Active Contour Models Driven by Hyperbolic Mean Curvature Flow for Image Segmentation

TL;DR

This work tackles the vulnerability of parabolic mean curvature flow–driven active contours to high-noise conditions by introducing hyperbolic mean curvature flow–driven ACMs (HMCF-ACMs) with an adaptive acceleration field. By deriving both Lagrangian and Eulerian formulations, the authors show that HMCF-ACMs evolve contours through normal motion and are numerically equivalent to wave equations when represented with a signed distance function, enabling stable, curvature-aware updates. They implement a high-accuracy, efficient numerical scheme using spectral (DCT) discretization and a weighted RK4 integrator, supported by stability analysis via Fourier methods. Extensive experiments on natural and medical images demonstrate superior noise robustness, lower parameter sensitivity, and improved segmentation accuracy compared with PMCF-ACMs, underscoring the practical potential of HMCF-ACMs for robust image segmentation and possible integration as a learnable prior in deep networks.

Abstract

Parabolic mean curvature flow-driven active contour models (PMCF-ACMs) are widely used for image segmentation, yet they suffer severe degradation under high-intensity noise because gradient-descent evolutions exhibit the well-known zig-zag phenomenon. To overcome this drawback, we propose hyperbolic mean curvature flow-driven ACMs (HMCF-ACMs). This novel framework incorporates an adjustable acceleration field to autonomously regulate curve evolution smoothness, providing dual degrees of freedom for adaptive selection of both initial contours and velocity fields. We rigorously prove that HMCF-ACMs are normal flows and establish their numerical equivalence to wave equations through a level set formulation with signed distance functions. An efficient numerical scheme combining spectral discretization and optimized temporal integration is developed to solve the governing equations, and its stability condition is derived through Fourier analysis. Extensive experiments on natural and medical images validate that HMCF-ACMs achieve superior performance under high-noise conditions, demonstrating reduced parameter sensitivity, enhanced noise robustness, and improved segmentation accuracy compared to PMCF-ACMs.

Figures (13)

• Figure 1: The curve $C(t) = \{ (x, y) \in \Omega \mid \phi(x, y, t) = 0 \}$ propagates in the normal direction. For convex parts, the curvature $\kappa \textgreater 0$, while for concave parts, the curvature $\kappa \textless 0$.
• Figure 2: Curvature-driven spiral evolution under the HMCF. Left to right: initial contours (red circles), intermediate evolution stages, and final result. Acceleration fields $\vec{F}^k(x,y,0)$ at iterations $k = \{0, 2, 10, 22\}$.
• Figure 3: Curvature-driven star evolution under the HMCF. Acceleration fields $\vec{F}^k(x,y,0)$ at iterations $k = \{0, 20, 75, 100\}$.
• Figure 4: Segmentation results (red curves) and initial contours (green curves) for C-V (rows 1--3) and HMCF-C-V (rows 4--6) under multi-type noise. Left-to-right blocks: Gaussian (0.05--0.25), salt&pepper (0.05--0.25) and periodic (100--255, frequency 0.2) noise levels.
• Figure 5: Variation of Dice metric with curvature parameters ($b$ for HMCF-C-V, $\mu$ for C-V) under different noise conditions. Left column: HMCF-C-V model; Right column: C-V model. Rows 1-3 correspond to Gaussian, salt&pepper, and periodic noise with increasing intensities, respectively.

Theorems & Definitions (1)

• Remark 1: SDF Maintenance