In this thesis, efficient threshold dynamics methods are developed and analysed for free interface problems including wetting dynamics and image segmentation.
The threshold dynamics method developed by Merriman, Bence and Osher (MBO) is an efficient method for simulating the motion by mean curvature flow. Direct generalization of MBO-type methods to the wetting problem with interfaces intersecting the solid boundary is not easy because solving the heat equation in a general domain with a wetting boundary condition is not as efficient as it is with the original MBO method. The dynamics of the contact point also follows a different law compared with the dynamics of the interface away from the boundary. In this thesis, we develop an efficient volume preserving threshold dynamics method for simulating wetting on rough surfaces. This method is based on minimization of the weighted surface area functional over an extended domain that includes the solid phase. The method is simple, stable with O(NlogN) complexity per time
step and is not sensitive to the inhomogeneity or roughness of the solid boundary. The convergence of the threshold dynamics method is rigorously analysed.
To further improve the efficiency of the algorithm, we propose an efficient boundary integral scheme for threshold dynamics via non-uniform fast Fourier transform (NUFFT). The first step is carried out by the evaluation of a boundary integral via NUFFT, and the second step is performed via a root finding algorithm along the normal directions of a discrete set of points on the interface. Unlike most existing methods where a volume discretization is needed for the whole computational domain, our scheme requires the discretization of physical space only in a small neighborhood of the interface and thus is mesh free. The algorithm is spectrally accurate in space for smooth interfaces and has O(NlogN) complexity with N the total number of discrete points near the interface when the time step t is not very small. The performance of the algorithm is illustrated via several numerical examples in both two and three dimensions.
We also propose an efficient threshold dynamics method for multi-phase image segmentation. The algorithm is based on minimizing piecewise constant Mumford-Shah functional in which the contour length (or perimeter) is approximated by a non-local multi-phase energy. The minimization problem is solved by an iterative method. Each iteration consists of computing simple convolutions followed by a thresholding step. The algorithm is easy to implement and has the optimal complexity O(NlogN) per iteration. We also show that the iterative algorithm has the total energy decaying property. We present some numerical results to show the efficiency of our method.