Shape Prior Segmentation Guided by Harmonic Beltrami Signature

TL;DR

This work tackles robust 2D image segmentation when only partial shape information is available. It integrates the Harmonic Beltrami Signature (HBS) as a shape prior into a quasi-conformal, topology-preserving segmentation framework, using a two-stage process that aligns the Beltrami coefficient $\mu$ with the HBS $B$ via $||\mu-B||_2$ minimization. Key contributions include extending HBS to serve as a priors-based constraint, a two-subproblem optimization (deformation and prior-normalization), and extensive experiments showing improved accuracy, noise robustness, and the ability to segment complex or multi-component shapes without preprocessing. The approach offers a principled and practical means to inject high-level shape knowledge into low-level segmentation, with implications for medical imaging and other domains where shape priors are valuable.

Abstract

This paper presents a novel shape prior segmentation method guided by the Harmonic Beltrami Signature (HBS). The HBS is a shape representation fully capturing 2D simply connected shapes, exhibiting resilience against perturbations and invariance to translation, rotation, and scaling. The proposed method integrates the HBS within a quasi-conformal topology preserving segmentation framework, leveraging shape prior knowledge to significantly enhance segmentation performance, especially for low-quality or occluded images. The key innovation lies in the bifurcation of the optimization process into two iterative stages: 1) The computation of a quasi-conformal deformation map, which transforms the unit disk into the targeted segmentation area, driven by image data and other regularization terms; 2) The subsequent refinement of this map is contingent upon minimizing the $L_2$ distance between its Beltrami coefficient and the reference HBS. This shape-constrained refinement ensures that the segmentation adheres to the reference shape(s) by exploiting the inherent invariance, robustness, and discerning shape discriminative capabilities afforded by the HBS. Extensive experiments on synthetic and real-world images validate the method's ability to improve segmentation accuracy over baselines, eliminate preprocessing requirements, resist noise corruption, and flexibly acquire and apply shape priors. Overall, the HBS segmentation framework offers an efficient strategy to robustly incorporate the shape prior knowledge, thereby advancing critical low-level vision tasks.

Figures (15)

• Figure 1: Quasi-conformal maps infinitesimal circles to ellipses. The Beltrami coefficient measures the distortion or dilation of the ellipse under the QC map.
• Figure 2: The illustration of HBS. (a) shows the shape $\Omega$ and conformal maps $\Phi_1$ and $\Phi_2$; (b) is the conformal welding $f = \Phi_1^{-1} \circ \Phi_2$; (c) is the Harmonic extension $H$ of conformal welding $f$; (d) is the GHBS $B$ corresponding to $H$.
• Figure 3: (a) is the original image $I$; (b) is the initial template $J_{1,0}$; (c) is the deform map $f_\mu$, which transforms a standard square grid into the illustrated grid; (d) is $I \circ f_\mu$, we can find that the bear is almost put into the unit disk; (e) is deformed template $J_{c_1,c_2} \circ f_\mu^{-1}$, where $c_1$ and $c_2$ are determined by the mean color of $I \circ f_\mu |_\mathbb{D}$ and $I \circ f_\mu |_{\mathbb{D}^c}$ respectively; (f) is the segmentation result, the green line is the boundary of target domain $f_\mu(\mathbb{D})$.
• Figure 4: The illustration of why the HBS can guide segmentation. Left:$\Omega^*$ is the ideal segmentation result, and $\mu^*$ is the HBS of $\Omega^*$. Middle:$\Omega_{prior}$ is the reference shape, $\Omega_B$ is the shape obtained by translating, rotating and scaling $\Omega_{prior}$ to maximum the similarity to $\Omega^*$, and $\mu_B$ is the HBS of $\Omega_{prior}$ and $\Omega_B$. Right:$\Omega$ is the segmentation result, and $\mu$ is the Beltrami coefficient of the deformation map corresponding to $\Omega$.
• Figure 5: Segmentation results of binary images. For each row, the sequence from left to right includes the original image, template shape, template's HBS, the segmentation result without HBS, and the segmentation result with HBS.

Theorems & Definitions (6)

• Theorem 1: Measurable Riemannian Mapping Theorem
• Definition 1
• Theorem 2
• Theorem 3
• Definition 2
• Theorem 5