Topology preserving Image segmentation using the iterative convolution-thresholding method

TL;DR

TP-ICTM addresses topology deviations in variational segmentation by embedding topology constraints into the ICTM framework using a topology-preserving correction guided by simple points and digital topology. The approach represents regions with binary indicators, approximates the interface via heat-kernel convolution, and alternates prediction and topology-preserving correction to maintain fixed $n$-connected foreground and $M$-connected background components. It also provides a monotone objective decay and stability guarantees. Empirical results on Chan-Vese and Locally Implicit Fitting variants show improved topology preservation, robustness to noise and complex patterns, with competitive computational efficiency and easy integration with other models.

Abstract

Variational models are widely used in image segmentation, with various models designed to address different types of images by optimizing specific objective functionals. However, traditional segmentation models primarily focus on the visual attributes of the image, often neglecting the topological properties of the target objects. This limitation can lead to segmentation results that deviate from the ground truth, particularly in images with complex topological structures. In this paper, we introduce a topology-preserving constraint into the iterative convolution-thresholding method (ICTM), resulting in the topology-preserving ICTM (TP-ICTM). Extensive experiments demonstrate that, by explicitly preserving the topological properties of target objects-such as connectivity-the proposed algorithm achieves enhanced accuracy and robustness, particularly in images with intricate structures or noise.

Figures (14)

• Figure 1: Examples of non-simple and simple points. The removal of the blue point in cases (a) and (b) will change the topology of all black points and the removal of the blue point in case (c) will not affect the topology property.
• Figure 1: First row: selected snapshots of the ICTM for CV. Second row: selected snapshots of the TP-ICTM for CV. See Section \ref{['sec:diffiteration']}.
• Figure 2: Monotonic decrease of the objective functional with some selected snapshots during the iteration process. See Section \ref{['sec:decay']}.
• Figure 3: Initial guess of a two discs image for parameter tuning in Section \ref{['sec:paratuning']}.
• Figure 4: Fix $\tau_1 = 0.001$ and $\tau_2 = 0.001$ with $\lambda = 0.005, 0.01,$ and $0.05$. From (a) to (c), the number of iterations required is $70$, $128$, and $418$. See Section \ref{['sec:paratuning']}.

Theorems & Definitions (11)

• Definition 1
• Definition 2: Topology Number
• Definition 3: Simplicity Function
• Example 2.1
• Remark 2.2
• Remark 4.1
• Theorem 5.1: Stability
• Proof 1
• Remark 5.2
• Remark 6.1