Conformable Convolution for Topologically Aware Learning of Complex Anatomical Structures

TL;DR

The paper addresses the challenge of preserving topology in medical image segmentation, where pixel-wise accuracy alone can miss important connectivity. It introduces Conformable Convolution, an adaptive layer that learns topology-guided sampling offsets, and the Topological Posterior Generator (TPG) that uses persistent homology to identify regions of topological significance. The approach is architecture-agnostic and demonstrated to improve topology metrics such as connectivity while maintaining or improving pixel-level accuracy across three diverse datasets. This work provides a practical framework for topology-aware segmentation that can be integrated into existing models to enhance reliability in clinical analyses.

Abstract

While conventional computer vision emphasizes pixel-level and feature-based objectives, medical image analysis of intricate biological structures necessitates explicit representation of their complex topological properties. Despite their successes, deep learning models often struggle to accurately capture the connectivity and continuity of fine, sometimes pixel-thin, yet critical structures due to their reliance on implicit learning from data. Such shortcomings can significantly impact the reliability of analysis results and hinder clinical decision-making. To address this challenge, we introduce Conformable Convolution, a novel convolutional layer designed to explicitly enforce topological consistency. Conformable Convolution learns adaptive kernel offsets that preferentially focus on regions of high topological significance within an image. This prioritization is guided by our proposed Topological Posterior Generator (TPG) module, which leverages persistent homology. The TPG module identifies key topological features and guides the convolutional layers by applying persistent homology to feature maps transformed into cubical complexes. Our proposed modules are architecture-agnostic, enabling them to be integrated seamlessly into various architectures. We showcase the effectiveness of our framework in the segmentation task, where preserving the interconnectedness of structures is critical. Experimental results on three diverse datasets demonstrate that our framework effectively preserves the topology in the segmentation downstream task, both quantitatively and qualitatively.

Figures (5)

• Figure 1: Our proposed layer comprises two modules: (a) Topological Posterior Generation: receives the input feature map $\phi_{in}$ from the previous layer and generates $\phi_{post}$. (b) Conformable Convolution: receives $\phi_{post}$, generates offsets with the first convolution layer for the adaptive kernel of the second convolution. The topology-aware features are extracted and passed through Batch Norm and ReLU layers. The proposed module depicts a layer that can be used at different positions in architectures such as UNet.
• Figure 2: An example visualization on how PH applies a filtering function $f_{\tau}(.)$ with changing $\tau$ ($\tau_1$, $\tau_2$, $\tau_3$) to the original image with vessel and noise , obtaining a nested set of cubical complexes Q ($Q_1$, $Q_2$, $Q_3$). As $\tau$ increases from $\tau_1$ to $\tau_2$, vessel is first born at $Q_1$ and noise is later born at $Q_2$ . Both of them die at $Q_3$, as $\tau$ further raises to $\tau_3$.
• Figure 3: Visualization of Topological Priors in each layer of UNet + Conform.
• Figure 4: Demonstration of the Gaussian dilation process on real and zoomed-in feature map: (a) $\phi_{pr}$ in a vessel feature map; (b) a zoomed-in synthetic feature map, depicting $\phi_{pr}$ emphasizing on regions of high topological interests, (c) the effect of Gaussian dilation in dilating the topologically significant regions; (d) the impact of Gaussian dilation on the vessel feature map.
• Figure 5: Qualitative Segmentation Results corresponding to \ref{['tabs:layers']}. $error_{\beta_{0}}$(highlighting disconnected components) are in red squares, while $error_{\beta_{1}}$(highlighting holes) are in red circles.