Alpha divergences based mass transport models for image matching problems
Pengwen Chen Changfeng Gui
Inverse Problems & Imaging 2011, 5(3): 551-590 doi: 10.3934/ipi.2011.5.551
Registration methods could be roughly divided into two groups: area-based methods and feature-based methods. In the literature, the Monge-Kantorovich (MK) mass transport problem has been applied to image registration as an area-based method. In this paper, we propose to use Monge-Kantorovich (MK) mass transport model as a feature-based method. This novel image matching model is a coupling of the MK problem with the well-known alpha divergence from the probability theory. The optimal matching scheme is the one which minimizes the weighted alpha divergence between two images. A primal-dual approach is employed to analyze the existence and uniqueness/non-uniqueness of the optimal matching scheme. A block coordinate method, analogous to the Sinkhorn matrix balancing method, can be used to compute the optimal matching scheme. We also derive a distance function for image morphing. Similar to elastic distances proposed by Younes, the geodesic under this distance function has an explicit expression.
keywords: Gaussian Kernel functions image morphing Image registration block coordinate ascent methods. image matching problems Sinkhorn matrix balancing Monge-Kantorovich mass transport Alpha divergences
Spike solutions to a nonlocal differential equation
Changfeng Gui Zhenbu Zhang
Communications on Pure & Applied Analysis 2006, 5(1): 85-95 doi: 10.3934/cpaa.2006.5.85
In this paper we consider a nonlocal differential equation, which is a limiting equation of one dimensional Gierer-Meinhardt model. We study the existence of spike steady states and their stability. We also construct a single-spike quasi-equilibrium solution and investigate the dynamics of spike-like solutions.
keywords: stability. spike solution Gierer-Meinhardt model
On some problems related to de Giorgi’s conjecture
Changfeng Gui
Communications on Pure & Applied Analysis 2003, 2(1): 101-106 doi: 10.3934/cpaa.2003.2.101
We study the uniqueness of minimizers for the Allen-Cahn energy and the nonexistence of monotone stationary solutions for the Allen-Cahn equation with double well potentials of different depths.
keywords: symmetry. minimizing Phase transition
Properties of translating solutions to mean curvature flow
Changfeng Gui Huaiyu Jian Hongjie Ju
Discrete & Continuous Dynamical Systems - A 2010, 28(2): 441-453 doi: 10.3934/dcds.2010.28.441
In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.
keywords: asymptotic behavior Elliptic equation mean curvature flow convex solution gradient estimate.
Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians
Ze Cheng Changfeng Gui Yeyao Hu
Discrete & Continuous Dynamical Systems - A 2019, 39(3): 1345-1358 doi: 10.3934/dcds.2019057

It is known that the supercritical Hardy-Littlewood-Sobolev (HLS) systems with an integer power of Laplacian admit classic solutions. In this paper, we prove that the supercritical HLS systems with fractional Laplacians $ (-Δ)^s $, $ s∈(0,1) $, also admit classic solutions.

keywords: Fractional Laplacian bifurcation theory asymptotic analysis

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