Topology-Preserving Deep Image Segmentation (1906.05404v1)

Abstract: Segmentation algorithms are prone to make topological errors on fine-scale structures, e.g., broken connections. We propose a novel method that learns to segment with correct topology. In particular, we design a continuous-valued loss function that enforces a segmentation to have the same topology as the ground truth, i.e., having the same Betti number. The proposed topology-preserving loss function is differentiable and we incorporate it into end-to-end training of a deep neural network. Our method achieves much better performance on the Betti number error, which directly accounts for the topological correctness. It also performs superiorly on other topology-relevant metrics, e.g., the Adjusted Rand Index and the Variation of Information. We illustrate the effectiveness of the proposed method on a broad spectrum of natural and biomedical datasets.

• The paper introduces TopoNet, a framework that integrates a differentiable topological loss based on persistent homology to significantly reduce Betti number errors.
• It leverages persistence diagrams to compute gradients and align predicted segmentation with ground truth, ensuring correct topological structures.
• Empirical results demonstrate that TopoNet outperforms state-of-the-art methods on topology-sensitive metrics while maintaining pixel-wise accuracy across diverse datasets.

Overview of Topology-Preserving Deep Image Segmentation

The paper "Topology-Preserving Deep Image Segmentation" addresses a critical challenge in image segmentation—specifically, the frequent occurrence of topological errors such as broken connections in fine-scale structures. The authors propose a novel framework, TopoNet, which integrates a topological loss function into the training pipeline of deep neural networks to enforce the correct topological alignment between predicted and ground truth segmentations.

Methodology

At the core of the approach is a differentiable, continuous-valued topological loss based on the Betti number, a critical topological invariant that quantifies the number of connected components and handles in a given space. The primary challenge lies in converting this discrete measure into a format that is compatible with the differentiable nature of deep learning frameworks. This is achieved by employing persistent homology—a concept from computational topology that summarizes topological features of a space across multiple scales via persistence diagrams.

The authors use these persistence diagrams to develop a matching criterion between the predicted segmentation's topology and that of the ground truth. This criterion forms the novel topological loss, which minimizes the squared differences between corresponding persistent dots of the prediction and ground truth diagrams. Notably, the derivation of gradients for this loss facilitates its incorporation into backpropagation, maintaining the end-to-end trainability of the network.

Empirical Results

The efficacy of TopoNet is demonstrated across a variety of datasets, including natural and biomedical image segmentation tasks such as those in the CREMI, ISBI12, ISBI13, DRIVE, Road, and CrackTree datasets. The paper reports substantial performance improvements on topology-sensitive metrics, including the Betti number error, Adjusted Rand Index (ARI), and Variation of Information (VOI), without sacrificing pixel-wise segmentation accuracy.

Quantitative results affirm that TopoNet outperforms state-of-the-art segmentation methods, particularly in maintaining topological correctness which is imperative for applications requiring structural integrity, such as neuron image segmentation. Notably, TopoNet significantly reduces Betti number errors compared to baseline models, thereby ensuring that key topological features like connectivity and loop structures are preserved.

Implications and Speculations

Practically, this method promises significant advancements in applications where topological integrity is vital, such as medical imaging, where the accurate delineation of structures can directly influence diagnostic outcomes. Theoretically, this work pushes the boundaries of integrating topological insights with deep learning, highlighting a symbiotic relationship between these domains. The concept of incorporating persistent homology into the learning process opens up new possibilities in the field of topological data analysis, especially in enhancing machine learning models’ ability to learn from and adapt to complex, structured data.

Future work may explore the scalability of TopoNet for 3D segmentation tasks, where topological preservation becomes even more critical. Moreover, the methodology presents a pathway to extend similar concepts to other domains such as graph-based learning, where topological properties are indispensable.

In conclusion, this paper contributes a significant methodological advancement to the domain of computer vision by addressing the often overlooked but crucial aspect of topological correctness in segmentation tasks.