11institutetext: Guilin University of Electronic Technology University,
Guilin, China
11email<EMAIL_ADDRESS> Corresponding author: Zetao
Jiang,11email<EMAIL_ADDRESS>22institutetext: City University of
Hong Kong, Hong Kong SAR of China
22email<EMAIL_ADDRESS>33institutetext: Baidu Research,
Beijing, China. 44institutetext: ZTE Corporation, Shenzhen, China
44email<EMAIL_ADDRESS>
# Infrared Image Super-Resolution via Heterogeneous Convolutional WGAN

Yongsong Huang 11 0000-0003-3114-9206 Zetao Jiang Qingzhong Wang 11
0000-0002-0914-2131 22 33 0000-0003-1562-8098 Qi Jiang 11 Guoming Pang 44

###### Abstract

Image super-resolution is important in many fields, such as surveillance and
remote sensing. However, infrared (IR) images normally have low resolution
since the optical equipment is relatively expensive. Recently, deep learning
methods have dominated image super-resolution and achieved remarkable
performance on visible images; however, IR images have received less
attention. IR images have fewer patterns, and hence, it is difficult for deep
neural networks (DNNs) to learn diverse features from IR images. In this
paper, we present a framework that employs heterogeneous convolution and
adversarial training, namely, heterogeneous kernel-based super-resolution
Wasserstein GAN (HetSRWGAN), for IR image super-resolution. The HetSRWGAN
algorithm is a lightweight GAN architecture that applies a plug-and-play
heterogeneous kernel-based residual block. Moreover, a novel loss function
that employs image gradients is adopted, which can be applied to an arbitrary
model. The proposed HetSRWGAN achieves consistently better performance in both
qualitative and quantitative evaluations. According to the experimental
results, the whole training process is more stable.

###### Keywords:

Super-resolution . Infrared image . Image processing . Heterogeneous kernel-
based convolution . Generative adversarial networks

##  1 Introduction

Image super-resolution (SR) reconstruction is a very active topic in computer
vision as it offers the promise of overcoming some of the limitations of low-
cost imaging sensors. Infrared (IR) image super-resolution plays an important
role in the military and medical fields and many other areas of vision
research. A major problem with IR thermal imaging is that IR images are
normally low resolution since the size and precision of IR sensors can be
limited. Image super-resolution is a promising and low-cost way to improve the
resolution and quality of IR images. Generally, image super-resolution methods
based on deep learning can be classified into two categories, namely, models
based on generative adversarial networks (GANs)[20, 16] and models based on
deep neural networks (DNNs)[26, 12, 8, 5, 6, 21, 18, 27], both of which have
achieved satisfying results on visible images. These methods can achieve a
good peak signal-to-noise ratio (PSNR). However, they do not consider the
visual characteristics of the human eye. The human eye is more sensitive to
contrast differences with a lower spatial frequency. The sensitivity of the
human eye to differences in brightness contrast is higher than its sensitivity
to color, and the perception of a region by the human eye is affected by the
surrounding areas. Situations in which the results of the evaluation are
inconsistent with the subjective feeling of a viewer therefore often occur. We
recommend using the structural similarity index (SSIM). The learning-based
SISR algorithm learns a mapping between low-resolution (LR) and high-
resolution (HR) image patches. The prior knowledge used is either explicit or
implicit, depending upon the learning strategy. The super-resolution
convolutional neural network (SRCNN)[4] algorithm introduced deep learning
methods to SISR. A faster model, the faster super-resolution convolutional
neural network (FSRCNN)[6], improved upon the SRCNN model and has also been
applied to SISR. The efficient subpixel convolutional neural network (ESPCN)
algorithm[21] and information multi-distillation network (IMDN)[12] were also
proposed to further improve the computational efficiency. A significant
advance in the generation of visually pleasing results is the super-resolution
generative adversarial network (SRGAN)[16]. A large number of SR methods have
been presented, most of which are designed for natural images. Fewer methods
have been designed for infrared images. GANs provide a powerful framework for
generating plausible-looking natural images. However, they have problems with
instability[11, 25]. Wasserstein generative adversarial networks (WGAN)[1] was
proposed as a solution to this problem. Given the issues that there are few
infrared image features and that super-resolution reconstruction is difficult,
the building units of the neural network and the loss functions that provide
better constraints each play an important role in improving the performance of
the GAN.

In this paper, we propose a novel approach for infrared image super-
resolution. We revisited the key components of SRGAN and improved the model in
two ways. First, we improved the network structure by introducing the
heterogeneous kernel-based residual block, which has fewer parameters than
previous algorithms, and it is easier to train. HetConv enables multiscale
extraction of image features by combining convolutional kernels of different
sizes. Second, we developed an improved loss function: the gradient cosine
similarity loss function. The traditional loss function does not consider the
characteristics of infrared images, and the gradient cosine similarity loss
function takes the image gradient as an important feature for better-
supervised training. The experimental datasets are publicly available[10], and
the experimental effects can be validated.

The remainder of this paper is organized as follows. The related works are
presented in Section 2. We describe the HetSRWGAN architecture and the
gradient cosine similarity loss function in Section 3. A quantitative
evaluation of new datasets, as well as visual illustrations, is provided in
Section 4. The paper concludes with a conclusion in Section 5.

##  2 Related Works

###  2.1 Generative Adversarial Networks

Generative adversarial networks[7] were proposed by Goodfellow, based on game
theory. In a pioneering work, C. Ledig et al[16] used SRGAN to learn the
mapping from LR to HR images in an end-to-end manner, achieving performance
superior to that of previous work. A low-resolution image $I^{LR}$ is input to
a generator network to generate the reconstructed image $I^{SR}$, while a
discriminator network takes the high-resolution images $I^{HR}$ and $I^{SR}$
as input to determine which is the real image and which is the reconstructed
image.

###  2.2 HetConv: Heterogeneous Kernel-Based Convolutions

The heterogeneous kernel-based convolutions algorithm was proposed by
Pravendra Singh[22]. Pravendra Singh et al presented a novel deep learning
architecture in which the convolution operation uses heterogeneous kernels.
Compared to standard convolution operations, the proposed HetConv reduces the
number of calculations (FLOPs) and parameters while still maintaining the
presentation efficiency. HetConv is especially different from the depthwise
convolutional filter used to perform depthwise convolution (DWC)[3], the
pointwise convolutional filter used to perform pointwise convolution (PWC)[24]
and the groupwise convolutional filter used to perform groupwise convolution
(GWC)[15]. In HetConv, a variable $P$ is used to control how much of the
normal convolution kernel is retained in the operation. In addition, the total
reduction is $R$ for $K\times K$ kernels. The number of calculations of
HetConv is compared with that of the normal convolution, as shown in Eq 1.

$R_{{HetConv}}=\frac{1}{P}+\frac{(1-1/P)}{K^{2}}$ (1)

According to the characteristics of the heterogeneous kernel-based
convolutions, we used a skip connection when designing the generator network
structure. The HetSRWGAN structure is shown in Figure 1.

##  3 HetSRWGAN

###  3.1 HetSRWGAN Architecture

Our main goal was to improve the overall visual performance of SR. In this
section, we describe our improved network architecture. The main difference
between the GAN and WGAN[1] is that the sigmoid function and batch
normalization (BN)[13] layer of the discriminator network are removed. The
entire neural network is stabilized by gradient punishment[1]. It has been
shown that removing the BN layer improves performance and reduces
complexity[25, 18]. Further, the removal of the BN layer contributes to
improving the robustness of the network and reduces the computational
complexity and memory consumption. We replaced the original basic block with a
heterogeneous kernel-based residual block (HetResidual block), which includes
HetConv, as depicted in Section 3.2. The HetResidual block is the basic
network building unit. This block requires fewer parameters than the original
basic block, improves network performance, and reduces computational
complexity. More parameters may lead to a higher probability of mode
collapse[25, 11], so reducing the total number of parameters is beneficial.
For the discriminator network, we deepened the network structure and
experimentally demonstrated that this modification improves image quality. The
detailed experimental results are given in Section 4. According to the
characteristics of the heterogeneous kernel-based convolutions, we used a skip
connection when designing the generator network structure.

Figure 1: Architecture of heterogeneous kernel-based super-resolution
Wasserstein GAN with the corresponding kernel size ($k$), number of feature
maps ($n$), stride ($s$) for each convolutional layer, padding ($p$) and
number of the normal convolution kernel ($P$) (Best viewed in color).

###  3.2 Heterogeneous Kernel-Based Residual Block

Kaiming He et al[9] first proposed the residual block structure and solved
some of the problems caused by deep neural networks by introducing a skip
connection and combination. The heterogeneous kernel-based residual block is
shown in detail in Figure 2. The relevant formula is analyzed as follows:

$\mathbf{y}_{i}=h\left(\mathbf{x}_{i}\right)+\mathcal{F}\left(\mathbf{x}_{i},\mathcal{W}_{i}\right)$
(2)

$\mathbf{x}_{i+1}=\mathcal{F}\left(\mathbf{x}_{i},\mathcal{W}_{i}\right)+h\left(\mathbf{x}_{i}\right)$
(3)

where $\mathcal{F}$ stands for the heterogeneous kernel-based residual block
processing. Since $h\left(\mathbf{x}_{l}\right)$ is an identity map, Eq 3 can
be derived:

$\mathbf{x}_{i+1}=\mathcal{F}\left(\mathbf{x}_{i},\mathcal{W}_{i}\right)+\mathbf{x}_{i}$
(4)

Figure 2: Architecture of Heterogeneous Kernel-Based Residual Block.

###  3.3 Gradient Cosine Similarity Loss Function

To make the reconstructed image $I^{SR}$ obtained from the generator network
closer to the high-resolution image $I^{HR}$, it is necessary to provide a
neural network loss function with effective constraints. We chose the spatial
gradient of the image as the feature that measures the similarity between two
images. When there is an edge in the image, there must be a high gradient
value. Conversely, when there is a relatively smooth region in an image, the
gray value changes little, and the corresponding gradient is also small. Using
the gradient as a feature not only captures contours, images, and some texture
information but also further weakens the effects of lighting. The gradient of
an image at a pixel point $(x,y)$ is a vector with direction and size. $G_{x}$
is the gradient of $I$ in direction $X$, and $G_{y}$ is the gradient of $I$ in
direction $Y$ direction. The gradient vector $\boldsymbol{v}$ can be expressed
as Eq 5.

$\boldsymbol{v}=\left[\boldsymbol{G}_{\boldsymbol{x}},\boldsymbol{G}_{\boldsymbol{y}}\right]^{\boldsymbol{T}}$
(5)

The infrared images in the dataset are RGB images, which are three-channel
images[2]. The gradient between the high-resolution three-channel image
$I^{HR}$ and the super-resolution reconstructed three-channel image $I^{SR}$
can be expressed as Eqs 6 and 7.

$\mathbf{I}_{G}^{HR}=\left(\mathbf{I}_{G_{r}}^{HR},\mathbf{I}_{G_{g}}^{HR},\mathbf{I}_{G_{b}}^{HR}\right)$
(6)

$\mathbf{I}_{G}^{SR}=\left(\mathbf{I}_{G_{r}}^{HR},\mathbf{I}_{G_{g}}^{HR},\mathbf{I}_{G_{b}}^{HR}\right)$
(7)

$\mathbf{I}_{G}^{HR}$ indicates the gradient vector of the high-resolution
image. The subscript of $G_{g}$ indicates the $green$ channel of the high-
resolution image. Other subscripts indicate different image channels of $red$
and $blue$. For super-resolution reconstructed images $I^{SR}$, the subscript
indicates the same. We use the cosine similarity to measure the similarity
between these two vectors, as shown in Eq 8.

$\cos_{sim}(\mathbf{X},\mathbf{Y})=\frac{\mathbf{X}\cdot\mathbf{Y}}{\|\mathbf{X}\|\cdot\|\mathbf{Y}\|}$
(8)

$\mathbf{X}$ and $\mathbf{Y}$ represent two matrices that can be multiplied by
points. The high-resolution image gradient $\mathbf{I}_{G}^{HR}$ and the SR
image gradient $\mathbf{I}_{G}^{SR}$ can be calculated according to algorithm
1.

We calculate the cosine similarity by stretching the two matrices into a one-
dimensional vector. Likewise, the similarity between the high-resolution image
gradient $\mathbf{I}_{G}^{HR}$ and the SR image gradient $\mathbf{I}_{G}^{SR}$
can be calculated according to algorithm 1. The generator loss function of the
SRGAN and WGAN includes content loss and adversarial loss. The generator loss
function of HetSRWGAN is shown in Eq 9:

Algorithm 1 Gradient Cosine Similarity Loss Function

Input: $I^{SR}$ , $I^{HR}$
Output: Gradient Cosine Similarity

1:Infrared images can be processed into RGB images[2].

2:while not convergent do

3:
$I^{HR}\longrightarrow\left(\mathbf{I}_{G_{r}}^{HR},\mathbf{I}_{G_{g}}^{HR},\mathbf{I}_{G_{b}}^{HR}\right)$

4:
$I^{SR}\longrightarrow\left(\mathbf{I}_{G_{r}}^{SR},\mathbf{I}_{G_{g}}^{SR},\mathbf{I}_{G_{b}}^{SR}\right)$
$\triangleright$ Gradient matrix.

5:
${\bf{X}^{{}^{\prime}}}={\left[{{\bf{I}}_{{G_{r}}}^{HR},{\bf{I}}_{{G_{g}}}^{HR},{\bf{I}}_{{G_{b}}}^{HR}}\right]_{1\times
m}}$

6:
${\bf{Y}^{{}^{\prime}}}={\left[{{\bf{I}}_{{G_{r}}}^{SR},{\bf{I}}_{{G_{g}}}^{SR},{\bf{I}}_{{G_{b}}}^{SR}}\right]_{1\times
m}}$ $\triangleright$ Matrix compression.

7:
$F_{\cos}\left(\bf{X}^{{}^{\prime}},\bf{Y}^{{}^{\prime}}\right)=\frac{\bf{X}^{{}^{\prime}}\cdot{{\bf{Y}^{{}^{\prime}}}^{T}}}{\left\|\bf{X}^{{}^{\prime}}\right\|\cdot\left\|\bf{Y}^{{}^{\prime}}\right\|}$
$\triangleright$ Cosine similarity.

8:return
${F_{\cos}}\left({{\bf{X}^{{}^{\prime}}},{\bf{Y}^{{}^{\prime}}}}\right)$

$\operatorname{Loss}^{SR}=l_{X}^{SR}+\lambda
l_{Gen}^{SR}+\mu\left(1-F_{cos}\right)$ (9)

where $l_{X}^{SR}$ and $l_{Gen}^{SR}$ represent the content loss and
adversarial loss, respectively.

##  4 Experiments and Evaluations

###  4.1 Training Details

Following SRGAN, all experiments were performed with a scaling factor of (4,
applied to the 2x2 image) between LR and HR images. We used the PSNR and
structural similarity index (SSIM) to evaluate the reconstructed images.
Super-resolved images were generated using the reference methods, including
SRMD, IMDN, DPSR, DBPN, SRCNN, FSRCNN, ESPCN, SRGAN, and super-resolution
Wasserstein GAN (SRWGAN). The generator was trained using the loss function
presented in Eq 9 with $\lambda=0.001$ and $\mu=0.001$. The learning rate was
set to 0.0001. We observed that a larger batch size benefits training a deeper
network. We set the batch size to 64. For optimization, we used Adam [14] with
$\beta_{1}=0.9$ in the generator. For the WGAN, we used the Asynchronous SGD
(ASGD)[19] in the discriminator. We implemented our models with the PyTorch
framework and trained them using NVIDIA TITAN X (Pascal) GPUs.

Figure 3: (a): Changes of loss function with the number of iterations on the
dataset CVC-09-1K (b): CVC-09-1K Dataset Training Average PSNR, (c): CVC-09-1K
Dataset Training Average SSIM

For training, we primarily used the CVC-09: FIR Sequence Pedestrian
Dataset[23]. In CVC-09, a sequence is composed of two sets of images, the day
and night sets, a designation which refers to the time of day at which they
were acquired. The first set contains 5990 frames, the second set contains
5081 frames, and each sequence was divided into training and testing sets. We
performed experiments on two datasets, namely, fusionA-22 and fusionC-22,
which contain images obtained by fusing infrared and visible light, using the
methods of literature[17] and literature[28], respectively[10]. An image after
the fusion of IR and visible light images will have better visual quality, and
it will be easier to distinguish details such as characters in the image. The
fused image also maintains significant information from the infrared image but
makes the performance of the algorithm more easily visualized.

Table 1: Quantitative evaluation of SR algorithms: Average PSNR/SSIM for scale factors $\times 4$. SRGAN1 has model collaps. Algorithm | Params $\downarrow$ | FLOPs $\downarrow$ | PSNR/$dB$ $\uparrow$ | SSIM $\uparrow$
---|---|---|---|---
fusionA-22 | fusionC-22 | fusionA-22 | fusionC-22
HetSRWGAN(Ours) | 0.496M | 0.095G | 30.302 | 31.987 | 0.858 | 0.883
SRMD[27] | 1.552M | 0.063G | 33.210 | 33.850 | 0.834 | 0.852
IMDN[12] | 0.893M | 91.70G | 29.725 | 30.057 | 0.735 | 0.751
DPSR[26] | 2.995M | 0.052G | 32.692 | 31.662 | 0.825 | 0.810
DBPN[8] | 10.41M | 0.106G | 17.438 | 17.934 | 0.816 | 0.842
SRWGAN | 0.956M | 0.132G | 28.319 | 28.520 | 0.799 | 0.805
SRGAN1[16] | 0.956M | 0.132G | 5.150 | 30.444 | 0.278 | 0.871
SRCNN[5] | 0.148M | 0.182M | 29.437 | 30.170 | 0.754 | 0.789
FSRCNN[6] | 0.013M | 0.077M | 30.624 | 31.094 | 0.797 | 0.822
ESPCN[21] | 0.061M | 0.001G | 30.814 | 31.607 | 0.789 | 0.819

###  4.2 Performance of the Final Networks

We compared the performance of three different super-resolution reconstruction
algorithms based on generative adversarial networks. Since the GAN cannot
simply use the loss function to judge the network training situation, we
selected the image after the end of each batch of training to calculate the
PSNR and SSIM values. When there are too many model parameters, mode collapse
will occur. As the number of iterations increased, SRWGAN was more robust.
SRGAN experiences mode collapse. Although the SRWGAN introduces gradient
punishment to solve the problem that the network cannot be trained in the
later stages, using cross-entropy as a loss function requires considerable
time to adjust parameters and still cannot guarantee the stability of the
model. Therefore, the loss function will have a negative value, which will
cause the curve to be discontinuous. There was no situation where convergence
or instability was not possible. The results are shown in Figure 3.

The total number of parameters for HetSRWGAN was reduced by 496657 compared to
that for SRGAN, a reduction of 52% (Table 1). The significantly reduced total
number of parameters helps to reduce the computational complexity of the model
and improve robustness.

Figure 4: Time efficiency comparison of all reconstruction methods. The same
colour means the same method.The horizontal axis represents the time required
for one training session, and the vertical axis represents an objective
indicator after the model converges. Figure 5: Super-resolution image
reconstruction effect comparison schematic diagram. From left to right:
original HR image, HetSRWGAN, SRGAN, ESPCN, FSRCNN, SRCNN, SRWGAN, IMDN,
Corresponding PSNR and SSIM are shown below the figure. Red indicates the
best. [$\times 4$ upscaling]

The SRMD model obtains better performance based on the PSNR; however, it has a
large number of parameters, resulting in long training and inference times and
greater memory consumption. (Table 1, Figure 4) The objective evaluation
indices of the average PSNR and average SSIM were calculated. DNNs have a good
effect in reconstructing visible images, but because of the features of
single-frame infrared images with few features and high redundancy, the
reconstruction effect is not good. (Table 1)

SRGAN does not provide control of the generation process, and there is mode
collapse (see Figure 1). The new loss function and HetResidual block make the
models faster to train and converge. The HetSRWGAN takes 24 seconds to train
each batch, and the average SSIM is 0.858 and 0.883. (see Table 1) Compared
with other methods, HetSRWGAN has the best time efficiency and average SSIM.
Figure 5 shows the reconstructions produced by different algorithms.

Figure 5 shows that our proposed HetSRWGAN outperformed previous approaches in
both sharpness and amount of detail. Previous GAN-based methods sometimes
introduce artifacts. For example, SRGAN adds noise to the entire image.
HetSRWGAN removes these artifacts and produces natural results.

##  5 Conclusions

Our proposed HetSRWGAN method can be well used for infrared image super-
resolution reconstruction. We proposed a novel architecture composed of
several heterogeneous kernel-based residual blocks without BN layers. A
gradient cosine similarity loss function was developed, which can provide
stronger supervision of image details, such as edges, and the reconstructed
high-resolution images contain more details and realistic textures.

## Acknowledgement

This research supported by the Nature Science Foundation of China grants
No.61876049, and No.61762066.

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# Radio Proper Motions of the Energetic Pulsar PSR J1813$-$1749

Sergio A. Dzib Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany  Luis F. Rodríguez Instituto de Radioastronomía y
Astrofísica, Universidad Nacional Autónoma de México, Morelia, Michoacán
58089, Mexico  Mesoamerican Centre for Theoretical Physics, Universidad
Autónoma de Chiapas, Tuxtla Gutiérrez, Chiapas 29050, Mexico

###### Abstract

PSR J1813–1749 has peculiarities that make it a very interesting object of
study. It is one of the most energetic and the most scattered pulsar known. It
is associated with HESS J1813–178, one of the brightest and most compact TeV
sources in the sky. Recently, Ho et al. used archival X-ray Chandra
observations separated by more than 10 years and determined that the total
proper motion of PSR J1813–1749 is $\sim 66$ mas yr-1, corresponding to a
velocity of $\sim$1900 km s-1 for a distance of 6.2 kpc. These results would
imply that this pulsar is the fastest neutron star known in the Galaxy and, by
estimating the angular separation with respect to the center of the associated
supernova remnant, has an age of only $\sim 300$ years, making it one of the
youngest pulsars known. Using archival high-angular-resolution VLA
observations taken over 12 years we have estimated the radio proper motions of
PSR J1813–1748 to be much smaller:
($\mu_{\alpha}\cdot\cos(\delta),\,\mu_{\delta}$)=($-5.0\pm 3.7,\,-13.2\pm
6.7$) mas yr-1, or a total proper motion of $14.8\pm 5.9$ mas yr-1. The
positions referenced against quasars make our results reliable. We conclude
that PSR J1813–1749 is not a very fast moving source. Its kinematic age using
the new total proper motion is $\sim 1350$ years. This age is consistent
within a factor of a few with the characteristic age of the pulsar and with
the age estimated from the broadband spectral energy distribution of HESS
J1813–178, as well as the age of the associated SNR.

pulsars:individual (CXOU J181335.1$-$174957) — ISM: individual objects (HESS
J1813-178, G12.82$-$0.02) — ISM: supernova remnants — techniques:
interferometric

††facilities: VLA††software:  CASA (McMullin et al., 2007).

##  1 Introduction

Figure 1: Background: VLA image of PSR J1813–1749 as observed in February
2006. Contours: PSR J1813–1749 as observed in December 2017 at X-band. Contour
levels are -3, 3, 6, and 9 times 7 $\mu$Jy beam-1, the noise level on this
epoch. The blue ellipse indicates the expected position of the radio source in
December 2017, following the proper motion measured by Ho et al. (2020), the
ellipse semi-major axis sizes consider the propagated errors.

PSR J1813–1749 (=CXOU J181335.1$-$174957) is the most scattered and the second
most energetic pulsar in the Milky Way (Halpern et al., 2012; Camilo et al.,
2021). First discovered and characterized using Chandra X-ray observations
(Gotthelf & Halpern, 2009; Halpern et al., 2012), it has a spin-down rate of
$\dot{P}=1.265\times 10^{-13}$, corresponding to a spin-down luminosity of
$\dot{E}=5.6\times 10^{37}$ erg s-1, values only below those measured for the
Crab pulsar (e.g., Halpern et al., 2012).

First attempts to detect the radio pulsed emission at low frequencies (1–2
GHz) from this pulsar failed (Helfand et al., 2007; Halpern et al., 2012; Dzib
et al., 2018). Recently, Camilo et al. (2021) finally detected the pulsed
radio emission at higher frequencies (4–10 GHz) and showed that the pulses are
highly scattered. The fact that the scattering is more severe at lower
frequencies probably explains the early failed attempts to detect the radio
pulsed emission. Camilo et al. (2021) show that the pulsed emission is
consistent with the radio continuum source detected by Dzib et al. (2010) and
Dzib et al. (2018) with the Karl G. Jansky Very Large Array (VLA) at similar
frequencies. Based on the high column density at X-rays and the large
dispersion measure, Camilo et al. (2021) also place a lower limit to its
distance of 6.2 kpc, that could be as large as 12 kpc.

The young and relatively compact ($\sim 2^{\prime}$ diameter) shell-type radio
supernova remnant (SNR) G12.82-0.02 (Brogan et al., 2005) and pulsar wind
nebula (PWN) observed at X-rays (Helfand et al., 2007; Funk et al., 2007;
Gotthelf & Halpern, 2009) have been associated to PSR J1813–1749. SNR
G12.82–0.02 and the PWN are associated with one of the brightest and most
compact objects discovered by the HESS Galactic Plane Survey (Aharonian et
al., 2005), the TeV source HESS J1813–178. This HESS source has been
associated with continuum high-energy emission from X-rays to gamma rays
(Reimer et al., 2008; Ubertini et al., 2005; Abdo et al., 2009; Albert et al.,
2006).

Recently, using archival X-ray observations, Ho et al. (2020) determined large
proper motions for PSR J1813–1749 of
($\mu_{\alpha}\cdot\cos(\delta),\,\mu_{\delta}$)= (${-64\pm 9},\,-14\pm 7$)
mas yr-1. As the pulsar is at an angular distance of $\sim 20^{\prime\prime}$
from the center of the SNR G12.82–0.02 the large proper motions would indicate
a young age of around 300 years, making it one of the youngest pulsars known.
This age, while consistent at the lower end of the age range of 285 to 2500
years for SNR G12.82–0.02 (Brogan et al., 2005), is however in conflict with
the age estimated for HESS J1813–178 of 2500 years (Zhu et al., 2018) and the
characteristic age of the pulsar of 5600 years (Halpern et al., 2012). As
discussed by Camilo et al. (2021), the total proper motion of $\sim 66$ mas
yr-1 would imply a tangential velocity of the order of 2,000 km s-1 at 6.2
kpc, the lower limit of the distance. This velocity is larger than that of any
well-measured velocity for a neutron star (cf., Deller et al. 2019). The total
proper motion of PSR J1814-1749 is an interesting subject to study, and in
this paper we present the proper motions measured with archival high angular
resolution observations taken with the VLA by us and by other groups.

##  2 VLA Observations

For our astrometric study we looked for VLA observations with high angular
resolution. The VLA provides the finest angular resolution in its most
extended configurations A and B. We also restricted our search to the C-band
(4–8 GHz) and X-band (8–12 GHz) which provide the best sensitivity, and where
PSR J1813–1749 has been previously detected. For the best astrometry it is
also recommended that the observations are phase referenced to the same quasar
(gain calibrator), as this provides nearly absolute astrometry.

We found two observational campaigns with all the above criteria and where the
target source has been detected. These observations have been previously
reported by Dzib et al. (2010) and Dzib et al. (2018). The first is one
observation done in 2006 with the historical VLA at 4.86 GHz, using the
A-configuration, under project AL673. The second, includes a series of 12
observations, 5 centered at a main frequency of 6.0 GHz and 7 at 10.0 GHz that
were made as part of project 17B-028. The observations were done in the
B-configuration covering the period from September 2017 to February 2018. We
also found a third observational campaign done in October 2012 under project
12B-278, at the mean frequency of 9.0 GHz, using the VLA in its A
configuration. However, this third campaign used a different gain calibrator.
In section 2.1 we discuss how we corrected for this limitation. All
observations were calibrated and imaged using the CASA software. Positions
were determined from the image using the CASA task imfit. Fluxes and other
emission properties were already given and discussed by Dzib et al. (2010) and
Dzib et al. (2018), and in this work we focus on the astrometry. Basic
properties of the image and position of the target source over time are listed
in Table LABEL:tab:vlapsr

Table 1: Image results, and positions of PSR J1813$-$1749.

Epoch | $\nu$ | VLA | Gain | Beam size | Noise | R.A. (J2000) | Dec. (J2000)
---|---|---|---|---|---|---|---
(yyyy.mm.dd) | (GHz) | Conf. | Calibrator |  [${}^{\prime\prime}\times^{\prime\prime};\,\,^{\circ}$] |  ($\mu$Jy bm-1) |  (18${}^{h}13^{m}$) |  (–17${}^{\circ}49^{\prime}$)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
2006 Feb. 25 | 4.8 | A | J1811-2055 |  $0.77\times 0.41;$ –9 | 19 |  35.s177 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$003 |  57.${}^{\prime\prime}62\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$08
2012 Oct. 05a | 9.0 | A | J1733-1304 |  $0.45\times 0.22;$ 19 | 11 |  35.s174 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$009 |  57.${}^{\prime\prime}56\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$13
2017 Sept. 18 | 6.0 | B | J1811-2055 |  $1.51\times 0.74;$ 15 | 8 |  35.s172 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$003 |  57.${}^{\prime\prime}85\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$08
2017 Sept. 18 | 10.0 | B | J1811-2055 |  $0.95\times 0.46;$ 18 | 6 |  35.s176 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$002 |  57.${}^{\prime\prime}72\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$06
2017 Dec. 11 | 6.0 | B | J1811-2055 |  $1.74\times 0.77;$ 33 | 9 |  35.s173 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$003 |  57.${}^{\prime\prime}74\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$05
2017 Dec. 11 | 10.0 | B | J1811-2055 |  $1.18\times 0.49;$ 35 | 7 |  35.s175 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$002 |  57.${}^{\prime\prime}77\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$05
2018 Jan. 8 | 6.0 | B | J1811-2055 |  $1.49\times 0.77;$ 23 | 7 |  35.s172 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$002 |  57.${}^{\prime\prime}78\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$05
2018 Jan. 8 | 10.0 | B | J1811-2055 |  $0.98\times 0.47;$ 26 | 7 |  35.s178 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$002 |  57.${}^{\prime\prime}73\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$05
2018 Jan. 13 | 6.0 | B | J1811-2055 |  $1.76\times 0.76;$ 32 | 8 |  35.s174 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$003 |  57.${}^{\prime\prime}75\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$06
2018 Jan. 13 | 10.0 | B | J1811-2055 |  $1.19\times 0.47;$ 35 | 8 |  35.s175 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$004 |  57.${}^{\prime\prime}64\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$09
2018 Jan. 21 | 10.0 | B | J1811-2055 |  $1.40\times 0.51;$ 31 | 7 |  35.s176 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$004 |  57.${}^{\prime\prime}88\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$09
2018 Jan. 28 | 6.0 | B | J1811-2055 |  $1.44\times 0.77;$ 16 | 7 |  35.s171 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$001 |  57.${}^{\prime\prime}80\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$04
2018 Jan. 28 | 10.0 | B | J1811-2055 |  $0.93\times 0.48;$ 21 | 6 |  35.s174 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$002 |  57.${}^{\prime\prime}72\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$04
2018 Feb. 4 | 10.0 | BnA | J1811-2055 |  $1.16\times 0.25;$ -57 | 8 |  35.s171 $\pm 0\hbox to0.0pt{.\hss}^{\rm s}$006 |  57.${}^{\prime\prime}70\pm 0\hbox to0.0pt{.\hss}^{\prime\prime}$05

a Data averaged from the two observed dates: Oct. 4 and Oct. 6 2012. The given
position for this epoch is already corrected for the systematic offset
discussed in sub-section 2.1.

###  2.1 Astrometric correction for the 2012 observations

The observations of 2006 and 2017-18 were made with the same gain calibrator
(J1811$-$2055). This typically assures an astrometric precision of order
$\simeq 0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}01$ for the case of
observations made at centimeter wavelengths in the A-configuration (Boboltz et
al., 2007; Perreault, 2019). We can estimate from our data the expected
precision as follows. We will use the ten determinations of position given in
Table 1 for the epochs between 2017 December 11 and 2018 February 4. Over this
brief period of time we do not expect significant proper motions. The
positional error scales linearly with the angular size of the synthesized
beam. Since all these observations were made in the B configuration we expect
positional errors about three times larger than in A configuration, that is,
about 30 mas. Furthermore, since this is a southern source, we expect the beam
size to be about twice bigger in declination that in right ascension (see
values in Table LABEL:tab:vlapsr). We indeed find that the root-mean-square
errors in right ascension and declination for these ten observations are 31.8
and 63.8 mas, respectively, approximately as expected. We conclude that the
positional error is given approximately by the angular size of the synthesized
beam over 20, that is $\theta$/20.

We were very interested in using the 2012 observations of project 12B$-$278 to
derive a measurement intermediate in time. However, these observations used a
different gain calibrator (J1733$-$1304) and this can introduce systematic
position errors of order $\simeq 0\hbox to0.0pt{.\hss}{{}^{\prime\prime}}1$ or
more unless a correction is applied to the positions. The 12B$-$278
observations were made during 2012 October 04 and 06 and we concatenated the
data for an average epoch of 2012.762. The observations were made in the A
configuration in band X (8.0-10.0 GHz), with 16 spectral windows of 128 MHz
width each.

Table 2: Gaia sources used to correct the astrometry of the 12B-078
positionsa.

Source | Type | R.A. (J2000) | Dec. (J2000) | $\mu_{\alpha}\cdot\cos{(\delta)}$ | $\mu_{\delta}$
---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6)
CXOUJ181314.2-175343 | Wolf-Rayet | $18^{\rm h}13^{\rm m}14\mbox{${}^{\rm s}\mskip-7.6mu.\,$}200$ | $-17^{\circ}53^{\prime}43\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}46$ | $-0.91\pm 0.07$ | $-1.99\pm 0.05$
2MASSJ18131908-1752585 | O8-O9If | $18^{\rm h}13^{\rm m}19\mbox{${}^{\rm s}\mskip-7.6mu.\,$}079$ | $-17^{\circ}52^{\prime}58\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}49$ | $-1.11\pm 0.13$ | $-1.81\pm 0.10$
$[$MFD2008$]$ 15 | LBV | $18^{\rm h}13^{\rm m}20\mbox{${}^{\rm s}\mskip-7.6mu.\,$}984$ | $-17^{\circ}49^{\prime}47\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}02$ | $-0.57\pm 0.07$ | $-1.98\pm 0.06$
CXOUJ181322.4-175350 | Wolf-Rayet | $18^{\rm h}13^{\rm m}22\mbox{${}^{\rm s}\mskip-7.6mu.\,$}492$ | $-17^{\circ}53^{\prime}50\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}32$ | $-0.78\pm 0.07$ | $-2.08\pm 0.05$
UCAC2 25154559 | B0-B3 star | $18^{\rm h}13^{\rm m}24\mbox{${}^{\rm s}\mskip-7.6mu.\,$}432$ | $-17^{\circ}52^{\prime}56\mbox{${}^{\prime\prime}\mskip-7.6mu.\,$}78$ | $-0.85\pm 0.06$ | $-2.16\pm 0.05$

a Positions and proper motions are from the Gaia eDR3 data release (Gaia
Collaboration et al., 2021). The positions of Gaia are accurate to 1 mas or
better. The proper motions are given in mas yr-1.

The final images of the two epochs of project 12B-278 and the 5 C-band epochs
of project 17B-028 were compared. In addition to the source associated with
PSR J1813$-$1749 we found five compact sources in common that are also
detected in the Gaia survey (Gaia Collaboration et al., 2016, 2021). The
highly accurate Gaia positions and proper motions of these five sources are
given in Table LABEL:tab:gaia. We corrected the radio positions of these
sources with the Gaia proper motions and used them to determine a systematic
offset between the positions obtained in the 17B-028 and 12B-278 projects.
This offset (17B-028 – 12B-78) is ${\Delta RA=0\hbox to0.0pt{.\hss}^{s}0110\pm
0\hbox to0.0pt{.\hss}^{s}0091}$; ${\Delta DEC=-0\hbox
to0.0pt{.\hss}^{{}^{\prime\prime}}035\pm 0\hbox
to0.0pt{.\hss}^{{}^{\prime\prime}}125}$. After adding this offset to the
positions of the 12B$-$278 project, we obtain a final position for PSR
J1813$-$1749 at this epoch:

$\displaystyle RA(J2000)=18^{h}~{}13^{m}~{}35\hbox to0.0pt{.\hss}^{s}180\pm
0\hbox to0.0pt{.\hss}^{s}009;$ $\displaystyle
DEC(J2000)=-17^{\circ}~{}49^{\prime}57\hbox
to0.0pt{.\hss}^{{}^{\prime\prime}}52\pm 0\hbox
to0.0pt{.\hss}^{{}^{\prime\prime}}13.$

We note that the error in the final position is dominated by the offset
correction applied.

##  3 Results

We have performed linear least square fits to the positions of PSR J1813–1749,
listed in Table LABEL:tab:vlapsr, to determine its proper motions. The values
obtained are ($\mu_{\alpha}\cdot\cos{\delta},\,\mu_{\delta}$)=(${-5.0\pm
3.7,\,-13.2\pm 6.7}$) mas yr-1. The positions of PSR 1813–1749 as a function
of time, and the best fit to its motion are shown in Figure 2.

Figure 2: Right ascension (top) and declination (bottom) of the PSR J1813–1749
radio emission as a function of time. The dashed lines are least-squares
linear fits to the data. The parameters of the fits are given in the text.

##  4 Discussion and Conclusions

The proper motions of PSR J1813–1749 determined from the X-ray observations
are ($\mu_{\alpha,X-rays}\cdot\cos{\delta},\,\mu_{\delta,X-rays}$)= ($-64\pm
9,\,-14\pm 7$) mas yr-1. By comparing them with our results, we clearly notice
a difference in the R.A. being at radio significantly smaller. Proper motions
measured at radio have as major advantage over the X-ray measurements that the
positions are registered against highly accurate positions measured for
quasars. The observations presented in this work also have a somewhat larger
time baseline, 12 years, than that of the X-ray observations presented by Ho
et al. (2020) of 10 years. This difference does not seem too significant, but
it should be emphasized that the accuracy of proper motion determinations
improves as the time interval to the 3/2 power (Dzib et al., 2017). The
motions reported at X-rays would be evident, between the first and last radio
observations, in positions offsets of –0.′′80=0.s05 and –0.′′17 in R.A. and
Dec., respectively. The offset in R.A. is clearly not present, see also Fig.1.

The large proper motion has also been questioned given the strong implications
for the nature of the pulsar since it implies a young kinematic age and a fast
tangential velocity larger than any other known pulsar (see also the
discussion by Camilo et al., 2021).

Both radio and X-ray emission are tracing the pulsar itself or material very
close to it. However, given the discussion above, the measured motions at
radio frequencies appear to be more reliable. We believe that most of the
position shift in the X-ray image could be due to a change in the brightness
structure of the PWN very near the pulsar, a possibility mentioned by Ho et
al. (2020). Such structure changes have been observed in the Crab pulsar
Nebula (Weisskopf et al., 2011).

The total proper motion of PSR J1813–1749 from the radio is ${14.8\pm 5.9}$
mas year-1. The lower limit of the distance to PSR J1813–1749 is 6.2 kpc and
can be as large as 12.0 kpc. Then, the tangential velocity ranges from
${435\pm 174}$ km s-1 to ${842\pm 336}$ km s-1. These velocities are in the
range of velocities estimated for other pulsars (i.e., Deller et al., 2019).
It should be noted that the error in the proper motion is large enough to
accept a stationary pulsar as a possible solution.

As noted by Ho et al. (2020), PSR J1813–1749 is offset about
$20^{\prime\prime}$ from the center of the SNR G12.82–0.02 (see also Figure 1
in Dzib et al., 2018). To reach this shift the kinematic age of the pulsar is
1351${{}^{+896}_{-385}}$ years. This age discards that this is a very young
pulsar and it is in better agreement with the ages estimated for the pulsar of
5600 years (Halpern et al., 2012) and for HESS J1813–178 of 2500 years (Zhu et
al., 2018) and for SNR G12.82–0.02 (Brogan et al., 2005).

To calculate the kinematic age we have assumed that the original position of
the exploding star was at the geometric center of the supernova remnant.
However, some SNRs have shown nonuniform expansion (e.g., Borkowski et al.,
2014) and the geometric center of the present-day structure does not
necessarily coincide with the center of the explosion.

VLA observations have proven to be an excellent tool to determine proper
motions, and in the case of PSR J1813–1749, are at the moment the best option.
Even VLBI observations will have difficulties measuring a value for this
source. The angular size of the radio source is estimated to be 0.′′034
(Camilo et al., 2021) due to broadening from plasma scattering. The angular
resolution of VLBA observations at 5 GHz is $\sim 0\hbox
to0.0pt{.\hss}^{\prime\prime}004$, so the source will be resolved. The total
flux density of the source is $\sim 100\,\mu$Jy (Dzib et al., 2018); if
resolved very little flux density will fall in a synthesized beam and the
emission will be hard to detect with standard VLBA observations. Furthermore,
astrometry of resolved sources is problematic. To better constrain the proper
motion of PSR J1813–1749, future VLA observations, as those presented here,
will be required.

We thank an anonymous referee for valuable comments. L.F.R. acknowledges the
financial support of DGAPA, UNAM (project IN108920), and CONACyT, México. The
National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc.

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