Correspondence<EMAIL_ADDRESS>
# Phenotype-preserving metric design for high-content image reconstruction by
generative inpainting

Vaibhav Sharma  Center for Advanced Systems Understanding (CASUS), Helmholtz-
Zentrum Dresden-Rossendorf e. V. (HZDR), Görlitz, Germany  Artur Yakimovich
Center for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden-
Rossendorf e. V. (HZDR), Görlitz, Germany  Artificial Intelligence for Life
Sciences CIC, Dorset, United Kingdom  Institute of Computer Science,
University of Wrocław, Wrocław, Poland

###### Abstract

In the past decades, automated high-content microscopy demonstrated its
ability to deliver large quantities of image-based data powering the
versatility of phenotypic drug screening and systems biology applications.
However, as the sizes of image-based datasets grew, it became infeasible for
humans to control, avoid and overcome the presence of imaging and sample
preparation artefacts in the images. While novel techniques like machine
learning and deep learning may address these shortcomings through generative
image inpainting, when applied to sensitive research data this may come at the
cost of undesired image manipulation. Undesired manipulation may be caused by
phenomena such as neural hallucinations, to which some artificial neural
networks are prone. To address this, here we evaluate the state-of-the-art
inpainting methods for image restoration in a high-content fluorescence
microscopy dataset of cultured cells with labelled nuclei. We show that
architectures like DeepFill V2 and Edge Connect can faithfully restore
microscopy images upon fine-tuning with relatively little data. Our results
demonstrate that the area of the region to be restored is of higher importance
than shape. Furthermore, to control for the quality of restoration, we propose
a novel phenotype-preserving metric design strategy. In this strategy, the
size and count of the restored biological phenotypes like cell nuclei are
quantified to penalise undesirable manipulation. We argue that the design
principles of our approach may also generalise to other applications.

###### keywords:

High-content fluorescence microscopy, Deep Learning, Inpainting, Sample
preparation, Artefact, Metric

##  1 INTRODUCTION

In the past several decades microscopy took a key role in biomedical discovery
and diagnostics [1, 2]. Yet, obtaining images of microscopic entities requires
sample preparation and staining procedures which are often complex and involve
multiple steps [3]. This complexity may lead to the occurrence of sample
preparation artefacts (SPA) at various stages of the process [1, 2]. Some
common causes of SPA include incomplete fixation of tissue samples, improper
embedding of tissue samples, inefficient removal of dehydration artefacts,
inefficient removal of staining artefacts, inclusion of dust particles and
excessive tissue damage during sample preparation. The presence of SPAs in
micrographs may significantly influence downstream data analysis and
interpretation leading to misinterpretation of the biological phenotype. While
in some cases these issues can be avoided by improving the handling, as well
as imaging of artefact-free samples selectively, in cases of high-content
screening (HCS) microscopy SPAs are often inevitable due to the use of
automated liquid handling and other high-content sample preparations
procedures [4, 5].

With the advent of digital image processing, deep learning (DL) and generative
models, the removal of SPAs from acquired HCS images may be attained by image
inpainting [6]. In this task, missing or corrupted regions are filled in by a
trained generative neural network using context data from the source image
itself. However, owing to effects like neural hallucinations [7] such image
reconstruction tasks may introduce undesired alterations and manipulations of
their own. While any generative model manipulates the image to a certain
extent, not all alterations are critical. It is therefore crucial to measure
how these alterations would affect the perception of the biomedical
observation (phenotype) by the image analysis system.

In this work, we employ an open HCI dataset [8] containing SPA and assess the
ability of the state-of-the-art generative DL algorithms to reconstruct the
images through generative inpainting. To ensure the scientific or diagnostic
accuracy of the results, we propose a phenotype-preserving metric allowing us
to assess the quality of inpainting from the phenotypic point of view. Due to
the nature of the dataset, the phenotype-preserving metric we explore in this
work focuses on the detection and measurement of cell nuclei. Yet, we argue
that similar design principles may be applied to other phenotypes.

##  2 RELATED WORK

Biomedical image reconstruction has been an active area of research for
several decades. Introduced in the 1960s, one of the earliest approaches for
biomedical image reconstruction was the filtered back projection method [9].
This method is widely used in computed tomography and has been improved over
the years with variations such as iterative reconstruction and adaptive
statistical iterative reconstruction [10]. Another popular approach is based
on the use of compressed sensing techniques. These techniques allow the
reconstruction of images from a small number of measurements, which can reduce
the amount of data acquisition time required for imaging. Compressed sensing
techniques have been applied to various types of biomedical images, including
magnetic resonance imaging [11], computed tomography [12, 13], and positron
emission tomography [14], and microscopy [15]. Notably in microscopy,
denoising and reconstruction is often associated with enhancing the resolution
[16, 17].

In addition to these general techniques, there are also specific methods
developed for certain types of biomedical images. For example, in ultrasound
imaging, time-reversal methods have been used for image reconstruction [18].
In optical coherence tomography, methods based on Fourier domain signal
processing have been used [19]. However, these techniques are often based on
assumptions, which may not hold. Additionally, compressed sensing
reconstruction can be computationally expensive, particularly for large
volumes of data. To counter the above limitations, DL approaches have also
been used for biomedical image reconstruction. Specifically, convolutional
neural networks (CNNs) have been used for image superresolution, denoising,
and artefact reduction [20, 21, 22].

However, the assessment of the quality of biomedical image reconstruction is
currently limited to well-established computer vision metrics, which may not
capture the intricacy required for scientific and diagnostic accuracy. Metrics
such as peak signal-to-noise ratio (PSNR) [23] and structural similarity index
measure (SSIM) [24] are often used to assess quality, yet are known to lack
complexity [25]. One approach for circumventing this is to evaluate the metric
on multiple scales [26]. Other approaches aim to introduce subjective quality
assessment involving human judgement [27]. In this work, we propose a novel
metric that incorporates both subjective and objective quality assessment of
HCI microscopy image reconstruction. The main criterion of the metric we
propose is to preserve the biological phenotype from the perspective of image
quantification. We argue that a conceptually similar metric design can be
proposed for other biomedical image modalities.

##  3 METHODS

###  3.1 Code Availability

The Python source code developed in this work is available under GPLv3 open-
source license at
github.com/casus/PhIRM.

###  3.2 Computational Setup

All the inpainting models used in this work have been trained using high-
performance computing. The Hemera HPC system has been used for this purpose.
All the experiments have been carried out using a single 32 GB NVIDIA V100 GPU
with 8 (eight) CPU cores. The maximum memory per CPU was set to 1443 MB.

###  3.3 Sample preparation Artefacts Masks Generation

The multispectral properties of the dataset presented [8] in this work have
been used to generate ground truth images containing exclusively nuclei or
SPA. The current methodology involves the generation of artefact masks from
2160x2160 pixel TIFF images obtained from the CFP channel. Otsu thresholding
[28] has been implemented to obtain a threshold value, which is subsequently
multiplied by 0.7 to enhance the quality of the masks. We observed that Otsu’s
thresholding often eliminates small artefacts or boundary pixels of large
artefacts, and hence the multiplication factor was introduced. The resultant
image is binarised and subjected to morphological operations in the form of
opening and closing.

###  3.4 Artificial Masks Generation

To understand the influence of mask shape on inpainting performance we have
created two kinds of artificial masks: rectangular masks and irregular masks.
In the case of the first, we created rectangular mask datasets which contain
images with varying mask area: 10-20%, 20-30%, 30-40% and 40-50%. For example,
A 10-20% rectangular mask dataset contains images having 10-20% of the total
image area covered by a rectangular mask. This is done by placing white
rectangles of a particular area range (e.g.: 10-20% of the entire image area)
at random regions inside a black image of size 256x256.

To train models with irregular-shaped masks of random sizes, we used an
approach to draw masks based on randomly positioned lines with a predefined
minimum and maximum number of vertices. Our approach rotates angles and
produces thick lines by joining these vertices. Then it puts circles in the
intersections of the two lines to guarantee their smoothness. This algorithm
has been adapted from the DeepFillV2 architecture [29].

###  3.5 Data Augmentation

To increase the size of our dataset we have developed an image patch (zoomed-
image) generator that extracts smaller zoomed and cropped images from a larger
image of size 2160x2160 pixels. The generator generates patches of size
256x256, starting from the left-hand side of the image and moving towards the
right until the entire image is covered. After normalization using the min-max
method, the resulting intensity range is 0 to 255. All 256x256-sized nuclei
and mask images used in this work have been created using this zoomed image
generator.

###  3.6 Model Fine-tuning

To ensure good performance of the Context Encoder, DeepFill V2 and Edge
Connect pre-trained networks, these models were fine-tuned using microscopy
data. For this we have generated and augmented dataset containing 36112 data
points with masks constructed as described in respective sections above.
Context Encoder was fine-tuned for 200 epochs with learning rate 0.0002.
DeepFill V2 was fine-tuned for 20 epochs with learning rate 0.0001. Edge
Connect was fine-tuned for 68 epochs with learning rate 0.0001. All networks
were trained until convergence.

###  3.7 Phenotype-preserving Image Reconstruction Metric Design

In the case of nuclear images, PhIRM can be designed to contain the following
components: nuclear count difference defined in Equation (1), nuclear area
difference defined in Equation (2) and artefact area difference defined in
Equation (3). These values can be computed as follows:

$NCD=\begin{cases}0\cdot 0&\text{i}f\alpha=0\\\ 1\cdot
1^{\alpha}&\text{i}f\alpha>0\\\ 2^{|\alpha|}&\text{i}f\alpha<0,\\\
\end{cases}$ (1)

where $\alpha$ (default value is: 1.1) is the difference in number of nuclei
between the original and reconstructed image.

$NAD=\omega_{NAD}\cdot(A_{nuc\ out}-A_{nuc\ in}),$ (2)

where $A_{nuc\ out},A_{nuc\ in}$ is the area of nuclei in the original and
reconstructed image, and $\omega_{NAD}$ (default value is: 0.0002) is the
respective weight.

$AAD=\omega_{AAD}\cdot(A_{art\ out}-A_{art\ in}),$ (3)

where $A_{art\ out},A_{art\ in}$ is the area of artefacts in the original and
reconstructed image, and $\omega_{AAD}$ (default value is: 0.001) is the
respective weight. These components can be computed in the following
algorithm.

Algorithm 1. PhIRM Factors Calculation
1\. Apply Otsu thresholding [28] to the input image to produce a binary mask
image.
2\. Perform connected components analysis on the binary image.
3\. Avoid components with an area of less than 50 px.
4\. Compute of mean and maximum values for each component.
5\. Identify components as artefacts if the maximum value computed in step 4
equals 255 and the mean value is greater than or equal to 210.
Otherwise identification as nuclei.
6\. Identify components as single nuclei if the maximum value computed in step
4 is less than 255 and the area of the component is less than 2200.
Otherwise, identify as a patch containing two overlapping nuclei.
7\. Store the total number of nuclei, total nuclei area, and total artefact
area.

After these steps are applied to both the source and inpainted image, the
difference in the number of nuclei, total nuclei area, and total artefact area
between the two images can be computed. These attributes are then used to
calculate the final score defined in the following Equation (4).

$PhIRM=\frac{10-(NCD+NAD+AAD)}{10}.$ (4)

##  4 RESULTS

Figure 1:  Metric validation set for comparison to expert opinion. (a)
examples of images in the validation set with respective alterations. (b)
Comparison of the expert opinion to the PhIRM metric measured on this test
image set. (c) Comparison of peak signal-to-noise ratio (PSNR) and structural
similarity index metric (SSIM) on the same set.

###  4.1 Metric design and validation

HCI microscopy containing SPA may be corrected computationally using
generative inpainting. However, the assessment of the scientific accuracy of
such corrected images is problematic. To address that we proposed Phenotype-
preserving Image Reconstruction Metric (PhIRM). At its core, PhIRM is
assessing whether the phenotype-relevant information is still preserved upon
image inpainting. In the case of high-content fluorescence microscopy of cell
nuclei imaged in the presence or absence of SPA [8], the preservation of the
phenotype constitutes factors including the area of the nuclei, their count,
as well as the area of the remaining artefact. These factors were taken into
account and heuristically weighted for this specific phenotype and task (see
Methods).

Next, to assess how PhIRM compares to expert assessment we have constructed a
test set containing the original image together with images manipulated in a
manner to corrupt the phenotype. Specifically, manipulations created images
with the missing nuclear area, missing nuclei or introduced artefacts (Fig
1a). We then asked four image analysis experts (including one senior author of
this work) to assess the difference between the original and manipulated
images from the perspective of phenotype consistency. Remarkably, despite the
judgement discrepancies in the extreme cases, overall expert opinion showed a
good agreement with PhIRM (Fig 1b).

To understand how PhIRM would compare to existing image reconstruction metrics
like SSIM and PSNR we measured these on the same set of images. (Fig 1c). Our
comparison suggested that while PSNR somewhat correlated with the expert
opinion, SSIM completely failed to capture the differences. Notably, in
extreme cases like very low consistency or very high consistency, PhIRM seems
to capture the differences better than PSNR. Interestingly, these were the
cases experts had a rather low consensus on as well. We attributed these
observations to an eventuality that both experts and PSNR would likely
penalise differences that may not be crucial for phenotypic measurements.
PhIRM at the same time would focus exclusively on aspects decisive for the
phenotypic accuracy.

###  4.2 High-content image reconstruction by generative inpainting

Figure 2:  State-of-the-art image inpainting architectures performance on
high-content microscopy (a) example of images with square masks covering
30-40% of the area with respective inpainting results using pre-trained
Context Encoder, DeepFill V2 and Edge Connect networks. (b) PhIRM metric
output for Context Encoder, DeepFill V2 and Edge Connect networks depending on
the area covered by the mask. Figure 3:  Edge Connect architecture performance
on irregularly-shaped masks and sample preparation artefacts. (a) Examples of
Edge Connect reconstruction on irregularly-shaped masks, accompanied by
phenotype-preserving image reconstruction metric (PhIRM). (b) Examples of Edge
Connect reconstruction on sample preparation artefact masks, accompanied by
PhIRM.

To measure the ability of generative inpainting to restore high-content images
we selected three state-of-the-art pre-trained models with varying expressive
capacity of the deep artificial neural networks (DNN) they are based on.
Namely, we selected Context Encoder [30], DeepFill V2 [29] and Edge Connect
[31]. Next, we fine-tuned these models on our dataset and evaluated their
performance using PhIRM. It is worth noting, however, that Context Encoder was
not designed to perform inpainting with irregularly shaped masks. Therefore,
to ensure a fair comparison, we have first created a synthetic set of images
with rectangular masks (Fig 2a). To understand how the area of the mask
impacts performance we have used four distinct ranges 10-20%, 20-30%, 30-40%,
40-50% (Fig 2b). Our comparison showed that Deep Fill V2 and Edge Connect
performed significantly better than the Context Encoder, especially with the
largest masks. On mask area above 30% Edge Connect outperformed Deep Fill V2.

Finally, we compared the performance of Edge Connect architecture on
rectangular masks to the performance on irregularly-shaped masks (Fig 3a) and
the masks derived from high-content image SPAs (Fig 3b). Remarkably, despite
quite obvious differences in shapes, with PhIRM between 0.86 and 0.99 the
performance remained high for both types of masks. Similarly to the case of
rectangular masks, performance seemed to be stronger connected to the area
obstructed, rather then shape of the mask. Based on this we concluded that the
state-of-the-art Edge Connect architecture may be employed for reconstructive
inpainting of high-content fluorescent microscopy images obstructed by SPAs
without significant alteration of nuclear phenotype.

##  5 CONCLUSIONS

This research highlights the potential of using generative inpainting for
microscopy image reconstruction. While generative inpainting bears a great
promise to microscopy reconstruction, it inevitably creates artefacts. To
ensure the scientific accuracy of such reconstructed images we propose a novel
metric - phenotype-preserving image reconstruction metric (PhIRM). The metric
score ranges from 0.0 to 1.0, with higher scores indicating better inpainting
quality. Compared to traditional metrics like PSNR and SSIM, the proposed
metric demonstrates higher relevance and sensitivity. PhIRM aims to take into
account only the changes crucial for the preservation of the biological
phenotype. In the specific case of quantification of fluorescent cell nuclei
in the presence of sample preparation artefacts in high-content imaging, these
changes are cell count, cell area and artefact area. We demonstrate that the
metric we proposed is in good agreement with expert opinion.

Noteworthy, in order to make PhIRM useful for other applications, a different
set of phenotype factors may need to be constructed. While this may seem to
limit the application of PhIRM in an off-the-shelf manner, we argue that the
construction of PhIRM for each individual application should be rather
straightforward for a subject matter expert. For example, should cell cycle
detection be important for the phenotype, one could include the difference in
the total fluorescence intensity of the marker.

Furthermore, we employed the PhIRM metric to evaluate the state-of-the-art
inpainting architectures including Context Encoder[30], DeepFill V2 [29] and
Edge Connect [31]. We demonstrate that both visually and according to the
PhIRM values DeepFill V2 and Edge Connect outperformed an older Context
Encoder architecture. Additionally, we showed that for our particular task
Edge Connect performed consistently better. Remarkably, we showed that, in the
case of Edge Connect inpainting, the size of the damaged area significantly
affects the inpainting quality. However, the shape of the damaged regions had
only a minor impact. In summary, in this work, we argue that if accompanied by
an adequate metric generative inpainting may be useful for image
reconstruction in HCI microscopy.

###### Acknowledgements.

We thank Dr. Vardan Andriasyan and Anthony Petkidis for their expert opinion
on microscopy image evaluation. This work was partially funded by the Center
for Advanced Systems Understanding (CASUS) which is financed by Germany’s
Federal Ministry of Education and Research (BMBF) and by the Saxon Ministry
for Science, Culture, and Tourism (SMWK) with tax funds on the basis of the
budget approved by the Saxon State Parliament.

## References

  * [1] Lang, P., Yeow, K., Nichols, A., and Scheer, A., “Cellular imaging in drug discovery,” Nature Reviews Drug Discovery 5(4), 343–356 (2006).
  * [2] Niazi, M. K. K., Parwani, A. V., and Gurcan, M. N., “Digital pathology and artificial intelligence,” The lancet oncology 20(5), e253–e261 (2019).
  * [3] Scriven, D. R., Lynch, R. M., and Moore, E. D., “Image acquisition for colocalization using optical microscopy,” American Journal of Physiology-Cell Physiology 294(5), C1119–C1122 (2008).
  * [4] Boutros, M., Heigwer, F., and Laufer, C., “Microscopy-based high-content screening,” Cell 163(6), 1314–1325 (2015).
  * [5] Bray, M.-A., Fraser, A. N., Hasaka, T. P., and Carpenter, A. E., “Workflow and metrics for image quality control in large-scale high-content screens,” Journal of biomolecular screening 17(2), 266–274 (2012).
  * [6] Elharrouss, O., Almaadeed, N., Al-Maadeed, S., and Akbari, Y., “Image inpainting: A review,” Neural Processing Letters 51, 2007–2028 (2020).
  * [7] Liu, H., Zheng, X., Han, J., Chu, Y., and Tao, T., “Survey on gan-based face hallucination with its model development,” IET Image Processing 13(14), 2662–2672 (2019).
  * [8] Sharma, V. and Yakimovich, A., “High-content multi-spectral fluorescence microscopy sample preparation artefacts,” (Jan. 2023).
  * [9] Deans, S. R., [The Radon transform and some of its applications ], Courier Corporation (2007).
  * [10] Prakash, P., Kalra, M. K., Digumarthy, S. R., Hsieh, J., Pien, H., Singh, S., Gilman, M. D., and Shepard, J.-A. O., “Radiation dose reduction with chest computed tomography using adaptive statistical iterative reconstruction technique: initial experience,” Journal of computer assisted tomography 34(1), 40–45 (2010).
  * [11] Lustig, M., Donoho, D., and Pauly, J. M., “Sparse mri: The application of compressed sensing for rapid mr imaging,” Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine 58(6), 1182–1195 (2007).
  * [12] Choi, K. and Brady, D. J., “Coded aperture computed tomography,” in [Adaptive Coded Aperture Imaging, Non-Imaging, and Unconventional Imaging Sensor Systems ], 7468, 99–108, SPIE (2009).
  * [13] Brady, D. J., Choi, K., Marks, D. L., Horisaki, R., and Lim, S., “Compressive holography,” Optics express 17(15), 13040–13049 (2009).
  * [14] Knoll, F., Koesters, T., Otazo, R., Block, T., Feng, L., Vunckx, K., Faul, D., Nuyts, J., Boada, F., and Sodickson, D. K., “Joint reconstruction of simultaneously acquired mr-pet data with multi sensor compressed sensing based on a joint sparsity constraint,” EJNMMI physics 1, 1–2 (2014).
  * [15] Pavillon, N. and Smith, N. I., “Compressed sensing laser scanning microscopy,” Optics express 24(26), 30038–30052 (2016).
  * [16] Gustafsson, M. G., Shao, L., Carlton, P. M., Wang, C. R., Golubovskaya, I. N., Cande, W. Z., Agard, D. A., and Sedat, J. W., “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophysical journal 94(12), 4957–4970 (2008).
  * [17] Müller, M., Mönkemöller, V., Hennig, S., Hübner, W., and Huser, T., “Open-source image reconstruction of super-resolution structured illumination microscopy data in imagej,” Nature communications 7(1), 10980 (2016).
  * [18] Lerosey, G., De Rosny, J., Tourin, A., Derode, A., Montaldo, G., and Fink, M., “Time reversal of electromagnetic waves,” Physical review letters 92(19), 193904 (2004).
  * [19] Huang, D., Swanson, E. A., Lin, C. P., Schuman, J. S., Stinson, W. G., Chang, W., Hee, M. R., Flotte, T., Gregory, K., Puliafito, C. A., et al., “Optical coherence tomography,” science 254(5035), 1178–1181 (1991).
  * [20] Weigert, M., Schmidt, U., Boothe, T., Müller, A., Dibrov, A., Jain, A., Wilhelm, B., Schmidt, D., Broaddus, C., Culley, S., et al., “Content-aware image restoration: pushing the limits of fluorescence microscopy,” Nature methods 15(12), 1090–1097 (2018).
  * [21] Qiao, C., Li, D., Guo, Y., Liu, C., Jiang, T., Dai, Q., and Li, D., “Evaluation and development of deep neural networks for image super-resolution in optical microscopy,” Nature Methods 18(2), 194–202 (2021).
  * [22] Dong, C., Loy, C. C., He, K., and Tang, X., “Image super-resolution using deep convolutional networks,” IEEE transactions on pattern analysis and machine intelligence 38(2), 295–307 (2015).
  * [23] Elbadawy, O., El-Sakka, M. R., and Kamel, M. S., “An information theoretic image-quality measure,” in [Conference Proceedings. IEEE Canadian Conference on Electrical and Computer Engineering (Cat. No. 98TH8341) ], 1, 169–172, IEEE (1998).
  * [24] Wang, Z., Bovik, A. C., Sheikh, H. R., and Simoncelli, E. P., “Image quality assessment: from error visibility to structural similarity,” IEEE transactions on image processing 13(4), 600–612 (2004).
  * [25] Ponomarenko, N., Silvestri, F., Egiazarian, K., Carli, M., Astola, J., and Lukin, V., “On between-coefficient contrast masking of dct basis functions,” in [Proceedings of the third international workshop on video processing and quality metrics ], 4, Scottsdale USA (2007).
  * [26] Wang, Z., Simoncelli, E. P., and Bovik, A. C., “Multiscale structural similarity for image quality assessment,” in [The Thrity-Seventh Asilomar Conference on Signals, Systems & Computers, 2003 ], 2, 1398–1402, Ieee (2003).
  * [27] Farrell, J. E., “Image quality evaluation,” Colour imaging: vision and technology , 285–313 (1999).
  * [28] Otsu, N., “A threshold selection method from gray-level histograms,” IEEE transactions on systems, man, and cybernetics 9(1), 62–66 (1979).
  * [29] Yu, J., Lin, Z., Yang, J., Shen, X., Lu, X., and Huang, T. S., “Free-form image inpainting with gated convolution,” in [Proceedings of the IEEE/CVF international conference on computer vision ], 4471–4480 (2019).
  * [30] Pathak, D., Krahenbuhl, P., Donahue, J., Darrell, T., and Efros, A. A., “Context encoders: Feature learning by inpainting,” in [Proceedings of the IEEE conference on computer vision and pattern recognition ], 2536–2544 (2016).
  * [31] Nazeri, K., Ng, E., Joseph, T., Qureshi, F. Z., and Ebrahimi, M., “Edgeconnect: Generative image inpainting with adversarial edge learning,” arXiv preprint arXiv:1901.00212 (2019).


# Multi-hop Links Quality Analysis of 5G Enabled Vehicular Networks

Shikuan Li1, Zipeng Li1, Xiaohu Ge1, Jing Zhang1, Minho Jo2  1School of
Electronic Information and Communications, Huazhong University of Science and
Technology, China  2Department of Computer Convergence Software, Korea
University, South Korea  Corresponding author: Xiaohu Ge. Email:
<EMAIL_ADDRESS>
###### Abstract

With the emerging of the fifth generation (5G) mobile communication systems,
millimeter wave transmissions are believed to be a promising solution for
vehicular networks, especially in vehicle to vehicle (V2V) communications. In
millimeter wave V2V communications, different vehicular networking services
have different quality requirements for V2V multi-hop links. To evaluate the
quality of different V2V wireless links, a new link quality indicator is
proposed in this paper considering requirements of the real-time and the
reliability in V2V multi-hop links. Moreover, different weight factors are
configured to reflect the different requirements of different types of
services on real-time and reliability in the new quality indicator. Based on
the proposed link quality indicator, the relationship between V2V link quality
and one-hop communication distance under different vehicle densities is
analyzed in this paper. Simulation results indicate that the link quality is
improved with the increasing of vehicle density and there exists an optimal
one-hop communication distance for the link quality when the vehicle density
is fixed.

##  I Introduction

In recent years, the vehicular network is considered to be one of the most
promising technologies for intelligent transportation system (ITS) [1, 2, 3].
A secure and efficient Vehicle-to-Vehicle (V2V) communications is the key to
support road safety and traffic efficiency applications in ITS. At the same
time, with the emerging of fifth generation (5G) mobile communication system
[4, 5, 6], millimeter wave communications have been considered to be an
effective technology in V2V communications. Millimeter wave links are adopted
for wireless multi-hop communications among vehicles. Therefore, different
information can be transmitted and shared among vehicles, which enabled all
kinds of services and applications in vehicular networks [7, 8]. These
services and applications can mainly be classified into two types. The first
is the safety services which require an extremely low latency, such as the
vehicle collision warning, obstacle detection and driving assistances. The
other is the infotainment services which prefer link reliability to the
latency, such as file transmissions, social entertainments and online
services.

In V2V communications, how to select the best communication link to meet
different requirements of all kinds of services has become a hot topic
recently. Many schemes have been proposed to reduce transmission delay in V2V
communications [9, 10, 11, 12]. In [9], the multi-hop transmission delay was
studied under different probability distributions of network nodes.
Mathematical expressions of the average communication delay among vehicles
were derived for interference-limited cases and noise-limited cases. Delay-
optimal routing algorithms in V2V communications were investigated to select
the communication link with the lowest transmission delay [10, 11]. Safety-
related messages broadcasting delay in V2V communications was analyzed in
[12]. The analytic result indicated that broadcasting delays were different
when one-hop transmission ranges among vehicles were different, so vehicles
can broadcast the safety-related messages more efficiently by selecting an
optimal transmission range. But this result was only available in messages
broadcasting scenarios and only the broadcasting delay was analyzed. However,
some vehicular services focused more on the V2V link reliability than the time
delay and can tolerate a large transmission delay. How to improve the
connectivity probability has been investigated in several works [13, 14, 15,
16, 17]. An analytical model supporting multi-hop relay based on V2V
communications was proposed to analyze the relationship between connectivity
probabilities and hops under different scenarios [13]. In [14], an analytical
model was proposed to analyze the successful transmission probabilities of
safety messages in linear multi-hop V2V links. The connectivity probabilities
of different communication modes, i.e. the unit disk model and the log-normal
shadowing model were compared in [15]. An analytical mode taking account of
the vehicle mobility and the path loss of wireless channel was proposed to
simulate real scenarios of vehicular networks. The connectivity characteristic
of platoon-based V2V communication scenarios was studied in [16], [17].
Analysis results showed that the connectivity probability can be significantly
improved when there are platoons in the vehicular network.

However, most of these studies only considered the delay-optimal V2V links or
the V2V links with the best connectivity probability. Very few study was done
to optimize the transmission delay and multi-hop connectivity in one
analytical model of vehicle networks. By taking both the transmission delay
and the connectivity into consideration, in this paper we proposed a new link
quality indicator to evaluate the V2V multi-hop link quality. Moreover, the
relationships between the link quality and network parameters are
investigated. Based on the proposed link quality indicator, optimal V2V multi-
hop links could be realized in 5G enabled vehicular networks. The
contributions and novelties of this paper are summarized as follows.

  * •

Considering the multi-hop V2V transmission delay and connectivity, a new link
quality indicator was proposed to evaluate the V2V multi-hop link quality.
Different kinds of vehicular service requirements are reflected by modifying
the weight factors of the transmission delay and connectivity in the link
quality indicator.

  * •

The multi-hop transmission delay and the connectivity probability of
millimeter wave based V2V communication are analyzed. The analytical
expression of the V2V link quality indicator is derived.

  * •

Simulation results indicate that V2V multi-hop link strategies depend on V2V
transmission delay, connectivity probability and weight factors. These results
provide some useful guidelines for selection strategies of V2V communication
links in vehicular networks.

The remainder of this paper is organized as follows. Section II describes the
system model and V2V link quality evaluation model. The multi-hop V2V
transmission delay and connectivity probability with different communication
strategies are investigated in Section III. In Section IV, simulations results
show the impact of V2V transmission delay, connectivity probability and weight
factors on the V2V link quality. Finally, Section V concludes this paper.

##  II System Model

###  II-A System scenarios

As shown in Fig. 1, we consider the scenario of a single lane in urban area,
where vehicles on the single lane can communicate with each other by
millimeter wave communications. Assuming that there is a vehicle which wants
to send some messages to another vehicle on the single lane. These messages
include the driving statuses of vehicles, safety-related messages, voice
messages, etc. Without loss of generality, we assume that the vehicle
${V_{a}}$ have some messages to be sent to the vehicle ${V_{b}}$ where is $L$
meters away from ${V_{a}}$. In most cases, since the millimeter wave
communication range is generally less than 200 meters [3], ${V_{a}}$ can not
communicate with ${V_{b}}$ directly. Hence a multi-hop V2V link need to be
performed to transmit these messages when $L$ is larger than 200 meters.

Figure 1: System model.

According to the traffic flow model[13], the probability that there are
vehicles on the meter road is expressed as

$P(n,L)=\frac{{{{(\rho L)}^{n}}{e^{-\rho L}}}}{{n!}},$ (1)

where $\rho$ is the traffic density, defined as the number of cars per meter.
Assuming that $m$ is the total number of cars between ${V_{a}}$ and ${V_{b}}$
and $m$ is subject to Poisson distribution with density of $\rho$. Without
loss of generality, ${V_{a}}$ is denoted as node ${n_{0}}$ and the ${i^{th}}$
vehicle away from vehicle ${V_{a}}$ is denoted as the node ${n_{i}}$.
Obviously, vehicle ${V_{b}}$ is denoted as node ${n_{m}}$ and the distance
between node ${n_{i}}$ and ${n_{0}}$ is denoted as ${x_{i}}$ and ${x_{m}}=L$.

###  II-B V2V multi-hop link quality indicator

To ensure requirements of messages satisfying the transmission delay and the
transmission success probability, a multi-hop V2V link with the best link
quality should be selected. To evaluate the link quality, the multi-hop link
quality indicator ${Q^{*}}$ is defined as

${Q^{*}}=\alpha{\rm P}+\beta D,$ (2)

where $P\in\left[{0,1}\right]$ represents the connect probability of ${V_{a}}$
and ${V_{b}}$, the weight factor $\alpha$ is used to indicate requirement
extent of the connect probability, the weight factor $\beta$ is used to
indicate requirement extent of transmission delay. Weight factors $\alpha$ and
$\beta$ should meet the formula requirement as
$\left\\{{\alpha+\beta=1\left|{\alpha,\beta\in\left[{0,1}\right]}\right.}\right\\}$.
Symbol $D$ is the indicator of transmission delay, the definition of $D$ is
given by

$D=1-\frac{{T\left(\phi\right)}}{{{T_{\max}}}},$ (3)

where $T\left(\phi\right)$ is the message transmission delay on the multi-hop
link $\phi$, ${T_{\max}}$ is the max transmission delay that can be tolerated.
The value range of $D$ in equation (3) is $D\in\left({-\infty,1}\right)$. When
$D$ is a negative number it means that the message transmission delay on the
multi-hop link $\phi$ is larger than ${T_{\max}}$. If vehicle ${V_{a}}$ send
the message to ${V_{b}}$ through a multi-hop link $\phi$, the link quality of
multi-hop link $\phi$ is given as
${Q^{*}}\left(\phi\right)\in\left({-\infty,1}\right]$. Substituting equation
(3) into equation (2), ${Q^{*}}\left(\phi\right)$ is derived by

${Q^{*}}\left(\phi\right)=\alpha
P+\beta\left({1-\frac{{T\left(\phi\right)}}{{{T_{\max}}}}}\right).$ (4)

As for different services in vehicular networks, different transmission delay
and connectivity requirements are needed. Therefore, selecting a multi-hop
link with optimal link quality to satisfy these requirements will make the
vehicular network more efficient and reliable. For a fixed $\alpha$ and
$\beta$, the multi-hop link with a larger link quality ${Q^{*}}$ means that
the performance of this multi-hop link is more suitable for a specific type of
message dissemination. In Section III, transmission delay and link
connectivity probability will be evaluated in details, impacts of network
parameters on link quality will be investigated to help selecting the optimal
V2V link under different vehicular network scenarios.

##  III Transmission delay and connectivity probability of V2V links

###  III-A Transmission delay of V2V links

To satisfy different requirements of vehicular network services on messages
transmission delay and transmission success probability, relay vehicles should
be selected accurately to optimizing the link quality in V2V multi-hop links.
Without loss of generality, vehicle nodes
$\left\\{{{c_{1}},{c_{2}},\cdots,{c_{k-1}}}\right\\}$ are denoted as relay
nodes which are selected from
$\left\\{{{n_{1}},{n_{2}},\cdots,{n_{m-1}}}\right\\}$ to transmit messages.
The transmission delay of V2V multi-hop link
$\phi\left\\{{{c_{0}},{c_{1}},\cdots,{c_{k}}}\right\\}$ is expressed as

${T_{\phi}}=\left({k-1}\right){T_{pro}}+\sum\limits_{i=1}^{k}{{T_{hop}}_{j}},$
(5)

where ${T_{hop}}_{j}$ represents the transmission delay of messages between
relay node ${c_{j-1}}$ and ${c_{j}}$, ${T_{pro}}$ represents the processing
time by relay node ${c_{j}}$.

  1. a)

One hop transmission delay in V2V links

Due to the characteristic of millimeter wave, the path loss $PL$ in any hop of
$\phi$ should not be ignored and is given as [4]

$PL[dB]({r_{j}})=69.6+20.9\log({r_{j}})+\xi,\xi\sim
N\left({0,{\sigma^{2}}}\right),$ (6)

where ${r_{j}}$ is denoted as the distance between vehicle node ${c_{j-1}}$
and ${c_{j}}$, $\xi$ is the shadow fading coefficient , and $\sigma$ is the
standard deviation of shadow fading, which is estimated as 5 dB. The transmit
power plus antenna gains of vehicle nodes is denoted as ${P_{tx}}$ and
${N_{0}}$ represents the Gaussian white noise power density. The single hop
transmission is regarded as success if the received signal-to-noise ratio
(SNR) of relay vehicle is larger than a given threshold $\theta$. Hence, the
transmission success probability in one hop is defined as

${P_{hop}}_{j}={\rm
P}\left({PL\leq{P_{tx}}(dB)-\theta(dB)-{N_{0}}{W_{mmWave}}(dB)}\right),$ (7)

where ${W_{mmWave}}$ is the millimeter wave bandwidth. Substituting (6) into
(7), the success probability of one hop transmission in a single slot is
derived by equation(8), where $erf()$ is the error function and

$\eta\left({{r_{j}}}\right)={P_{tx}}-\theta-{N_{0}}{W_{mmWave}}-69.6-20.9{\log_{10}}{r_{j}}.$

$\begin{gathered}\begin{array}[]{l}{P_{hopj}}={\rm
P}(\xi\leq{P_{tx}}-\theta-{N_{0}}{W_{mmWave}}-69.6-20.9{\log_{10}}{r_{j}})\\\
\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}=\frac{1}{2}\left({1+erf\left({\frac{{\eta\left({{r_{j}}}\right)}}{{\sqrt{2}\sigma}}}\right)}\right)\end{array}\end{gathered}$
(8)

Without loss of generality, let ${t_{slot}}$ be the time cost of a single
slot, due to the large bandwidth of millimeter wave, we can assume that for a
successful transmission, the messages can be sent within a single time slot,
so the one hop transmission delay ${T_{hop}}_{j}$ can be derived by using the
time cost of a time slot over the success transmission probability expressed
as

$\begin{array}[]{l}{T_{hopj}}=\frac{{{t_{slot}}}}{{{P_{hopj}}}}\\\ \\\ {\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}=\frac{{2{t_{slot}}}}{{1+erf\left({\frac{{\eta\left({{r_{j}}}\right)}}{{\sqrt{2}\sigma}}}\right)}}\end{array},$
(9)

  1. b)

Connection strategy in V2V links

The transmission delay on multi-hop link $\phi$ shows that the total hop
number $k$ is a key network parameter which determines the transmission delay.
As for a larger hop number, the distance between two adjacent relay vehicles
is supposed to be shorter, which means that an optimal link connection
strategy should be selected to determine the one hop communication distance.
For any relay vehicle ${c_{j}}$ whose distance from ${n_{0}}$ is ${x_{j}}$ in
the V2V multi-hop links, the connection strategy which determines to select
the next relay vehicle ${c_{j+1}}$ is configured by

${c_{j+1}}=\mathop{argmin}\limits_{{n_{v}}\in\left\\{{{n_{0}},\cdots,{n_{m}}}\right\\}}\left|{{x_{v}}-\left({{x_{j}}+r}\right)}\right|,$
(10)

where $r$ is the average one hop communication distance, ${x_{v}}$ is the
distance between ${n_{v}}$ and vehicle ${n_{0}}$. The connection strategy is
that for a relay vehicle ${c_{j}}$, the vehicle node ${n_{v}}$ is always
selected as the next relay vehicle ${c_{j+1}}$ based on the result of (10).

  1. c)

Multi-hop link time delay in V2V links

When all relay vehicles follow the connection strategy in (10), the
transmission delay ${T_{\phi}}$ in (5) is simplified as

$T\left(\phi\right)=E\left({k{T_{hopj}}}\right)+\left({E\left(k\right)-1}\right){T_{pro}},$
(11)

where $E\left(k\right)=\left\lceil{L/r}\right\rceil$ represents the
expectation of the hop number in the multi-hop link $\phi$ when the average
one hop communication distance is $r$. $\left\lceil{L/r}\right\rceil$
represents the minimum integer that greater than or equal to $L/r$.
Substituting (8) into (11), the total transmission delay $T\left(\phi\right)$
on the V2V multi-hop $\phi$ is derived by

$T\left(\phi\right)=\frac{{2\left\lceil{{\raise 2.15277pt\hbox{$\scriptstyle
L$}\kern-1.00006pt/\kern-1.49994pt\lower 1.07639pt\hbox{$\scriptstyle
r$}}}\right\rceil{t_{slot}}}}{{\left({1+{\rm
M}\left(r\right)}\right)}}+\left\lceil{{\raise 2.15277pt\hbox{$\scriptstyle
L$}\kern-1.00006pt/\kern-1.49994pt\lower 1.07639pt\hbox{$\scriptstyle
r$}}-1}\right\rceil{T_{pro}},$ (12)

where ${\rm
M}\left(r\right)=erf\left({\frac{{\eta\left(r\right)}}{{\sqrt{2}\sigma}}}\right)$
and
$\eta\left(r\right)={P_{tx}}-\theta-{N_{0}}{W_{mmWave}}-69.6-20.9{\log_{10}}r$.
Based on (12), the transmission delay $T\left(\phi\right)$ is determined by
the distance $L$ between vehicle ${V_{a}}$ and ${V_{b}}$, one hop average
communication distance $r$, the transmit power ${P_{tx}}$ and the SNR
threshold $\theta$.

###  III-B V2V transmission connectivity probability

In this paper, the unit disk communication model is introduced to determine
the connection probability between the relay vehicle node ${c_{j}}$ and
${c_{j+1}}$. For the unit disk model, any vehicle pair is able to be connected
through millimeter wave communications if the distance between each pair is
less than the coverage radius. Thus, the connectivity probability between
vehicle node ${c_{j}}$ and ${c_{j+1}}$ is expressed as

${P_{c}}=\left\\{{\begin{array}[]{*{20}{c}}{1,}&{{r_{v}}\leq{R_{v}}}\\\
{0,}&{otherwise}\end{array}}\right.,$ (13)

where ${r_{v}}$ represents the distance of vehicle node ${c_{j}}$ and
${c_{j+1}}$, the coverage radius of millimeter wave is fixed as ${R_{v}}$.

Since the number of vehicles is subject to Poisson distribution with the
parameter $\rho$, the distances ${r_{i}}\left({i=1,2,\cdots,m}\right)$ between
adjacent vehicle nodes are subject to an exponential distribution with the
parameter $\rho$. Hence, the cumulative distribution function (cdf) of
${r_{i}}$ is given by

$P\left\\{{{r_{i}}\leq{r_{0}}}\right\\}=1-{e^{-\rho{r_{0}}}},{r_{0}}\geq 0.$
(14)

Due to the memoryless property of Poisson distribution, the vehicle number
which between two connected vehicle nodes in link $\phi$ is also subject to
the Poisson distribution with the same parameter $\rho$. Without loss of
generality, let $s$ be the vehicle number within one hop communication
distance ${r_{v}}$, the expectation of vehicle number can be expressed as
$E\left(s\right)=\rho{r_{v}}$. Obviously, ${r_{v}}$ is the sum of $s$
independent exponential variables with the same parameter $\rho$, which means
that ${r_{v}}$ is subject to the Erlang distribution with shape parameter $s$.
Therefore, the cumulative distribution function (cdf) of ${r_{v}}$ is given by

$P\left\\{{{r_{v}}\leq{r_{0}}}\right\\}=F\left({{r_{0}}}\right)=1-{e^{-\rho{r_{0}}}}\sum\limits_{i=0}^{s-1}{\frac{{{{\left({\rho{r_{0}}}\right)}^{i}}}}{{i!}}},{r_{0}}\geq
0.$ (15)

According to (13) and (15), the connectivity probability $P$ of V2V multi-hop
link $\phi\left\\{{{c_{0}},{c_{1}},\cdots,{c_{k}}}\right\\}$ between vehicle
${V_{a}}$ and ${V_{b}}$ is derived by (16), where
${r_{v}}\left({v=1,2,\cdots,k}\right)$ is the one hop communication distance,
$r=E\left({{r_{v}}}\right)$ is the average one hop communication distance.

$\begin{gathered}\begin{array}[]{l}P={\mathop{\rm
P}\nolimits}\left({{r_{1}}\leq{R_{v}}}\right){\mathop{\rm
P}\nolimits}\left({{r_{2}}\leq{R_{v}}}\right)\cdots{\mathop{\rm
P}\nolimits}\left({{r_{k}}\leq{R_{v}}}\right)\\\ \\\ {\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}=\prod\limits_{i=1}^{{\rm
E}\left(k\right)}{{\mathop{\rm
P}\nolimits}\left({E\left({{r_{v}}}\right)\leq{R_{v}}}\right)}\\\ \\\ {\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}={\left({1-{e^{-\rho{R_{v}}}}\sum\limits_{i=0}^{\left\lceil{\rho
r}\right\rceil-1}{\frac{{{{\left({\rho{R_{v}}}\right)}^{i}}}}{{i!}}}}\right)^{\left\lceil{L/r}\right\rceil}}\end{array}\end{gathered}$
(16)

Since the expressions of V2V multi-hop transmission delay $T\left(\phi\right)$
and connectivity probability $P$ have been derived, substituting (12) and (16)
into (4), the V2V multi-hop link quality ${Q^{*}}\left(\phi\right)$ is derived
by (17).

$\displaystyle\begin{array}[]{l}{Q^{*}}\left(\phi\right)=\alpha
P+\beta\left({1-\frac{{T\left(\phi\right)}}{{{T_{\max}}}}}\right)\\\ \\\
=\alpha{\left({1-{e^{-\rho{R_{v}}}}\sum\limits_{i=0}^{\left\lceil{\rho
r}\right\rceil-1}{\frac{{{{\left({\rho{R_{v}}}\right)}^{i}}}}{{i!}}}}\right)^{\left\lceil{L/r}\right\rceil}}+\beta\left({1-\frac{{\frac{{2\left\lceil{{\raise
1.07639pt\hbox{$\scriptstyle L$}\kern-0.50003pt/\kern-0.74997pt\lower
0.5382pt\hbox{$\scriptstyle r$}}}\right\rceil{t_{slot}}}}{{\left({1+{\rm
M}\left(r\right)}\right)}}+\left\lceil{{\raise 1.50694pt\hbox{$\scriptstyle
L$}\kern-0.70004pt/\kern-1.04996pt\lower 0.75346pt\hbox{$\scriptstyle
r$}}-1}\right\rceil{T_{pro}}}}{{{T_{\max}}}}}\right)\end{array}$ (17)

##  IV Numerical Results and Discussion

In this section, the V2V multi-hop link quality with different vehicles
densities and different average one hop communication distances are
numerically analyzed. The different requirements of different vehicular
network services on messages transmission delay and transmission success
probability are reflected by weight factors. Some default parameters are
configured as follows: $\rho=0.05\sim 0.25$, $L=1000m$, ${R_{v}}=100m$,
${t_{slot}}=50\mu s$, ${T_{pro}}=20\mu s$, ${T_{\max}}=20ms$,
${P_{tx}}=30dBm$, ${W_{mmWave}}=200MHz$, ${N_{0}}=-174dBm/Hz$ [4, 13].

Figure 2: V2V multi-hop link quality with respect to vehicles density $\rho$
and average one hop communication distance $r$ ( $\alpha=\beta=0.5$ ).

Fig. 2 plots the V2V multi-hop link quality as a function of vehicles density
$\rho$ and average one hop communication distance $r$ for the situation of
fixed weight factors $\alpha=\beta=0.5$. As shown in Fig. 2, multi-hop link
quality ${Q^{*}}$ is a monotonically increasing function of vehicle density
$\rho$. When the vehicle density is fixed, with the increase of average
communication distance, multi-hop link quality ${Q^{*}}$ first increases and
then deceases, which implies there exists an optimal average communication
distance conducing to the best link quality.

Figure 3: V2V multi-hop link quality with respect to different average one hop
communication distance $r$ ( $\alpha=\beta=0.5$ ).

Fig. 3 shows V2V multi-hop link quality with respect to different average one
hop communication distance $r$. Without loss of generality, the weight factors
$\alpha$ and $\beta$ are both configured as $\alpha=\beta=0.5$ in Fig. 3. it
is shown that there exists an optimal average one hop communication distance
to achieve the best link quality. The reason is that when the vehicle density
is fixed, with a relatively small value of $r$, the relay vehicles will
communicate with a vehicle that has a relatively close distance. Hence, this
V2V multi-hop link will have a large of hop numbers. In this case, a
relatively small value of $r$ conduces to a large connectivity probability.
But the transmission delay increases with the increase of the hop number.
Hence, the quality of this link is relatively low. On the other hand, a multi-
hop link with a relatively large value of $r$ leads to a low transmission
delay but the connectivity probability is relatively low. Hence, there exists
a link with an optimum average distance which leads to the best link quality
by trading off the transmission delay and connectivity probability. It is
worth noting that with a different vehicle density, the optimum average
distance is different. This result is helpful to find the best V2V
communication strategy in vehicular networks.

Figure 4: V2V multi-hop link quality with respect to different vehicle density
$\rho$ ( $\alpha=\beta=0.5$ ).

In Fig. 4, the effect of vehicles density $\rho$ on V2V multi-hop link quality
${Q^{*}}$ is investigated. The weight factors are both fixed at
$\alpha=\beta=0.5$. As shown in Fig. 4, when one hop communication distance
$r$ is fixed, the link quality increases with the increase of the vehicle
density. The reason is that a large vehicle density implies there are more
optional vehicles to be relay vehicle nodes. Hence, the connectivity
probability increases with the increase of vehicle density.

Figure 5: V2V multi-hop link quality with respect to different weight factors
$\alpha$ and $\beta$ ( $\rho=0.07$ ).

Fig. 5 shows V2V multi-hop link quality with respect to different weight
factors $\alpha$ and $\beta$. Without loss of generality, the vehicle density
is configured as $\rho=0.07$ in Fig. 5, V2V multi-hop links have different
link qualities when weight factors of transmission delay and transmission
success probability are different. These results indicate that with different
requirements of different vehicular network services on messages transmission
delay and transmission success probability, the multi-hop link quality and the
optimal average communication distance are different.

##  V Conclusion

In this paper, a new link quality indicator has been proposed to evaluate
multi-hop link quality by taking the transmission delay and the connectivity
both into consideration. Furthermore, the analytical expressions of V2V multi-
hop transmission delay and connectivity probability have been derived. Based
on these, relationships between the link quality and some network parameters
have been analyzed and numerical results shows that V2V link quality increases
with the increase of the vehicle density, and there exists a optimal
communication distance corresponding to the requirement of vehicular network
services. These results indicated that when network parameters changes, the
V2V multi-hop link quality can be improved by adjusting the link strategies.
These results can also provide some useful guidelines for selection strategies
of V2V communication links in vehicular networks.

## Acknowledgment

The author would like to acknowledge the support from the NSFC Major
International Joint Research Project under the grant 61210002, the Hubei
Provincial Science and Technology Department under the grant 2016AHB006, the
Fundamental Research Funds for the Central Universities under the grant
2016AHB006. This research is partially supported by EU FP7-PEOPLE-IRSES,
project acronym CROWN (grant no. 610524).

## References

  * [1] S. h. Sun, J. l. Hu, Y. Peng, X. m. Pan, L. Zhao and J. y. Fang, “Support for vehicle-to-everything services based on LTE," IEEE Wireless Communications., vol. 23, no. 3, pp. 4–8, June. 2016.
  * [2] G. Karagiannis et al., “Vehicular Networking: A Survey and Tutorial on Requirements, Architectures, Challenges, Standards and Solutions," IEEE Communications Surveys & Tutorials., vol. 13, no. 4, pp. 584–616, Fourth Quarter. 2011.
  * [3] X. Ge, H. Cheng, G. Mao, Y. Yang and S. Tu, “Vehicular Communications for 5G Cooperative Small-Cell Networks," IEEE Transactions on Vehicular Technology., vol. 65, no. 10, pp. 7882–7894, Oct. 2016.
  * [4] A. Ghosh et al., “Millimeter-Wave Enhanced Local Area Systems: A High-Data-Rate Approach for Future Wireless Networks," IEEE Journal on Selected Areas in Communications., vol. 32, no. 6, pp. 1152–1163, June. 2014.
  * [5] A. Mastrosimone and D. Panno, “New challenge: Moving network based on mmWave technology for 5G era," 2015 International Conference on Computer, Information and Telecommunication Systems (CITS)., Gijon, 2015, pp. 1–5.
  * [6] X. Ge, S. Tu, T. Hao, Q. Li and G. Mao, “Energy Efficiency of Small Cell Backhaul Networks Based on Gauss-Markov Mobile Models," IET Networks., vol. 4, no. 2, pp. 158–167, Mar. 2015.
  * [7] G. Araniti, C. Campolo, M. Condoluci, A. Iera and A. Molinaro, “LTE for vehicular networking: a survey," IEEE Communications Magazine., vol. 51, no. 5, pp. 148–157, May. 2013.
  * [8] M. L. Sichitiu and M. Kihl, “Inter-vehicle communication systems: a survey," IEEE Communications Surveys & Tutorials., vol. 10, no. 2, pp. 88–105, Second Quarter. 2008.
  * [9] D. Hu, J. Wu and P. Fan, “Minimizing End-to-End Delays in Linear Multihop Networks," IEEE Transactions on Vehicular Technology., vol. 65, no. 8, pp. 6487–6496, Aug. 2016.
  * [10] O. Choi, S. Kim, J. Jeong, H. W. Lee and S. Chong, “Delay-Optimal Data Forwarding in Vehicular Sensor Networks," IEEE Transactions on Vehicular Technology., vol. 65, no. 8, pp. 6389–6402, Aug. 2016.
  * [11] K. Stamatiou and M. Haenggi, “Optimal Spatial Reuse in Poisson Multi-Hop Networks," 2010 IEEE Global Telecommunications Conference GLOBECOM 2010., Miami, FL, 2010, pp. 1–6.
  * [12] X. Li, B. j. Hu, H. Chen, B. Li, H. Teng and M. Cui, “Multi-hop delay reduction for safety-related message broadcasting in vehicle-to-vehicle communications," IET Communications., vol. 9, no. 3, pp. 404–411, Feb. 2015.
  * [13] W. Zhang et al., “Multi-Hop Connectivity Probability in Infrastructure-Based Vehicular Networks," IEEE Journal on Selected Areas in Communications., vol. 30, no. 4, pp. 740–747, May. 2012.
  * [14] X. Li, B. j. Hu, H. Chen and J. Ye, “Analysis of connectivity probability and hop count for multi-hop broadcasting in vehicular networks," 2013 International Conference on Connected Vehicles and Expo (ICCVE)., Las Vegas, NV, 2013, pp. 551–555.
  * [15] J. Zhao, Y. Chen and Y. Gong, “Study of connectivity probability based on cluster in vehicular Ad Hoc networks," 2016 8th International Conference on Wireless Communications & Signal Processing (WCSP)., Yangzhou, 2016, pp. 1–5.
  * [16] C. Shao, S. Leng, Y. Zhang, A. Vinel and M. Jonsson, “Performance Analysis of Connectivity Probability and Connectivity-Aware MAC Protocol Design for Platoon-Based VANETs," IEEE Transactions on Vehicular Technology., vol. 64, no. 12, pp. 5596–5609, Dec. 2015.
  * [17] S. I. Sou and O. K. Tonguz, “Enhancing VANET Connectivity Through Roadside Units on Highways," IEEE Transactions on Vehicular Technology., vol. 60, no. 8, pp. 3586–3602, Oct. 2011.