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\section{Related Works}
\textbf{Medical Image Segmentation.} U-Net~\cite{ronneberger2015u} is the foundation of modern medical image segmentation. Numerous variations have refined this backbone through skip connections~\cite{zhou2019unet++}, attention mechanisms~\cite{oktay2018attention}, multi-scale feature extraction~\cite{ibtehaz2020multiresunet}, and residual~\cite{milletari2016v} or densely connected designs~\cite{jegou2017one}. These advances consistently enhance feature representation and spatial detail.
%, they remain fundamentally rooted in standard Euclidean geometry.
A persistent challenge in medical image segmentation is robustness to noise, artifacts, and acquisition variability~\cite{nazir2024recent, kumar2025denoising}. Several studies highlight that U-Net like networks experience substantial performance drops under noise perturbations or distribution shift~\cite{kervadec2019boundary, alom2019recurrent,valanarasu2021kiu}. Other studies expose the vulnerability of these networks to adversarial noise and provide robustness assessment frameworks~\cite{asgari2018vulnerability,wang2024susceptibility,daza2021towards}. To mitigate these issues, the community has explored stronger data augmentation~\cite{zhou2019models}, adversarial robustness training~\cite{asgari2018vulnerability,wang2024susceptibility}, model architecture enhancement~\cite{daza2021towards}, defining new loss functions~\cite{gonzalez2025robust} and domain adaptation mechanisms such as unsupervised feature learning~\cite{kamnitsas2017unsupervised, chen2020unsupervised}. Robustness can also be addressed through classical denoising objectives~\cite{kaur2018review,wang2021image}, which requires separate training stages. While these methods improve stability, they typically require additional objectives, adversarial modules, or task-specific tuning. Holistic end-to-end robustness strategies such as nnU-Net~\cite{isensee2018no, isensee2021nnu} attempt to reduce sensitivity to acquisition variability and noise through extensive normalization and augmentation policies.
Despite such progress, U-Net remains sensitive to noise. This motivates us to explore an alternative geometry for U-Net that offers an inherent robustness to noise.

\textbf{Hyperbolic Neural Networks.} Owing to its strong ability to embed hierarchical and tree-like structures \cite{sarkar2011low, nickel2017poincare, ganea2018entailment, van2025low}, hyperbolic geometry has seen a surge in popularity within deep learning literature \cite{peng2021hyperbolic, mettes2024hyperbolic}, leading to numerous applications in NLP \cite{dhingra2018embedding, he2024language, mandica2024hyperbolic}, graph learning \cite{liu2019hyperbolic, chami2019hyperbolic, zhang2021hyperbolic} and, particularly, computer vision \cite{khrulkov2020hyperbolic, guo2022clipped, ermolov2022hyperbolic, desai2023hyperbolic, poppi2025hyperbolic, pal2025compositional}. Commonly, these approaches restrict the use of hyperbolic geometry to the final layers of the model. For example, \citeauthor{atigh2022hyperbolic} use a Euclidean segmentation model to obtain pixel-level features, which are projected to hyperbolic space and classified using a hyperbolic classifier. Hyperbolic learning has also shown great early potential in the medical domain.
%Recently, hyperbolic deep learning has also found its way into the medical domain.
\citeauthor{ayubcha2024improved, gonzalez2025hyperbolic} map image features from Euclidean ResNets to the hyperboloid and perform medical classification. Similarly, \citeauthor{wang2023vison} map pixel features from a Euclidean vision transformer to the Poincar\'e ball and perform hyperbolic multi-lesion segmentation in diabetic retinopathy. Several approaches use the recent advances in hyperbolic multimodal learning to align the representations of medical data, such as images, brain activations or WSI hierarchies, with the representations from textual descriptions \cite{qiao2025hyden, pate2025fine, huang2025hyperpath, baek2025mnm}. Current literature however focuses on hyperbolic geometry for the classification or semantic space only. This work strives to make the full architecture of U-Net hyperbolic. 
%All of these works have in common that the application of hyperbolic geometry is constrained to either the embedding space or to the final few layers of the network, whereas we focus on fully hyperbolic neural networks.
We note that the hyperbolic networks of \citeauthor{lensink2022fully} are based on the hyperbolic telegraph equation, which is unrelated to the hyperbolic geometry that is used in our work.

%Several fully hyperbolic architectures acting on the Poincar\'e ball model or the hyperboloid model of hyperbolic space have been proposed. Most notably, 
\citeauthor{ganea2018hyperbolic} proposed the first hyperbolic version of hyperbolic multinomial logistic regression on the Poincar\'e ball model, which was extended to a hyperbolic linear layer by \citeauthor{shimizu2021hyperbolic}. \citeauthor{chen2022fully} proposed an alternative architecture in the hyperboloid model of hyperbolic space. Early results with fully hyperbolic networks in image classification have shown improved robustness compared to their Euclidean and partially hyperbolic counterparts \cite{van2023poincare,bdeir2024fully}. However, despite the importance of robustness in segmentation, no prior work has explored the potential of such architectures for this domain. We propose the first fully hyperbolic version of a U-Net for image segmentation and demonstrate that it achieves significantly increased robustness.