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 .

A vector-valued phase field model for polycrystalline solidification using operator splitting method

2019 ◽  
Vol 163 ◽  
pp. 37-49
Author(s):  
Jian-kun Ren ◽  
Yun Chen ◽  
Bin Xu ◽  
Ming-yue Sun ◽  
Dian-zhong Li
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Operator Splitting ◽  
Splitting Method ◽  
Phase Field Model ◽  
Operator Splitting Method ◽  
Vector Valued

Application of High Dimensional B-Spline Interpolation in Solving the Gyro-Kinetic Vlasov Equation Based on Semi-Lagrangian Method

2017 ◽  
Vol 22 (3) ◽  
pp. 789-802 ◽  
Author(s):  
Xiaotao Xiao ◽  
Lei Ye ◽  
Yingfeng Xu ◽  
Shaojie Wang
Keyword(s):  
Vlasov Equation ◽  
Spline Function ◽  
Spline Interpolation ◽  
Operator Splitting ◽  
Splitting Method ◽  
High Dimensional ◽  
Time Step ◽  
B Spline ◽  
Operator Splitting Method ◽  
Computation Efficiency

AbstractThe computation efficiency of high dimensional (3D and 4D) B-spline interpolation, constructed by classical tensor product method, is improved greatly by precomputing the B-spline function. This is due to the character of NLT code, i.e. only the linearised characteristics are needed so that the unperturbed orbit as well as values of the B-spline function at interpolation points can be precomputed at the beginning of the simulation. By integrating this fixed point interpolation algorithm into NLT code, the high dimensional gyro-kinetic Vlasov equation can be solved directly without operator splitting method which is applied in conventional semi-Lagrangian codes. In the Rosenbluth-Hinton test, NLT runs a few times faster for Vlasov solver part and converges at about one order larger time step than conventional splitting code.

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 The Sum and Difference of Two Lognormal Random Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
C. F. Lo
Keyword(s):  
Stochastic Process ◽  
Probability Distributions ◽  
Operator Splitting ◽  
Splitting Method ◽  
Unified Approach ◽  
New Approach ◽  
Numerical Examples ◽  
Operator Splitting Method ◽  
Approximate Probability ◽  
Analytical Series

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum ofNlognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

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