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Thesis (Ph. D.)--Columbia University, 1991. Includes bibliographical references (leaf 48).

In this paper, we propose a novel iterative convolution-thresholding method (ICTM) that is applicable to a range of variational models for image segmentation. A variational model usually minimizes an energy functional consisting of a fidelity term and a regularization term. In the ICTM, the interface between two different segment domains is implicitly represented by their characteristic functions. The fidelity term is usually written as a linear functional of the characteristic functions and the regularized term is approximated by a functional of characteristic functions in terms of heat kernel convolution. This allows us to design an iterative convolution-thresholding method to minimize the approximate energy. The method is simple, efficient and enjoys the energy-decaying property. Numerical experiments show that the method is easy to implement, robust and applicable to various image segmentation models.

Inspired by works of Castéras (Pac J Math 276:321–345, 2015), Li and Zhu (Calc Var Partial Differ Equ 58:1–18, 2019), Sun and Zhu (Calc Var Partial Differ Equ 60:1–26, 2021), we propose a heat flow for the mean field equation on a connected finite graph G=(V,E). Namely ∂_tϕ(u)=Δu−Q+ρ\frac{e^u}{∫_V e^u dμ} u(⋅,0)=u_0, where Δ is the standard graph Laplacian, ρ is a real number, Q:V→R is a function satisfying ∫_V Qdμ=ρ, and ϕ:R→R is one of certain smooth functions including ϕ(s)=es. We prove that for any initial data u_0 and any ρ∈R, there exists a unique solution u:V×[0,+∞)→R of the above heat flow; moreover, u(x, t) converges to some function u_∞:V→R uniformly in x∈V as t→+∞, and u_∞ is a solution of the mean field equation Δu_∞−Q+ρ\frac{e^{u_∞}}{∫_V e^{u_∞}dμ}=0. Though G is a finite graph, this result is still unexpected, even in the special case Q≡0. Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz–Simon type inequality and use it to conclude the convergence of the heat flow.