\section{Introduction}
\label{sec:intro}
Tree-like structures are ubiquitous in living organisms and serve vital functions, for example as blood vessels, airways, or neuronal networks.
Understanding how these branched systems develop, function, and change under pathological conditions requires detailed structural information.
Image-based analysis has become a key tool for investigating such processes at the subcellular to organ scale, using diverse modalities such as MRI, micro-CT, electron microscopy, or light-sheet microscopy~\cite{cheng2024,Walek2023,soubeyrand2023,todorov2020,Obenaus2017}.
To analyze these structures, image data are processed to extract simplified network representations—typically skeletons or graphs—from which features such as overall topology, segment lengths, and branching patterns can be quantified.
For example, \citet{liu2021assessment} showed that altered topology of the microvasculature plays an important role in hepatocellular carcinoma (HCC).
%role in diseases such as preeclampsia or ~\cite{liu2021assessment,kraeker2023abstract}
However, obtaining accurate and complete reconstructions remains challenging; manual or semi-automated tracing is still often the method of choice, particularly in cardiovascular imaging \cite{Pampols-Perez2025piezo2,RiosCoronado2025}.

Topological correctness refers to accurately preserving the connectivity of the biological network.
Many vessel-graph extraction pipelines perform foreground-background segmentation of the image first \cite{tetteh2019_deepvesselnet,todorov2020,Wittmann_2025_CVPR}, and then skeletonize the foreground mask using TEASAR or variants of the Lee algorithm \cite{voreen2009,Drees2021voreenSkel,Bumgarner2022vesselvio}.
However, these approaches struggle to achieve topological accuracy: 
% Challenges arise from the complex, tree-like morphology of vascular networks. 
Variable imaging contrast can render vessel segments faint or discontinuous, prompting segmentation methods to produce false splits; at the same time, branches running in close proximity often lead to false merges that incorrectly connect distinct vessels.
In the subsequent skeletonization step, such false merge errors create artificial cycles or spurious branching points, as illustrated in the left part of \figureref{fig:our_approach_summary}. 
%
Topological losses~\cite{lux2025topograph,kirchhoff2024,Shit2021cldice} or simple heuristics like thinning of ground-truth masks may help with this issue to some extent -- However, segmentation as a modular step remains fundamentally ill-suited for modelling the task of topologically correct vessel graph reconstruction.
%
This limitation is not specific to foreground-background segmentation, but also holds for instance segmentation: 
Although the vasculature forms a globally connected system, imaging typically covers only a restricted anatomical region, yielding multiple disjoint trees; 
while instance segmentation can, in principle, model the separation of different vessel trees, it is not designed to prevent false merges \emph{within} a single tree.

\begin{figure}
    \centering
    %\small
    \includegraphics[width=0.9\linewidth]{figures/intro_figure_direction_vectors_corrected.png}
    %\vspace{-10pt}
    \caption{\textbf{Segmentation-and-skeletonization vs.\ Vesselpose.}
Traditional segment-and-skeletonize pipelines often produce incorrect skeletons, especially when distinct vessels lie in close proximity. In contrast, Vesselpose leverages pixel-wise direction vectors to robustly reconstruct vascular trees, naturally handling closely apposed branches as well as multiple distinct trees.}
    \label{fig:our_approach_summary}
    \vspace{-20pt}
\end{figure}

Therefore, alternative strategies are needed to achieve reliable and topologically accurate vascular graph reconstructions.
Earlier work used combinatorial optimization to assemble vessel trees from small centerline tracklets \cite{tueretken2010,tueretken2011}, yielding globally optimal tree reconstructions w.r.t.\ some objective under topological constraints, thereby ensuring topological correctness.
However, even with heuristics and relaxed constraints, Integer Linear Programming (ILP)-based methods \cite{tueretken2016,robben2014,robben2016,rempfler2016minimum} remain computationally expensive and do not scale to large vascular networks.
More recently, image-to-graph frameworks \cite{vesselformer2024,trexplorer2024, trexplorer_super2025}, inspired by DETR \cite{carion2020eccv,zhu2021deformable}, have emerged as a promising direction.
Most of them have been validated primarily on synthetic datasets, where
Vesselformer \cite{vesselformer2024} still produces notable topological errors, while Trexplorer \cite{trexplorer2024} suffers from duplicate branching and premature tracking termination. Trexplorer-Super \cite{trexplorer_super2025} addresses these issues and extends evaluation to real datasets, yet its training and evaluation remain restricted to single-tree structures, whereas real vascular volumes typically contain multiple disjoint trees.
Thus, a more general and computationally feasible solution is still required—one that can robustly extract topologically meaningful graphs from multi-tree vascular networks.

At the same time, issues of topological correctness have been addressed very successfully for the highly related tasks of cell segmentation and tracking in 3D(+t) microscopy data \cite{cellpose2021,Malin-Mayor2023}.
Here, the community has moved away from the traditional segment-then-fix paradigm, replacing deep learning (DL)-based binary segmentation with models that predict pixel-wise shape properties that encode topologically relevant information \cite{hirsch2020_an_auxil_task_for_learn_nucle_segme_in_3d,patchperpix,Sheridan2023lsd}.
Most prominently, Cellpose \cite{cellpose2021,cellpose2022} predicts vector fields pointing toward object centers;
similarly, for cell tracking through time, \citet{Malin-Mayor2023} predict pixel-wise direction vectors that point backward in time to the center of the same or mother cell in the previous frame.
Iteratively following these vectors reconstructs complete cell lineages and ultimately traces each cell back to its origin.
This approach leverages the biological prior that cells divide but do not merge, ensuring a unique predecessor.
These advances highlight the value of predicting pixel-wise topological information, suggesting a promising direction that has not yet been extended to vascular tree reconstruction.

Building on these insights, we propose a method that extracts topologically plausible vessel trees from 3D images using a heuristic solver guided by pixel-wise predictions.
We train a network to predict direction vectors that point toward the vessel centerline while being biased in the rootward direction, leveraging the anatomical prior that vessel diameter typically increases toward the root.
This prior enables robust orientation along the tree and naturally suits vascular and airway networks.
By defining the flow from endpoints toward the root, we circumvent directional ambiguities at branching points, resulting in a well-defined direction vector at each location.

The predicted binary mask and direction vectors serve then as input to a novel skeletonization objective that reconstructs the tree structure by following the learned vector field.
%
In summary, our main contributions are as follows:
\begin{itemize}
    \item We present a DL-based method that predicts voxel-wise directional vectors from 3D vascular images, which a fast heuristic solver then assembles into a consistent vessel centerline graph.
    \item We introduce meaningful and easily interpretable topology-aware evaluation metrics such as false splits and false merges for graphs, proposing a tailor-made \emph{assignment strategy} (cf.\ \citet{metrics_reloaded2024}) based on hierarchical graph-matching.
    \item We outperform current state-of-the-art on synthetic and real datasets and extend evaluation to a widely used multi-tree dataset \cite{tetteh2019_deepvesselnet} and a real 3D micro-CT dataset, achieving superior topological accuracy and reconstruction quality.
\end{itemize}
The code for the model and evaluation, along with trained models and prediction results, is publicly available at \url{https://github.com/Kainmueller-Lab/Vesselpose}.

% \begin{figure}
%     \centering
%     \includegraphics[width=1.0\linewidth]{figures/graphical_abstract_with_enumeration.pdf}
%     \caption{\textbf{Blood vessel reconstruction and evaluation.}
%     (a) A U-Net predicts vessel foreground and voxel-wise direction vectors from the raw image.
%     (b) A modified TEASAR algorithm extracts a skeleton graph.
%     (c)~Predicted skeletons are evaluated against ground-truth using hierarchical graph matching as assignment strategy, which yields topologically meaningful error metrics.
%     }
%     \label{fig:vessel_tree_prediction}
%     \vspace{-15pt}
% \end{figure}

\begin{figure}
    \centering
    \includegraphics[width=0.9\linewidth]{figures/graphical_abstract_v2.pdf}
    \caption{\textbf{Blood vessel reconstruction and evaluation.}
    (a) A U-Net predicts vessel foreground and voxel-wise direction vectors from the raw image.
    (b) A modified TEASAR algorithm extracts a skeleton graph.
    (c)~Predicted skeletons are evaluated against ground-truth using hierarchical graph matching as assignment strategy, which yields topologically meaningful error metrics.
    }
    \label{fig:vessel_tree_prediction}
    \vspace{-15pt}
\end{figure}