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A modularized algorithmic framework for interface related optimization problems using characteristic functions

Dong Wang, Shangzhi Zeng, Jin Zhang

TL;DR

This paper considers the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction and shows that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer.

Abstract

In this paper, we consider the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction. In particular, the problem focuses on interface related optimization problems where the interface is implicitly described by characteristic functions of the corresponding domains. Under such representation and discretization, the problem is then formulated into a discretized optimization problem where the objective function is concave with respect to characteristic functions and convex with respect to state variables. We show that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer. Extensive numerical examples are performed to support the theory.

A modularized algorithmic framework for interface related optimization problems using characteristic functions

TL;DR

This paper considers the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction and shows that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer.

Abstract

In this paper, we consider the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction. In particular, the problem focuses on interface related optimization problems where the interface is implicitly described by characteristic functions of the corresponding domains. Under such representation and discretization, the problem is then formulated into a discretized optimization problem where the objective function is concave with respect to characteristic functions and convex with respect to state variables. We show that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer. Extensive numerical examples are performed to support the theory.
Paper Structure (15 sections, 12 theorems, 108 equations, 6 figures, 4 algorithms)

This paper contains 15 sections, 12 theorems, 108 equations, 6 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Algorithmic framework
  3. Convergence results
  4. Convergence analysis for Global $\theta$ $\&$ Global $u$ case
  5. Convergence analysis for Global $\theta$ $\&$ Local $u$ case
  6. Convergence analysis for inexact Stationay $\theta$ $\&$ Local $u$ case
  7. Analysis on updating $u$ towards local minimum
  8. Analysis on updating $\theta$ with one-step acceleration
  9. Interpretation for the theoretical validity of ICTM
  10. Summary of the convergence results
  11. Numerical experiments
  12. Application to the implicit surface reconstruction problem
  13. Application to the Chan--Vese model for image segmentation
  14. Application to the Local Intensity Fitting (LIF) model for image segmentation
  15. Conclusion and discussions

Key Result

lemma 1

For any $\bar{\theta} \in \mathcal{S}$, let $\bar{u}$ be a local minimum of problem that is, there exists $\epsilon > 0$ such that Then $\bar{u} \in C$.

Figures (6)

  • Figure 1: A diagram to several level sets of the distance function $\tilde{d}$ to a discrete point cloud. See Section \ref{['sec:surface']}.
  • Figure 2: Results obtained from \ref{['eq:iscpg']} with different step size $\alpha_u = 0.5, 1, 2, 4, 8$. See Section \ref{['sec:surface']}.
  • Figure 3: Change of objective function value with respect to iteration steps with different step size $\alpha_u = 0.5, 1, 2, 4, 8$. See Section \ref{['sec:surface']}.
  • Figure 4: Results obtained from \ref{['eq:iscthresholding']} for different point clouds. See Section \ref{['sec:surface']}.
  • Figure 5: Left: the initial guesses for the iteration. Middle: the converged results. Right: curves for the objective function values with respect to iteration step. In two images, $\lambda$ are set to be $0.03$ and $0.1$ respectively. See Section \ref{['sec:cv']}.
  • ...and 1 more figures

Theorems & Definitions (20)

  • lemma 1
  • proof
  • proposition 1
  • lemma 2
  • lemma 3
  • proof
  • proposition 2
  • proof
  • theorem 1
  • proof
  • ...and 10 more