activation threshold

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonin Chambolle ◽  
Vito Crismale
Keyword(s):  
Brittle Fracture ◽  
Asymptotic Analysis ◽  
Phase Field ◽  
Symmetric Operator ◽  
Field Approximation ◽  
Field Variable ◽  
Activation Threshold ◽  
Linearized Elasticity ◽  
The One ◽  
Limit Energy

AbstractWe study the Γ-limit of Ambrosio–Tortorelli-type functionals {D_{\varepsilon}(u,v)}, whose dependence on the symmetrised gradient {e(u)} is different in {\mathbb{A}u} and in {e(u)-\mathbb{A}u}, for a {\mathbb{C}}-elliptic symmetric operator {\mathbb{A}}, in terms of the prefactor depending on the phase-field variable v. The limit energy depends both on the opening and on the surface of the crack, and is intermediate between the Griffith brittle fracture energy and the one considered by Focardi and Iurlano [Asymptotic analysis of Ambrosio–Tortorelli energies in linearized elasticity, SIAM J. Math. Anal. 46 2014, 4, 2936–2955]. In particular, we prove that G(S)BD functions with bounded {\mathbb{A}}-variation are (S)BD.

scholarly journals Phase-field model of precipitation processes with coherency loss

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Tian-Le Cheng ◽  
You-Hai Wen
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Ostwald Ripening ◽  
Phase Field Model ◽  
Energy Criterion ◽  
Field Variable ◽  
Ginzburg Landau ◽  
Underlying Mechanisms ◽  
Flood Fill ◽  
Coherency Loss

AbstractA phase-field model is proposed to simulate coherency loss coupled with microstructure evolution. A special field variable is employed to describe the degree of coherency loss of each particle and its evolution is governed by a Ginzburg-Landau type kinetic equation. For the sake of computational efficiency, a flood-fill algorithm is introduced that can drastically reduce the required number of field variables, which allows the model to efficiently simulate a large number of particles sufficient for characterizing their statistical features during Ostwald ripening. The model can incorporate size dependence of coherency loss, metastability of coherent particles, and effectively incorporate the underlying mechanisms of coherency loss by introducing a so-called differential energy criterion. The model is applied to simulate coarsening of Al3Sc precipitates in aluminum alloy and comprehensively compared with experiments. Our results clearly show how the particle size distribution is changed during coherency loss and affects the coarsening rate.

scholarly journals Spectral Approximation of Fractional PDEs in Image Processing and Phase Field Modeling

2017 ◽  
Vol 17 (4) ◽  
pp. 661-678 ◽  
Author(s):  
Harbir Antil ◽  
Sören Bartels
Keyword(s):  
Image Processing ◽  
Phase Field ◽  
Differential Operators ◽  
Mathematical Tool ◽  
Phase Field Models ◽  
Field Modeling ◽  
Regularity Properties ◽  
Fractional Pdes ◽  
Fractional Differential ◽  
Fractional Differential Operators

AbstractFractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets and sharp transitions across interfaces are of interest. The numerical solution of corresponding model problems via a spectral method is analyzed. Its efficiency and features of the model problems are illustrated by numerical experiments.

scholarly journals On second order nonlocal boundary value problem at resonance

2016 ◽  
Vol 56 (1) ◽  
pp. 143-153 ◽  
Author(s):  
Katarzyna Szymańska-Dębowska
Keyword(s):  
Boundary Value Problem ◽  
Bounded Variation ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Function Of Bounded Variation ◽  
Nonlocal Boundary Value Problem ◽  
At Resonance ◽  
Problem At Resonance

Abstract This work is devoted to the existence of solutions for a system of nonlocal resonant boundary value problem $$\matrix{{x'' = f(t,x),} \hfill & {x'(0) = 0,} \hfill & {x'(1) = {\int_0^1 {x(s)dg(s)},} }} $$ where f : [0, 1] × ℝk → ℝk is continuous and g : [0, 1] → ℝk is a function of bounded variation.

A Survey on Solar Image Segmentation Techniques

2014 ◽  
Vol 945-949 ◽  
pp. 1899-1902
Author(s):  
Yuan Yuan Fan ◽  
Wei Jiang Li ◽  
Feng Wang
Keyword(s):  
Image Processing ◽  
Pattern Recognition ◽  
Image Analysis ◽  
Image Segmentation ◽  
Image Retrieval ◽  
Fundamental Issue ◽  
Solar Image ◽  
Segmentation Methods

Image segmentation is one of the basic problems of image processing, also is the first essential and fundamental issue in the solar image analysis and pattern recognition. This paper summarizes systematically on the image segmentation techniques in the solar image retrieval and the recent applications of image segmentation. Then the merits and demerits of each method are discussed in this paper, in this way we can combine some methods for image segmentation to reach the better effects in astronomy. Finally, according to the characteristics of the solar image itself, the more appropriate image segmentation methods are summed up, and some remarks on the prospects and development of image segmentation are presented.

Near-Band IR Small Scene Star Image Segmentation Based on Partial Histogram Grey Level

2014 ◽  
Vol 496-500 ◽  
pp. 1834-1839
Author(s):  
Zhe Wang ◽  
Zhe Yan ◽  
Wei Tan
Keyword(s):  
Image Processing ◽  
Image Segmentation ◽  
Real Time ◽  
Grey Level ◽  
Star Image ◽  
Star Images

The near-band IR star images segmentation and recognition is key technique in day time star navigation. Due to the scene of near-band IR star imaging relative small and stellar with high star grade are limited. Pertinence and dynamic grey level threshold is necessary for image processing arithmetic. In order to enhance near-band IR star images segmentation and recognition in real-time, this paper present the process of partial histogram grey level threshold and improve for actually near-band IR star images with scene of no more than 1.5°×1.5°. It can reduce the calculation of near-band IR star images with adjustable threshold. And get rid of disturbance of small imaging square stars and noise points.

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