This chapter surveys recent numerical advances in the phase field method for geometric
surface evolution and related geometric nonlinear partial differential equations (PDEs) …

We present two graph-based algorithms for multiclass segmentation of high-dimensional
data on graphs. The algorithms use a diffuse interface model based on the Ginzburg …

Variational methods, which have been tremendously successful in image segmentation,
work by minimizing a given objective functional. The objective functional usually consists of …

Field-guided parameterization methods have proven effective for quad meshing of surfaces;
these methods compute smooth cross fields to guide the meshing process and then …

Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously
challenging to solve. Recently, Liu et al.(in: Proceedings of the National Academy of …

We show how auction algorithms, originally developed for the assignment problem, can be
utilized in Merriman, Bence, and Osher's threshold dynamics scheme to simulate multi …

Active contour models have been widely used in image segmentation, and the level set
method (LSM) is the most popular approach for solving the models, via implicitly …

We present a novel probabilistic approach for generating multi-fidelity data while accounting
for errors inherent in both low-and high-fidelity data. In this approach a graph Laplacian …

Abstract Deep Convolutional Neural Networks (DCNNs) can well extract the features from
natural images. However, the classification functions in the existing network architecture of …

This paper is a republication of an MMS paper [AL Bertozzi and A. Flenner, Multiscale
Model. Simul., 10 (2012), pp. 1090--1118] describing a new class of algorithms for …