11institutetext:  Université Clermont Auvergne, CNRS, SIGMA Clermont, Institut
Pascal, F-63000 22institutetext: Univ Lyon, INSA‐Lyon, Université Claude
Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220,
U1294,F‐69100

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

Sophie Carneiro-Esteves 1122 Antoine Vacavant 1122 Odyssée Merveille 22

###### 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 explicity, 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:

B

lood 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 connectivity 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 [22, 20, 5].

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 [34, 24, 29]. 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 [31, 25]. 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 [32, 27, 19, 11, 6, 9, 26]. 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 [23, 12], 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.

Figure 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 $\text{G}_{\text{reco}}$ with a residual U-Net architecture. (3)
finally, the trained model is iteratively applied on a vascular segmentations
with disconnections.

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 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, $G_{\text{reco}}$, 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 0.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 $G_{\text{reco}}$. We also
generated a synthetic dataset composed of 20 synthetic vascular trees with
OpenCCO [13] that was also used as a training dataset for $G_{\text{reco}}$.
In 3D, we used the Bullitt[1] and IXI111https://brain-development.org/ixi-
dataset/ [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 $G_{\text{reco}}$.

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 $G_{\text{reco}}$ 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 $4$. 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
$\epsilon_{\beta_{0}}=\left|\frac{{\beta_{0}-{\beta_{0}}_{\text{gt}}}}{{{\beta_{0}}_{\text{gt}}}}\right|$,
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. 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

####  3.2.1 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 $G_{\text{reco}}$ 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 $G_{\text{reco},s=X}$
(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, $G_{\text{reco}}$ 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 0.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
$G_{\text{reco}}$ iteratively to reconnect disconnections 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.

| $s=6$ |  | $s=8$ |  | $s=10$ |  | $s=12$
---|---|---|---|---|---|---|---
Training Test | DSC $\uparrow$ | ASSD $\downarrow$ |  $\epsilon_{\beta_{0}}$ $\downarrow$ |  | DSC $\uparrow$ | ASSD $\downarrow$ |  $\epsilon_{\beta_{0}}$ $\downarrow$ |  | DSC $\uparrow$ | ASSD $\downarrow$ |  $\epsilon_{\beta_{0}}$ $\downarrow$ |  | DSC $\uparrow$ | ASSD $\downarrow$ | $\epsilon_{\beta_{0}}\downarrow$
| 0.979 | 0.202 | 96.811 |  | 0.974 | 0.22 | 107.367 |  | 0.97 | 0.232 | 122.198 |  | 0.963 | 0.243 | 132.617
Before $\text{G}_{\text{reco}}$ |  $\pm$ 0.004 |  $\pm$ 0.06 |  $\pm$ 67.065 |  |  $\pm$ 0.004 |  $\pm$ 0.071 |  $\pm$ 71.883 |  |  $\pm$ 0.005 |  $\pm$ 0.057 |  $\pm$ 83.665 |  |  $\pm$ 0.007 |  $\pm$ 0.054 |  $\pm$ 86.553
| 0.983 | 0.074 | 11.429 |  | 0.98 | 0.085 | 14.095 |  | 0.978 | 0.101 | 15.485 |  | 0.974 | 0.116 | 17.619
$G_{\text{reco},s=6}$ |  $\pm$ 0.003 |  $\pm$ 0.012 |  $\pm$ 8.783 |  |  $\pm$ 0.003 |  $\pm$ 0.015 |  $\pm$ 10.92 |  |  $\pm$ 0.004 |  $\pm$ 0.019 |  $\pm$ 10.758 |  |  $\pm$ 0.004 |  $\pm$ 0.022 |  $\pm$ 13.8
| 0.985 | 0.067 | 15.461 |  | 0.983 | 0.077 | 17.301 |  | 0.981 | 0.09 | 19.365 |  | 0.977 | 0.103 | 23.039
$G_{\text{reco},s=8}$ |  $\pm$ 0.002 |  $\pm$ 0.013 |  $\pm$ 11.256 |  |  $\pm$ 0.003 |  $\pm$ 0.014 |  $\pm$ 12.69 |  |  $\pm$ 0.004 |  $\pm$ 0.019 |  $\pm$ 14.269 |  |  $\pm$ 0.004 |  $\pm$ 0.02 |  $\pm$ 18.927
| 0.984 | 0.078 | 14.61 |  | 0.981 | 0.089 | 16.829 |  | 0.979 | 0.101 | 18.82 |  | 0.975 | 0.117 | 22.056
$G_{\text{reco},s=10}$ |  $\pm$ 0.003 |  $\pm$ 0.015 |  $\pm$ 11.223 |  |  $\pm$ 0.003 |  $\pm$ 0.016 |  $\pm$ 13.051 |  |  $\pm$ 0.004 |  $\pm$ 0.023 |  $\pm$ 14.516 |  |  $\pm$ 0.004 |  $\pm$ 0.017 |  $\pm$ 17.864
| 0.982 | 0.089 | 16.232 |  | 0.98 | 0.102 | 18.561 |  | 0.977 | 0.118 | 19.776 |  | 0.974 | 0.126 | 20.612
$G_{\text{reco},s=12}$ |  $\pm$ 0.003 |  $\pm$ 0.015 |  $\pm$ 12.213 |  |  $\pm$ 0.003 |  $\pm$ 0.017 |  $\pm$ 13.75 |  |  $\pm$ 0.004 |  $\pm$ 0.024 |  $\pm$ 14.034 |  |  $\pm$ 0.004 |  $\pm$ 0.019 |  $\pm$ 16.189

Table 1: Quantitative results of applying $G_{\text{reco}}$ trained on the
OpenCCO dataset with several values of $s$ and applied on the Drive dataset
with several values of $s$.

####  3.2.2 Convergence of the proposed approach

We propose to apply $G_{\text{reco}}$ 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 $G_{\text{reco}}$ 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 $G_{\text{reco},s=X}$
($X\in{6,8,10,12}$) 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 $G_{\text{reco}}$ iteratively converges as shown by
Figure 2(a). Interestingly this convergence occurs even though
$G_{\text{reco}}$ was applied to images with disconnection sizes different
from those it was trained on. This highlights the robustness of our approach.

Figure 2: Results of applying $\text{G}_{\text{reco}}$, trained on OpenCCO, on
the DRIVE annotations artificially disconnected with sizes
$s\in\\{6,8,10,12\\}$. (a) convergence curves displaying the $\ell_{2}$ norm
of the difference of the last two consecutive results. (b-d) quantitative
results of our models applied with an increasing number of iterations.

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
$G_{\text{reco}}$ 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
0.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. 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
$G_{\text{reco},{\text{IXI}}}$, 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 0.A.

|  | Variational approach |  | Deep Learning
---|---|---|---|---
|  Training Test | DSC | ASSD | $\epsilon_{\beta_{0}}$ | AUC |  | DSC | ASSD | $\epsilon_{\beta_{0}}$ | AUC
|  | 0.758 | 2.017 | 98.54 | 0.838 |  | 0.811 | 1.155 | 34.04 | 0.901
| Segmentation |  $\pm$ 0.025 |  $\pm$ 0.452 |  $\pm$ 91.88 |  $\pm$ 0.025 |  |  $\pm$ 0.015 |  $\pm$ 0.181 |  $\pm$ 26.86 |  $\pm$ 0.023
|  | 0.767 | 2.423 | 9.003 | 0.850 |  | 0.809 | 1.192 | 9.111 | 0.903
| ${\text{G}_{\text{reco}_{\text{CCO}}}}$ |  $\pm$ 0.023 |  $\pm$ 0.582 |  $\pm$ 11.39 |  $\pm$ 0.025 |  |  $\pm$ 0.016 |  $\pm$ 0.19 |  $\pm$ 7.606 |  $\pm$ 0.022
|  | 0.768 | 2.332 | 6.609 | 0.851 |  | 0.810 | 1.198 | 11.37 | 0.903
2D | ${\text{G}_{\text{reco}_{\text{STARE}}}}$ |  $\pm$ 0.023 |  $\pm$ 0.533 |  $\pm$ 5.817 |  $\pm$ 0.025 |  |  $\pm$ 0.015 |  $\pm$ 0.183 |  $\pm$ 9.095 |  $\pm$ 0.022
| p-values | $\thicksim 10^{-6}$ | 0.057 | $\thicksim 10^{-6}$ | $\thicksim 10^{-6}$ |  | $0.832$ | $0.466$ | $\thicksim 10^{-6}$ | 0.811
|  | 0.476 | 3.587 | 26.56 | - |  | 0.756 | 1.488 | 3.203 | -
| Segmentation |  $\pm$ 0.02 |  $\pm$ 0.42 |  $\pm$ 10.26 | - |  |  $\pm$ 0.015 |  $\pm$ 0.211 |  $\pm$ 1.689 | -
|  | 0.495 | 4.154 | 5.618 | - |  | 0.740 | 1.552 | 1.697 | -
3D | ${\text{G}_{\text{reco}_{\text{IXI}}}}$ |  $\pm$ 0.019 |  $\pm$ 0.427 |  $\pm$ 2.411 | - |  |  $\pm$ 0.014 |  $\pm$ 0.21 |  $\pm$ 1.029 | -
| p-values | $\thicksim 10^{-4}$ | $\thicksim 10^{-6}$ | $\thicksim 10^{-17}$ | - |  | $\thicksim 10^{-5}$ | $0.226$ | $\thicksim 10^{-10}$ | -

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 ${\text{G}_{\text{reco}_{\text{STARE}}}}$ in 2D and the
segmentation and ${\text{G}_{\text{reco}_{\text{IXI}}}}$ in 3D.

(a) Annotation (b) Segmentation (c) ${\text{G}_{\text{reco},\text{CCO}}}$ (d)
${\text{G}_{\text{reco},\text{STARE}}}$

(e) Annotation (f) Segmentation (g) ${\text{G}_{\text{reco},\text{CCO}}}$ (h)
${\text{G}_{\text{reco},\text{STARE}}}$

Figure 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).

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 $>90\%$ 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 learns 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
VascuSynth222https://vascusynth.cs.sfu.ca/Welcome.html 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.

##  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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##  Appendix 0.A Qualitative results

(a) Annotation (b) Disconnected annotation with $s=12$ (c)
$\text{G}_{\text{reco},s=6}$

(d) $\text{G}_{\text{reco},s=8}$ (e) $\text{G}_{\text{reco},s=10}$ (f)
$\text{G}_{\text{reco},s=12}$

Figure 0.A.1: Qualitative results of the influence of the disconnection size
experiment. $G_{\text{reco}}$ was trained with a disconnection size $X$ (with
$X\in{6,8,10,12}$), denoted $G_{\text{reco},s=X}$, on the OpenCCO dataset. A
drive dataset annotation (a) was disconnected with $s=12$ to obtain (b),
followed by a single application of $G_{\text{reco},s=X}$ on (b), resulting in
(c-f). Blue arrows indicate successful reconnections, pink arrows highlight
incorrect reconnections, and green arrows denote differences between the
results.

Figure 0.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.

Figure. 0.A.2 shows the evolution of the applying our post-processing several
times on a disconnected image from the DRIVE dataset. In the blue and red box
of , we can observe that some disconnections that were 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 in the green box. Interestingly, these false
reconnections often occur when noise fragments are present near a vessel, and
$G_{\text{reco}}$ use them to create realistic vessels.

Figure 0.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 0.A.2: Evolution of a DRIVE disconnected manual annotation (with
$s=12$) through successive application of $\text{G}_{\text{reco},s=8}$.

(a) Annotation (b) Segmentation (c) ${\text{G}_{\text{reco},\text{IXI}}}$

(d) Annotation (e) Segmentation (f) ${\text{G}_{\text{reco},\text{IXI}}}$

Figure 0.A.3: Results of our post-processing in 3D. (a) the annotation of a
Bullit 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
$s=8$.

##  Appendix 0.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
https://github.com/creatis-myriad/plug-and-play-reco-regularization.

  * •

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
$\operatorname{P}(i)=\frac{2^{p-i}}{2^{p}-1}$, 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
$\mathcal{N}(\frac{s}{i+1},0)$, where $s$ is the mean disconnection size
parameter of our post-processing. The thiner 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 draw
from a Normal distribution $\mathcal{N}(\frac{N}{2},\frac{N}{4})$ 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 $\mathcal{N}(3,1)$) is centered
on $c$.

    * –

$n^{\prime}$ pixels are randomly selected inside the disk and added
($n^{\prime}\hookrightarrow\mathcal{N}(\frac{N}{2},\frac{N}{4})$) to the
segmentation.