THESIS
2018

xvi, 94 pages : illustrations ; 30 cm

**Abstract**
In this thesis, we introduce connections between diffuse interface models, total
variation models, and the threshold dynamics scheme. We show that it is a
natural extension of many continuum diffuse interface models to the discrete
graph based models. The Ginzburg-Landau functional is close connected to
total variation minimization via classical Laplacian operator and is often used to
describe two terms with a sharp interface between them. The Merriman-Bence-Osher (MBO) algorithm is efficient in minimizing these diffuse interface models
on graphs.

In Chapter 2, we review the classical diffuse interface models. Such continuum
PDEs involve a front propagating by mean curvature on each of its point. We
also study the definition of mean curvature flow as well as discrete version of...[

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In this thesis, we introduce connections between diffuse interface models, total
variation models, and the threshold dynamics scheme. We show that it is a
natural extension of many continuum diffuse interface models to the discrete
graph based models. The Ginzburg-Landau functional is close connected to
total variation minimization via classical Laplacian operator and is often used to
describe two terms with a sharp interface between them. The Merriman-Bence-Osher (MBO) algorithm is efficient in minimizing these diffuse interface models
on graphs.

In Chapter 2, we review the classical diffuse interface models. Such continuum
PDEs involve a front propagating by mean curvature on each of its point. We
also study the definition of mean curvature flow as well as discrete version of
mean curvature flow on arbitrary graphs. Since the diffuse interface models and
mean curvature flow are all closely related to the MBO scheme, we also review
the classical MBO method and extend it for arbitrary graphs.

In Chapter 3, an algorithm based on point integral method is introduced. Inspired
by diffuse interface models that have been used in variety of problems, like
fluid dynamics and image processing, we introduce an algorithm to solve heat
equation over a point cloud which is believed to lie on a low dimensional manifold
for most natural data sets. Due to the complicated geometrical structure of the
manifold, it is not easy to solve PDEs on arbitrary manifold. However, utilizing
the point integral method with proper kernel function [1], we can solve heat
equation on the point cloud efficiently. We also use the Merriman-Bence-Osher
(MBO) [2] scheme to speed up the scheme.

In Chapter 4, we proposed an efficient iterative thresholding 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(N log N) 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.

In Chapter 5, a novel weighted nonlocal total variation (WNTV) method is proposed.
Compared to the classical nonlocal total variation methods, our method
modifies the energy functional to introduce a weight to balance between the
labeled sets and unlabeled sets. With extensive numerical examples in semi-supervised
clustering, image inpaiting and image colorization, we demonstrate
that WNTV provides an effective and efficient method in many image processing
and machine learning problems.

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