scholarly journals Fast and Efficient Numerical Finite Difference Method for Multiphase Image Segmentation

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Yibao Li ◽  
Sungha Yoon ◽  
Jian Wang ◽  
Jintae Park ◽  
Sangkwon Kim ◽  
...  
Keyword(s):  
Image Segmentation ◽  
Goodness Of Fit ◽  
Operator Splitting ◽  
Gradient Flow ◽  
Flow Equation ◽  
Solution Algorithm ◽  
Transition Width ◽  
Time Step ◽  
The Stability ◽  
Euler Methods

We present a simple numerical solution algorithm for a gradient flow for the Modica–Mortola functional and numerically investigate its dynamics. The proposed numerical algorithm involves both the operator splitting and the explicit Euler methods. A time step formula is derived from the stability analysis, and the goodness of fit of transition width is tested. We perform various numerical experiments to investigate the property of the gradient flow equation, to verify the characteristics of our method in the image segmentation application, and to analyze the effect of parameters. In particular, we propose an initialization process based on target objects. Furthermore, we conduct comparison tests in order to check the performance of our proposed method.

scholarly journals An Accurate and Practical Explicit Hybrid Method for the Chan–Vese Image Segmentation Model

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1173
Author(s):  
Darae Jeong ◽  
Sangkwon Kim ◽  
Chaeyoung Lee ◽  
Junseok Kim
Keyword(s):  
Image Segmentation ◽  
Governing Equation ◽  
Hybrid Method ◽  
Operator Splitting ◽  
Splitting Method ◽  
Phase Field Model ◽  
Gradient Flow ◽  
Time Step ◽  
Euler’S Method ◽  
Operator Splitting Method

In this paper, we propose a computationally fast and accurate explicit hybrid method for image segmentation. By using a gradient flow, the governing equation is derived from a phase-field model to minimize the Chan–Vese functional for image segmentation. The resulting governing equation is the Allen–Cahn equation with a nonlinear fidelity term. We numerically solve the equation by employing an operator splitting method. We use two closed-form solutions and one explicit Euler’s method, which has a mild time step constraint. However, the proposed scheme has the merits of simplicity and versatility for arbitrary computational domains. We present computational experiments demonstrating the efficiency of the proposed method on real and synthetic images.

scholarly journals Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Geosciences ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 29 ◽  
Author(s):  
Paul M. Delgado ◽  
V. M. Krushnarao Kotteda ◽  
Vinod Kumar
Keyword(s):  
Fixed Point ◽  
Asymptotic Solution ◽  
Relative Error ◽  
Operator Splitting ◽  
Splitting Method ◽  
Approximate Solutions ◽  
Time Step ◽  
Operator Splitting Method ◽  
The Stability ◽  
Stability And Consistency

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .

scholarly journals An Efficient Multi-Scale Local Binary Fitting-Based Level Set Method for Inhomogeneous Image Segmentation

2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Dengwei Wang
Keyword(s):  
Image Segmentation ◽  
Level Set ◽  
Operator Splitting ◽  
Reaction Diffusion ◽  
Initial Contour ◽  
Time Step ◽  
Detection Mechanism ◽  
Diffusion Term ◽  
Implementation Phase ◽  
Level Set Function

An efficient level set model based on multiscale local binary fitting (MLBF) is proposed for image segmentation. By introducing multiscale idea into the LBF model, the proposed MLBF model can effectively and efficiently segment images with intensity inhomogeneity. In addition, by adding a reaction diffusion term into the level set evolution (LSE) equation, the regularization of the level set function (LSF) can be achieved, thus completely eliminating the time-consuming reinitialization process. In the implementation phase, in order to greatly improve the efficiency of the numerical solution of the level set segmentation model, we introduce three strategies: The first is the additive operator splitting (AOS) solver which is used for breaking the restrictions on time step; the second is the salient target detection mechanism which is used to achieve full automatic initialization of the LSE process; the third is the sparse filed method (SFM) which is used to restrict the groups of pixels that need to be updated in a small strip region. Under the combined effect of these three strategies, the proposed model achieves very high execution efficiency in the following aspects: contour location accuracy, speed of evolution convergence, robustness against initial contour position, and robustness against noise interference.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

scholarly journals Coupled THM and Matrix Stability Modeling of Hydroshearing Stimulation in a Coupled Fracture-Matrix Hot Volcanic System

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Yong Xiao ◽  
Jianchun Guo ◽  
Hehua Wang ◽  
Lize Lu ◽  
John McLennan ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Thermal Conduction ◽  
Dominant Role ◽  
Injection Well ◽  
Time Step ◽  
Volcanic System ◽  
Induced Stress ◽  
The Matrix ◽  
Matrix Stability ◽  
The Stability

A coupled thermal-hydraulic-mechanical (THM) model is developed to simulate the combined effect of fracture fluid flow, heat transfer from the matrix to injected fluid, and shearing dilation behaviors in a coupled fracture-matrix hot volcanic reservoir system. Fluid flows in the fracture are calculated based on the cubic law. Heat transfer within the fracture involved is thermal conduction, thermal advection, and thermal dispersion; within the reservoir matrix, thermal conduction is the only mode of heat transfer. In view of the expansion of the fracture network, deformation and thermal-induced stress model are added to the matrix node’s in situ stress environment in each time step to analyze the stability of the matrix. A series of results from the coupled THM model, induced stress, and matrix stability indicate that thermal-induced aperture plays a dominant role near the injection well to enhance the conductivity of the fracture. Away from the injection well, the conductivity of the fracture is contributed by shear dilation. The induced stress has the maximum value at the injection point; the deformation-induced stress has large value with smaller affected range; on the contrary, thermal-induced stress has small value with larger affected range. Matrix stability simulation results indicate that the stability of the matrix nodes may be destroyed; this mechanism is helpful to create complex fracture networks.

Efficient Two-dimensional Acoustic Wave Finite-Difference Numerical Simulation in Strongly Heterogeneous Media Using the Adaptive Mesh Refinement (AMR) Technique

Geophysics ◽  
2021 ◽  
pp. 1-76
Author(s):  
Chunli Zhang ◽  
Wei Zhang
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Adaptive Mesh Refinement ◽  
Heterogeneous Media ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Grid Spacing ◽  
Time Step ◽  
Velocity Models ◽  
The Stability

The finite-difference method (FDM) is one of the most popular numerical methods to simulate seismic wave propagation in complex velocity models. If a uniform grid is applied in the FDM for heterogeneous models, the grid spacing is determined by the global minimum velocity to suppress dispersion and dissipation errors in the numerical scheme, resulting in spatial oversampling in higher-velocity zones. Then, the small grid spacing dictates a small time step due to the stability condition of explicit numerical schemes. The spatial oversampling and reduced time step will cause unnecessarily inefficient use of memory and computational resources in simulations for strongly heterogeneous media. To overcome this problem, we propose to use the adaptive mesh refinement (AMR) technique in the FDM to flexibly adjust the grid spacing following velocity variations. AMR is rarely utilized in acoustic wave simulations with the FDM due to the increased complexity of implementation, including its data management, grid generation and computational load balancing on high-performance computing platforms. We implement AMR for 2D acoustic wave simulation in strongly heterogeneous media based on the patch approach with the FDM. The AMR grid can be automatically generated for given velocity models. To simplify the implementation, we employ a well-developed AMR framework, AMReX, to carry out the complex grid management. Numerical tests demonstrate the stability, accuracy level and efficiency of the AMR scheme. The computation time is approximately proportional to the number of grid points, and the overhead due to the wavefield exchange and data structure is small.

Ba-, Si- and vacancy-rich phlogopites from the talc-bearing sulfide ore deposit of La Creuse, Beaujolais, France

2018 ◽  
Vol 82 (5) ◽  
pp. 1187-1210
Author(s):  
Marie-Lola Pascal ◽  
Michel Fonteilles ◽  
Véronique Tournis ◽  
Benoît Baptiste ◽  
Jean-Louis Robert ◽  
...  
Keyword(s):  
Goodness Of Fit ◽  
Monoclinic Space Group ◽  
Excess Charge ◽  
Formula Unit ◽  
X Ray Diffraction ◽  
Ore Deposit ◽  
Sulfide Ore ◽  
The Common ◽  
The Stability ◽  
High Level

ABSTRACTBa-rich and Si-rich phlogopites occur in the talc-bearing rocks of the La Creuse sulfide ore deposit in Beaujolais, France. They form a group of compositions completely separated from the common Al-rich phlogopites that occur in the surrounding talc-free metasiltites and metarhyolites, with higher Ba and Mg and lower Al contents. The Ba-rich phlogopites have a relatively narrow compositional range (0.24 to 0.80 Ba per formula unit, for 44 valencies) with high and constant Si (5.8 atoms per formula unit, apfu) and Mg + Fe (5.6 apfu), probably buffered by the presence of talc. Compared to low-Al phlogopites from talc-free rocks, the excess charge introduced by the BaK–1 substitution is compensated by interlayer vacancies. Such a high level of interlayer vacancy (0.56 pfu), related to the talc-producing metasomatic conditions, is essential for the stability of this special group of Ba-rich and Si-rich phlogopites.Single crystal X-ray diffraction analyses were performed. Ba-rich and Si-rich phlogopite is monoclinic, space group C2/m, (R = 5.31%) with a = 5.3185(5), b = 9.2136(9), c = 10.1349(11) Å and β = 100.131(11)°. The occupancies of Mg/Fe and K/Ba were refined exploring different vacancies. The solutions giving the best R factor (4.77%) and goodness-of-fit (1.06) are obtained with 15% < vacancy < 40% at the interlayer site.

scholarly journals MHDSTS: a new explicit numerical scheme for simulations of partially ionised solar plasma

2018 ◽  
Vol 615 ◽  
pp. A67 ◽  
Author(s):  
P. A. González-Morales ◽  
E. Khomenko ◽  
T. P. Downes ◽  
A. de Vicente
Keyword(s):  
Numerical Scheme ◽  
Operator Splitting ◽  
Conservation Equation ◽  
Ambipolar Diffusion ◽  
Point Of View ◽  
Integration Time ◽  
Solar Photosphere ◽  
Time Step ◽  
Diffusion Term ◽  
Fluid Approximation

The interaction of plasma with magnetic field in the partially ionised solar atmosphere is frequently modelled via a single-fluid approximation, which is valid for the case of a strongly coupled collisional media, such as solar photosphere and low chromosphere. Under the single-fluid formalism the main non-ideal effects are described by a series of extra terms in the generalised induction equation and in the energy conservation equation. These effects are: Ohmic diffusion, ambipolar diffusion, the Hall effect, and the Biermann battery effect. From the point of view of the numerical solution of the single-fluid equations, when ambipolar diffusion or Hall effects dominate can introduce severe restrictions on the integration time step and can compromise the stability of the numerical scheme. In this paper we introduce two numerical schemes to overcome those limitations. The first of them is known as super time-stepping (STS) and it is designed to overcome the limitations imposed when the ambipolar diffusion term is dominant. The second scheme is called the Hall diffusion scheme (HDS) and it is used when the Hall term becomes dominant. These two numerical techniques can be used together by applying Strang operator splitting. This paper describes the implementation of the STS and HDS schemes in the single-fluid code MANCHA3D. The validation for each of these schemes is provided by comparing the analytical solution with the numerical one for a suite of numerical tests.

A New Image Segmentation Method Based on Improved Fast Implicit Level Set Scheme in X/γ-Ray Inspection System

2011 ◽  
Vol 103 ◽  
pp. 705-710 ◽  
Author(s):  
Yu Jie Li ◽  
Hui Min Lu ◽  
Li Feng Zhang ◽  
Shi Yuan Yang ◽  
Serikawa Seiichi
Keyword(s):  
Image Segmentation ◽  
Level Set ◽  
Active Contours ◽  
Operator Splitting ◽  
Inspection System ◽  
Segmentation Method ◽  
Piecewise Constant ◽  
Γ Ray ◽  
Luggage Inspection ◽  
Segmentation Of Images

Digital X/γ-ray imaging technology has been widely used to help people deliver effective and reliable security in airports, train stations, and public buildings. Nowadays, luggage inspection system with digital radiographic/computed tomography (DR/CT) represents a most advanced nondestructive inspection technology in aviation system, which is capable of automatically discerning interesting regions in the luggage objects with CT subsystem. In this paper, we propose a new model for active contours to detect luggage objects in the system, in order to facilitate people to identify the things in luggage. The proposed method is based on techniques of piecewise constant and piecewise smooths Chan-Vese Model, semi-implicit additive operator splitting (AOS) scheme for image segmentation. Different from traditional models, the fast implicit level set scheme (FILS) is ordinary differential equation (ODE). Characterized by no need of any pre-information of topology of images and efficient segmentation of images with complex topology, the FILS scheme is fast more than traditional level set scheme 30 times. At the same time, it performs well in image segmentation of DR images in our experiments.

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