# Restoring Connectivity in Vascular Segmentation using a Learned Post-Processing Model

Sophie Carneiro-Esteves, Antoine Vacavant, Odyssee Merveille

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## To cite this version:

Sophie Carneiro-Esteves, Antoine Vacavant, Odyssee Merveille. Restoring Connectivity in Vascular Segmentation using a Learned Post-Processing Model. 2024. hal-04542858

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# Restoring Connectivity in Vascular Segmentation using a Learned Post-Processing Model

Sophie Carneiro-Esteves, Antoine Vacavant, Odyssee Merveille

► To cite this version:

Sophie Carneiro-Esteves, Antoine Vacavant, Odyssee Merveille. Restoring Connectivity in Vascular Segmentation using a Learned Post-Processing Model. Topology- and Graph-Informed Imaging Informatics 2024, MICCAI 2024 Workshop, Oct 2024, Marrakech, Morocco. hal-04542858v2

HAL Id: hal-04542858

https://hal.science/hal-04542858v2

Submitted on 22 Nov 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Restoring Connectivity in Vascular Segmentation using a
Learned Post-Processing Model

Sophie Carneiro-Esteves¹,², Antoine Vacavant¹,², and Odyssee Merveille²

¹ Université Clermont Auvergne, CNRS, SIGMA Clermont, Institut Pascal, F-63000

² Univ Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220, U1294,F-69100

**Abstract.** Accurate segmentation of vascular networks is essential for computer-aided tools designed to address cardiovascular diseases. Despite more than thirty years of research, it remains a challenge to obtain vascular segmentation results that preserve the connectivity of the underlying vascular network. Yet connectivity is one of the key feature of these tools. In this work, we propose a post-processing algorithm aiming to reconnect vascular structures that have been disconnected by a segmentation algorithm. Connectivity being a complex property to model explicitly, we propose to learn this geometric feature either through synthetic data or annotations of the application of interest. The resulting post-processing model can be used on the output of any supervised or unsupervised vascular segmentation algorithm. We show that this post-processing effectively restores the connectivity of vascular networks both in 2D and 3D images, leading to improved overall segmentation results.

**Keywords:** Blood vessels, segmentation, connectivity, deep learning, post-processing.

## 1 Introduction

Blood vessel segmentation is a crucial step for various tasks such as blood flow simulation or 3D modeling, and enhancing our understanding of vascular networks physiology, pathologies. However, segmenting blood vessels is challenging due to their thin and tortuous nature, making them easily altered by noise and artefacts. This often results in fragmented blood vessel segmentations which is a major problem for most downstream tasks.

For over thirty years, methods have been introduced to enhance both the quality and the connec-
tivity of blood vessels segmentation. Several unsupervised filtering approaches were first proposed.
Vesselness filters [16] aim at enhancing the signal from blood vessels and decrease the one from
other non-tubular structures. These filters are designed to detect blood vessels at different scales,
employing either a Gaussian-scale paradigm and Hessian-based features extraction [8, 30], or a
mathematical morphology approach using paths as structuring elements [21]. These filters are
usually the first step of more complex segmentation pipelines [5, 20, 22].

However, determining hyperparameters for these filters can be challenging and there is no guar-
antee on the connectivity of the vascular tree. Alternative unsupervised methods, such as tracking [3]
or minimal path methods [18] can ensure the structure connectivity. Nevertheless, these methods
require a time-consuming user interaction to define seed points. All these approaches are further
limited by having to explicitly model blood vessels.

Supervised methods, and in particular deep learning-based ones, offer the power to represent
complex phenomena by learning implicit functions, provided there are sufficient annotations on the
# Restoring Connectivity in Vascular Segmentations using a Learned Post-Processing Model

Sophie Carneiro-Esteves¹,²[0009-0000-1098-8030], Antoine Vacavant¹[0000-0001-9616-3282], and Odyssee Merveille²[0000-0002-9918-3761]

¹ Université Clermont Auvergne, CNRS, SIGMA Clermont, Institut Pascal, F-63000 antoine.vacavant@uca.fr

² Univ Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220, U1294,F-69100 {sophie.carneiro,odyssee.merveille}@creatis.insa-lyon.fr

**Abstract.** Accurate segmentation of vascular networks is essential for computer-aided tools designed to address cardiovascular diseases. Despite more than thirty years of research, it remains a challenge to obtain vascular segmentation results that preserve the connectivity of the underlying vascular network. Yet connectivity is one of the key features of these tools. In this work, we propose a post-processing algorithm aiming to reconnect vascular structures that have been disconnected by a segmentation algorithm. Connectivity being a complex property to model explicitly, we propose to learn this geometric feature either through synthetic data or annotations of the application of interest. The resulting post-processing model can be used on the output of any supervised or unsupervised vascular segmentation algorithm. We show that this post-processing effectively restores the connectivity of vascular networks both in 2D and 3D images, leading to improved overall segmentation results.

**Keywords:** Blood vessels, segmentation, connectivity, deep learning, post-processing.

## 1 Introduction

Blood vessel segmentation is a crucial step for various tasks such as blood flow simulation or 3D modeling, and enhancing our understanding of vascular networks physiology, and pathologies. However, segmenting blood vessels is challenging due to their thin and tortuous nature, making them easily altered by noise and artifacts. This often results in fragmented blood vessel segmentations which is a major problem for most downstream tasks.

For over thirty years, methods have been introduced to enhance both the quality and the connectivity of blood vessel segmentation. Several unsupervised filtering approaches were first proposed. Vesselness filters [16] aim at enhancing the signal from blood vessels and decreasing the one from other non-tubular structures. These filters are designed to detect blood vessels at different scales,
<figure><img src="image_3.png" /><figcaption>Fig. 1: Pipeline of our method. (1) a dataset is generated containing pairs of connected and disconnected vascular structures. (2) this dataset is used to train a model $G_{reco}$ with a residual U-Net architecture. (3) finally, the trained model is iteratively applied on a vascular segmentations with disconnections.</figcaption></figure>

target application. Several approaches dedicated to vascular segmentation were proposed [24,29,34]. More recently, approaches were developed to improve vascular connectivity of the segmentation results. Classic vesselness filters were used to help the network model tubular shapes [25, 31]. Alternative approaches focused on adapting the segmentation architecture to facilitate the learning of a function that preserves connectivity. Attention modules [36] were incorporated into architectures such as U-Net [24], and a topology-aware feature synthesis network was proposed to correct the prediction topology based on the Euler characteristic [17]. Proxy tasks were also introduced to help the model focus on the structure topology such as the centerline extraction or distance-map computation [15]. Many works proposed dedicated loss functions to improve the result connectivity [6, 9, 11, 19, 26, 27, 32]. All these connectivity-preserving strategies assume that a large annotated dataset is available, which is rarely the case in vascular imaging applications.

Another research direction was explored consisting in the design of post-processing techniques dedicated to the reconnection of vascular segmentations results. Various algorithms have been suggested, relying on centerlines [7], graphs [12, 23], and contour completion processes [37]. These approaches are complex to use due to their high dependence on parameter selection, and none of them provide the code necessary to reproduce or compare their results.

In our previous work [2], we proposed a strategy to train the reconnecting model and used it to develop an unsupervised plug-and-play segmentation approach. We recognized the potential for the reconnecting term to be applied more broadly as a post-processing step for any type of vascular segmentation result. In the present article, we thoroughly investigate this idea. In particular, we
employing either a Gaussian-scale paradigm and Hessian-based features extrac-
tion [8,30], or a mathematical morphology approach using paths as structuring
elements [20]. These filters are usually the first step of more complex segmen-
tation pipelines [5, 21, 22].

However, determining hyperparameters for these filters can be challenging and there is no guarantee on the connectivity of the vascular tree. Alternative unsupervised methods, such as tracking [3] or minimal path methods [18] can ensure the structure connectivity. Nevertheless, these methods require a time-consuming user interaction to define seed points. All these approaches are further limited by having to explicitly model blood vessels.

Supervised methods, and in particular deep learning-based ones, offer the power to represent complex phenomena by learning implicit functions, provided there are sufficient annotations on the target application. Several approaches dedicated to vascular segmentation were proposed [24,29,34]. More recently, approaches were developed to improve the vascular connectivity of the segmentation results. Classic vesselness filters were used to help the network model tubular shapes [25,31]. Alternative approaches focused on adapting the segmentation architecture to facilitate the learning of a function that preserves connectivity. Attention modules [36] were incorporated into architectures such as U-Net [24], and a topology-aware feature synthesis network was proposed to correct the prediction topology based on the Euler characteristic [17]. Proxy tasks were also introduced to help the model focus on the structure topology such as the centerline extraction or distance-map computation [15]. Many works proposed dedicated loss functions to improve the result connectivity [6, 9, 11, 19, 26, 27, 32]. All these connectivity-preserving strategies assume that a large annotated dataset is available, which is rarely the case in vascular imaging applications.

Another research direction was explored consisting of the design of post-
processing techniques dedicated to the reconnection of vascular segmentation
results. Various algorithms have been suggested, relying on centerlines [7], graphs
[12, 23], and contour completion processes [37]. These approaches are complex
to use due to their high dependence on parameter selection, and none of them
provide the code necessary to reproduce or compare their results.

In [2], they proposed a strategy to train the reconnecting model and used it
to develop an unsupervised plug-and-play segmentation approach. We recognized
the potential for the reconnecting term to be applied more broadly as a post-
processing step for any type of vascular segmentation result. In the present
article, we thoroughly investigate this idea.

In this article, we conduct a comprehensive investigation of this concept.
We specifically examine the properties of the reconnecting term, focusing on
the impact of the disconnection size parameter and its convergence behavior.
Furthermore, we demonstrate the practicality and versatility of this new post-
processing strategy by applying it to outputs from various segmentation methods
and datasets, and comparing it with a recently developed reconnecting post-
processing approach.
analyse the properties of the reconnecting term, including the influence of the disconnection size parameter, and its convergence. Additionally, we demonstrate the usefulness and versatility of this novel post-processing strategy by applying it to outputs from various segmentation approaches and datasets.

## 2 Proposed method

In our previous work [2], we developed a model, $G_{reco}$, based on a residual U-Net [14], that learns to reconnect disconnected vessel-like structures from a binary segmentation result. This model is trained on pairs of images containing connected and disconnected vessel-like structures (see top of Fig. 1).

We proposed an algorithm capable of generating random and realistic disconnections from any binary vascular structure. This algorithm is described in details in Appendix B. We demonstrated that our reconnecting term can be learned either based on manual annotations from the dataset of interest, or solely from synthetic images. This bypasses the need for an annotated dataset while still yielding satisfying results. To control the reconnection power of our model, we generated disconnections with sizes drawn from a Gaussian distribution with mean $s$ and standard deviation $\sigma$. In this work, we propose to use this reconnecting model as a post-processing step by applying it iteratively until most disconnections have been filed (see bottom of Fig. 1). The code of our approach is available at https://github.com/creatis-myriad/plug-and-play-reco-regularization.

## 3 Experiments

In this section, we first present the experimental set up used in the following experiments. Next, we analyse the impact of various key elements of the method through an ablation study. Finally, we demonstrate its pertinence on segmentations resulting from different methods.

### 3.1 Experimental set up

To analyse and demonstrate the effectiveness of our method, we tested our framework in both 2D and 3D and used synthetic and real datasets. In 2D, we used the DRIVE [33] dataset composed of 40 retinophotographies and their manual vascular annotations, and the STARE [10] dataset composed of 20 manual annotations of retinophotographies. STARE was used to train our reconnecting model and DRIVE was used as a test dataset to apply $G_{reco}$. We also generated a synthetic dataset composed of 20 synthetic vascular trees with OpenCCO [13] that was also used as a training dataset for $G_{reco}$. In 3D, we used the Bullitt [1] and IXI<fnref n="3" /> [35] datasets composed of 33 and 22 brain magnetic resonance angiography (MRA) respectively and their manual vascular annotations. IXI was used to train our reconnecting model while Bullitt was used as a test dataset to apply $G_{reco}$.

Our disconnection algorithm was used on the DRIVE, STARE and IXI to generate disconnected vascular trees with several mean disconnection size of $s$ ($s \in \{6, 8, 10, 12\}$). We experimentally set the standard deviation $\sigma = 4$ for DRIVE and STARE and to $\sigma = 2$ for IXI.

The backbone architecture for $G_{reco}$ is a residual U-Net model [14] trained for 1000 epochs in 2D and 3000 epochs in 3D. We used an Adam optimizer with a learning rate of $10^{-3}$. We employed a

<footnote n="3">³ https://brain-development.org/ixi-dataset/</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1: Pipeline of our method. (1) a dataset is generated containing pairs of connected and disconnected vascular structures. (2) this dataset is used to train a model $G_{reco}$ with a residual U-Net architecture. (3) finally, the trained model is iteratively applied on vascular segmentations with disconnections.</figcaption></figure>

## 2 Proposed method

In [2], a model, $G_{reco}$, based on a residual U-Net [14], is learned to reconnect disconnected vessel-like structures from a binary segmentation result. This model is trained on pairs of images containing connected and disconnected vessel-like structures (see top of Fig. 1).

The reconnecting term can be learned either based on manual annotations from the dataset of interest, or solely from synthetic images. This approach eliminates the need for an annotated dataset while still producing satisfactory results. When using synthetic images for training, an algorithm capable of generating random yet realistic disconnections from any binary vascular structure was proposed. To control the reconnection power of the model, disconnections are generated with various sizes drawn from a Gaussian distribution with mean $s$ and standard deviation $\sigma$.

In this work, we propose to use this reconnecting model as a standalone post-processing step applicable to any binary segmentation result.

Intuitively, the size of the disconnections $s$ of the training dataset should be adjusted to match the disconnection sizes in the segmentation. A small value may be insufficient to reconnect a vessel with a large gap, whereas a large value increases the risk of false reconnections. To address both small and large disconnections while minimizing false reconnections, we propose applying $G_{reco}$ iteratively with a small value of $s$. This approach reconnects vessels gradually rather than attempting long-range reconnections in a single step (see bottom
weighted Dice loss function as presented in [2] and a batch size of 4. In 2D, the models were trained with an 80% split for training and 20% for validation, while in 3D, the split was 90% for training and 10% for validation. The final model was selected based on the best validation loss achieved during the training.

There is no single metric that can reflect the quality of vascular segmentations. Depending on the specifics of the clinical questions, different properties of the vascular segmentation may be prioritized. Because we focus on connectivity we chose the three following differents metrics: the classic Dice coefficient (DSC) evaluates the global quality of the segmentation, the Average Symmetric Surface Distance (ASSD) evaluates the segmentation without volumetric bias, and the error ratio of the number of connected components $\epsilon_{\beta_0}$ evaluates the segmentation connectivity. This error ratio is defined as $\epsilon_{\beta_0} = \left| \frac{\beta_0 - \beta_{0gt}}{\beta_{0gt}} \right|$, with $\beta_0$ the number of connected components of the segmentation and $\beta_{0gt}$ the number of connected component of the annotation. The error ratio was preferred over the value of $\beta_0$ as $\beta_0$ is usually larger than 1 in the DRIVE and Bullitt annotations. We finally included the Area Under the Curve (AUC), another metric evaluating the global segmentation quality, for the 2D experiment, as it is a metric traditionally used in DRIVE benchmarks.

## 3.2 Ablation study

**Influence of the size of the disconnections** The training dataset is a key element of our framework as it defines the concept of what should be connected. The main parameter of this training dataset is the mean size of disconnections, denoted as $s$, which has been added to the segmentations. The model $G_{reco}$ is designed to reconnect these disconnections. Intuitively, the size of the disconnections $s$ should be tuned to reflect the size of the disconnections present in the segmentations. A small value may not be sufficient to reconnect a vessel with a large gap, while a large value increases the probability of connecting vessels that should not be connected. However, we aim at designing a post-processing that works on a large range of disconnection sizes and so our approach should not be too sensitive to this value. To assess this sensitivity, we trained four different models, denoted $G_{reco,s=X}$ (with $X \in \{6, 8, 10, 12\}$), on the OpenCCO dataset which have been disconnected with a mean disconnection size $X$. We then tested these 4 models on the annotations of the DRIVE dataset that have also been disconnected with increasing values of $s$ ($s = 6, s = 8, s = 10$ or $s = 12$). Hence each model will be tested on disconnections sizes it has not been trained for. In this experiment, $G_{reco}$ has only been applied once to the segmentation results. The results are presented in Table 1.

Overall, all models perform well and there is no significant drop in metrics when applied to data exhibiting disconnection sizes significantly different from those in the training dataset. As expected, the Dice coefficient do not show a significant improvement after applying our post-processing as the reconnexion fragments only represent a small portion of the overall vessels. However, $\epsilon_{\beta_0}$ and the ASSD significantly decreases, indicating that our post-processing successfully reconnected fragments of vessels. We also observe that the size of the disconnections in the training dataset seems to correlate well with the reconnections that occur. As expected, models trained on larger disconnections tend to perform better on large disconnections (see Appendix A), while still facing challenges in reconnecting smaller disconnections. The model trained with disconnections with a size of $s = 8$, seems a good compromise between efficacy in reconnecting and avoiding excessive false connections.

In this experiment, we chose to evaluate our models after applying them only once, to better understand their behavior. However, our goal is to use $G_{reco}$ iteratively to reconnect disconnections
of Fig. 1). The code of our approach is available at https://github.com/creatis-
myriad/plug-and-play-reco-regularization.

3 Experiments

**3.1 Experimental set up**

To analyse and demonstrate the effectiveness of our method, we tested our frame-
work in both 2D and 3D and used both synthetic and real datasets. In 2D, we
used the DRIVE [33] dataset composed of 40 retinophotographies and their
manual vascular annotations, and the STARE [10] dataset, which includes 20
annotated retinophotographies. The STARE dataset was used to train our re-
connecting model, while the DRIVE dataset served as the test set for applying
G<sub>reco</sub>. Additionnaly, we generated a synthetic dataset comprising 20 synthetic
vascular trees with OpenCCO [13], which was also employed as a training dataset
for G<sub>reco</sub>. In 3D, we used the Bullitt [1] and IXI<sup>3</sup> [35] datasets composed of 33
and 22 brain magnetic resonance angiographies (MRA) respectively and their
manual vascular annotations. IXI served to train our reconnecting model while
Bullitt was used as a test set to apply G<sub>reco</sub>.

The disconnection algorithm was used on the DRIVE, STARE and IXI datasets to generate disconnected vascular trees with several mean disconnection sizes ($s \in \{6, 8, 10, 12\}$). We experimentally set the standard deviation $\sigma = 4$ for DRIVE and STARE and to $\sigma = 2$ for IXI.

The backbone architecture for $G_{\text{reco}}$ is a residual U-Net model [14] trained for 1000 epochs in 2D and 3000 epochs in 3D. We used an Adam optimizer with a learning rate of $10^{-3}$. We employed a weighted Dice loss function as presented in [2] and a batch size of 32 in 2D and 4 in 3D. In 2D, the models were trained with an 80% split for training and 20% for validation, while in 3D, the split was 90% for training and 10% for validation. The final model was selected based on the best validation loss achieved during the training.

We selected three metrics to evaluate the segmentations. The classic Dice coefficient (DSC) assesses the overall quality of the segmentation. The Average Symmetric Surface Distance (ASSD) evaluates the segmentation geometry by measuring the distances between each element of the segmentation contour and the corresponding contour in the annotation. Finally, the error ratio of the number of connected components $\epsilon_{\beta_0}$ evaluates the segmentation connectivity. This error ratio is defined as $\epsilon_{\beta_0} = \left| \frac{\beta_0 - \beta_{0\text{gt}}}{\beta_{0\text{gt}}} \right|$, with $\beta_0$ the number of connected components of the segmentation and $\beta_{0\text{gt}}$ the number of connected component of the annotation. The error ratio was preferred over the value of $\beta_0$ as $\beta_0$ is usually larger than 1 in the DRIVE and Bullitt annotations.

**3.2 Ablation study**

**Influence of the size of the disconnections** The training dataset is a key element of our framework as it defines the concept of reconnection. The main

<footnote n="3">³ https://brain-development.org/ixi-dataset/</footnote>
<table><caption>Table 1: Quantitative results of applying $G_{reco}$ trained on the OpenCCO dataset with several values of s and applied on the Drive dataset with several values of s.</caption><thead><tr><th rowspan="2">Training</th><th colspan="3">s = 6</th><th colspan="3">s = 8</th><th colspan="3">s = 10</th><th colspan="3">s = 12</th></tr><tr><th>DSC ↑</th><th>ASSD ↓</th><th>εβ₀ ↓</th><th>DSC ↑</th><th>ASSD ↓</th><th>εβ₀ ↓</th><th>DSC ↑</th><th>ASSD ↓</th><th>εβ₀ ↓</th><th>DSC ↑</th><th>ASSD ↓</th><th>εβ₀ ↓</th></tr></thead><tbody><tr><td>Before G<sub>reco</sub></td><td>0.979<br>± 0.004</td><td>0.202<br>± 0.06</td><td>96.811<br>± 67.065</td><td>0.974<br>± 0.004</td><td>0.22<br>± 0.071</td><td>107.367<br>± 71.883</td><td>0.97<br>± 0.005</td><td>0.232<br>± 0.057</td><td>122.198<br>± 83.665</td><td>0.963<br>± 0.007</td><td>0.243<br>± 0.054</td><td>132.617<br>± 86.553</td></tr><tr><td>G<sub>reco,s=6</sub></td><td>0.983<br>± 0.003</td><td>0.074<br>± 0.012</td><td><b>11.429</b><br>± 8.783</td><td>0.98<br>± 0.003</td><td>0.085<br>± 0.015</td><td><b>14.095</b><br>± 10.92</td><td>0.978<br>± 0.004</td><td>0.101<br>± 0.019</td><td><b>15.485</b><br>± 10.758</td><td>0.974<br>± 0.004</td><td>0.116<br>± 0.022</td><td><b>17.619</b><br>± 13.8</td></tr><tr><td>G<sub>reco,s=8</sub></td><td><b>0.985</b><br>± 0.002</td><td><b>0.067</b><br>± 0.013</td><td>15.461<br>± 11.256</td><td><b>0.983</b><br>± 0.003</td><td><b>0.077</b><br>± 0.014</td><td>17.301<br>± 12.69</td><td><b>0.981</b><br>± 0.004</td><td><b>0.09</b><br>± 0.019</td><td>19.365<br>± 14.269</td><td><b>0.977</b><br>± 0.004</td><td><b>0.103</b><br>± 0.02</td><td>23.039<br>± 18.927</td></tr><tr><td>G<sub>reco,s=10</sub></td><td>0.984<br>± 0.003</td><td>0.078<br>± 0.015</td><td>14.61<br>± 11.223</td><td>0.981<br>± 0.003</td><td>0.089<br>± 0.016</td><td>16.829<br>± 13.051</td><td>0.979<br>± 0.004</td><td>0.101<br>± 0.023</td><td>18.82<br>± 14.516</td><td>0.975<br>± 0.004</td><td>0.117<br>± 0.017</td><td>22.056<br>± 17.864</td></tr><tr><td>G<sub>reco,s=12</sub></td><td>0.982<br>± 0.003</td><td>0.089<br>± 0.015</td><td>16.232<br>± 12.213</td><td>0.98<br>± 0.003</td><td>0.102<br>± 0.017</td><td>18.561<br>± 13.75</td><td>0.977<br>± 0.004</td><td>0.118<br>± 0.024</td><td>19.776<br>± 14.034</td><td>0.974<br>± 0.004</td><td>0.126<br>± 0.019</td><td>20.612<br>± 16.189</td></tr></tbody></table>

that could not necessarily be addressed in a single iteration. We analysed the interest of this iterative approach and evaluate its convergence in the next experiment.

**Convergence of the proposed approach** We propose to apply $G_{reco}$ iteratively to reconnect vessels gradually instead of attempting to reconnect across a long range all at once. Our intuition suggests that this approach should limit the creation of false reconnections. In this experiment, we aim first at validating this hypothesis. Secondly, considering the limited number of disconnections in an image, we anticipate that applying $G_{reco}$ iteratively will converge to a fixed-point image where all disconnections are filled. In this section we thus explore the experimental convergence of our framework.

We used the same 4 reconnection models $G_{reco,s=X}$ ($X \in \{6, 8, 10, 12\}$) and applied each one on the Drive dataset which have been disconnected with several mean disconnection sizes ($s \in \{6, 8, 10, 12\}$). The results are presented in Figure 2.

We observe that applying $G_{reco}$ iteratively converges as shown by Figure 2(a). Interestingly this convergence occurs even though $G_{reco}$ was applied to images with disconnection sizes different from those it was trained on. This highlights the robustness of our approach.

Figures 2 (b-d) show that the size of the disconnections in the training dataset have an impact on the quality of the reconnections made on the segmentation results. The model trained on the dataset created with the parameter $s = 8$ appeared to be a good compromise between significant reconnections (small $\epsilon_{\beta_0}$) and limiting false reconnections (small ASSD and high DSC values). Moreover applying iteratively $G_{reco}$ on the segmentations improve the connectivity of the vascular network and converges to a fixed-point image, regardless of the trained disconnections sizes. Qualitative results are discussed in Appendix A.

Applying our method on artificially disconnected images let us analyse precisely the behavior of our reconnecting term but this introduced a bias since we used the same algorithm to generate disconnections in both the training and test datasets. The next section will explore the behavior of our reconnecting approach in a more realistic scenario.

## 3.3 Applications

In this experiment, we applied our post-processing to real 2D and 3D segmentation results obtained from both an unsupervised variational approach and a supervised deep learning approach.
<figure><img src="image_3.png" /><figcaption>Fig. 2: Qualitative results of the influence of the disconnection size experiment. $G_{reco}$ was trained with a disconnection size $X$ (with $X \in 6, 8, 10, 12$), denoted $G_{reco,s=X}$, on the OpenCCO dataset. A drive dataset annotation (a) was disconnected with $s = 12$ to obtain (b), followed by a single application of $G_{reco,s=X}$ on (b), resulting in (c-f). Blue arrows indicate successful reconnections, pink arrows highlight incorrect reconnections, and green arrows denote differences between the results.</figcaption></figure>

parameter of this training dataset is the mean size of a disconnection, denoted as $s$. We trained four different models, denoted $G_{reco,s=X}$ (with $X \in 6, 8, 10, 12$), on the OpenCCO dataset which has been disconnected with a mean disconnection size $X$. We then tested these 4 models on the annotations of the DRIVE dataset that have also been disconnected with increasing values of $s$ ($s = 6, s = 8, s = 10$ or $s = 12$). Hence each model will be tested on disconnection sizes it has not been trained for. In this experiment, $G_{reco}$ has only been applied once to the segmentation results. The results are presented in Table 1.

As expected, the Dice coefficient of all models does not show a significant improvement after applying our post-processing, as the reconnection fragments represent only a small portion of the overall vessels. However both $\epsilon_{\beta_0}$ and ASSD significantly decrease compared to the initial segmentation (Before $G_{reco}$), indicating that our post-processing successfully reconnected fragments of vessels.
<figure><img src="image_4.png" /><figcaption>Fig. 2: Results of applying $G_{\text{reco}}$, trained on OpenCCO, on the DRIVE annotations artificially disconnected with sizes $s \in \{6, 8, 10, 12\}$. (a) convergence curves displaying the $l_2$ norm of the difference of the last two consecutive results. (b-d) quantitative results of our models applied with an increasing number of iterations.</figcaption></figure>

Specifically, we used the variational segmentation method proposed by Chan *et al.* [4] with a total variation [28] regularization. A state-of-the-art U-Net architecture was used as the supervised approach as detailed in [2]. The variational segmentation yields disconnected and noisy results whereas the supervised one produces more complete and connected segmentations. We chose to evaluate our framework on these two different types of results to highlight the versatility of our post-processing method. We conducted experiments both on 2D and 3D images. In 2D, we trained our reconnecting model $G_{\text{reco}}$ either on the synthetic OpenCCO dataset, denoted as $G_{\text{reco,CCO}}$, or the STARE dataset, denoted as $G_{\text{reco,STARE}}$, that have both been disconnected with a mean disconnection size set to $s = 8$. We ran both segmentation strategies (variational and deep-learning) on the DRIVE dataset and applied our post-processing.

In 3D, we trained our reconnecting model on the IXI dataset, denoted as $G_{\text{reco,IXI}}$, that have been disconnected with a mean disconnection size set to $s = 8$. We ran both segmentation strategies on the Bullitt dataset and applied our post-processing. Results are summarized in Table 2, Figure 3 and in Appendix A.

We observe that, in general, our post-processing either slightly increases the DSC and ASSD values of the segmentations or does not significantly change them (p-value < 0.05). The slight decrease in the ASSD of the 3D variational approach and the DSC of the 3D deep learning approach can be attributed to some false reconnection of aligned artifacts. The overall stability of the DSC and ASSD metrics is expected, because our post-processing primarily involves adding a few pixels to reconnect vessels. These additional pixels represent only a small fraction of the total number of true positive pixels in the image, thus minimally impacting the DSC or ASSD. By contrast, we anticipate a drastic decrease in the connectivity metric $\epsilon_{\beta_0}$, since our post-processing affects the connectivity of the segmentations. This is indeed observed. Specifically, we note a decrease of > 90%
<table><caption>Table 1: Results of applying $G_{reco}$ trained on the OpenCCO dataset with several values of $s$, and applied on the Drive dataset with several values of $s$.</caption><thead><tr><th rowspan="2">Training</th><th colspan="2">s = 6</th><th colspan="2">s = 8</th><th colspan="2">s = 10</th><th colspan="2">s = 12</th></tr><tr><th>DSC ↑</th><th>ASSD ↓</th><th>εβ₀ ↓</th><th>DSC ↑</th><th>ASSD ↓</th><th>εβ₀ ↓</th><th>DSC ↑</th><th>ASSD ↓</th><th>εβ₀ ↓</th></tr></thead><tbody><tr><td>Before G<sub>reco</sub></td><td>0.979<br>± 0.004</td><td>0.202<br>± 0.06</td><td>96.811<br>± 67.065</td><td>0.974<br>± 0.004</td><td>0.22<br>± 0.071</td><td>107.367<br>± 71.883</td><td>0.97<br>± 0.005</td><td>0.232<br>± 0.057</td><td>122.198<br>± 83.665</td><td>0.963<br>± 0.007</td><td>0.243<br>± 0.054</td><td>132.617<br>± 86.553</td></tr><tr><td>G<sub>reco,s=6</sub></td><td>0.983<br>± 0.003</td><td>0.074<br>± 0.012</td><td><b>11.429</b><br>± 8.783</td><td>0.98<br>± 0.003</td><td>0.085<br>± 0.015</td><td><b>14.095</b><br>± 10.92</td><td>0.978<br>± 0.004</td><td>0.101<br>± 0.019</td><td><b>15.485</b><br>± 10.758</td><td>0.974<br>± 0.004</td><td>0.116<br>± 0.022</td><td><b>17.619</b><br>± 13.8</td></tr><tr><td>G<sub>reco,s=8</sub></td><td><b>0.985</b><br>± 0.002</td><td><b>0.067</b><br>± 0.013</td><td>15.461<br>± 11.256</td><td><b>0.983</b><br>± 0.003</td><td><b>0.077</b><br>± 0.014</td><td>17.301<br>± 12.69</td><td><b>0.981</b><br>± 0.004</td><td><b>0.09</b><br>± 0.019</td><td>19.365<br>± 14.269</td><td><b>0.977</b><br>± 0.004</td><td><b>0.103</b><br>± 0.02</td><td>23.039<br>± 18.927</td></tr><tr><td>G<sub>reco,s=10</sub></td><td>0.984<br>± 0.003</td><td>0.078<br>± 0.015</td><td>14.61<br>± 11.223</td><td>0.981<br>± 0.003</td><td>0.089<br>± 0.016</td><td>16.829<br>± 13.051</td><td>0.979<br>± 0.004</td><td>0.101<br>± 0.023</td><td>18.82<br>± 14.516</td><td>0.975<br>± 0.004</td><td>0.117<br>± 0.017</td><td>22.056<br>± 17.864</td></tr><tr><td>G<sub>reco,s=12</sub></td><td>0.982<br>± 0.003</td><td>0.089<br>± 0.015</td><td>16.232<br>± 12.213</td><td>0.98<br>± 0.003</td><td>0.102<br>± 0.017</td><td>18.561<br>± 13.75</td><td>0.977<br>± 0.004</td><td>0.118<br>± 0.024</td><td>19.776<br>± 14.034</td><td>0.974<br>± 0.004</td><td>0.126<br>± 0.019</td><td>20.612<br>± 16.189</td></tr></tbody></table>

We also observe a correlation between the size of the disconnections in the training dataset and the reconnections that occur. Models trained on larger disconnections tend to perform better on large disconnections (see Figure. 2), while still facing challenges in reconnecting smaller disconnections. The model trained with disconnections with a size of $s = 8$, seems a good compromise, effectively reconnecting vessels while minimizing excessive false connections.

**Convergence of the proposed approach** In the previous experiment, we evaluated our models after a single application to better understand their behavior. However, our goal is to use $G_{reco}$ iteratively to address disconnections that can not be fully reconnected in a single iteration. In this experiment, we analyze the interest of this iterative approach. Additionally, given the limited number of disconnections in an image, we anticipate that applying $G_{reco}$ iteratively will converge to a fixed-point image where all disconnections are filled. Thus, we also explore the experimental convergence of our framework in this experiment.

We used the same 4 reconnecting models $G_{reco,s=X}$ ($X \in 6, 8, 10, 12$) and applied each one on the Drive dataset which has been disconnected with several mean disconnection sizes ($s \in 6, 8, 10, 12$). The results are presented in Figure 3 and in Figure 4.

We observe that the iterative application of $G_{reco}$ converges as shown in Figure 3 (a). Interestingly this convergence occurs even though $G_{reco}$ was applied to images with disconnection sizes different from those used in the training set. This highlights the robustness of our approach. In the blue boxes of Figure 4, we can observe that some disconnections, not successfully reconnected with a single iteration, are gradually reconnected with additional iterations. However, it is worth noting that some false reconnections may still occur, as shown by the green boxes. Interestingly, these false reconnections often occur when noise fragments are present near a vessel, and $G_{reco}$ uses them to create realistic vessels.

Applying our method on artificially disconnected images let us analyze precisely the behavior of our reconnecting term but this introduced a bias since we used the same algorithm to generate disconnections in both the training and
<table><caption>Table 2: Quantitative results obtained with our 2D and 3D reconnecting models on variational and deep learning segmentations. The p-values (from the t-test for normal distributions, or Wilcoxon test otherwise) are shown between the segmentation and G<sub>recoSTARE</sub> in 2D and the segmentation and G<sub>recoIXI</sub> in 3D.</caption><thead><tr><th rowspan="2">Training</th><th rowspan="2"></th><th colspan="4">Variational approach</th><th colspan="4">Deep Learning</th></tr><tr><th>DSC</th><th>ASSD</th><th>ε<sub>β0</sub></th><th>AUC</th><th>DSC</th><th>ASSD</th><th>ε<sub>β0</sub></th><th>AUC</th></tr></thead><tbody><tr><td rowspan="4">2D</td><td>Segmentation</td><td>0.758</td><td><b>2.017</b></td><td>98.54</td><td>0.838</td><td><b>0.811</b></td><td><b>1.155</b></td><td>34.04</td><td>0.901</td></tr><tr><td>G<sub>recoCCO</sub></td><td>± 0.025 ± 0.452 ± 91.88 ± 0.025</td><td>0.767</td><td>2.423</td><td>9.003</td><td>0.850</td><td>0.809</td><td>1.192</td><td><b>9.111</b> <b>0.903</b></td></tr><tr><td>G<sub>recoSTARE</sub></td><td>± 0.023 ± 0.582 ± 11.39 ± 0.025</td><td><b>0.768</b></td><td>2.332</td><td><b>6.609</b></td><td><b>0.851</b></td><td>± 0.016 ± 0.19 ± 7.606 ± 0.022</td><td>0.810</td><td>1.198</td><td>11.37 <b>0.903</b></td></tr><tr><td>p-values</td><td>~ 10<sup>-6</sup></td><td>0.057</td><td>~ 10<sup>-6</sup></td><td>~ 10<sup>-6</sup></td><td>0.832</td><td>0.466</td><td>~ 10<sup>-6</sup></td><td>0.811</td></tr><tr><td rowspan="4">3D</td><td>Segmentation</td><td>0.476</td><td><b>3.587</b></td><td>26.56</td><td>-</td><td><b>0.756</b></td><td><b>1.488</b></td><td>3.203</td><td>-</td></tr><tr><td>G<sub>recoIXI</sub></td><td>± 0.02 ± 0.42 ± 10.26</td><td><b>0.495</b></td><td>4.154</td><td><b>5.618</b></td><td>-</td><td>± 0.015 ± 0.211 ± 1.689</td><td>-</td><td>-</td></tr><tr><td>p-values</td><td>~ 10<sup>-4</sup></td><td>~ 10<sup>-6</sup></td><td>~ 10<sup>-17</sup></td><td>-</td><td>~ 10<sup>-5</sup></td><td>0.226</td><td>~ 10<sup>-10</sup></td><td>-</td></tr></tbody></table>

in 2D, and > 80% in 3D for the variational approach, and > 67% in 2D, and > 47% in 3D for the deep learning approach.

It is interesting to note that the model trained on the synthetic OpenCCO dataset yields slightly less inferior results, often due to the creation of false connections. Our reconnecting terms learn to reconnect only based on geometric features. Therefore, the closer the geometry of the vessels in the training dataset to that of the test dataset, the better the performance tends to be. Nonetheless, the drop of performance is quite small which makes our term very useful in a purely unsupervised context in 2D when no vascular annotation is available. In 3D, to the best of our knowledge, there is no synthetic vascular network generation software that yields vascular trees geometrically close enough to a real brain vascular network. In particular, the tortuosities of brain vascular network is much higher than what is possible to generate with softwares like VascuSynth<fnref n="4" /> or OpenCCO.

## 4 Conclusion

In this article, we introduced a novel vascular segmentation post-processing to favor vascular network connectivity. This post-processing can be used in an unsupervised or supervised context depending on the availability of vascular annotations on the dataset of interest. We conducted an extensive validation of our approach both in 2D and 3D and showed that our post-processing is robust to the size of disconnections, converges to a reconnected result when used iteratively and significantly improve the connectivity of segmentation results. This approach is purely based on geometric properties of the vessels in binary segmentations and thus some false reconnections may appear. Future work include taking into account the intensities of the underlying image to avoid these false reconnections.

This work was supported by Agence Nationale de la Recherche (ANR-22-CE45-0018, ANR-18-CE45-0018), LABEX PRIMES (ANR-11-LABX-0063). This work was granted access to the

<footnote n="4">⁴ https://vascusynth.cs.sfu.ca/Welcome.html</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 3: Results of applying $G_{reco}$, trained on OpenCCO, on the DRIVE annotations artificially disconnected with sizes $s \in \{6, 8, 10, 12\}$. (a) convergence curves displaying the $l_2$ norm of the difference of the last two consecutive results. (b-d) quantitative results of our models applied with an increasing number of iterations.</figcaption></figure>

test datasets. The next section will explore the behavior of our reconnecting approach in a more realistic scenario.

## 3.3 Applications

In this experiment, we applied our post-processing to real 2D and 3D segmentation results obtained from both an unsupervised variational approach and a supervised deep learning approach. Specifically, we used the variational segmentation method proposed by Chan *et al.* [4] with a total variation [28] regularization. A gold standard U-Net architecture was used as the supervised approach as detailed in [2]. The variational segmentation yields disconnected and noisy

<figure><img src="image_5.png" /><figcaption>Fig. 4: Evolution of a DRIVE disconnected manual annotation (with $s = 12$) through successive application of $G_{reco,s=8}$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3: Results of our post-processing applied to a DRIVE segmentation result from the variational (top row) and deep learning approach (bottom row). (b) and (f) depict segmentation results from variational and deep learning approaches respectively, before post-processing with $G_{\text{reco}}$. $G_{\text{reco}}$ is trained on either the synthetic OpenCCO dataset (c) and (g) or the real STARE dataset (d) and (h).</figcaption></figure>

HPC resources of IDRIS under the allocations 2022-AD011013887 and 2023-AD011014452 made by GENCI.

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<table><caption>Table 2: Quantitative results obtained with our 2D and 3D reconnecting models on variational and deep learning segmentations. The p-values (from the t-test for normal distributions, or Wilcoxon test otherwise) are shown between the segmentation and $G_{recoSTARE}$ in 2D and the segmentation and $G_{recoIXI}$ in 3D.</caption><thead><tr><th rowspan="2">training</th><th rowspan="2"></th><th colspan="3">Variational approach</th><th colspan="3">Deep Learning</th></tr><tr><th>DSC</th><th>ASSD</th><th>ε<sub>β0</sub></th><th>DSC</th><th>ASSD</th><th>ε<sub>β0</sub></th></tr></thead><tbody><tr><td rowspan="8">2D</td><td>Segmentation</td><td>0.758 ± 0.025</td><td>2.017 ± 0.452</td><td>98.543 ± 91.876</td><td><b>0.811</b> ± 0.015</td><td><b>1.155</b> ± 0.181</td><td>34.037 ± 26.858</td></tr><tr><td>PRW [23]</td><td>0.758 ± 0.025</td><td><b>2.012</b> ± 0.448</td><td>88.909 ± 85.873</td><td>0.808 ± 0.015</td><td>1.171 ± 0.177</td><td>22.017 ± 18.588</td></tr><tr><td>G<sub>recoCCO</sub></td><td>0.767 ± 0.023</td><td>2.423 ± 0.582</td><td>9.003 ± 11.388</td><td>0.809 ± 0.016</td><td>1.192 ± 0.19</td><td><b>9.111</b> ± 7.606</td></tr><tr><td>G<sub>recoSTARE</sub></td><td><b>0.768</b> ± 0.023</td><td>2.332 ± 0.533</td><td>6.609 ± 5.817</td><td>0.810 ± 0.015</td><td>1.198 ± 0.183</td><td>11.374 ± 9.095</td></tr><tr><td>p-values</td><td>~ 10<sup>-6</sup></td><td>0.057</td><td>~ 10<sup>-6</sup></td><td>0.832</td><td>0.466</td><td>~ 10<sup>-6</sup></td></tr><tr><td rowspan="4">3D</td><td>Segmentation</td><td>0.476 ± 0.02</td><td><b>3.587</b> ± 0.42</td><td>26.562 ± 10.264</td><td><b>0.756</b> ± 0.015</td><td><b>1.488</b> ± 0.211</td><td>3.203 ± 1.689</td></tr><tr><td>G<sub>recoIXI</sub></td><td><b>0.495</b> ± 0.019</td><td>4.154 ± 0.427</td><td><b>5.618</b> ± 2.411</td><td>0.74 ± 0.014</td><td>1.552 ± 0.21</td><td><b>1.697</b> ± 1.029</td></tr><tr><td>p-values</td><td>~ 10<sup>-4</sup> ~ 10<sup>-6</sup> ~ 10<sup>-17</sup> ~ 10<sup>-5</sup></td><td>~ 10<sup>-6</sup></td><td>~ 10<sup>-17</sup></td><td>0.226</td><td>~ 10<sup>-10</sup></td><td></td></tr></tbody></table>

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In 2D, we trained our reconnecting model $G_{reco}$ either on the synthetic OpenCCO dataset, denoted as $G_{reco,CCO}$, or the STARE dataset, denoted as $G_{reco,STARE}$, that have both been disconnected with a mean disconnection size set to $s = 8$. We ran both segmentation strategies (variational and deep-learning) on the DRIVE dataset and applied our post-processing.

In 3D, we trained our reconnecting model on the IXI dataset, denoted as $G_{reco,IXI}$, that have been disconnected with a mean disconnection size set to $s = 8$. We ran both segmentation strategies on the Bullitt dataset and applied our post-processing.

As previously mentioned, to the best of our knowledge, no other reconnecting post-processing approach provides open-access code. Despite this, for comparative purposes, we implemented the probability regularized walk (PRW) algorithm, as proposed in [23], and conducted a comparison of our results with it. This method integrates the probability output of a neural network and the directions of broken vessel segments into a regularized walk algorithm to reconnect them to the main component. The hyperparameters of this method have been optimized for each image using a grid search. Due to incomplete details in the original article [23], we made certain choices that might differ from those of the authors. These decisions were made to ensure a fair comparison. Our implementation can be accessed at https://github.com/creatis-myriad/plug-and-play-reco-regularization. Results are summarized in Table 2, Figure 5.

We observe that, in general, our post-processing either slightly increases the DSC and ASSD values of the segmentations or does not significantly change them (p-value < 0.05). The slight decrease in the ASSD of the 3D variational
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<figure><img src="image_6.png" /><figcaption>Fig. 5: Results of our post-processing applied to a DRIVE segmentation result from the variational (top row) and deep learning approach (bottom row). (b) and (g) depict segmentation results from variational and deep learning approaches respectively, before post-processing with $G_{\text{reco}}$. $G_{\text{reco}}$ is trained on either the synthetic OpenCCO dataset (d) and (i) or the real STARE dataset (e) and (j). (c) and (h) represent the post-processing proposed in [23].</figcaption></figure>

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higher than what is possible to generate with softwares like VascuSynth<fnref n="4" /> or
OpenCCO.

The PRW algorithm reconnects some vessels, as evidenced by the small de-
crease in ϵβ₀, but it is significantly less effective than our approach. PRW re-
lies on probability maps to determine the direction of reconnection. However,
we observed that the probability maps generated by both the variational and
deep learning approaches exhibit a low dynamic range and are nearly binary.
Consequently, the reconnection capability of PRW is limited on these types of
segmentation results. In contrast, our approach directly relies on binary images
to predict the reconnecting direction.

Additionally, qualitatively, PRW tends to produce more unrealistic reconnec-
tions, such as vessel enlargements, compared to those generated by our method.

4 Conclusion

In this article, we introduced a novel vascular segmentation post-processing to
favor vascular network connectivity. This post-processing can be used in an un-
supervised or supervised context depending on the availability of vascular anno-
tations on the dataset of interest. We conducted an extensive validation of our
approach both in 2D and 3D and showed that our post-processing is robust to
the size of disconnections, converges to a reconnected result when used itera-
tively, and significantly improves the connectivity of segmentation results. We
also compared our approach to a recent reconnecting post-processing algorithm,
demonstrating that our method reconnects significantly more vessels and does
so in a more realistic manner. However, our approach is purely based on the
vessel's geometric properties in binary segmentations and thus some false recon-
nections may appear. Future works include taking into account the intensities
of the underlying image to avoid these false reconnections.

5 Acknowledgments

This work was supported by Agence Nationale de la Recherche (ANR-22-CE45-
0018, ANR-18-CE45-0018), LABEX PRIMES (ANR-11-LABX-0063). This work
was granted access to the HPC resources of IDRIS under the allocations 2022-
AD011013887 and 2023-AD011014452 made by GENCI.

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<footnote n="4">⁴ https://vascusynth.cs.sfu.ca/Welcome.html</footnote>
A Qualitative results

Figure A.1 compares the results of applying our post-processing model trained with several values of s, on a Drive image that has been disconnected with a specific disconnection size.

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<figure><img src="image_6.png" /><figcaption>Fig. A.1: Qualitative results of the influence of the disconnection size experiment. G_reco was trained with a disconnection size X (with X ∈ 6, 8, 10, 12), denoted G_reco,s=X, on the OpenCCO dataset. A drive dataset annotation (a) was disconnected with s = 12 to obtain (b), followed by a single application of G_reco,s=X on (b), resulting in (c-f). Blue arrows indicate successful reconnections, pink arrows highlight incorrect reconnections, and green arrows denote differences between the results.</figcaption></figure>
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Figure A.3 shows the result of our post-processing applied on an image of the Bullitt dataset. The blue arrows point to correct reconnections while the green arrow show a false reconnexion due to noise.

<figure><img src="image_7.png" /><figcaption>Fig. A.2: Evolution of a DRIVE disconnected manual annotation (with $s = 12$) through successive application of $G_{reco,s=8}$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. A.3: Results of our post-processing in 3D. (a) the annotation of a Bullitt dataset image, (b) a segmentation result from a 3D UNet (c) the post-processing result with our reconnecting term trained on the IXI dataset with $s = 8$.</figcaption></figure>
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## B Disconnection generation algorithm

In this section, we provide a more detailed description of the algorithm we have proposed for generating realistic disconnections in vascular segmentations. The code for this algorithm is available at.

- We compute the segmentation centerline and distance map. The distance map is defined such that each pixel of the segmentation has a value representing the minimum distance to the background. The value of each centerline pixel corresponds to the local radius of the vessel.

- We compute the segmentation centerline and distance map. The distance map is defined such that each pixel of the segmentation has the value of the minimum distance to the background. The value of each centerline pixel correspond to the local radius of the vessel.

- For each disconnection

* We draw a value $i \in [1, p]$ from the distribution defined by the probability $P(i) = \frac{2^{p-i}}{2^p-1}$, where $p$ is the maximum radius value in the image. Here $i$ corresponds to the radius of a vessel in which the disconnection will occur.

* Randomly select a centerline pixel $x$ with a radius of $i$.

* The disconnection size $d$ is drawn in a Normal distribution $N(\frac{s}{i+1}, 0)$, where $s$ is the mean disconnection size parameter of our post-processing. The thinner the vessel the longer the disconnection will be.

* We generate the disconnection by randomly removing $n$ pixels from the segmentation within a disk of radius $d$ centered on $x$. Here $n$ is drawn from a Normal distribution $N(\frac{N}{2}, \frac{N}{4})$ with $N$ the number of pixels in the disk.

- For each artefact

* The center $c$ of the artefact is randomly selected outside of the segmentation mask.

* A disk of radius $r$ (where $r$ is drawn from $N(3, 1)$) is centered on $c$.

* $n'$ pixels are randomly selected inside the disk and added $(n' \leftarrow N(\frac{N}{2}, \frac{N}{4}))$ to the segmentation.