Abstract
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.
Highlights
We consider a practical and efficient numerical method to solve the following equation: zφ(x, zt t) 2εΔφ(x, t) −π sin(2πφ(x, t)), ε x ∈ Ω, t > 0, (1)where φ(x, t) is a phase-field function in space x and time t and ε is a positive constant
We present a simple numerical solution algorithm for the gradient flow for the Modica–Mortola functional and numerically investigate its dynamics
We analyze the stability of scheme for time step size to our method and examine the goodness of fit of linear relationship for various ε values. e results indicate that proper phase separations are achieved via our simple explicit method. e proposed numerical algorithm can be applied to multiphase image segmentation problems
Summary
We consider a practical and efficient numerical method to solve the following equation: zφ(x, zt t). Bogosel et al [5] proposed an efficient phase-field method based on a multiphase Γ-convergence. Inspired by the work of Jung et al [1], the authors in [27] presented a hybrid numerical method for multiphase image segmentation using a phasefield model. Huska et al [29] presented an extension of the Mumford–Shah model for multiphase image segmentation using alternating directions methods of multipliers. In this study, we propose an explicit segmentation scheme which directly updates the state of the given system at a later time level from the state of the system at the current time level without applying any iterative methods. E main purpose of this paper is to present a simple explicit numerical solution algorithm to the gradient flow for the Modica–Mortola functional.
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