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

TL;DR

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.

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.

Figures (6)

• Figure 1: Visual comparison of segmentation results on a noisy image with different time steps $\tau$. The parameter is set to $\tilde{\lambda} = 0.6$. (a) The noisy input image. (b) The initial square contour. (c)--(l) Comparison between the traditional threshold dynamics (TD) method and the proposed median filter (MF) scheme with decreasing time steps $\tau \in \{9, 7, 5, 3, 1\} \times 10^{-4}$. It can be observed that the MF scheme (d, f, h, j, l) consistently achieves accurate segmentation regardless of the time step size. In contrast, the TD method demonstrates significant instability: at intermediate time steps (e, g, i), it fails to handle topological changes (splitting); and at a small time step (k), it suffers from the severe pinning effect, where the contour fails to evolve from the initialization (compare k with b).
• Figure 2: Visual verification of sharpness. Top row: 3D evolution of the interface level set function $\phi$ showing the sharpening process. Bottom row: Corresponding histograms of pixel values showing the transition from a continuous distribution to a binary state. The columns represent the progression from initialization (left) to the final geometric steady state (right).
• Figure 3: The simulation is performed with a time step $\tau=5 \times 10^{-4}$, and the effective parameter $\tilde{\lambda}=0.6$. The sequence illustrates the robust evolution of the contour, effectively capturing geometric details and handling topological changes, finally converging to a sharp binary solution at iteration $k=90$.
• Figure 4: Numerical verification of the proposed method using the LIF model. (a)--(c) The algorithm successfully overcomes intensity inhomogeneity by utilizing local intensity information to guide the contour. (d) The energy decreases monotonically, empirically confirming the unconditional stability claimed in Theorem \ref{['thm:unconditional_stability']} for the median-filter-based iteration.
• Figure 5: Numerical verification of the proposed framework on the Flow Contraction problem. (a) Domain setup with a full-height inlet transitioning to a restricted outlet of height $1/3$. (b) Initial random state. (c) The algorithm successfully forms a smooth converging channel. (d) The energy curve confirms unconditional stability.

Theorems & Definitions (15)

• Lemma 3.1: Monotonicity
• Proof 1
• Definition 4.1: Relaxed Energy Functional
• Theorem 4.2: Equivalence via Threshold Decomposition
• Proof 2
• Lemma 4.3: Existence and Uniqueness
• Proof 3
• Theorem 4.4: Generic Binary Enforcement
• Proof 4
• Definition 4.5: Movement Limiter