A median-filter-based framework for interface optimal design problems

TL;DR

This paper introduces a median-filter-based numerical framework for interface optimization problems that overcomes discretization issues, guarantees energy stability, and converges to binary solutions across various applications.

Contribution

It extends binary schemes into a continuous level-set framework using weighted quantiles, improving accuracy and stability in interface evolution problems.

Findings

Effectively eliminates the pinning effect in interface evolution.

Guarantees unconditional energy stability of the iterative scheme.

Accurately converges to binary solutions in multiple models.

Abstract

We present a robust and efficient numerical framework based on a median filter scheme for solving a broad class of interface-related optimization problems, from image segmentation to topology optimization. A key innovation of our work is the extension of the binary scheme into a continuous level-set scheme via a weighted quantile interpretation. Unlike traditional binary iterative convolution-thresholding method (ICTM), this continuous median filter scheme effectively overcomes the pinning effect caused by spatial discretization, achieving accurate interface evolution even with small time steps. We also provide a rigorous theoretical analysis, proving the unconditional energy stability of the iterative scheme. Furthermore, we prove that for a wide class of data fidelity terms, the convex relaxation inherently enforces a binary solution, justifying the effectiveness of the method without explicit penalization. Numerical experiments on the Chan--Vese model, the local intensity fitting (LIF) model, and topology optimization in Stokes flow demonstrate that the proposed efficient continuous framework effectively eliminates the pinning effect, guarantees unconditional energy stability, and accurately converges to binary solutions.