# Multi-Scale Body-Part Mask Guided Attention for Person Re-identification

Honglong Cai Zhiguan Wang Jinxing Cheng
Suning R&D Center USA
{honglong.cai, doris.wang<EMAIL_ADDRESS>
###### Abstract

Person re-identification becomes a more and more important task due to its
wide applications. In practice, person re-identification still remains
challenging due to the variation of person pose, different lighting,
occlusion, misalignment, background clutter, etc. In this paper, we propose a
multi-scale body-part mask guided attention network (MMGA), which jointly
learns whole-body and part-body attention to help extract global and local
features simultaneously. In MMGA, body-part masks are used to guide the
training of corresponding attention. Experiments show that our proposed method
can reduce the negative influence of variation of person pose, misalignment
and background clutter. Our method achieves rank-1/mAP of 95.0%/87.2% on the
Market1501 dataset, 89.5%/78.1% on the DukeMTMC-reID dataset, outperforming
current state-of-the-art methods.

##  1 Introduction

Figure 1: Examples of challenges in person re-identification and how our
attention mechanism can handle the challenges. (a-b) occlusions, (c-d)
inaccurate bounding-box detection, (e-f) variation of pose. The second to
forth images in each group are the global attention map in original image,
upper-body attention map and bottom-body attention map generated by our
proposed MMGA network.

Person re-identification (re-ID) aims at identifying the presence of same
person in different cameras with different backgrounds, poses and positions.
It is still a challenging task due to large variations on persons like pose,
occlusion, clothes, background clutter and detection failure which are shown
in Figure 1.

Low-level features like colors, shapes, contours and local descriptors are
used to train traditional re-ID models with low accuracy [8, 12]. Nowadays,
with the fast development of deep neural networks, deep features of human
image learned through convolutional neural network (CNN) is demonstrated to
have better discrimination and robustness to represent the image, which has
made significant improvement on the re-ID problem [6, 24, 26, 33]. The
features learned from deep learning network should capture the most salient
clues that can represent identities of different persons. However, most of the
existing deep learning methods learn features from the whole image that
contains not only the human body parts, but also the background regions [40,
37]. The background regions containing clutter and occlusions may lead to a
misalignment problem . To address this issue, some recent works [31, 38, 45,
6, 34, 18] show that locating the significant body parts and learning the
discriminative features from these informative regions can reduce the negative
effects of clutter and occlusions, and thus improve the re-ID accuracy.

Visual attention has shown its success in re-ID tasks [42, 22, 29, 19], as the
mechanism conforms to the human visual system that a whole image is not likely
to be processed in its entirety at once, but only the salient parts of the
whole visual space are focused when and where needed. Visual attention module
can help to extract dynamic features from salient parts mostly like human body
parts in a image by guiding the learning towards informative image regions
[29]. Given the human body information, attention maps where regions of
interest are presented have much stronger responses on body region compared
with background regions [22, 42]. Inspired by this, whole human body mask has
been used to guide the attention model training [29]. The whole human body is
segmented from the background region and then attention network is guided by
this binary mask. However, learning global attention through the whole human
body mask may suppress local informative body part regions that have stronger
responses.

To relieve this dilemma, we introduce a multi-scale body-part mask guided
attention network. We split body masks into upper-body masks and bottom-body
masks and use them to guide the training of upper-body attention and bottom-
body attention respectively. As shown in Figure 1, our proposed network can
learn whole-body attention, upper-body attention and bottom-body attention.
Moreover, comparing to [29] , which needs mask in both training and inference
phase, our proposed method needs mask only in training phase, and thus saves
significant time in inference phase.

Our experiments indicate that multi-scale body-part mask guided attention
network can significantly improve the accuracy of person re-identification and
it still has space for improvement. We conclude our paper with qualitative
results and demonstrate the potential of the method.

The contributions of our work can be summarized as follows:

  * •

We propose a mask guided attention method to address person re-identification
problem. In our proposed method, mask is used to guide attention training only
in the training phase. We don’t need mask in inference phase which makes our
method particularly efficient.

  * •

We creatively use the masks of different parts of body to guide attention
learning in training phase. In this paper, we separate person body into upper
part and bottom part and use them to guide the training of upper and bottom
attention respectively.

##  2 Related Work

This section provides an overview of closely related work in deep person re-
identification. Deep learning has been successfully used to improve the
performance of person re-identification. Both verification and identification
models are applied in deep person re-identification [5, 20, 33, 1].

Part-Based Model: Recently, a number of new methods have been designed to
capture richer and finer visual cues by jointly learning from not only whole-
body images but also body-part images, which pushed the re-ID performances to
a new level. According to different partition strategies, the part-body based
methods can be classified into three categories: 1) Human body parts are
generated through predefined fixed-height horizontal stripes. In work [45, 31,
9, 38], they both equally slice the feature maps of input image in vertical
orientation. 2) The human body-part regions are detected and the local
features are generated jointly. [43] proposed a method that can estimate body
parts in a feature space through ROI pooling, while the local features can be
generated at the same time. 3) Human body can be divided into different parts
according to the keypoints detected by off-the-shelf pose estimation model. In
work [42], 18 human body joints are obtained through pose estimation model and
5 body regions are defined according to the body joints. These methods all
focus on the part partition scheme for the local feature extraction network.
In our proposed method, we generate the body-part mask and use the local mask
to guide our attention model, as human vision may not focus on the whole human
body but only on some body parts in a image.

Attention mechanism in Re-ID: One difficulty person re-identification
suffering is misalignment, as the bounding-boxes detected by the detection
algorithm may not be accurate enough, for example only partial profile of
pedestrian may appear at the corner of the image. To overcome this issue,
attention mechanism is proposed to aid the network to learn where to ’look’
at. Attention mechanism plays a more and more important role in computer
vision field [22, 35, 25, 39, 44, 19, 42, 36, 4]. In [22], a harmonious
attention model is introduced to combine both soft and hard attention
mechanism. A multi-directional attention model is proposed in [25], which can
extracts attentive features through masking different levels of features with
attention map. Given that most of the attention based methods generate global
attention through whole-body images, the local attention learning from each
body part is ignored, which may lead to suboptimal performance when person
images suffer from large pose variations, misalignment, local occlusion, etc.
To address this issue, [19] proposes a spatiotemporal attention model which
can automatically discover a diverse set of distinctive body parts. In our
proposed model, the local attention can be learned through local mask, thus
the re-ID performance can be further improved.

Mask mechanism in Re-ID: Human body mask obtained from image segmentation
models can be used to handle the background clutter problem. With deep
learning based image segmentation algorithms including Mask RCNN [13], JPPNet
[23], Dense Pose [11], etc., human body mask can be extracted well and the
background region can be almost removed. However, there are only a few works
[29, 3, 17] introducing semantic segmentation into re-ID task. This scarcity
is due to large computational complexity involved in semantic segmentation for
human mask. In our work, we just utilize the mask to guide the training of our
attention model so that the mask is only needed in training phase. After we
get the learning metric, mask is no longer needed to extract features which is
time-saving compared with other mask based re-ID algorithms.

##  3 Our Proposed Method

Figure 2: Split mask from the middle-line may suffer misalignment problem. The
first row shows the original images, the second row shows whole-body masks.
The left pair of the third row is the result of divide whole-body mask into
upper-half part and bottom-half part according to the middle line. The right
pair is upper-body mask and bottom-body mask. Figure 3: Structure of our
proposed network. Backbone is resnet50. There are two attention modules in the
network. The first attention module is guided by whole-body mask. The second
attention module has three branches which are guided by whole-body mask,
upper-body mask and bottom-body mask respectively.

In this paper, we propose a multi-scale body-part mask guided attention
network. The overview of our proposed network is shown in Figure 3. The detail
structure of our body-part guided attention modules is shown in Figure 4. Our
attention modules are guided not only by whole-body masks but also by
different body-part masks. In this paper, body masks are separated into upper-
body masks and bottom-body masks.

Figure 4: Structure of our mask guided attention branch. $(a)$ is the channel
attention. $(b)$ is the spatial attention. Channel attention consists of one
average pooling layer and two linear layers. Spatial attention is composed of
three convolutional layers. Spatial attention is guided by body masks.

There are two attention modules in our proposed network. The first attention
module is guided only by whole-body masks. The second attention module has
three branches. The first branch is guided by whole-body mask, the second
branch is guided by upper-body mask, and the third branch is guided by bottom-
body mask.

###  3.1 Overall Architecture

Variety of network structures are used in person re-identification [14, 32,
36]. In our work, we employ resnet50 [14] as backbone network with some
modifications following the recent works [31, 38, 9]. The last spatial down-
sampling operation is removed from resnet50. In this way, we can get a larger
feature map. We also remove original global average pooling (GAP) layer and
fully connected layer from resnet50. At the end of our backbone network,
global average pooling layer and linear layer are employed to reduce the
dimension of the output feature. The size of whole-body features is reduced to
1024, while the size of the upper-body features and the bottom-body features
is reduced to 512.

###  3.2 Attention Module

Inspired by [4, 22], our attention module is composed of spatial attention and
channel attention. The spatial attention aims to define the importance of
pixels. The channel attention is used to set weights to different channels.
Body masks are used to guide the training of spatial attention as shown in
Figure 4.

The input to our attention module is a 3-D tensor
$\boldsymbol{X}\in{R^{h_{in}\times w_{in}\times c_{in}}}$. The output of our
attention module $\boldsymbol{A}\in{R^{h_{out}\times w_{out}\times c_{out}}}$.

Spatial Attention: Our Spatial attention consists of three convolutional
layers. The input channel size of the first convolutional layer is $c_{in}$.
The output channel size of it is $c_{in}/s$ where $s$ is the spatial attention
reduction hyperparameter. The kernel size of the first convolutional layer is
$1$. For the second convolutional layer, the input channel size is $c_{in}/s$,
the output channel size is $c_{in}/s^{2}$, and the kernel size is $1$. The
input channel size of the last convolutional layer is $c_{in}/s^{2}$, the
output channel size is $1$, while the kernel size is 1. After three
convolutional layers, a $Sigmoid$ function is applied to the spatial attention
output. Inspired by [14, 36], we add $1.0$ to $Sigmoid$ function. The output
of the spatial attention module is $\boldsymbol{S}\in{R^{h_{out}\times
w_{out}\times 1}}$.

There is a down-sampling operation in stage3 of the backbone resnet50. An
average pooling layer is applied in the first attention module between the
first convolutional layer and the second convolutional layer so that the
output of the first attention module has the same dimension as the output of
stage3 of the backbone network. Average pooling is not applied in the second
attention module.

Channel Attention: Our channel attention module is composed of one average
pooling layer over spatial features and two linear layers.

We first apply an average pooling over spatial pixels in each channel as
below:

$\boldsymbol{C}_{1}=\frac{1}{h_{in}\times
w_{in}}\sum^{h_{in}}_{i=1}\sum^{w_{in}}_{j=1}\boldsymbol{X}_{i,j,1:c_{in}}$
(1)

Then two linear layers are applied to perform squeeze-expansion. The input
size of the first linear layer is $c_{in}$ and the output size is $c_{in}/r$
where $r$ is the channel attention reduction hyperparameter. The input size of
the second linear layer is $c_{in}/r$, while the output size of it is
$c_{out}$. After these two linear layers, a $Sigmoid$ function is applied to
map the channel attention to range $0$ \- $1$. The channel attention is
further unsqueezed to 3-dimension attention features,
$\boldsymbol{C}\in{R^{1\times 1\times c_{out}}}$.

Combine Spatial-Channel Attention: Spatial attention and channel attention are
combined by element-wise multiplication:

$\boldsymbol{A}=\boldsymbol{S}\times\boldsymbol{C}$ (2)

where $\boldsymbol{A}\in{R^{h_{out}\times w_{out}\times c_{out}}}$ is the
output of our attention module.

###  3.3 Multi-scale Body-part Mask Guided Attention

In this paper, we propose a multi-scale body-part mask guided attention
network. Our proposed network can not only extract global informative features
but also pay attention to local discriminative features. In our proposed
network, whole-body attention, upper-body attention and bottom-body attention
are guided by whole-body masks, upper-body masks and bottom-body masks
respectively as shown in Figure 3.

Recently, a number of re-ID methods combine global features and local features
to improve the accuracy of person re-identification. In order to extract local
features, images or feature maps are divided into fix-height strips. But it is
difficult to handle misalignment problem when inaccurate bounding-box occurs
as shown in Figure 2. In our proposed network, images and feature maps are not
divided into fix-height strips. Instead, local attention is directly guided by
local masks so that our network can accurately locate informative local part.
This mechanism can significantly reduce the influence of background clutter,
occlusion, inaccurate detection, etc. Our attention module can accurately
locate the informative part and thus help the network to extract whole-body
features, upper-body features and bottom-body features separately.

We design two attention modules in our proposed method. The first attention
module is used to filter background influence. The second attention module
help the network extract whole-body features, upper-body features and bottom-
body features accurately. The first attention module takes the output from
stage 2 of the backbone resnet50. The output of the attention module is
performed element-wise multiplication with the output features from stage 3 of
the backbone resent50. The result of the element-wise multiplication is the
input of the second attention module. There are 3 branches in the second
attention module. The first branch is guided by whole-body masks. It acts as
whole-body attention and help extract whole-body features. The second
attention is upper-body attention. Its training is guided by upper-body masks
to help extract upper body features. The third attention is bottom-body
attention. Its training is guided by bottom-body mask to help extract bottom
body features. The output of these three attention branches are performed
element-wise product multiplication with the output features from stage 4 of
backbone resnet50

###  3.4 Loss Function

To learn the attention map, mask guided attention loss is introduced. We also
employ softmax loss for classification and hard triplet loss for metric
learning. The total loss is the sum of mask guided attention loss, softmax
loss and batch-hard triplet loss.

Mask Guided Attention Loss: Mask is used to guide the training of spatial
attention. To compute the attention loss, the output of spatial attention is
normalized as below:

$\boldsymbol{S}^{norm}=\frac{\boldsymbol{S}-\min(\boldsymbol{S})}{\max(\boldsymbol{S})-\min(\boldsymbol{S})}$
(3)

where $\boldsymbol{S}$ represents the spatial attention and
$\boldsymbol{S}^{norm}$ is the normalized spatial attention. Then attention
loss can be calculated through Root Mean Squared Error (RMSE):

$L=\sqrt{\frac{\sum^{h}_{i=1}\sum^{w}_{j=1}\lVert\boldsymbol{M}_{i,j}-\boldsymbol{S}^{norm}_{i,j}\rVert^{2}}{n_{batch}}}$
(4)

where $\boldsymbol{M}$ is the resized human body mask, and $n_{batch}$ is the
batch size at training stage.

The total Mask-guided attention loss is the sum of four parts.

$L_{att}=L_{att}^{1,w}+L_{att}^{2,w}+\lambda_{0}L_{att}^{2,u}+\lambda_{0}L_{att}^{2,b}$
(5)

where $L_{att}^{1,w}$ is the attention loss of the first attention module.
$L_{att}^{2,w}$ is the attention loss of the first branch in the second
attention module. $L_{att}^{2,u}$ is the upper-body mask guided attention loss
in the second attention module. $L_{att}^{2,b}$ is the bottom-body mask guided
attention loss in the second attention module. $\lambda_{0}$ is used to
balance the whole-body mask guided attention loss and part-body mask guided
attention loss.

Softmax Loss: At the end of our proposed network, global average pooling and
linear layer are performed to reduce the dimensions of whole body features,
upper-body features and bottom-body features to 1024, 512, 512 respectively.
Upper-body features and bottom-body features are concatenated together to form
a 1024-dimension local body-part features. Then softmax loss can be used to
compute the whole body loss as well as the body-part loss through these whole-
body features and local body-part features. The loss functions are expressed
as below:

$L_{softmax}^{w}=-\frac{1}{n_{batch}}\sum^{n_{batch}}_{i=1}{log\frac{\exp({\boldsymbol{W}_{y_{i}}^{w}}^{T}\boldsymbol{f}^{w}_{i})}{\sum^{N}_{j=1}\exp(\boldsymbol{W}^{w}_{j}\boldsymbol{f}^{w}_{i})}}$
(6)

$L_{softmax}^{l}=-\frac{1}{n_{batch}}\sum^{n_{batch}}_{i=1}{log\frac{\exp({\boldsymbol{W}_{y_{i}}^{l}}^{T}\boldsymbol{f}^{l}_{i})}{\sum^{N}_{j=1}\exp(\boldsymbol{W}^{l}_{j}\boldsymbol{f}^{l}_{i})}}$
(7)

Equation (6) illustrates the whole-body softmax loss where
$\boldsymbol{f}^{w}$ represents 1024 dimension whole-body features and
${\boldsymbol{W}^{w}_{k}}$ is the whole-body weight vector for identity $k$.
Equation (7) illustrates the local-body softmax loss with $\boldsymbol{f}^{l}$
representing the local-body features which is a concatenation of 512
dimensions of upper-body features and 512 dimensions of bottom-body features.
${\boldsymbol{W}^{l}_{k}}$ representing the part-body weight vector for
identity $k$. $N$ is the number of identities in the training dataset.

Triplet Loss: Triplet loss and its variation [28, 15, 27] is widely used in
person re-identification. Batch-hard triplet loss is also applied in our
proposed method to improve the accuracy.

Whole-body features, upper-body features and bottom-body features are
concatenated together and normalized. Dimension of the new features is 2048.
The new features $\boldsymbol{f}^{all}$ are also used at inference stage.

The triplet loss can be formulated as below:

$\begin{split}L_{triplet}=\frac{1}{n_{batch}}\sum^{P}_{i=1}\sum^{K}_{a=1}[m&+\max_{p=1...K}\lVert{\boldsymbol{f}_{a}^{all}}^{(i)}-{\boldsymbol{f}_{p}^{all}}^{(i)}\rVert_{2}\\\
&-\min_{\begin{subarray}{c}n=1...K\\\ j=1...P\\\ j\neq
i\end{subarray}}\lVert{\boldsymbol{f}_{a}^{all}}^{(i)}-{\boldsymbol{f}_{n}^{all}}^{(j)}\rVert_{2}]_{+}\end{split}$
(8)

where ${\boldsymbol{f}_{a}^{all}}^{(i)}$, ${\boldsymbol{f}_{p}^{all}}^{(i)}$,
and ${\boldsymbol{f}_{n}^{all}}^{(i)}$ are the concatenated and normalized
features of anchor, positive and negative samples respectively. P is the
number of identities in each mini-batch and K is the number of images for each
identity.

By combining all the above loss, our final loss function for the end to end
multi-scale body-part mask guided attention network is as below:

$Loss=L_{softmax}^{w}+L_{softmax}^{l}+\lambda_{1}L_{triplet}+\lambda_{2}L_{att}$
(9)

where $\lambda_{1}$ and $\lambda_{2}$ are used to balance different loss.

##  4 Experiments

Figure 5: The top-5 ranking list for the query images on Market-1501 and
DukeMTMC-reID by MMGA. The retrieved images are from the gallery set and not
from the same camera as the query images. The first column is query images.
Whole-body attention maps on original images, whole-body attention maps,
upper-body attention maps, bottom-body attention maps from our MMGA are in the
second to fifth columns. The right side shows the retrieved top-5 ranking
images with our proposed MMGA method. All top-5 are correct.

###  4.1 Dataset and Evaluation Protocol

We choose two person re-ID benchmarks for evaluation, that is, Market-1501
[47] and DukeMTMC-reID [48]. It is necessary to introduce these datasets and
protocol we use.

DukeMTMC-reID: DukeMTMC-reID is a subset of the DukeMTMC dataset for image-
based re-identification. It consists of 36411 images of 1812 identities from 8
different cameras. There are 1404 identities appearing in more than two
cameras and 408 identities (distractor ID) appearing in only one camera. 16522
images of 702 persons are divided into training set. 19889 images of the
remaining identities are divided into testing set, with 2228 in query set and
17661 in gallery set.

Market-1501: Market-1501 contains 32668 images of 1501 identities. A total of
six cameras were used, including 5 high-resolution cameras and one low-
resolution camera. 12936 images of 751 identities are divided into training
set. 19732 images of the remaining 750 identities are divided into testing
set. 3368 images are in query set. The maximum number of query images is 6 for
an identity.

Protocols: We adopt Cumulative Matching Characteristic (CMC) and mean average
precision (mAP) to evaluate our method. Rank-1, rank-5, rank-10 and mAP
results are reported. All results reported in this paper are under single-
query setting. During evaluation, following the evaluation method in MGN [38],
we extract the features of the original images and the horizontally flipped
images. The average of these features are used as the final features. Re-
ranking is not used in this paper.

###  4.2 Implementation Details

Body Mask: 20 different body parts (like head, hand, arm, leg, etc) can be
segmented with JPPNet [23, 10]. According to these body part masks, whole
human body is divided into upper body-part group and bottom body-part group.
The reason why we only classify body parts into these two groups is that body
limb masks cannot be generated so accurately with JPPNet model. Many wrong
masks appear or some body-part masks are missing when upper/bottom body is
further subdivided. However, if the body-part mask can be labeled by hand or
generated by a more accurate model, we believe that a better performance can
be achieved as attention model can focus on more local details of human body
parts .

We resize the masks to $24\times 8$, the same size as the output of attention
modules and use them to guide attention training.

Data pre-processing: We follow commonly used data augmentation methods in
person re-identification. All images are resized to $384\times 128$.
Horizontal random flipping and random erasing [50] is used in the training
phase.

Network settings: Resnet50 with pretrained parameters on ImageNet [7] is
adopted as the backbone network. In our attention module, we set the spatial
attention reduction hyperparameter $s$ equals 8 and the channel attention
reduction hyperparameter $r$ equals 8.

Loss: The margin of batch hard triplet loss is 0.3. 24 identities are sampled
in each mini-batch and 4 images are sampled for each identities. The number of
images in each mini-batch is 96. The loss hyperparameters are
$\lambda_{0}=0.5$, $\lambda_{1}=2.0$ and $\lambda_{2}=0.1$.

Optimization: We adopt SGD optimizer with a weight decay of $5\times 10^{-4}$.
The initial learning rate for the backbone network is 0.05, and the initial
learning rate of other parameters is 0.1. The learning rate is decayed by a
factor of 0.5 for every 90 epochs. Our model is totally trained on 900 epochs.
Our model is implemented on Pytorch platform and trained with two NVIDIA
1080Ti GPUs.

###  4.3 Ablation Study

Table 1: Ablation study on Market-1501 and DukeMTMC-reID. WMGA refers to the method that attention is guided only by whole-body masks. DMGA refers to the method that divides whole-body masks from middle into upper part and bottom part and these masks are used to guide attention training. MMGA refers to our multi-scale body-part mask guided attention method. Model | Market-1501 | DukeMTMC-reID
---|---|---
| Rank 1 | mAP | Rank 1 | mAP
Baseline(%) | 92.7 | 83.5 | 86.9 | 73.9
Baseline+Att(%) | 93.4 | 85.0 | 87.5 | 75.9
WMGA(%) | 94.3 | 86.4 | 88.3 | 77.3
DMGA(%) | 94.4 | 86.9 | 88.6 | 77.8
MMGA(%) | 95.0 | 87.2 | 89.5 | 78.1
Table 2: Comparison of results on Market-1501 and DukeMTMC-reID. Model | Market-1501 | DukeMTMC-reID
---|---|---
| Rank 1 | Rank 5 | Rank 10 | mAP | Rank 1 | mAP
BoW+kissme [47] | 44.4 | 63.9 | 72.2 | 20.8 | 25.1 | 12.2
WARCA [16] | 45.2 | 68.1 | 76.0 | – | – | –
SVDNet [30] | 82.3 | 92.3 | 95.2 | 62.1 | 76.7 | 56.8
PAN [49] | 82.8 | – | – | 63.4 | 71.6 | 51.5
PAR [46] | 81.0 | 92.0 | 94.7 | 63.4 | – | –
MultiLoss [21] | 83.9 | – | – | 64.4 | – | –
TripletLoss [15] | 84.9 | 94.2 | – | 69.1 | – | –
MultiScale [5] | 88.9 | – | – | 73.1 | 79.2 | 60.6
MLFN [2] | 90.0 | – | – | 74.3 | 81.0 | 62.8
HA-CNN [22] | 91.2 | – | – | 75.7 | 80.5 | 63.8
AACN [41] | 85.9 | – | – | 66.9 | 76.8 | 59.3
MSCAN [29] | 83.8 | – | – | 74.3 | – | –
AlignedReID [45] | 91.0 | 96.3 | – | 79.4 | – | –
Deep-Person [1] | 92.3 | – | – | 79.5 | 80.9 | 64.8
PCB+RPP [31] | 93.8 | 97.5 | 98.5 | 81.6 | 83.3 | 69.2
HPM [9] | 94.2 | 97.5 | 98.5 | 82.7 | 86.6 | 74.3
Attention-Driven [42] | 95.0 | 98.3 | 99.1 | 86.5 | 86.4 | 74.6
MGN [38] | 95.7 | – | – | 86.9 | 88.7 | 78.4
MMGA (Ours) | 95.0 | 98.2 | 98.9 | 87.2 | 89.5 | 78.1

We perform comprehensive ablation studies on Market-1501 and DukeMTMC-reID.

Baseline: In Table 1, Baseline is trained only on backbone resnet50 without
attention module. After the backbone resnet50, a global average pooling layer
and a linear layer is applied. A 1024-dimension global features are extracted.
Classification loss and triplet loss are combined to train the model. The
hyperparameter for balancing the weight of triplet loss and classification
loss is the same as mentioned in implementation details section. Rank-1/mAP of
baseline is 92.7%/83.5% on Market-1501 dataset and 86.9%/73.9% on DukeMTMC-
reID dataset.

Baseline+Att: In Table 1, Baseline+Att denotes the model that two attention
modules are applied to the baseline. The attention modules are not guided by
body masks in training stage and there is only whole-body attention branch in
the second attention module. Results are listed in Table 1. On Market-1501
dataset, rank-1/mAP is 93.4%/85.0% and on DukeMTMC-reID dataset, rank-1/mAP is
87.5%/75.9%. Attention mechanism can improve the accuracy by +0.7%/+1.5% for
rank1/mAP on Market-1501 dataset and +0.6%/+2.0% on DukeMTMC-reID dataset.

Whole-body Mask-guided attention (WMGA): We evaluate the effect of mask-guided
attention mechanism. WMGA refers to the method that attention modules are
guided by whole-body masks. Part-body masks are not used in WMGA. WMGA
achieves rank-1/mAP=94.3%/86.4% on Market-1501 and rank-1/mAP=88.3%/77.3% on
DukeMTMC-reID. WMGA improves the accuracy of Baseline+Att by a large margin.
The experiments illustrate that using masks to guide the training of attention
can significantly improve the accuracy of person re-identification.

Multi-scale Body-part Mask guided Attention (MMGA) versus WMGA: In our
proposed method (MMGA), not only whole-body masks are used but also part-body
masks are used to guide the training of attention modules. Thanks to MMGA
method, our network can predict in detail where the person is and locate
upper-body and bottom-body accurately. This mechanism can significantly reduce
the impact of background clutter, varient of pose and misalignment. The
results of MMGA are listed in Table 1. MMGA achieves 95.0%/87.2% in rank-1/mAP
on Market-1501 outperforming WMGA by +0.7%/+0.8%. On DukeMTMC-reID, MMGA
achieves 89.5%/78.1% in rank-1/mAP improving the accuracy of WMGA by
+1.2%/+0.8%.

Ours (MMGA) versus Divided-Mask guided attention (DMGA): We also divided
whole-body masks into upper-half parts and bottom-half parts according to the
middle line and use this kind of masks to guide the training of our attention
module (DMGA). The upper-half masks and bottom-half masks can somewhat act as
upper-body masks and bottom-body masks. However, dividing masks from the
middle can not locate the upper-body part and bottom-body part accurately,
especially for misalignment images. DMGA achieves 94.4%/86.9% in rank-1/mAP on
Market-1501 and 88.6%/77.8% on DukeMTMC-reID. Experiments show that the
results of DMGA are less than MMGA by -0.6%/-0.3% on Market-1501 and
-0.9%/-0.3% on DukeMTMC-reID.

The above ablation study illustrates that attention mechanism can help feature
extraction of person re-identification and thus using multi-scale body-part
mask to guide attention training can significantly improve the accuracy of
person re-identification.

In Figure 5, we list four inaccurate person detection images from query set on
Market-1501 and DukeMTMC-reID. Our proposed method (MMGA) can locate where the
informative regions are and learn upper-body attention as well as bottom-body
attention correctly. Attention maps of MMGA network are shown in Figure 5. The
top-5 retrieve results for these four inaccurate detection images are all
correct.

###  4.4 Comparison with State-of-the-Art

We compare our proposed multi-scale body-part mask guided attention method
with current state-of-the-art methods on Market-1501 and DukeMTMC-reID.

Strip-based methods [38, 31, 9] divide features into several horizontal
stripes to extract local features but it is difficult to handle misalignment
problem. [42] uses key-points to divide original images into several parts so
that they can exploit global features and local features. Attention-based
methods [22, 42] use attention mechanism to handle background clutter and
extract important features. [29] add masks into input to form four-channeled
inputs which is incompatible with the pre-trained resnet models. Our proposed
multi-scale part-body mask guided attention method uses multi-scale masks to
guide the training of different attention modules. Our mothod can locate the
important location and extract global features and local features accurately.
Our methods achieves the state-of-art performance. Results are shown in Table
2.

Market-1501: On Market-1501 dataset, our proposed MMGA method outperforms all
the state-of-the-art algorithms and exceeds the current best model MGN by
+0.3% in mAP. Our method achieves the second best result in rank-1.

DukeMTMC-reID: Our proposed MMGA achieves 89.5%/78.1% in rank-1/mAP on
DukeMTMC-reID. Though we achieve the second best result in mAP, 0.3% lower
than MGN, in rank-1, our method outperforms all other models and exceeds the
state-of-the-art method MGN by +0.8%. We should pay attention that our model
is smaller than MGN. In MGN, backbone resnet50 is divided into three branches
at stage 3 and these three branches do not share weights with each other.

##  5 Conclusion and Discussion

In this paper, we propose a multi-scale body-part mask guided attention
network. The training of attention modules is guided by whole-body masks,
upper-body masks and bottom-body masks. Our method can accurately locate the
important parts of human body. Experiments show that our proposed method can
significantly improve the accuracy of person re-identification and our method
achieves the state-of-the-art result. What’s more, in this paper, we only
divided body masks into upper-body masks and bottom-body masks. We believe if
we can divide masks into finer details, we can further improve the accuracy of
person re-identification. We leave this experiment as future work.

## References

  * [1] Xiang Bai, Mingkun Yang, Tengteng Huang, Zhiyong Dou, Rui Yu, and Yongchao Xu.  Deep-person: Learning discriminative deep features for person re-identification, 2017.
  * [2] Xiaobin Chang, Timothy M. Hospedales, and Tao Xiang.  Multi-level factorisation net for person re-identification.  2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Jun 2018.
  * [3] Di Chen, Shanshan Zhang, Wanli Ouyang, Jian Xi Yang, and Ying Tai.  Person search via a mask-guided two-stream cnn model.  In ECCV, 2018.
  * [4] Long Chen, Hanwang Zhang, Jun Xiao, Liqiang Nie, Jian Shao, Wei Liu, and Tat-Seng Chua.  Sca-cnn: Spatial and channel-wise attention in convolutional networks for image captioning.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5659–5667, 2017.
  * [5] Y. Chen, X. Zhu, and S. Gong.  Person re-identification by deep learning multi-scale representations.  In 2017 IEEE International Conference on Computer Vision Workshops (ICCVW), pages 2590–2600, Oct 2017.
  * [6] De Cheng, Yihong Gong, Sanping Zhou, Jinjun Wang, and Nanning Zheng.  Person re-identification by multi-channel parts-based cnn with improved triplet loss function.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1335–1344, 2016.
  * [7] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei.  ImageNet: A Large-Scale Hierarchical Image Database.  In CVPR09, 2009.
  * [8] Michela Farenzena, Loris Bazzani, Alessandro Perina, Vittorio Murino, and Marco Cristani.  Person re-identification by symmetry-driven accumulation of local features.  In 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 2360–2367. IEEE, 2010.
  * [9] Yang Fu, Yunchao Wei, Yuqian Zhou, Honghui Shi, Gao Huang, Xinchao Wang, Zhiqiang Yao, and Thomas Huang.  Horizontal pyramid matching for person re-identification.  arXiv preprint arXiv:1804.05275, 2018.
  * [10] Ke Gong, Xiaodan Liang, Dongyu Zhang, Xiaohui Shen, and Liang Lin.  Look into person: Self-supervised structure-sensitive learning and a new benchmark for human parsing.  In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017.
  * [11] Riza Alp Guler, George Trigeorgis, Epameinondas Antonakos, Patrick Snape, Stefanos Zafeiriou, and Iasonas Kokkinos.  Densereg: Fully convolutional dense shape regression in-the-wild.  2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jul 2017.
  * [12] Omar Hamdoun, Fabien Moutarde, Bogdan Stanciulescu, and Bruno Steux.  Person re-identification in multi-camera system by signature based on interest point descriptors collected on short video sequences.  In 2008 Second ACM/IEEE International Conference on Distributed Smart Cameras, pages 1–6. IEEE, 2008.
  * [13] Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross B. Girshick.  Mask r-cnn.  pages 2980–2988, 2017.
  * [14] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jun 2016.
  * [15] Alexander Hermans, Lucas Beyer, and Bastian Leibe.  In defense of the triplet loss for person re-identification.  arXiv preprint arXiv:1703.07737, 2017.
  * [16] Cijo Jose and François Fleuret.  Scalable metric learning via weighted approximate rank component analysis.  Lecture Notes in Computer Science, page 875–890, 2016.
  * [17] Mahdi M. Kalayeh, Emrah Basaran, Muhittin Gokmen, Mustafa E. Kamasak, and Mubarak Shah.  Human semantic parsing for person re-identification.  2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Jun 2018.
  * [18] Dangwei Li, Xiaotang Chen, Zhang Zhang, and Kaiqi Huang.  Learning deep context-aware features over body and latent parts for person re-identification.  2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jul 2017.
  * [19] Shuang Li, Slawomir Bak, Peter Carr, and Xiaogang Wang.  Diversity regularized spatiotemporal attention for video-based person re-identification.  2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Jun 2018.
  * [20] W. Li, R. Zhao, T. Xiao, and X. Wang.  Deepreid: Deep filter pairing neural network for person re-identification.  In 2014 IEEE Conference on Computer Vision and Pattern Recognition, pages 152–159, June 2014.
  * [21] Wei Li, Xiatian Zhu, and Shaogang Gong.  Person re-identification by deep joint learning of multi-loss classification.  Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, Aug 2017.
  * [22] Wei Li, Xiatian Zhu, and Shaogang Gong.  Harmonious attention network for person re-identification.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2285–2294, 2018.
  * [23] Xiaodan Liang, Ke Gong, Xiaohui Shen, and Liang Lin.  Look into person: Joint body parsing & pose estimation network and a new benchmark.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 2018\.
  * [24] Yutian Lin, Liang Zheng, Zhedong Zheng, Yu Wu, and Yi Yang.  Improving person re-identification by attribute and identity learning.  arXiv preprint arXiv:1703.07220, 2017.
  * [25] Xihui Liu, Haiyu Zhao, Maoqing Tian, Lu Sheng, Jing Shao, Shuai Yi, Junjie Yan, and Xiaogang Wang.  Hydraplus-net: Attentive deep features for pedestrian analysis.  2017 IEEE International Conference on Computer Vision (ICCV), pages 350–359, 2017.
  * [26] Tetsu Matsukawa and Einoshin Suzuki.  Person re-identification using cnn features learned from combination of attributes.  In 2016 23rd International Conference on Pattern Recognition (ICPR), pages 2428–2433. IEEE, 2016.
  * [27] Ergys Ristani and Carlo Tomasi.  Features for multi-target multi-camera tracking and re-identification.  2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Jun 2018.
  * [28] Florian Schroff, Dmitry Kalenichenko, and James Philbin.  Facenet: A unified embedding for face recognition and clustering.  2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jun 2015.
  * [29] Chunfeng Song, Yan Huang, Wanli Ouyang, and Liang Wang.  Mask-guided contrastive attention model for person re-identification.  In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018.
  * [30] Yifan Sun, Liang Zheng, Weijian Deng, and Shengjin Wang.  Svdnet for pedestrian retrieval.  2017 IEEE International Conference on Computer Vision (ICCV), Oct 2017.
  * [31] Yifan Sun, Liang Zheng, Yi Yang, Qi Tian, and Shengjin Wang.  Beyond part models: Person retrieval with refined part pooling (and a strong convolutional baseline).  In Proceedings of the European Conference on Computer Vision (ECCV), pages 480–496, 2018.
  * [32] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich.  Going deeper with convolutions.  2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jun 2015.
  * [33] Rahul Rama Varior, Mrinal Haloi, and Gang Wang.  Gated siamese convolutional neural network architecture for human re-identification.  In European Conference on Computer Vision, pages 791–808. Springer, 2016.
  * [34] Rahul Rama Varior, Bing Shuai, Jiwen Lu, Dong Xu, and Gang Wang.  A siamese long short-term memory architecture for human re-identification.  In European Conference on Computer Vision, pages 135–153. Springer, 2016.
  * [35] Cheng Wang, Qian Zhang, Chang Huang, Wenyu Liu, and Xinggang Wang.  Mancs: A multi-task attentional network with curriculum sampling for person re-identification.  In ECCV, 2018.
  * [36] Fei Wang, Mengqing Jiang, Chen Qian, Shuo Yang, Cheng Li, Honggang Zhang, Xiaogang Wang, and Xiaoou Tang.  Residual attention network for image classification.  2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jul 2017.
  * [37] Faqiang Wang, Wangmeng Zuo, Liang Lin, David Zhang, and Lei Zhang.  Joint learning of single-image and cross-image representations for person re-identification.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1288–1296, 2016.
  * [38] Guanshuo Wang, Yufeng Yuan, Xiong Chen, Jiwei Li, and Xi Zhou.  Learning discriminative features with multiple granularities for person re-identification.  In Proceedings of the 26th ACM International Conference on Multimedia, MM ’18, pages 274–282, New York, NY, USA, 2018. ACM.
  * [39] Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He.  Non-local neural networks.  2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Jun 2018.
  * [40] Tong Xiao, Hongsheng Li, Wanli Ouyang, and Xiaogang Wang.  Learning deep feature representations with domain guided dropout for person re-identification.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1249–1258, 2016.
  * [41] Jing Xu, Rui Zhao, Feng Zhu, Huaming Wang, and Wanli Ouyang.  Attention-aware compositional network for person re-identification.  2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Jun 2018.
  * [42] Fan Yang, Ke Yan, Shijian Lu, Huizhu Jia, Xiaodong Xie, and Wen Gao.  Attention driven person re-identification.  Pattern Recognition, 86:143 – 155, 2019.
  * [43] Hantao Yao, Shiliang Zhang, Yongdong Zhang, Jintao Li, and Qi Tian.  Deep representation learning with part loss for person re-identification.  IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 2019.
  * [44] Han Zhang, Ian J. Goodfellow, Dimitris N. Metaxas, and Augustus Odena.  Self-attention generative adversarial networks.  arXiv:1805.08318, 2018.
  * [45] Xuan Zhang, Hao Luo, Xing Fan, Weilai Xiang, Yixiao Sun, Qiqi Xiao, Wei Jiang, Chi Zhang, and Jian Sun.  Alignedreid: Surpassing human-level performance in person re-identification.  arXiv preprint arXiv:1711.08184, 2017.
  * [46] Liming Zhao, Xi Li, Yueting Zhuang, and Jingdong Wang.  Deeply-learned part-aligned representations for person re-identification.  2017 IEEE International Conference on Computer Vision (ICCV), Oct 2017.
  * [47] L. Zheng, L. Shen, L. Tian, S. Wang, J. Wang, and Q. Tian.  Scalable person re-identification: A benchmark.  In 2015 IEEE International Conference on Computer Vision (ICCV), pages 1116–1124, Dec 2015.
  * [48] Zhedong Zheng, Liang Zheng, and Yi Yang.  Unlabeled samples generated by gan improve the person re-identification baseline in vitro.  In Proceedings of the IEEE International Conference on Computer Vision, 2017.
  * [49] Zhedong Zheng, Liang Zheng, and Yi Yang.  Pedestrian alignment network for large-scale person re-identification.  IEEE Transactions on Circuits and Systems for Video Technology, page 1–1, 2018.
  * [50] Zhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li, and Yi Yang.  Random erasing data augmentation, 2017.


# Multi-band light curve analysis of the 40.5-minute period eclipsing double-
degenerate binary SDSS J082239.54$+$304857.19

Alekzander Kosakowski1, Mukremin Kilic1, Warren Brown2
1Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma,
440 W. Brooks St., Norman, OK, 73019 USA
2Smithsonian Astrophysical Observatory, 60 Garden St, Cambridge, MA 02138 USA

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the Apache Point Observatory BG40 broadband and simultaneous Gemini
$r$-band and $i$-band high-speed follow-up photometry observations and
analysis of the 40.5 minute period eclipsing detached double-degenerate binary
SDSS J082239.54$+$304857.19. Our APO data spans over 318 days and includes 13
primary eclipses, from which we precisely measure the system’s orbital period
and improve the time of mid-eclipse measurement. We fit the light curves for
each filter individually and show that this system contains a low-mass DA
white dwarf with radius $R_{A}=0.031\pm 0.006~{}{\rm R_{\odot}}$ and a
$R_{B}=0.013\pm 0.005~{}{\rm R_{\odot}}$ companion at an inclination of
$i=87.7\pm 0.2^{\circ}$. We use the best-fitting eclipsing light curve model
to estimate the temperature of the secondary star as $T_{\rm eff}=5200\pm
100~{}{\rm K}$. Finally, while we do not record significant offsets to the
expected time of mid-eclipse caused by the emission of gravitational waves
with our 1-year baseline, we show that a $3\sigma$ significant measurement of
the orbital decay due to gravitational waves will be possible in 2023, at
which point the eclipse will occur about $8$ seconds earlier than expected.

###### keywords:

stars: white dwarfs – stars:binaries: eclipsing – stars:individual:
J082239.54$+$304857.19 – stars:individual: J0822$+$3048

††pubyear: 2020††pagerange: Multi-band light curve analysis of the 40.5-minute
period eclipsing double-degenerate binary SDSS
J082239.54$+$304857.19–References

##  1 Introduction

Eclipsing binary systems provide rare opportunities to directly measure the
physical parameters of both of the stars in the system. If the primary and
secondary eclipses are both clearly visible, it is possible to test the
theoretical mass-radius relationship (Parsons et al., 2017). Furthermore, with
precisely-measured mid-eclipse times, it is also possible to measure the
effects of orbital decay due to the loss of orbital angular momentum from
gravitational wave emission and torques caused by tidal interaction (Piro,
2011; Benacquista, 2011; Fuller & Lai, 2013).

Even in eclipsing systems where the secondary eclipse is obscured by a
significantly brighter primary star, it is still possible to place constraints
on the properties of the hidden secondary star by using the information
contained within the primary eclipse through light curve fitting. Additional
information on these invisible companions can be obtained from radial velocity
measurements of the primary star, which provide information on the system’s
orbital period and mass ratio. Comparing results from light curve fitting with
evolutionary models and stellar atmosphere models allows for an independent
way to confirm the temperature and radii of both of the stars in the binary
system.

As of this work, there are only 14 known eclipsing double-degenerate systems.
With periods ranging from 7 to 354 minutes, these systems are: NLTT 11748
(Steinfadt et al., 2010), CSS 41177 (Drake et al., 2010; Parsons et al.,
2011), GALEX J171708.5$+$675712 (Vennes et al., 2011), J0651$+$2844 (Brown et
al., 2011), J0751$-$0141 (Kilic et al., 2014), J1152$+$0248 (Hallakoun et al.,
2016), J0822$+$3048 (Brown et al., 2017), J1539$+$5027 (Burdge et al., 2019a),
ZTF J1901$+$5309 (Coughlin et al., 2020), ZTF J0538$+$1953 (Burdge et al.,
2020a), ZTF J2029+1534 (Burdge et al., 2020a), ZTF J0722$-$1839 (Burdge et
al., 2020a), ZTF J1749$+$0924 (Burdge et al., 2020a), and ZTF J2243+5242
(Burdge et al., 2020b). Here we report on follow-up observation and analysis
of the relatively faint ($g_{0}=20.198\pm 0.023~{}{\rm mag}$), 40.5-minute
period double-degenerate eclipsing binary system SDSS J082239.54$+$304857.19
(hereafter: J0822$+$3048) using the APO 3.5-meter and Gemini North 8.1-meter
telescopes.

Originally discovered by Brown et al. (2017) as a part of an ongoing search
for extremely low mass ($M<0.3~{}{\rm M}_{\odot}$) white dwarfs (Brown et al.,
2020; Kosakowski et al., 2020), J0822$+$3048 is the seventh eclipsing double
white dwarf binary discovered. The authors used the MMT 6.5-meter telescope
with the blue-channel spectrograph to obtain radial velocity measurements of
the J0822$+$3048 system and showed that it contains a $M_{A}=0.304\pm
0.014~{}{\rm M_{\odot}}$ DA white dwarf and a degenerate companion with mass
$M_{B}=0.524\pm 0.050~{}{\rm M_{\odot}}$ on a 40.5 minute orbit. They
followed-up their spectroscopic observations with 68 minutes of broadband
photometry with a blue filter made of Schott BG40 filter glass (BG40 filter,
$340-600~{}{\rm nm}$) using the Apache Point Observatory (APO) 3.5-meter
telescope frame-transfer camera, Agile, and found two short ($\sim
60$-second), 0.2 mag deep eclipses in the light curve with a separation
consistent with the orbital period obtained through their radial velocity
fits. Based on these two eclipses, the authors placed weak constraints on the
absolute radii of the stars in the system.

We expand upon the discovery data with an additional 492 minutes of APO BG40
broadband filter data spread across two additional observing sessions for a
total BG40 filter baseline of over 318 days, as well as 209 minutes of
simultaneous $r$-band and $i$-band filter data from the 8.1-meter Gemini North
telescope using the high-speed camera, ’Alopeke. We use these data to further
constrain the component radii, orbital inclination, and mid-eclipse timing of
the system.

This paper is organized as follows. In section 2, we present the observations
and discuss the data reduction steps used to create our final light curves. In
section 3 we discuss our data analysis methods, and in section 4 we discuss
our results on the binary system parameters and conclude.

##  2 Observations and Data Reduction

###  2.1 Apache Point Observatory

J0822$+$3048 was originally observed on UT 2017 March 02 using the 3.5-meter
telescope at the Apache Point Observatory (APO) with the BG40 broadband filter
on the Agile frame-transfer camera (Mukadam et al., 2011) exposing for 68
minutes with 30-second back-to-back exposures. This discovery dataset captured
two primary eclipses, each containing only two data points.

We obtained follow-up data on UT 2017 November 16 and UT 2018 January 14 using
an identical setup to the discovery data with 30-second exposures. Our first
night of follow-up observations spanned 322 minutes. We excluded the final 81
minutes of data due to cloud coverage significantly affecting the light curve.
The remaining 241 minutes includes six primary eclipses. Our second night of
follow-up observations spanned 251 minutes and covers six primary eclipses.
One of these eclipses is lost due to instrument problems. Figure 1 shows our
calibrated light curves for our BG40 datasets. Our best-fitting model from our
Monte Carlo light curve fits to the APO BG40 data (discussed below) is
overplotted as a solid red line.

###  2.2 Gemini North

We supplemented the APO BG40 broadband filter data with simultaneous $r$\- and
$i$-band observations using the dual-channel frame-transfer camera, ’Alopeke
(Scott & Howell, 2018) on the 8.1-meter Gemini North telescope. The
observations were taken in eight, nearly back-to-back, observing blocks each
containing 100 back-to-back 15-second exposures on UT 2019 March 12 as a part
of the program GN-2019A-Q-119. These observations spanned 209 minutes and
included five primary eclipses. Cloud coverage affected the quality of the
data about 2 hours into the observations. Unfortunately, due to an issue with
the GPS timing synchronization between the telescope and the ’Alopeke
instrument at the time of observation, the Gemini data is systematically
shifted by about $-21$ seconds. We note that the relative frame timing is
unaffected by this systematic shift. Figure 2 shows our calibrated Gemini
$r$-band (top) and $i$-band (bottom) light curves. Our best-fitting model from
our Monte Carlo light curve fits to the Gemini $r$-band and $i$-band data
(discussed below) are overplotted as solid red lines.

###  2.3 Data Reduction

We used the IRAF package CCDRED to perform image reduction using a set of bias
images, dark images, and twilight flats, each taken on the same nights as our
observations. We performed relative aperture photometry using the IRAF package
DAOPHOT using a circular source aperture with radius based on the FWHM of each
image and a background annulus surrounding each source aperture. For our APO
Agile data, we used two nearby, relatively bright, nonvariable field stars to
calibrate the resulting light curve. For our Gemini ’Alopeke data, because the
’Alopeke instrument has a much smaller field of view than Agile, we only had
three nearby, non-variable, field stars of similar brightness available to
calibrate our target light curve. We detrended each light curve by fitting and
subtracting a third-order polynomial. Finally, we converted our APO data
timing system from beginning-of-exposure TAI to middle-of-exposure Barycentric
Dynamical Time (BJD_TDB, Eastman, Siverd, & Gaudi, 2010) and our Gemini data
timing from end-of-exposure TAI to middle-of-exposure BJD_TDB.

Figure 1: Calibrated light curves for APO BG40 broadband from UT 2017 March 02
(top), UT 2017 November 16 (middle), and UT 2018 January 14 (bottom). The
best-fit model based on light curve fitting to the combined APO data with
JKTEBOP discussed in the text is overplotted in red. Figure 2: Calibrated
light curves for Gemini $r$-band (top) and $i$-band (bottom) filters obtained
simultaneously on UT 2019 March 12. The best-fit model based on light curve
fitting with JKTEBOP discussed in the text is overplotted in red. The timing
shown is as recorded by the ’Alopeke instrument and is systematically offset
by about 21 seconds due to an instrument problem discussed in the text.

##  3 Initial Period Determination

The orbital period of the J0822$+$3048 system was originally determined using
radial velocity measurements based on the Balmer lines in the optical spectrum
and roughly confirmed through light curve fitting of the discovery light curve
containing two adjacent primary eclipses. We combined our new APO BG40
broadband data with the discovery dataset to create a master light curve
spanning just over 318 days and containing 13 primary eclipses. We use this
master light curve to perform light curve fitting and to determine the orbital
period of the J0822$+$3048 system.

Since the periods obtained through radial velocity measurements and the
eclipsing light curve of the discovery dataset are only roughly consistent, to
obtain an appropriate initial period estimate for light curve fitting, we used
the AstroPy implementation of the Lomb-Scargle periodogram to create a power
spectrum from all of the combined APO BG40 data using simple models with
varying numbers of sine-terms. We limited our period range to search between
40 and 41 minutes with a step size of about 1 ms. Our Lomb Scargle models each
returned an identical best-fit frequency at 35.55448746 cycles d-1. While we
do not estimate uncertainties on this initial measurement, this frequency is
only 0.0002 cycles d-1 ($\approx 0.01~{}{\rm s}$) greater than the original
frequency obtained using radial velocity observations of 35.55429140 cycles
d-1. We use this period as our initial value when performing light curve
fitting discussed in the next section.

##  4 System Parameters - Light Curve Fitting

###  4.1 APO BG40 Broadband fits

We modeled the system parameters using JKTEBOP (Southworth, 2013), which uses
Levenberg-Marquardt minimization to obtain best-fit parameter values. For the
BG40 dataset, we fit for the sum and ratio of the fractional system component
radii ($r=R/a$), inclination angle, stellar light ratio, and orbital period.
We chose to fix the mass ratio and initialized these parameters based on
values taken from the discovery paper, with exceptions for the orbital period,
which we initialized based on our previous Lomb-Scargle estimate.

We used a 4-parameter limb darkening law with coefficients for a $T_{\rm
eff}=14,000~{}{\rm K}$, $\log{g}=7.14$ He core white dwarf primary star and
$T_{\rm eff}=5,000~{}{\rm K}$, $\log{g}=8.00$ C/O core white dwarf secondary
star. Due to technical limitations in the JKTEBOP software restricting limb
darkening values to be greater than $-1.0$, we used the limb-darkening
coefficients of Gianninas et al. (2013) for the LSST $u$-, $g$-, $r$-, and
$i$-band filters and converted these to the BG40 broadband filter system using
equation 3 of Hallakoun et al. (2016). The Gianninas et al. (2013) intensity
functions are in good agreement with the updated Claret et al. (2020)
intensity functions so we expect this substitution to have minimal effect on
our results. Similarly, we used fixed gravity darkening coefficients from
Claret et al. (2020) for the $u$-, $g$-, $r$-, and $i$-band filters and once
again converted these to the BG40 system using equation 3 in Hallakoun et al.
(2016). The best-fitting models for our APO BG40 and Gemini $r$-band and
$i$-band fits are overplotted onto the calibrated light curve data and shown
in Figures 1 and 2, respectively.

Figure 3: Parameter distributions from Monte Carlo fits to the combined APO
BG40 broadband light curve. The diagonal contains the 1-D parameter
distributions split into 40 bins (black histogram) with a 1-D Gaussian KDE
overplotted as a blue shaded distribution. The off-diagonal plots contain the
2-D parameter distributions with 2-D Gaussian KDE contours overplotted.
Primary/Secondary median fits and $1\sigma$ ranges are marked with vertical
red/blue lines for single/double-peaked distributions. Due to the poorly
constrained light ratio and lack of visible secondary eclipses in the light
curve, the secondary star’s radius and the system’s inclination are not well-
constrained to a single peak and are strongly anti-correlated.

We used JKTEBOP’s Monte Carlo analysis to create parameter distributions and
estimate uncertainties for each of our fitted parameters. This is done by
creating a simulated light curve based on Gaussian perturbations to the best-
fit model light curve and performing Levenberg-Marquardt minimization to the
simulated light curve. Details for this Monte Carlo analysis method can be
found in Southworth, Maxted, & Smalley (2004) and Southworth et al. (2005). We
performed 15,000 of these Monte Carlo fits to the combined BG40 master light
curve and filtered out results that converged to unphysical values, such as
inclination angles $i<80^{\circ}$ that would not show eclipses in this system.
After filtering, we were left with over 13,600 successful fits from which we
created the resulting parameter distributions. Figure 3 shows the final
parameter distributions for our APO BG40 light curve fits. The diagonal shows
the 1-D histograms with a 1-D Gaussian Kernel Density Estimate (KDE)
overplotted as a blue-shaded distribution. We marked the locations of the
median fit and the $1\sigma$ range of the data if the distribution is single-
peaked. Because our 30-second exposures poorly sample the short primary
eclipses, and because our light curves do not show a clear secondary eclipses,
the secondary star’s radius and the system’s inclination are not well-
constrained to a single best value and are strongly anti-correlated. For these
double-peaked distributions, we fit a Gaussian to each peak separately and
report the resulting central value and width of each Gaussian as the ‘best’
fits. We overplot these best fits and their $1\sigma$ range as red and blue
vertical lines on top of their respective peaks. Best-fit values for each
parameter are reported above each histogram. The off-diagonal plots show 2-D
distributions of each Monte Carlo fit with individual results marked as black
points and 2-D Gaussian KDE overplotted as colored contours.

###  4.2 Gemini $r$-band & $i$-band fits

For our Gemini $r$-band and $i$-band fits, we performed 15,000 Monte Carlo
simulations fitting for sum and ratio of the fractional system component
radii, inclination, and light ratio. We initialized the parameters based on
the best-fitting parameters from the APO BG40 data. We chose to fix the period
at the best-fit result from the BG40 data fit due to the much longer baseline
of the APO data. Our Gemini $r$-band and $i$-band parameter distributions can
be seen in Figures 4 and 5 and follow the same organization as Figure 3.

While all peak values agree within their respective $1\sigma$ ranges across
each filter, we note that the large temperature difference between the primary
and secondary stars resulted in a $3\sigma$ detection of the cooler secondary
star in the system’s light ratio for the redder Gemini $r$\- and $i$-band
filters. This increased significance allowed the Gemini fits to strongly favor
one peak over another, essentially breaking the degeneracy between the
system’s inclination and the secondary star’s fractional radius. Best-fit
parameters and their uncertainties for all filters are presented in Table 1,
along with their variance-weighted mean values. We calculated absolute radius
values based on our light curve fitting using the orbital separation
$a=0.364\pm 0.008~{}{\rm R_{\odot}}$ from the discovery publication. Figure 6
shows the resulting phase-folded light curves using the period from our APO
BG40 dataset fits. We overplot the best-fit models created from the best-fit
parameters in Table 1 as a solid red line and zoom in to the regions
surrounding the primary and secondary eclipses.

Table 1: Best-fit parameters from the APO BG40 broadband and Gemini $r$-band and $i$-band datasets. Peak values for double-peaked distributions are reported together. Preferred solutions to the double-peaked parameters are bolded for clarity. We include the variance-weighted mean values across all filters for each parameter. Parameter | BG40 broadband |  $r$-band |  $i$-band | Mean Value
---|---|---|---|---
Light Ratio | 0.020${}^{+0.013}_{-0.010}$ | 0.016${}^{+0.006}_{-0.005}$ | 0.027${}^{+0.010}_{-0.009}$ |
$r_{B}+r_{A}$ | 0.118${}^{+0.013}_{-0.010}$ | 0.128${}^{+0.008}_{-0.007}$ | 0.122${}^{+0.011}_{-0.009}$ | 0.124$\pm$0.005
$r_{B}/r_{A}$ | 0.449$\pm$0.015 | 0.445$\pm$0.010 | 0.439$\pm$0.014 | 0.444$\pm$0.007
| 0.573$\pm$0.027 | 0.565$\pm$0.026 | 0.572$\pm$0.029 | 0.570$\pm$0.016
$i$ (∘) | 87.9$\pm$0.4 | 87.5$\pm$0.4 | 87.7$\pm$0.5 | 87.7$\pm$0.2
| 86.4$\pm$0.4 | 86.0$\pm$0.3 | 86.0$\pm$0.4 | 86.0$\pm$0.2
Period (d) | $0.0281258394\pm(1.5\times 10^{-9})$ |  |  |
$R_{A}$ (${\rm R}_{\odot}$) | 0.029$\pm$0.010 | $0.032\pm 0.009$ | $0.031\pm 0.010$ | 0.031$\pm$0.006
$R_{B}$ (${\rm R}_{\odot}$) | 0.013$\pm$0.009 | 0.014$\pm$0.008 | 0.013$\pm$0.008 | 0.013$\pm$0.005
| 0.017$\pm$0.009 | $0.018\pm 0.009$ | $0.018\pm 0.009$ | 0.018$\pm$0.005
Figure 4: Parameter distributions from Monte Carlo fits to the Gemini $r$-band
light curve. The diagonal contains the 1-D parameter distribution split into
40 bins (black histogram) with a 1-D Gaussian KDE overplotted as a blue shaded
distribution. The off-diagonal plots contain the 2-D parameter distributions
with 2-D Gaussian KDE contours overplotted. Primary/Secondary median fits and
$1\sigma$ ranges are marked with vertical red/blue lines for single/double-
peaked distributions. Figure 5: Parameter distributions from Monte Carlo fits
to the Gemini $i$-band light curve. The diagonal contains the 1-D parameter
distribution split into 40 bins (black histogram) with a 1-D Gaussian KDE
overplotted as a blue shaded distribution. The off-diagonal plots contain the
2-D parameter distributions with 2-D Gaussian KDE contours overplotted.
Primary/Secondary median fits and $1\sigma$ ranges are marked with vertical
red/blue lines for single/double-peaked distributions. Figure 6: Phase-folded
light curves for APO BG40 broadband (top), Gemini $r$-band (middle), and
Gemini $i$-band (bottom). Best-fit models based on JKTEBOP Monte Carlo results
are overplotted in red. Zoomed-in plots surrounding the primary and secondary
eclipses are included for each filter. The secondary eclipse is not seen in
any filter.

##  5 Estimating Effective Temperature and Radius of the Secondary Star

###  5.1 Temperature Estimate

We used the best-fitting parameters from our light curve fitting to estimate
the effective temperature of the secondary star. We first calculated the
system’s absolute magnitude using the extinction-corrected SDSS apparent
magnitudes and the distance from the discovery data obtained through
spectroscopic models. We then interpolated over the C/O core DA white dwarf
cooling models of Tremblay, Bergeron, & Gianninas
(2011)111http://www.astro.umontreal.ca/ bergeron/CoolingModels to a mass of
$M_{B}=0.524~{}{\rm M_{\odot}}$. Our interpolation resulted in effective
temperatures $T_{{\rm eff},r}=5210\pm 150~{}{\rm K}$ and $T_{{\rm
eff},i}=5180\pm 120~{}{\rm K}$ for the $r$-band and $i$-band, respectively. We
take the variance-weighted mean of these results and accept $T_{\rm
eff}=5200\pm 100~{}{\rm K}$ as the secondary white dwarf’s effective
temperature.

###  5.2 Radius Estimates

Between our light curve fitting results across three filters, the component
radii for the J0822$+$3048 system are fairly-well constrained to a single
solution. Here we compare our results to evolutionary model predictions for an
$M=0.304\pm 0.014~{}{\rm M_{\odot}}$ He core primary white dwarf and
$M=0.524\pm 0.050~{}{\rm M_{\odot}}$ C/O core secondary white dwarf.

For the primary white dwarf, we interpolated over the He core white dwarf
evolutionary tracks of Istrate et al. (2016), including elemental diffusion
and stellar rotation, and obtain a primary radius of $R_{A}=0.025\pm
0.001~{}{\rm R_{\odot}}$. This value roughly agrees within $1\sigma$ of our
estimate from the light curve fitting of $R_{A}=0.031\pm 0.006~{}{\rm
R_{\odot}}$.

For the secondary star, we interpolated over the evolutionary models for C/O
core composition, thick hydrogen layer ($q_{H}=10^{-4}$), white dwarfs
(Fontaine, Brassard, & Bergeron, 2001). This interpolation resulted in a
radius estimate of $R_{B}=0.014\pm 0.001~{}{\rm R_{\odot}}$. This is in
excellent agreement with our estimate from light curve fitting of
$R_{B}=0.013\pm 0.005~{}{\rm R_{\odot}}$. Our light curve fitting results
agree well with the mass-radius relation for white dwarfs and confirms that
our Gemini $r$\- and $i$-band fit results have identified the correct peak
where our APO BG40 fit failed. In addition, because the secondary star’s
radius and the system’s inclination were strongly anti-correlated, we are now
also able to select the correct inclination peak at $i=87.9\pm 0.4^{\circ}$ in
our APO BG40 distribution.

##  6 Eclipsing Timing Estimate and Orbital Decay

The orbit of compact double-degenerate systems decays due to the loss of
angular momentum (Landau & Lifshitz, 1958). While gravitational waves are
generally the dominant source of angular momentum loss in these compact
systems, torques caused by strong tidal interaction between the stars in
compact systems may also contribute significantly to the total angular
momentum loss (Piro, 2011; Benacquista, 2011; Fuller & Lai, 2013). Eclipse
timing measurements taken over long baselines have been used as a method to
directly measure the effects of orbital decay in these systems.

In the case of the 12-minute period eclipsing double-degenerate binary
J0651$+$2844 (Brown et al., 2011), Hermes et al. (2012) measured the system’s
mid-eclipse timing over a baseline of 13 months and show that the period of
the system is decaying at a rate of $\dot{P}=(-9.8\pm 2.8)\times 10^{-12}$ s
s-1. They showed that, while the system shows evidence for tidal interaction
in its ellipsoidal variations, a longer baseline is required to measure the
orbital decay contribution from the tidal interaction in the system.

Similarly, Burdge et al. (2019a) have used new and archival data to perform
mid-eclipse timing measurements of the 7-minute period eclipsing double-
degenerate binary J1539$+$5027\. They precisely measured the system’s orbital
decay with a 10 year baseline and showed that the orbital decay is consistent
with constant change in period $\dot{P}=(-2.373\pm 0.005)\times 10^{-11}$ s
s-1. Additionally, Burdge et al. (2019b) have identified a 20-minute non-
eclipsing double-degenerate binary system showing strong ellipsoidal variation
caused by tidal distortions. They used these ellipsoidal variations to measure
the orbital decay of the system caused by gravitational wave emission and
estimated the contribution to the decay from tidal effects. Finally, Burdge et
al. (2020b) have identified an 8.8-minute period eclipsing double-degenerate
binary system using ZTF archival data and show that the system is undergoing
rapid orbital decay. They estimate that tidal effects could contribute as much
as $7.5$ percent to the orbital decay of the system.

Here we measure the time of mid-eclipse for each of our APO epochs to prepare
for future orbital decay studies of the J0822$+$3048 system. Because of the
systematic offset in the timing of our Gemini data, we estimated the mid-
eclipse time only for the three epochs of APO BG40 data. For each epoch of
data, we performed 50,000 Monte Carlo fits using JKTEBOP to fit the light
curves for only the mid-eclipse time, using the best-fit parameters in Table 1
as initial parameters.

Table 2: Best-fit mid-eclipse times (barycentric dynamical time) for the APO
BG40 broadband data. $T_{0}$ (BJD_TDB)
---
$2457814.82095\pm 0.00005$
$2458073.88809\pm 0.00002$
$2458132.89610\pm 0.00003$
Figure 7: Mid-eclipse timing distributions from 50,000 Monte Carlo fits to
each of the three APO BG40 broadband light curves. Fits that converged greater
than 45 seconds from the median were excluded as unphysical.

We used the time of ingress and egress from the best-fit model light curve to
estimate the eclipse duration as $T_{eclipse}\approx 90$ seconds, with minimum
light lasting $\approx 20$ seconds. We therefore exclude results with mid-
eclipse timing greater than 45 seconds from the median fit value, as those
results place the middle of the eclipse outside of the observed range of the
eclipse itself. Figure 7 shows the resulting distribution for each epoch of
data. We fit a Gaussian to the central peak of each distribution and report
the central value and width as the best-fit and $1\sigma$ uncertainty. These
values are reported in Table 2. We calculated the offset of each observed mid-
eclipse time from its expected value by measuring the eclipse timing offset
from a linear projection based on the first epoch’s time of eclipse and the
orbital period of the system. We note that in our second and third APO BG40
data sets, the measured mid-eclipse timings are $+3.0$ and $+2.9$ seconds off
of the expected time assuming no orbital decay, but each agree within the
relatively large $\pm 4.3$ seconds $1\sigma$ range on the discovery dataset’s
mid-eclipse timing.

Despite not recording a significant offset in the measured mid-eclipse timing,
we revisited the decay of mid-eclipse timing due to gravitational waves using
the two new epochs of APO data discussed in this work. Figure 8 shows an
$(O-C)$ diagram with the best-fit mid-eclipse timing measurements to our two
new epochs of APO BG40 data (black data points with error bars) plotted as an
offset from the expected mid-eclipse timing assuming no orbital decay (black
dashed line) with a period of 0.0281258394 d. We exclude the discovery data
set due to its relatively large uncertainties. We plot the projected offsets
in mid-eclipse timing out to the expected launch date of the LISA mission in
2034, based on angular momentum loss solely due to the emission of
gravitational waves (Piro, 2011) as a red dashed line with shaded $1\sigma$
region dominated by the uncertainties in the masses of each star in the
system. We used these projected values and the mean uncertainty in our
measured values with 30-second exposure times and calculate that a $3\sigma$
significant mid-eclipse timing offset measurement will be possible in the year
2023, at which point the J0822$+$3048 system will eclipse $8.4\pm 0.7$ seconds
earlier than expected as measured from our second APO epoch.

Figure 8: Best-fit mid-eclipse timing measurements for J0822$+$3048 for our
two new epochs of APO data discussed in the text plotted as offsets to the
expected mid-eclipse timing, in seconds, based on the period of
$P=0.0281258394{\rm d}$ determined through light curve fitting. We include the
projected offsets based on Piro (2011) estimates of angular momentum loss
solely due to gravitational wave emission as a function of time as a red
dashed line with a shaded $1\sigma$ range up to the expected launch date of
the LISA mission in 2034. The dark grey shaded region represents the projected
offset from gravitational wave emission, but also includes uncertainty in our
initial time of eclipse measurement added in quadrature.

##  7 Conclusions

We have expanded upon the discovery APO BG40 light curve of J0822$+$3048 with
an additional 492 minutes of APO BG40 data and 209 minutes of simultaneous
Gemini $r$-band and $i$-band data. We analyzed these light curves and improved
the estimates for the absolute radii of both stars in the system using a
combination of light curve fitting and white dwarf evolutionary models. Our
fits resulted in variance-weighted mean component radii values of
$R_{A}=0.031\pm 0.006~{}{\rm R_{\odot}}$, $R_{B}=0.013\pm 0.005~{}{\rm
R_{\odot}}$, and system inclination $i=87.7\pm 0.2^{\circ}$. In addition, we
use the results from light curve fitting together with white dwarf cooling
models to estimate the secondary white dwarf’s effective temperature at
$T_{\rm eff}=5200\pm 100~{}{\rm K}$.

Finally, we have reported an accurate and precise orbital period for this
system and measured mid-eclipse times for each epoch of APO data for use with
future eclipsing timing variability studies. We show that, with 30-second
exposures, a $3\sigma$ significant mid-eclipse offset measurement will be
possible during the year 2023, at which point the mid-eclipse time will be
offset by $-8.4\pm 0.7$ seconds due to the loss of angular moment from the
emission of gravitational waves. With the expected launch of the LISA mission
in 2034, we predict that J0822$+$3048 will show an $83.7\pm 7.3~{}{\rm s}$
offset in mid-eclipse timing by the time LISA launches. While J0822$+$3048
falls just below the LISA 4-year sensitivity curve with a signal-to-noise
ratio of $S/N\approx 3.6$, with its precise period and sky position known, a
gravitational wave detection may be possible.

While we have placed constraints on the parameters of the stars in the system,
there is still room for improvement. Higher-quality data may provide the first
direct detection of the secondary eclipse, allowing for absolute measurements
on the secondary’s radius and temperature. Additionally, reduced exposure
times will allow for increased significance in future orbital decay
measurements and are therefore also desired.

## Acknowledgements

This work was supported in part by the NSF under grant AST-1906379. This
research is based in part on observations obtained with the Apache Point
Observatory 3.5-meter telescope, which is owned and operated by the
Astrophysical Research Consortium

Some of the Observations in the paper made use of the High-Resolution Imaging
instrument(s) ‘Alopeke. ‘Alopeke was funded by the NASA Exoplanet Exploration
Program and built at the NASA Ames Research Center by Steve B. Howell, Nic
Scott, Elliott P. Horch, and Emmett Quigley. ‘Alopeke was mounted on the
Gemini North telescope of the international Gemini Observatory, a program of
NSF’s NOIRLab, which is managed by the Association of Universities for
Research in Astronomy (AURA) under a cooperative agreement with the National
Science Foundation. on behalf of the Gemini partnership: the National Science
Foundation (United States), National Research Council (Canada), Agencia
Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia,
Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia,
Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science
Institute (Republic of Korea).

This research made use of Astropy,222http://www.astropy.org a community-
developed core Python package for Astronomy (Astropy Collaboration et al.,
2013, 2018).

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Benacquista (2011) Benacquista M. J., 2011, ApJL, 740, L54
  * Brown et al. (2011) Brown W. R., Kilic M., Hermes J. J., Allende Prieto C., Kenyon S. J., Winget D. E., 2011, ApJL, 737, L23
  * Brown et al. (2017) Brown W. R., Kilic M., Kosakowski A., Gianninas A., 2017, ApJ, 847, 10
  * Brown et al. (2020) Brown W. R., Kilic M., Kosakowski A., Andrews J. J., Heinke C. O., Agüeros M. A., Camilo F., et al., 2020, ApJ, 889, 49
  * Burdge et al. (2019a) Burdge K. B., Coughlin M. W., Fuller J., Kupfer T., Bellm E. C., Bildsten L., Graham M. J., et al., 2019a, Natur, 571, 528
  * Burdge et al. (2019b) Burdge K. B., Fuller J., Phinney E. S., van Roestel J., Claret A., Cukanovaite E., Gentile Fusillo N. P., et al., 2019b, ApJL, 886, L12
  * Burdge et al. (2020a) Burdge K. B., Prince T. A., Fuller J., Kaplan D. L., Marsh T. R., Tremblay P.-E., Zhuang Z., et al., 2020a, arXiv, arXiv:2009.02567
  * Burdge et al. (2020b) Burdge K. B., Coughlin M. W., Fuller J., Kaplan D. L., Kulkarni S. R., Marsh T. R., Prince T. A., et al., 2020b, arXiv, arXiv:2010.03555
  * Claret et al. (2020) Claret A., Cukanovaite E., Burdge K., Tremblay P.-E., Parsons S., Marsh T. R., 2020, A&A, 634, A93
  * Coughlin et al. (2020) Coughlin M. W., Burdge K., Phinney E. S., van Roestel J., Bellm E. C., Dekany R. G., Delacroix A., et al., 2020, MNRAS, 494, L91
  * Drake et al. (2010) Drake A. J., Beshore E., Catelan M., Djorgovski S. G., Graham M. J., Kleinman S. J., Larson S., et al., 2010, arXiv, arXiv:1009.3048
  * Eastman, Siverd, & Gaudi (2010) Eastman J., Siverd R., Gaudi B. S., 2010, PASP, 122, 935
  * Fontaine, Brassard, & Bergeron (2001) Fontaine G., Brassard P., Bergeron P., 2001, PASP, 113, 409
  * Fuller & Lai (2013) Fuller J., Lai D., 2013, MNRAS, 430, 274. doi:10.1093/mnras/sts606
  * Gianninas et al. (2013) Gianninas A., Strickland B. D., Kilic M., Bergeron P., 2013, ApJ, 766, 3
  * Hallakoun et al. (2016) Hallakoun N., Maoz D., Kilic M., Mazeh T., Gianninas A., Agol E., Bell K. J., et al., 2016, MNRAS, 458, 845
  * Hermes et al. (2012) Hermes J. J., Kilic M., Brown W. R., Winget D. E., Allende Prieto C., Gianninas A., Mukadam A. S., et al., 2012, ApJL, 757, L21
  * Istrate et al. (2016) Istrate A. G., Marchant P., Tauris T. M., Langer N., Stancliffe R. J., Grassitelli L., 2016, A&A, 595, A35
  * Kilic et al. (2014) Kilic M., Hermes J. J., Gianninas A., Brown W. R., Heinke C. O., Agüeros M. A., Chote P., et al., 2014, MNRAS, 438, L26
  * Kosakowski et al. (2020) Kosakowski A., Kilic M., Brown W. R., Gianninas A., 2020, ApJ, 894, 53
  * Landau & Lifshitz (1958) Landau, L. D., & Lifshitz, E. M., 1958, The Classical Theory of Fields, (Oxford: Oxford Pergamon Press)
  * Mukadam et al. (2011) Mukadam A. S., Owen R., Mannery E., MacDonald N., Williams B., Stauffer F., Miller C., 2011, PASP, 123, 1423
  * Steinfadt et al. (2010) Steinfadt J. D. R., Kaplan D. L., Shporer A., Bildsten L., Howell S. B., 2010, ApJL, 716, L146
  * Parsons et al. (2011) Parsons S. G., Marsh T. R., Gänsicke B. T., Drake A. J., Koester D., 2011, ApJL, 735, L30
  * Parsons et al. (2017) Parsons S. G., Gänsicke B. T., Marsh T. R., Ashley R. P., Bours M. C. P., Breedt E., Burleigh M. R., et al., 2017, MNRAS, 470, 4473
  * Piro (2011) Piro A. L., 2011, ApJL, 740, L53
  * Scott & Howell (2018) Scott N. J., Howell S. B., 2018, SPIE, 10701, 107010G
  * Southworth, Maxted, & Smalley (2004) Southworth J., Maxted P. F. L., Smalley B., 2004, MNRAS, 351, 1277. doi:10.1111/j.1365-2966.2004.07871.x
  * Southworth et al. (2005) Southworth J., Smalley B., Maxted P. F. L., Claret A., Etzel P. B., 2005, MNRAS, 363, 529. doi:10.1111/j.1365-2966.2005.09462.x
  * Southworth (2013) Southworth J., 2013, A&A, 557, A119
  * Tremblay, Bergeron, & Gianninas (2011) Tremblay P.-E., Bergeron P., Gianninas A., 2011, ApJ, 730, 128
  * Vennes et al. (2011) Vennes S., Thorstensen J. R., Kawka A., Németh P., Skinner J. N., Pigulski A., Ste&sacute, et al., 2011, ApJL, 737, L16