#  PAniC-3D: Stylized Single-view 3D Reconstruction
from Portraits of Anime Characters

Shuhong Chen*† Kevin Zhang* Yichun Shi† Heng Wang† Yiheng Zhu† Guoxian Song†
Sizhe An† Janus Kristjansson* Xiao Yang† Matthias Zwicker*
University of Maryland - College Park, MD, USA*
ByteDance†

###### Abstract

We propose PAniC-3D, a system to reconstruct stylized 3D character heads
directly from illustrated (p)ortraits of (ani)me (c)haracters. Our anime-style
domain poses unique challenges to single-view reconstruction; compared to
natural images of human heads, character portrait illustrations have hair and
accessories with more complex and diverse geometry, and are shaded with non-
photorealistic contour lines. In addition, there is a lack of both 3D model
and portrait illustration data suitable to train and evaluate this ambiguous
stylized reconstruction task. Facing these challenges, our proposed PAniC-3D
architecture crosses the illustration-to-3D domain gap with a line-filling
model, and represents sophisticated geometries with a volumetric radiance
field. We train our system with two large new datasets (11.2k Vroid 3D models,
1k Vtuber portrait illustrations), and evaluate on a novel AnimeRecon
benchmark of illustration-to-3D pairs. PAniC-3D significantly outperforms
baseline methods, and provides data to establish the task of stylized
reconstruction from portrait illustrations.

Figure 1: Overview of contributions. Our (A) PAniC-3D system is able to
reconstruct a 3D radiance field directly from a line-based portrait
illustration. We gather a new (B) Vtuber illustration dataset and (C) Vroid 3D
models dataset in order to cross the illustration-render domain gap and
supervise reconstruction. To evaluate, we provide a new (D) AnimeRecon
benchmark of paired illustrations and 3D models, establishing the novel task
of stylized single-view reconstruction of anime characters. (Art attributions
in suppl.)

##  1 Introduction & Related Work

With the rise of AR/VR applications, there is increased demand for not only
high-fidelity human avatars, but also non-photorealistic 3D characters,
especially in the “anime” style. Most character designers typically create
concept illustrations first, allowing them to express complex and highly
diverse characteristics like hair, accessories, eyes, skins, headshapes, etc.
Unfortunately, the process of developing illustrated concept art into an
AR/VR-ready 3D asset is expensive, requiring professional 3D artists trained
to use expert modeling software. While template-based creators democratize 3D
avatars to an extent, they are often restricted to 3D assets compatible with a
specific body model.

We propose PAniC-3D, a system to automatically reconstruct a stylized 3D
character head directly from illustrated (p)ortraits of (ani)me (c)haracters.
We formulate our problem in two parts: 1) implicit single-view head
reconstruction, 2) from across an illustration-3D domain gap. To summarize our
contributions:

  * •

PAniC-3D: a system to reconstruct the 3D radiance field of a stylized
character head from a single line-based portrait illustration.

  * •

The Vroid 3D dataset of 11.2k character models and renders, the first such
dataset in the anime-style domain to provide 3D assets with multiview renders.

  * •

The Vtuber dataset of 1.0k reconstruction-friendly portraits (aligned, front-
facing, neutral-expression) that bridges the illustration-render domain gap
through the novel task of line removal from drawings.

  * •

The AnimeRecon benchmark with 68 pairs of aligned 3D models and corresponding
illustrations, enabling quantitative evaluation of both image and geometry
metrics for stylized reconstruction.

###  1.1 Implicit 3D Reconstruction

While there has been much work on mesh-based reconstruction from images [23],
these systems are not expressive enough to capture the extreme complexity and
diversity of topology of our 3D characters. Inspired by the recent successes
in generating high-quality 3D radiance fields [4, 5, 25, 39], we instead turn
to implicit representations. However, to achieve high-quality results, recent
implicit reconstruction work such as PixelNerf [40] tend to operate solely
from 2D images, due to the lack of publicly-available high-quality 3D data.
Some implicit reconstruction systems like Pifu [31] employing complex 3D
assets have shown reasonable success using point-based supervision, but
require careful point sampling techniques and loss balancing.

There is also a body of work on sketch-based modeling, where 3D
representations are recovered from contour images. For example, Rui _et al_.
[24] use a multi-view decoder to predict sketch-to-depth and normals, which
are then used for surface reconstruction. Song _et al_. [44] additionally try
to compensate multi-view drawing discrepancies by learning to realign the
inputs. While related to our single-view portrait reconstruction problem,
these methods require multi-view sketches that are difficult for character
artists to draw consistently, and cannot handle color input.

For our case with complex high-quality 3D assets, we demonstrate the
superiority of differentiable volumetric rendering for reconstruction. We
build off of recent unconditional generative work (EG3D [4]), formulating the
problem of reconstruction as a conditional generation, proposing several
architecture improvements, and applying direct 2.5D supervision signal as
afforded by our 3D dataset.

###  1.2 Anime-style 3D Avatars and Illustrations

It is a fairly common task for 3D character artists to produce a 3D model from
a portrait illustration; however from the computer graphics standpoint, this
stylized reconstruction setup adds additional ambiguity to an already ill-
posed problem. In addition, while there’s work in the popular anime/manga
domain using 3D character assets (for pose estimation [18], re-targetting [20,
17], and reposing [22], etc.), there’s a lack of publicly-available 3D
character assets with multi-view renders that allow scalable training (Tab.
1). In light of these issues, we propose AnimeRecon (Fig. 1d) to formalize the
stylization task with a paired illustration-to-3D benchmark, and provide the
Vroid dataset (Fig. 1c) of 3D assets to enable large-scale training.

Within the problem of stylized reconstruction, we solve the task of contour
removal from illustrations. There is much work on line extraction [38, 21],
sketch simplification [33, 34], reconstruction from lines [11, 24], line
exploits for artistic imagery [6, 41], and scratch-line removal [27, 32, 29,
35, 8]; however, the removal of lines from line-based illustration has seen
little focus. We examine this contour deletion task in the context of adapting
drawings to render-like images more conducive to 3D reconstruction; we find
that naive image-to-image translation [45, 19] is unsuited to the task, and
propose a simple yet effective adversarial training setup with facial feature
awareness. Lastly, we provide a Vtuber dataset (Fig. 1b) of portraits to train
and evaluate contour removal for 3D reconstruction.

3D datasets | Vroid | AniRec. | AC | CoNR | ADD
---|---|---|---|---|---
3D avail. | Y | Y | - | - | -
renders avail. | Y | Y | Y | - | -
multiview | Y | Y | - | Y | Y
paired illustr. | - | Y | - | - | -
Table 1:  3D anime datasets comparison. Our new Vroid dataset is the first to make 3D anime models and multiview renders available. The AniRecon benchmark allows quantitative evaluation of both 2D image and 3D geometry metrics. Others left-to-right: AnimeCeleb [20], CoNR [22], Anime Drawings Dataset [18].  Portraits | Vtuber | AC | AP | iCF | BP
---|---|---|---|---|---
illustrations | Y | - | Y | Y | Y
aligned face | Y | Y | Y | Y | -
front-facing | Y | Y | - | - | -
neutral expr. | Y | Y | - | - | -
segmented | Y | Y | - | - | Y
face kpts. | Y | - | - | - | Y
Table 2:  Anime image datasets comparison. Our new Vtuber dataset allows us to
examine the domain gap between line-based illustrations and 3D renders, and is
filtered/standardized specifically for characteristics desirable in 3D head
reconstruction. Others left-to-right: AnimeCeleb [20], AnimePortraits [3],
iCartoonFace [43], BizarrePose [7].

##  2 Methodology

PAniC-3D is composed of two major components (Fig. 1a): a 3D reconstructor
directly supervised to predict a radiance field from a given front render, and
an illustration-to-render module that translates images to the reconstructor’s
training distribution. The two parts are trained independently, but are used
sequentially at inference.

###  2.1 3D Reconstruction Module

The 3D reconstruction module Fig. 3 is trained with direct supervision to
invert a frontal render into a volumetric radiance field. We build off of
recent unconditional generative work (EG3D [4]), formulating the problem of
reconstruction as that of conditional generation, proposing several
architecture improvements, and applying direct 2.5D supervision signal as
afforded by our 3D dataset.

Conditional input: The given front orthographic view to reconstruct is resized
and appended to the intermediate feature maps of the Stylegan2 backbone used
in EG3D [4]. In addition, at the earliest feature map we give the model high-
level domain-specific semantic information about the input by concatenating
the penultimate features of a pre-trained Resnet-50 anime tagger. The tagger
provides high-level semantic features appropriate for conditioning the
generator; it was pretrained by prior work [7] on 1062 relevant classes such
as blue_hair, cat_ears, twin_braids, etc.

Feature pooling: As the spatial feature maps are to be reshaped into a 3D
triplane as in EG3D [4], we found it beneficial to pool a fraction of each
feature map’s channels along the image axes (see Fig. 3 left). This simple
technique helps distribute information along common triplane axes, improving
performance on geometry metrics.

Multi-layer triplane: As proposed in concurrent work [1], we improve the EG3D
triplane by stacking more channels along each plane (see Fig. 3 center). The
method may be interpreted as a hybrid between a triplane and a voxel grid
(they are equivalent if the number of layers equals the spatial size). Setting
three layers per plane allows better spatial disambiguation when bilinearly
sampling the volume, and particularly helps our model generate more plausible
backs of heads (a challenge not faced by EG3D).

Losses: We take full advantage of the ground-truth 2.5D representations
afforded to us by our available 3D assets. Our reconstruction losses include:
RGB $L_{1}$, LPIPS [42], silhouette $L_{1}$, and depth $L_{2}$; these are
applied to the front, back, right, and left orthographic views, as shown in
Fig. 3. A discriminative loss is applied to improve the detail quality, in
addition to maintaining the generation orientation. We also keep the R1 and
density regularization losses from EG3D training. Our 2.5D representations and
adversarial setup allow us to surpass similar single-view reconstructors such
as PixelNerf [40] which work only with color losses.

Post-processing: We leverage our assumption that front-orthographic views are
given as input, by stitching the given input onto the generated radiance field
at inference. The xy-coordinates of each pixel’s intersection within the
volume are used to sample the input as a uv-texture map; we cast few
additional rays from each intersection to test for visibility from the front,
and apply the retexturing accordingly. This simple yet effective method
improves detail preservation from the input at negligible cost.

###  2.2 Illustration-to-Render Module

In order to remove non-realistic contour lines present in the input
illustration, but absent in a diffusely-lit radiance field, we design an
illustration-to-render module (Fig. 4). Assuming access to unpaired
illustrations and renders (our Vtuber and Vroid datasets, respectively), the
shallow network re-generates pixel colors near lines in the drawing in order
to adversarially match the render image distribution.

Similar to unpaired image-to-image models like CycleGAN and UGATIT [45, 19],
we also impose a small identity loss; while this may seem counter-productive
for our infilling case where identity is preserved in non-generated regions,
we found that this stabilizes the GAN training. Note that our setup also
differs from other infilling models, in that we inpaint to match a
distribution different from the input.

Following prior work extracting sketches from line-based animations [6], we
use the simple difference of gaussians (DoG) operator in order to prevent
double-line extraction around each stroke.

While most lines present in the drawing should be removed, certain lines
around key facial features must be preserved as they indeed appear in
renderings (eyes, mouth, nose, etc.). We employ an off-the-shelf anime facial
landmark detector [16] to create convex hulls around critical structures,
where infilling is disallowed.

We show that this line removal module indeed achieves a more render-like look;
it performs image translation more accurately than baseline methods when
evaluated over our AnimeRecon pairs (Tab. 4), and removes line artifacts from
the ultimate radiance field renders (Fig. 6).

Figure 2: (a) No-line diffuse render, (b) real-lined illustration, (c) Blender
Freestyle [14], (d) RTSC suggestive contours [9]. Toon shaders over-draw (c,
cheeks) or miss lines (d, bowtie); it is non-trivial to model the artistic
line placement in real drawings (b). Thus, we train our Illustration-to-Render
module on real lines drawn by artists instead of synthetically-added lines.
Figure 3: Schematic of the 3D reconstruction module. The front-orthographic
input rendering is fed to a series of upsampling convolutions, with
intermediate feature pooling to help distribute information along common
triplane axes. The final feature stack is reshaped into a multi-layer
triplane, which is volumetrically rendered and super-resolutioned to the final
output. Reconstruction losses are applied to front, left, right, and back
views (right view omitted in figure), and adversarial loss is applied to a
random perspective view. (Art attributions in suppl.) Figure 4: Schematic of
the illustration-to-render module. We design a simple yet effective network to
cross the domain gap by removing illustration contour lines absent in a
diffuse 3D render, while retaining lines present around facial features like
the eyes and mouth. (Art attributions in suppl.) Figure 5: Qualitative
comparison of baselines. Illustrations with lines removed by our illustration-
to-render module are fed to various reconstruction frameworks; our PAniC-3D
system delivers plausible reconstructions, while other methods struggle to
preserve identity and predict reasonable geometry. Note that metrics (Tab. 3)
between the displayed ground-truth and prediction are restricted to an ROI
bounding box/rectangular prism (unshown) during evaluation. (Art attributions
in suppl.) Figure 6: The effect of our illustration-to-render module on
reconstruction. Without our proposed line-infilling method, the reconstruction
module trained on contour-less renders is unable to cross the illustration
domain gap. Notice the line artifacts along the shoulders, and improper
contours along the chins. (Art attributions in suppl.)

##  3 Data

Unless otherwise mentioned, we use 80-10-10 splits for training, validation,
and testing.

###  3.1 Vroid 3D Dataset

We collect a large dataset of 11.2k 3D anime characters from VroidHub, in
order to train both the reconstruction and image-to-image translation modules
of PAniC-3D. Unlike previous work in the 3D anime domain using MikuMikuDance
PMD/PMX models [18, 20, 17], Vroid VRM models conform to the GLTF2 standard
[15] with several extensions [37], allowing us to render using ModernGL [10,
12]. As the data is crowd-sourced, we filter against a variety of undesirable
properties, such as: texture corruption, too much or too little transparency,
extreme character sizes, missing bones, etc. The 11.2k renders we use
represent 70% retension of our original 16.0k scraped models.

All our image data are rendered with unit ambient lighting (_e.g_. using
diffuse surface color only), unit distance away from the “neck” bone common to
VRMs [37]. Our choice to model only diffuse lighting is motivated by the
frequent artistic decision to paint speculars as textures (Tab. 4 row 2, hair
and eye highlights on diffuse renders). Adding specular renderering to painted
highlights would introduce inconsistent lighting effects. Linear blend
skinning is used to lower characters’ arms 60-degrees from their resting
T-pose position. We supersample at 1024px resolution, before bilinear
downsampling to 512px for training and testing.

The dataset for PAniC-3D’s reconstruction module consists of both random
perspective views for adversarial training, and fixed orthographic views for
the input and reconstruction losses. Each 3D model is rendered from 8 random
perspective views (with uniformly-sampled 360-degree azimuth, normally-sampled
elevation of 20-degree standard deviation, and fixed 30-degree full field-of-
view); this yields a total of 89.6k perspective images, each with known camera
parameters. The four orthographic views are taken at fixed 90-degree angles
from the front, sides, and back. The unpaired 3D data for our image-to-image
translation module is simply the front orthographic view of each character
identity.

###  3.2 Vtuber Portraits Dataset

We gather 1004 portraits from the VirtualYoutuber Fandom Wiki as the unpaired
illustration dataset for our image-to-image module. These portraits were
manually filtered from 15.2k scraped images for desired properties, _e.g_.
high-resolution, front-facing, neutral-expression, uncropped, etc. In order to
mimic the image distribution of the orthographic front-view Vroid renders, we
white out the backgrounds using the character segmenter from Chen _et al_. [7]
and select the largest connected component. In addition, we align the facial
keypoints of each illustration (extracted with a pretrained YOLOv3 [16, 28])
to match the height and scale distributions of keypoint detections on the
Vroid renders.

While we could add lines artificially, we found toon shaders [14, 9] to poorly
model artistic judgements needed to place lines on intricate characters (Fig.
2). To avoid these artifacts, we designed a data-driven Illustr2Render module
to train on real lines drawn by artists.

###  3.3 AnimeRecon Benchmark

We collect a benchmark set for stylized reconstruction by finding 68
characters with both 3D models and closely-aligned illustrations.
Specifically, we source from the 3D mobile game Genshin Impact (for which we
can match character portraits from the Fandom Wiki) and the virtual talent
agency Hololive (from which several members have both 3D avatars and a Fandom
Wiki portrait). As the raw 3D data from both sources comes in MMD format, we
convert to VRM using the DanSingSing converter; the rendering process is the
same as that of Vroid.

The portraits are aligned to their corresponding front-view orthographic
renders by a manually-decided mixture of YOLOv3 facial keypoints [16, 28] and
ORB detections. We segment out backgrounds using the same model [7] and
procedure as with Vtuber portraits. In order to maintain separation from
training data, we remove all Hololive identities present in this benchmark set
from the Vtuber portraits set.

Inevitably, the illustrations do not all align perfectly with their
corresponding renders, and many Genshin illustrations are cropped when
aligned. In order to perform more reasonable evaluation, we additionally label
a rectangular region of interest (ROI) for each aligned pair, within which the
alignment is appropriate; all 2D image and 3D geometry metrics reported are
restricted to the ROI when calculated.

|  | front |  |  | back |  |  | 360 |  |  | geom. |
---|---|---|---|---|---|---|---|---|---|---|---|---
| CLIP | LPIPS | PSNR | CLIP | LPIPS | PSNR | CLIP | LPIPS | PSNR | CD | F-1@5 | F-1@10
PAniC-3D (full) | 94.66 | 19.37 | 16.91 | 85.08 | 30.02 | 15.51 | 84.68 | 25.25 | 15.98 | 1.33 | 37.73 | 65.50
no feature pooling | 94.64 | 19.26 | 16.95 | 84.06 | 30.23 | 15.53 | 84.30 | 25.42 | 15.96 | 1.38 | 35.26 | 65.51
no multi-layer triplane | 94.39 | 19.99 | 16.94 | 84.28 | 30.37 | 15.41 | 84.49 | 26.13 | 15.82 | 1.44 | 34.55 | 65.95
no side/back loss | 94.54 | 19.93 | 16.86 | 85.50 | 32.18 | 14.73 | 83.98 | 27.34 | 15.39 | 1.56 | 36.82 | 58.67
no 2.5D loss | 94.10 | 20.76 | 17.72 | 83.27 | 32.51 | 15.11 | 84.58 | 26.19 | 15.65 | 1.37 | 36.42 | 63.88
PixelNerf [40] | 91.07 | 22.96 | 16.76 | 81.02 | 32.01 | 15.74 | 80.42 | 24.86 | 16.31 | 1.45 | 35.07 | 65.50
EG3D+Img2SG [2] | 85.90 | 30.78 | 13.92 | 77.78 | 39.84 | 12.68 | 79.78 | 31.10 | 13.86 | 2.05 | 11.88 | 23.55
EG3D+PTI [30] | 89.90 | 25.93 | 15.50 | 77.37 | 39.16 | 13.37 | 79.78 | 32.62 | 14.32 | 2.19 | 11.38 | 23.54
Pifu [31] | 75.12 | 41.62 | 11.94 | 75.01 | 43.63 | 12.51 | 73.62 | 32.45 | 13.81 | 1.35 | 37.37 | 68.13
Table 3:  Ablations and baselines. We evaluate and compare our system using
our new AnimeRecon benchmark of illustrations with their ground-truth 3D
reconstructions. Image reconstruction metrics are listed for the front
(reconstruction of the given input) and back orthographic views, as well as
averaged over twelve perspective spin-around views. On the right, chamfer
distances and F-1 scores [13] capture the correctness of reconstructed meshes.
For fair comparison, lines are preprocessed out of all input images using our
illustration-to-render model, and evaluation is performed within our annotated
ROIs. The top two of each category are bolded.  Figure 7: 3D reconstruction
module ablations. Row 1: feature pooling helps distribute information across
the triplane, and correctly structures the hair as short. Rows 2+3: the multi-
layer triplane helps disambiguate the front/back, and side/back-view losses
significantly improve both geometry and texture. (Art attributions in suppl.)

##  4 Results & Evaluation

In this section, we provide a breakdown of reconstruction and image-to-image
translation performance, with both qualitative and quantitative comparison to
other baselines.

###  4.1 3D Reconstruction Results

####  4.1.1 Metrics

As shown in Tab. 3, we evaluate over our new AnimeRecon benchmark using both
2D image and 3D geometry metrics. All the illustration inputs went through our
illustration-to-render module before being fed to the respective method; we
believe this is a fairer comparison, as all the reconstructors were trained on
renders.

Image metrics are measured by comparing the predicted radiance field’s
integrated image with the ground-truth render from the same camera viewpoint.
We show such measures for the front orthographic view (which should match the
input), the back orthographic view, and an average of 12 perspective cameras
circling the character at 30-degree intervals. For the front and back views,
we restrict evaluation to our AnimeRecon ROI (Fig. 1d); for the 12 circling
views, we crop to the horizontal bounding box strip.

We show standard color metrics like PSNR for completeness, but as there are
inevitably imperfections on the AnimeRecon illustration-render pairs (even
within the ROI), perceptual metrics like CLIP image cosine similarity [26] and
LPIPS [42] are generally more relevant to quality.

The geometry metrics are on the right of Tab. 3. We extract meshes from both
the ground-truth 3D asset and the predicted radiance field (through marching
cubes), and delete faces with vertices outside the rectangular prism defined
by the ROI annotation (see Fig. 1d top). We then sample 10k points randomly
from each mesh subset, to compute the point-cloud chamfer distance and F-1
scores at 5cm and 10cm [13]. To put F-1 in perspective, the average Vroid head
width is 25.5cm (real heads are 14cm). The F-1 are low overall, due to larger
proportions and protruding hair/accessories with large surface area.

Though Mesh R-CNN also proposes the normal consistency metric [13], it is
difficult to obtain for our data, as there are assets with broken ground truth
normals (zero values, inverted meshes). We can calculate rough consistency by
re-estimating all normals, resulting in: Ours 75.6, PixelNerf 77.5, Img2SG
73.0, PTI 72.7, Pifu 77.3. We observed that these approximate measures don’t
match well with visual quality, but include them here for completeness.

####  4.1.2 Baselines

Comparisons are shown between our method and several other implicit
reconstruction methods (Fig. 5). Of the methods shown, only Pifu [31] receives
point-wise signals for optimization; we see that this works to its detriment,
as the near-surface point sampling strategies bias the model towards certain
geometric structures (such as black eyelashes that often protrude from the
face). The other methods using volumetric rendering inherently weigh the
supervision signal such that the final rendered product is consistent.

Image2stylegan [2] and Pivotal Tuning Inversion [30] are optimization-based
methods of performing single-view reconstruction, the latter of which was used
in EG3D to demonstrate reconstruction by projection [4]. We train an EG3D with
similar hyperparameters as the Stylegan2 backbone used in our reconstruction
module, and allow the two respective baselines to optimize features/weights in
the prior to match the given input. Unfortunately, the EG3D prior is not
sufficient to regularize the optimization, leading to implausible results;
while PTI worked reasonably well for EG3D, it may not work as well on our
setup where the back of the head must also be projected.

Lastly we compare to our model with the PixelNerf [40] single-view setup. The
key difference in this comparison is that PixelNerf does not use adversarial
nor 2.5D reconstruction losses (for fairer comparison with us, we trained
PixelNerf with LPIPS in addition to $L_{1}$, resulting in significantly less
blurry results). We see that without these signals provided by our available
3D assets, the quality of details and geometry does not match up to PAniC-3D.

####  4.1.3 Ablations

From Tab. 3 and Fig. 7, we conclude that our architecture decisions improve
both the qualitative and quantitative performance of our reconstruction
system. It is shown that feature pooling is able to propagate information
along triplane axes to improve generated geometries, the multi-layer triplane
adds additional model capacity to further disambiguate locations for features,
and the addition of fixed side/back-view losses significantly improves
geometry and texture. Expectedly, removing 2.5D supervision also deteriorates
performance.

###  4.2 Illustration-to-Render Results

As shown in Tab. 4 and Fig. 6, our illustration-to-render model was able to
effectively remove contours that would not appear in renders. We evaluate in
Tab. 4 how closely the translated illustrations match their rendered
counterpart from the AnimeRecon benchmark (restricted to the ROI annotation),
and find that we are able to significantly outperform naive inpainting [36] as
well as off-the-shelf image-to-image translation systems [19, 45]. The others
struggled to retain key semantic structures like eyes/mouths, and often failed
to retain the original identity.

In Fig. 6, it is shown that removing lines indeed alleviates the effects of
domain transfer between illustrations and their output predicted radiance
field; artifacts like extra contours and bands along shoulders are
significantly reduced using our line removal strategy.

| LPIPS↓ | CLIP↑ | PSNR↑
---|---|---|---
Ours | 18.26 | 94.97 | 16.96
Telea [36] | 23.91 | 93.88 | 16.67
CycleGAN [45] | 21.39 | 93.81 | 15.23
UGATIT [19] | 39.64 | 85.48 | 13.18

Table 4:  Illustration-to-render results. Over our AnimeRecon benchmark, we
calculate perceptual and color metrics between translated illustrations and
their corresponding 3D renders, restricted to the ROI annotations. Our method
crosses the illustration-render domain gap better than both naive inpainting
(which struggles to retain facial features) and deep image2image models (which
fail to retain identity). LPIPS is scaled by 1e2.

##  5 Limitations & Future Work

From the figures comparing the ground-truth 3D model renders and generated
radiance fields, it is evident that this task is incredibly difficult.
Although PAniC-3D performs better than other baseline methods, there is still
a large gap in quality between our reconstructions and real character assets;
this is still particularly evident for the hind occluded regions, as well as
the areas around the ears connecting the front and back. Our model is usually
able to prevent faces from appearing on the backside, but still largely relies
on copying the visible portions of accessories to cover the occluded parts. In
the future, we may look into ways of decomposing the radiance field in an
object-centric manner, and generate accessories through a separate more
expressive process. An obvious extension of this work would be to model full-
body characters and exploit more opportunities given by the Vroid dataset,
which supports blendshape facial expressions, hair physics, full-body rigging,
and more.

In conclusion, we propose PAniC-3D, a system to reconstruct stylized 3D
character heads directly from illustrated portraits of anime characters. Our
anime-style domain poses unique challenges to single-view reconstruction when
compared to natural images of human heads, such as hair and accessories with
more complex and diverse geometry, and non-photorealistic contour lines.
Furthermore, there is a lack of both 3D model and portrait illustration data
suitable to train and evaluate this ambiguous stylized reconstruction task.
Facing these challenges, our proposed PAniC-3D architecture crosses the
illustration-to-3D domain gap with a line-filling model, and represents
sophisticated geometries with a volumetric radiance field. We train our system
with two large new datasets (11.2k Vroid 3D models, 1k Vtuber portrait
illustrations), and evaluate on a novel AnimeRecon benchmark of illustration-
to-3D pairs. PAniC-3D significantly outperforms baseline methods, and can
generate plausible fully-textured geometries from a single input drawing. We
hope that our proposed system and provided datasets may help establish the
stylized reconstruction task.

## Acknowledgements

The authors would like to thank Softmind Ltd. for sharing their DanSingSing
PMX-to-VRM converter, Geng Lin for adding functions to the converter, Srinidhi
Hegde for valuable discussion on related topics, and NVIDIA for additional GPU
support.

## References

  * [1] Panohead: Geometry-aware 3d full-head generative adversarial networks.  2022\.
  * [2] Rameen Abdal, Yipeng Qin, and Peter Wonka.  Image2stylegan: How to embed images into the stylegan latent space?  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 4432–4441, 2019.
  * [3] Gwern Branwen, Anonymous, and Danbooru Community.  Danbooru2019 portraits: A large-scale anime head illustration dataset.  https://www.gwern.net/Crops#danbooru2019-portraits, March 2019\.
  * [4] Eric R Chan, Connor Z Lin, Matthew A Chan, Koki Nagano, Boxiao Pan, Shalini De Mello, Orazio Gallo, Leonidas Guibas, Jonathan Tremblay, Sameh Khamis, et al.  Efficient geometry-aware 3d generative adversarial networks.  CVPR, 2022.
  * [5] Eric R Chan, Marco Monteiro, Petr Kellnhofer, Jiajun Wu, and Gordon Wetzstein.  pi-gan: Periodic implicit generative adversarial networks for 3d-aware image synthesis.  In CVPR, 2021.
  * [6] Shuhong Chen and Matthias Zwicker.  Improving the perceptual quality of 2d animation interpolation.  arXiv preprint arXiv:2111.12792, 2021.
  * [7] Shuhong Chen and Matthias Zwicker.  Transfer learning for pose estimation of illustrated characters.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 793–802, 2022.
  * [8] Zhiguo Cheng and Yuncai Liu.  A graph-based method to remove interferential curve from text image.  Machine Vision and Applications, 17:219–228, 2006.
  * [9] Doug DeCarlo et al.  Suggestive contours for conveying shape.  In SIGGRAPH, pages 848–855. 2003.
  * [10] Szabolcs Dombi.  Moderngl, high performance python bindings for opengl 3.3+.  GitHub repository.
  * [11] Marek Dvorožňák, Daniel Sỳkora, Cassidy Curtis, Brian Curless, Olga Sorkine-Hornung, and David Salesin.  Monster mash: a single-view approach to casual 3d modeling and animation.  ACM Transactions on Graphics (TOG), 39(6):1–12, 2020.
  * [12] Einar Forselv.  moderngl-window, a cross-platform windowing/utility library for moderngl.  GitHub repository.
  * [13] Georgia Gkioxari and Jitendra Malik.  Jj: Mesh r-cnn.  In Proceedings of the IEEE International Conference on Computer Vision, 2019.
  * [14] Stéphane Grabli et. al.  Programmable style for npr line drawing.  In Eurographics, 2004.
  * [15] The Khronos Group.  Gltf specification.  2022\.
  * [16] hysts.  Anime face detector.  https://github.com/hysts/anime-face-detector, 2021.
  * [17] Pramook Khungurn.  Talking head anime from a single image 2: More expressive.  2021\.
  * [18] Pramook Khungurn and Derek Chou.  Pose estimation of anime/manga characters: a case for synthetic data.  pages 1–6, 2016.
  * [19] Junho Kim, Minjae Kim, Hyeonwoo Kang, and Kwanghee Lee.  U-gat-it: Unsupervised generative attentional networks with adaptive layer-instance normalization for image-to-image translation.  arXiv preprint arXiv:1907.10830, 2019.
  * [20] Kangyeol Kim, Sunghyun Park, Jaeseong Lee, Sunghyo Chung, Junsoo Lee, and Jaegul Choo.  Animeceleb: Large-scale animation celebheads dataset for head reenactment.  2022\.
  * [21] Chengze Li, Xueting Liu, and Tien-Tsin Wong.  Deep extraction of manga structural lines.  ACM Transactions on Graphics (TOG), 36(4):1–12, 2017.
  * [22] Zuzeng Lin, Ailin Huang, Zhewei Huang, Chen Hu, and Shuchang Zhou.  Collaborative neural rendering using anime character sheets.  arXiv preprint arXiv:2207.05378, 2022.
  * [23] Shichen Liu, Tianye Li, Weikai Chen, and Hao Li.  Soft rasterizer: A differentiable renderer for image-based 3d reasoning.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 7708–7717, 2019.
  * [24] Zhaoliang Lun, Matheus Gadelha, Evangelos Kalogerakis, Subhransu Maji, and Rui Wang.  3d shape reconstruction from sketches via multi-view convolutional networks.  In 2017 International Conference on 3D Vision (3DV), pages 67–77. IEEE, 2017.
  * [25] Roy Or-El, Xuan Luo, Mengyi Shan, Eli Shechtman, Jeong Joon Park, and Ira Kemelmacher-Shlizerman.  Stylesdf: High-resolution 3d-consistent image and geometry generation.  CVPR, 2022.
  * [26] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al.  Learning transferable visual models from natural language supervision.  In International Conference on Machine Learning, pages 8748–8763. PMLR, 2021.
  * [27] N Shobha Rani, P Vineeth, and Deeptha Ajith.  Detection and removal of graphical components in pre-printed documents.  International Journal of Applied Engineering Research, 11(7):4849–4856, 2016.
  * [28] Joseph Redmon and Ali Farhadi.  Yolov3: An incremental improvement.  arXiv preprint arXiv:1804.02767, 2018.
  * [29] A Rehman, F Kurniawan, and T Saba.  An automatic approach for line detection and removal without smash-up characters.  The Imaging Science Journal, 59(3):177–182, 2011.
  * [30] Daniel Roich, Ron Mokady, Amit H Bermano, and Daniel Cohen-Or.  Pivotal tuning for latent-based editing of real images.  ACM Transactions on Graphics (TOG), 42(1):1–13, 2022.
  * [31] Shunsuke Saito, Zeng Huang, Ryota Natsume, Shigeo Morishima, Angjoo Kanazawa, and Hao Li.  Pifu: Pixel-aligned implicit function for high-resolution clothed human digitization.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2304–2314, 2019.
  * [32] Timothy K Shih, Louis H Lin, and Wonjun Lee.  Detection and removal of long scratch lines in aged films.  In 2006 IEEE International Conference on Multimedia and Expo, pages 477–480. IEEE, 2006.
  * [33] Edgar Simo-Serra, Satoshi Iizuka, and Hiroshi Ishikawa.  Mastering sketching: adversarial augmentation for structured prediction.  ACM Transactions on Graphics (TOG), 37(1):1–13, 2018.
  * [34] Edgar Simo-Serra, Satoshi Iizuka, and Hiroshi Ishikawa.  Real-time data-driven interactive rough sketch inking.  ACM Transactions on Graphics (TOG), 37(4):1–14, 2018.
  * [35] Domenico Tegolo and Francesco Isgro.  Scratch detection and removal from static images using simple statistics and genetic algorithms.  In Proceedings 2001 International Conference on Image Processing (Cat. No. 01CH37205), volume 1, pages 265–268. IEEE, 2001.
  * [36] Alexandru Telea.  An image inpainting technique based on the fast marching method.  Journal of graphics tools, 9(1):23–34, 2004.
  * [37] VRoid.  Vrm specification.  2022\.
  * [38] Holger Winnemöller, Jan Eric Kyprianidis, and Sven C Olsen.  Xdog: An extended difference-of-gaussians compendium including advanced image stylization.  Computers & Graphics, 36(6):740–753, 2012.
  * [39] Yang Xue, Yuheng Li, Krishna Kumar Singh, and Yong Jae Lee.  Giraffe hd: A high-resolution 3d-aware generative model.  In CVPR, 2022.
  * [40] Alex Yu, Vickie Ye, Matthew Tancik, and Angjoo Kanazawa.  pixelnerf: Neural radiance fields from one or few images.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4578–4587, 2021.
  * [41] Lvmin Zhang, Yi Ji, and Chunping Liu.  Danbooregion: An illustration region dataset.  In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XIII 16, pages 137–154. Springer, 2020.
  * [42] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang.  The unreasonable effectiveness of deep features as a perceptual metric.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 586–595, 2018.
  * [43] Yi Zheng, Yifan Zhao, Mengyuan Ren, He Yan, Xiangju Lu, Junhui Liu, and Jia Li.  Cartoon face recognition: A benchmark dataset.  In Proceedings of the 28th ACM International Conference on Multimedia, pages 2264–2272, 2020.
  * [44] Yue Zhong, Yonggang Qi, Yulia Gryaditskaya, Honggang Zhang, and Yi-Zhe Song.  Towards practical sketch-based 3d shape generation: The role of professional sketches.  IEEE Transactions on Circuits and Systems for Video Technology, 31(9):3518–3528, 2020.
  * [45] Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros.  Unpaired image-to-image translation using cycle-consistent adversarial networks.  In Proceedings of the IEEE international conference on computer vision, pages 2223–2232, 2017.


# Solar-like oscillations and ellipsoidal variations in TESS observations of
the binary 12 Boötis

Warrick H. Ball,1 Andrea Miglio,1,2,3 William J. Chaplin,1 Keivan G. Stassun,4
Rafael García,5 Lucia González-Cuesta,6,7 Savita Mathur,6,7 Thierry
Appourchaux,8 Othman Benomar,9,10,11 Derek L. Buzasi,12 Chen Jiang (姜晨),13
Cenk Kayhan,14 Sibel Örtel,15 Zeynep Çelik Orhan,15 Mutlu Yıldız,15 J. M. Joel
Ong (王加冕),16 Sarbani Basu16
1School of Physics and Astronomy, University of Birmingham, Edgbaston,
Birmingham B15 2TT, United Kingdom
2Dipartimento di Fisica e Astronomia, Universitá degli Studi di Bologna, Via
Gobetti 93/2, I-40129 Bologna, Italy
3INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via
Gobetti 93/3, I-40129 Bologna, Italy
4Department of Physics & Astronomy, Vanderbilt University, Nashville, TN
37235, USA
5Département d’Astrophysique/AIM, CEA/IRFU, CNRS/INSU, Univ. Paris-Saclay &
Univ. de Paris, 91191 Gif-sur-Yvette, France
6Instituto de Astrofísica de Canarias, La Laguna, Tenerife, Spain
7Dpto. de Astrofísica, Universidad de La Laguna, La Laguna, Tenerife, Spain
8Université Paris-Sud, Institut d’Astrophysique Spatiale, UMR 8617, CNRS,
Bâtiment 121, 91405 Orsay Cedex, France
9 Department of Astronomical Science, School of Physical Sciences, SOKENDAI,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
10National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan
11Center for Space Science, New York University Abu Dhabi, P.O. Box 129188,
Abu Dhabi, UAE
12Department of Chemistry and Physics, Florida Gulf Coast University, 10501
FGCU Blvd., Fort Myers, FL 33965 USA
13Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3,
37077 Göttingen, Germany
14Department of Astronomy and Space Sciences, Science Faculty, Erciyes
University, 38030 Melikgazi, Kayseri, Turkey
15Department of Astronomy and Space Sciences, Science Faculty, Ege University,
35100, Bornova, İzmir, Turkey
16Department of Astronomy, Yale University, P.O. Box 208101, New Haven, CT
06520-8101, USA

(Accepted 2022 August 01. Received 2022 July 29; in original form 2022 May 23)

###### Abstract

Binary stars in which oscillations can be studied in either or both components
can provide powerful constraints on our understanding of stellar physics. The
bright binary 12 Boötis (12 Boo) is a particularly promising system because
the primary is roughly 60 per cent brighter than the secondary despite being
only a few per cent more massive. Both stars have substantial surface
convection zones and are therefore, presumably, solar-like oscillators. We
report here the first detection of solar-like oscillations and ellipsoidal
variations in the TESS light curve of 12 Boo. Though the solar-like
oscillations are not clear enough to unambiguously measure individual mode
frequencies, we combine global asteroseismic parameters and a precise fit to
the spectral energy distribution (SED) to provide new constraints on the
properties of the system that are several times more precise than values in
the literature. The SED fit alone provides new effective temperatures,
luminosities and radii of $6115\pm 45\,\mathrm{K}$, $7.531\pm
0.110\,\mathrm{L}_{\odot}$ and $2.450\pm 0.045\,\mathrm{R}_{\odot}$ for 12 Boo
A and $6200\pm 60\,\mathrm{K}$, $4.692\pm 0.095\,\mathrm{L}_{\odot}$ and
$1.901\pm 0.045\,\mathrm{R}_{\odot}$ for 12 Boo B. When combined with our
asteroseismic constraints on 12 Boo A, we obtain an age of
$2.67^{+0.12}_{-0.16}\,\mathrm{Gyr}$, which is consistent with that of 12 Boo
B.

###### keywords:

stars: oscillations (including pulsations); stars: individual (12 Boo); stars:
binaries; asteroseismology

††pubyear: 2022††pagerange: Solar-like oscillations and ellipsoidal variations
in TESS observations of the binary 12 Boötis–References

##  1 Introduction

Binary stars have long provided important tests of widely-used one-dimensional
stellar models. Great attention is usually given to double-lined eclipsing
binaries, in which masses and radii can be measured, but astrometric double-
lined binaries also provide stellar masses, which are arguably stars’ most
important initial physical parameter.

Independently, the study of stellar oscillations— _asteroseismology_ —also
provides important tests of stellar physics. When we can identify multiple
modes and measure their frequencies, each mode provides a slightly different
average of some interior properties, which allows very precise measurements of
certain characteristics (e.g., the mean density) and potentially tests to
distinguish between theories of the stars’ interior physics (see e.g. Aerts,
2021, for a recent review). In cool stars like the Sun
($T_{\mathrm{eff}}\lesssim 6500\,\mathrm{K}$), near-surface convection drives
and damps oscillations across a wide range of frequencies. These oscillations
are known as _solar-like oscillations_ ; stars that show these oscillations
are _solar-like oscillators_. The modes in solar-like oscillators can easily
be identified by the regular patterns they follow (see e.g. García & Ballot,
2019, for a recent review). Their study has recently been revolutionised by
space-based photometry from CoRoT (Auvergne et al., 2009; CoRot Team, 2016),
_Kepler_ (Borucki et al., 2010) and K2 (Howell et al., 2014).

The asteroseismology of binary stars is thus particularly promising for
testing stellar models but solar-like oscillations have been measured in very
few main-sequence or early subgiant binary systems. Miglio et al. (2014)
predicted that _Kepler_ would detect solar-like oscillations in both
components of only a few main-sequence binaries, which has been borne out.
Main-sequence binaries in which _Kepler_ observed solar-like oscillations in
both components include 16 Cyg (KIC 12069424, KIC 12069449; Metcalfe et al.,
2012; Davies et al., 2015), HD 177412 (KIC 7510397; Appourchaux et al., 2015)
and HD 176465 (KIC 10124866; White et al., 2017). Halbwachs (1986) identified
HIP 92961 & 92962 (KIC 9139151 & KIC 9139163) as a pair with common proper
motions and Silva Aguirre et al. (2017) included both stars in their survey of
solar-like oscillators observed by _Kepler_. Aside from these main-sequence
binaries observed by _Kepler_ , ground-based radial velocity campaigns have
also observed solar-like oscillations in both components of $\alpha$ Cen
(Bouchy & Carrier, 2001; Kjeldsen et al., 2005).

With the launch of the Transiting Exoplanet Survey Satellite (TESS; Ricker et
al., 2015), we can now reverse the selection process and search for solar-like
oscillations in known main-sequence or subgiant binaries. TESS is less
sensitive than _Kepler_ , so detecting solar-like oscillations in main-
sequence and subgiant stars is limited to very bright stars with $G\lesssim 6$
(see, e.g., the non-detection of solar-like oscillations in AI Phe, Maxted et
al., 2020). For example, solar-like oscillations have been detected in TESS
observations of HD 221416 (Huber et al., 2019), $\nu$ Ind (Chaplin et al.,
2020), $\lambda^{2}$ For (Nielsen et al., 2020) 94 Aqr Aa (Metcalfe et al.,
2020), HD 38529 (Ball et al., 2020), HD 19916 (Addison et al., 2021), $\rho$
CrB (Metcalfe et al., 2021), $\alpha$ Men A (Chontos et al., 2021), $\gamma$
Pav, $\zeta$ Tuc and $\pi$ Men (Huber et al., 2022).

We report here the results of our search for solar-like oscillations in the
TESS light curve of 12 Boötis (d Boötis, HR 5304, HD 123999, TIC 418010485;
hereafter 12 Boo), a bright ($G=4.66$) binary system comprising two roughly
$1.4\,\mathrm{M}_{\odot}$ stars in a slightly eccentric ($e\approx 0.2$),
$9.6\,\mathrm{d}$ orbit. Campbell & Wright (1900) originally discovered the
radial velocity variations in 12 Boo and Harper (1914) subsequently measured
more radial velocities and fit orbital parameters. Merrill (1922) conducted an
interferometric survey of a number of binary stars but could not resolve the
orbit of 12 Boo. The binary nature of the system was not revisited until Abt &
Levy (1976) computed new orbital elements. de Medeiros & Udry (1999) measured
new radial velocities as part of a survey of evolved stars.

Boden et al. (2000) presented the first interferometric observations of the
orbit of the system and noted that the primary is about $0.5$ magnitudes
brighter in $V$ than the secondary despite being of about the same mass. They
recognised that this probably meant the primary has recently left the main-
sequence and the secondary is near the end of its main-sequence life. Boden et
al. (2005) refined their previous result and Tomkin & Fekel (2006)
independently derived consistent orbital parameters. Konacki et al. (2010)
presented an even more precise simultaneous fit of the interferometric
measurements combined with the radial velocities they measured as part of the
TATOOINE search for circumbinary planets. More recently, Behr et al. (2011)
reported more independent radial velocities and Kervella et al. (2017) made a
limited set of measurements of 12 Boo as a calibrator for their study of
$\alpha$ Cen.

Miglio et al. (2007) highlighted potential of 12 Boo as a test for stellar
models. In particular, they investigated how the detection of mixed modes,
which have oscillating components in the core and envelope, would distinguish
different possibilities for the extent of extra mixing at the boundary of the
main-sequence convective core. Miglio et al. (2007) also pointed out that even
the more modest goal of measuring the stars’ large frequency separations
$\Delta\nu$ would provide more precise radius estimates.

We present here our analysis of TESS light curve of 12 Boo, which includes the
first detection of solar-like oscillations and ellipsoidal variations in this
system. In Sec. 2, we detail our new analyses of the TESS light curve and
provide an updated fit of the spectral energy distribution incorporating new
photometry and parallax data from Gaia. In Sec. 3, we summarise our results
from fitting multiple sets of stellar models to the new constraints, including
asteroseismic constraints on 12 Boo A. We close in Sec. 4 with some
speculation on the future seismic potential of 12 Boo.

##  2 Observations

Figure 1: TESS light curves of 12 Boo from the PDCSAP pipeline (blue) and our
custom reduction (orange, offset). We subtracted a slowly-varying trend
(black) before computing the power spectrum (Fig. 2).

###  2.1 Light curve

TESS observed 12 Boo on Camera 2 during Sector 23 (2020-03-19 to 2020-04-15)
at a cadence of 2 min (i.e., TESS’s original short cadence). Our initial
inspection of the pipeline-reduced aperture photometry (Pre-search Data
Conditioning Simple Aperture Photometry, PDCSAP), shown by the blue points in
Fig. 1, immediately showed mmag-level variations matching the orbital period
of the system. These are ellipsoidal variations caused by the gravity of each
star slightly distorting its companion. As far as we know, this is the first
time ellipsoidal variations have been reported for 12 Boo.

The PDCSAP light curve, shown in blue in Fig. 1, offered only a marginal
detection of solar-like oscillations in 12 Boo A, so we experimented with
custom light curves, reduced from the short-cadence imagettes. The imagettes
include many cadences that were excluded from the PDCSAP reduction when
scattered light left too little data to derive cotrending basis vectors, which
the pipeline uses to remove systematic effects (Fausnaugh et al., 2021,
quality bit 16). We selected all pixels with a median flux greater than
$100\mathrm{e}^{-}\,\mathrm{s}^{-1}$, including one line of pixels above and
below the saturated columns as well as the columns before and after the
saturated ones. We then removed all data points taken while the spacecraft was
in safe mode (quality bit 2), pointing to Earth (quality bit 4) or
desaturating the reaction wheels (quality bit 32). We also removed data points
marked as impulsive outliers (quality bit 10). Finally, small gaps were filled
using the inpainting techniques described by García et al. (2014) and Pires et
al. (2015). Our custom light curve is shown in orange in Fig. 1, offset
downwards by $1\,\mathrm{ppt}$ for visibility. The ellipsoidal variations
remain clear and we recovered several days of data near the beginnings of
TESS’s orbits. These are the data that the PDCSAP pipeline excluded because
the scattered light prevents it from deriving cotrending basis vectors.

To produce the seismic light curve, we applied a triangular filter with a
window function of 5 days ($2.31\,\,\mu\mathrm{Hz}$). To reduce the border
effects at the beginning and the end of the series, the filter was computed
for the times covered in the original light curve after reflecting the
original light curve at each end by 2.5 days. The trend is shown by the black
line in Fig. 1 and effectively removes the ellipsoidal variations, as well as
slow systematic drifts at the beginnings and ends of each of TESS’s orbits.

Figure 2: Raw (black) and smoothed (orange) power spectrum of our custom light curve for 12 Boo, computed using a Lomb–Scargle periodogram after subtracting the slowly-varying trend (Fig. 1). Figure 3: Échelle diagram of the raw power spectrum (c.f. Fig. 2), folded on an approximate large separation of $\Delta\nu=41.7\,\mu\mathrm{Hz}$. Team | $\Delta\nu/\,\mu\mathrm{Hz}$ | $\nu_{\mathrm{max}}/\,\mu\mathrm{Hz}$
---|---|---
Benomar | $42.21_{-0.31}^{+0.38}$ | $696^{+17}_{-16}$
Jiang | $41.6\pm 0.5$ | $667\pm 5$
Mathur | $41.26\pm 1.47$ | $669\pm 40$
Combined | $41.7\pm 1.0$ | $678\pm 29$
Table 1: Large frequency separations $\Delta\nu$ and frequencies of maximum oscillation power $\nu_{\mathrm{max}}$ measured by three independent teams. | 12 Boo A | 12 Boo B | Source
---|---|---|---
| Literature values
$T_{\mathrm{eff}}/K$ | $6130\pm 100$ | $6230\pm 150$ | Boden et al. (2005)
$R/\,\mathrm{R}_{\odot}$ | $2.474\pm 0.095$ | $1.86\pm 0.15$ | Boden et al. (2005)
$L/\,\mathrm{L}_{\odot}$ | $7.76\pm 0.35$ | $4.69\pm 0.74$ | Boden et al. (2005)
| Constraints
$M/\,\mathrm{M}_{\odot}$ | $1.4109\pm 0.0028$ | $1.3677\pm 0.0028$ | Konacki et al. (2010)
$T_{\mathrm{eff}}/K$ | $6115\pm 45$ | $6200\pm 60$ | Sec. 2.3
$R/\,\mathrm{R}_{\odot}$ | $2.450\pm 0.045$ | $1.901\pm 0.045$ | Sec. 2.3
$L/\,\mathrm{L}_{\odot}$ | $7.531\pm 0.110$ | $4.692\pm 0.095$ | Sec. 2.3
$[\mathrm{Fe}/\mathrm{H}]$ | $-0.065\pm 0.101$ | $-0.065\pm 0.101$ | Sec. 2.4
| Stellar modelling results
$R/\,\mathrm{R}_{\odot}$ | $2.464^{+0.042}_{-0.025}$ | $1.881^{+0.056}_{-0.040}$ | Sec. 3
$\bar{\rho}/(\mathrm{g\thinspace cm}^{-3})$ | $0.133^{+0.004}_{-0.006}$ | $0.290^{+0.019}_{-0.026}$ | Sec. 3
$\log g$ | $3.802^{+0.010}_{-0.014}$ | $4.028^{+0.023}_{-0.028}$ | Sec. 3
$t/\,\mathrm{Gyr}$ | $2.67^{+0.12}_{-0.16}$ | $2.66^{+0.11}_{-0.15}$ | Sec. 3
Table 2: Various properties of 12 Boo. The upper set include constraints in
the stellar modelling (Sec. 3) and the lower set the results of the modelling.

###  2.2 Asteroseismic parameters

Fig. 2 shows the power spectrum of our detrended custom light curve after
subtracting the trend determined above. There is a clear excess of power
around $700\,\mu\mathrm{Hz}$, where the oscillations of the primary are
expected based on its previously measured mass, luminosity and radius (see
Table 2) combined with the $\nu_{\mathrm{max}}$ scaling relation (Brown et
al., 1991). There is no clear excess for the secondary, whose oscillations
should peak roughly around $1200\,\mu\mathrm{Hz}$ based on a similar
calculation using its properties from the literature.

Fig. 3 shows the echelle diagram around the obvious power excess, folded on a
large separation $\Delta\nu=41.7\,\mu\mathrm{Hz}$. This produces a roughly
vertical ridge, which is characteristic of solar-like oscillations, around
$40\,\mu\mathrm{Hz}$. Three teams (Benomar et al., 2012; Jiang et al., 2011;
Mathur et al., 2010) independently analysed the power spectrum to measure the
properties of the oscillations. The teams reported mutually consistent values
for the frequency of maximum oscillation power $\nu_{\mathrm{max}}$ and large
separation $\Delta\nu$, shown in Table 1.

The teams did not, however, reach a consensus on the identity of any
individual mode frequencies, at least partly because the properties of 12 Boo
A place it where empirical relations for identifying the angular degrees,
which use their horizontal offset in the echelle diagram $\epsilon$, are
uncertain (e.g. White et al., 2012). The identification is further confounded
by the potentially mixed modes, which would deviate from the simple asymptotic
spacing of high-order pressure modes, and the rotational splitting, which
would be around $1.2\,\mu\mathrm{Hz}$ if 12 Boo A’s rotation is aligned and
synchronised with its orbit. Boden et al. (2000) noted that, if the rotation
axes are aligned with the orbit, the rotation periods are slightly shorter
than the orbital period, presumably because the tidal torques are strongest at
periastron, when the stars are moving relatively quickly (see e.g. Hut, 1981).

We therefore proceeded to analyse 12 Boo A using $\nu_{\mathrm{max}}$ and
$\Delta\nu$ but no individual frequencies. Our combined mean values for
$\nu_{\mathrm{max}}$ and $\Delta\nu$ are the means of the three estimates. Our
consolidated variance is the mean of the variances plus the variance of the
means. We symmetrized the slightly asymmetric results by Benomar by taking the
mean of the $\pm 1\sigma$ limits as the central value. This gives
$\nu_{\mathrm{max}}=678\pm 29\,\mu\mathrm{Hz}$ and $\Delta\nu=41.7\pm
1.0\,\mu\mathrm{Hz}$, as shown in Table 1.

Figure 4: Spectral energy distribution (SED) of 12 Boo. Each black cross represents an observed passband flux at its central wavelength. The vertical bar indicates the measured uncertainty and the horizontal bar the wavelength range of the passband. The overall SED is shown in grey, with additional grey points showing the model value for each passband. The blue and orange curves show the SEDs of the primary and secondary, respectively. | $R_{A}$ | $R_{B}$ | ${T_{\mathrm{eff}}}_{,A}$ | ${T_{\mathrm{eff}}}_{,B}$
---|---|---|---|---
$R_{A}$ | $1.000$ | $0.801$ | $-0.930$ | $-0.656$
$R_{B}$ | $0.801$ | $1.000$ | $-0.701$ | $-0.909$
${T_{\mathrm{eff}}}_{A}$ | $-0.930$ | $-0.701$ | $1.000$ | $0.562$
${T_{\mathrm{eff}}}_{B}$ | $-0.656$ | $-0.909$ | $0.562$ | $1.000$
Table 3: Correlation coefficients in SED fit.

###  2.3 Spectral energy distribution

Reported luminosities of 12 Boo A and B pre-date the precise magnitudes and
parallaxes from Gaia. We therefore computed a new fit to the total spectral
energy distribution (SED) to determine the individual luminosities and
effective temperatures, using the methods described by Stassun & Torres (2016)
and Stassun et al. (2017); Stassun et al. (2018). The photometry comprises
fluxes in the $197\,\mathrm{nm}$, $237\,\mathrm{nm}$ and $274\,\mathrm{nm}$
(ultraviolet) passbands of TD-1 (Thompson et al., 1978), $UBV$ magnitudes from
Mermilliod (2006), $B_{\mathrm{T}}V_{\mathrm{T}}$ magnitudes from Tycho-2 (Høg
et al., 2000a, b) Strömgren $uvby$ magnitudes from Paunzen (2015),
$JHK_{\mathrm{S}}$ magnitudes from 2MASS (Skrutskie et al., 2006), $W1$–$4$
magnitudes from WISE (Wright et al., 2010), and Gaia $G$, $G_{\mathrm{BP}}$
and $G_{\mathrm{RP}}$ magnitudes from Early Data Release 3 (EDR3; Gaia
Collaboration et al., 2016, 2021; Riello et al., 2021). The two stars were
distinguished by using flux ratios derived from the $V$-band magnitude
difference in Boden et al. (2005) and the $H$-band magnitude difference in
Kervella et al. (2017).

The best fitting radii $R$ and effective temperatures $T_{\mathrm{eff}}$ are
shown in Table 2, along with the luminosities $L$ derived using the
Stefan–Boltzmann law. Fig. 4 shows the observed photometry along with the
total SED and the SEDs of the two stars. Because both stars are fit
simultaneously and are spectrally so similar, the fit parameters are strongly
correlated. We have included the correlation coefficients of the radii and
effective temperatures in Table 3.

###  2.4 Existing complementary data

We choose to use the masses measured by Konacki et al. (2010), whose radial
velocities have the smallest residuals (about $34.0$ and
$38.3\,\mathrm{m}\,\mathrm{s}^{-1}$ for 12 Boo A and B) and whose
interferometry has the greatest phase coverage. The composition of a
star—usually expressed through its metallicity $[\mathrm{Fe}/\mathrm{H}]$—is
an important constraint on its evolution. For 12 Boo, we use the spectroscopic
measurements by Balachandran (1990) and Lèbre et al. (1999) of $-0.03\pm 0.09$
and $-0.1\pm 0.1$, respectively, which we combine (as we did
$\nu_{\mathrm{max}}$ and $\Delta\nu$ above) to obtain
$[\mathrm{Fe}/\mathrm{H}]=-0.065\pm 0.101$. We assume both stars have surface
abundances that are the same within this uncertainty.

##  3 Stellar modelling

Four teams computed best-fitting stellar models of the primary using a number
of established methods (Ball et al., 2020; Jiang & Gizon, 2021; Çelik Orhan et
al., 2021) that fit predictions by various stellar models (Christensen-
Dalsgaard, 2008b; Demarque et al., 2008; Paxton et al., 2019) and stellar
oscillation programs (Antia & Basu, 1994; Christensen-Dalsgaard, 2008a;
Townsend & Teitler, 2013; Townsend et al., 2018) to the specified constraints.
Three of the teams also modelled the secondary. As constraints for both stars,
they used the masses $M$, metallicities $[\mathrm{Fe}/\mathrm{H}]$,
luminosities $L$ and effective temperatures $T_{\mathrm{eff}}$ in Table 2. For
12 Boo A, they also used the consolidated $\nu_{\mathrm{max}}$ and $\Delta\nu$
from Table 1.

We combined the results from each team using a linear opinion pool (Stone,
1961),111Though this is the usual modern reference for the idea, Bacharach
(1979) attributes the concept to LaPlace (1814). where the overall probability
distribution is taken as the equally-weighted sum of the probability
distribution from each modeller. Some teams reported strongly asymmetric
uncertainties, so we fit the results from each team using the lognormal
distribution defined by Hosking & Wallis (1997, Sec. A.8), which allows
arbitrary skewness and has the normal distribution as the limit when the shape
parameter is zero. We then interpolated the total cumulative distribution
function at the percentiles that correspond to the median and $\pm 1\sigma$
limits of a normal distribution, and report these as our central results and
(asymmetric) uncertainties.

The modelling results are also shown in Table 2. We note that the stars’ ages
are constrained to within about 6 per cent. This is principally because the
masses are very precisely known. Ages are often imprecise because of the
correlation with other parameters, including the mass. Because main-sequence
lifetimes $t_{\mathrm{MS}}$ scale roughly like $M^{-3}$, one naïvely expects
fractional age uncertainties at least about $3$ times larger than the fraction
mass uncertainties. This precision on the ages is not typical of
asteroseismology of individual stars, whose masses are less tightly
constrained.

##  4 Discussion and conclusions

We have analysed the TESS light curve of the binary star 12 Boo, with the aim
of combining individual seismic frequencies with the precise masses from the
stars’ observed mutual orbit. We were unable to robustly identify individual
frequencies but have used new photometric constraints, the parallax
measurement from Gaia and the global oscillation
properties—$\nu_{\mathrm{max}}$ and $\Delta\nu$—of 12 Boo A to revise the
properties of the system. Crucially, this includes the radius, which is
otherwise only weakly constrained in this non-eclipsing system.

The SED fit alone significantly improves the precision of the individual
components’ properties. We compare our results to those of Boden et al.
(2005), who appear to have most recently evaluated the total SED, and whose
values are also reported in Table 2. The dramatic improvement is driven by the
precise photometry from Gaia and the additional differential $H$-band
magnitude by Kervella et al. (2017). There is little improvement through the
Gaia EDR3 parallax of $27.484\pm 0.117\,\mathrm{mas}$, which is only slightly
more precise than the orbital parallax of $27.72\pm 0.15\,\mathrm{mas}$ that
Boden et al. (2005) derived. For 12 Boo B, Boden et al. (2005) estimated a
radius of $1.86\pm 0.15\,\mathrm{R}_{\odot}$, compared to our $1.901\pm
0.045\,\mathrm{R}_{\odot}$ from the SED fit. That is, our new radius is three
times as precise. Our radius of $1.881^{+0.056}_{-0.040}\,\mathrm{R}_{\odot}$
from detailed stellar modelling reflects this constraint. For 12 Boo A, our
SED fit gives a radius of $2.450\pm 0.045\,\mathrm{R}_{\odot}$, which is about
twice as precise as the estimate of $2.474\pm 0.095\,\mathrm{R}_{\odot}$ given
by Boden et al. (2005).

Correspondingly, we have improved the bolometric luminosity estimates for 12
Boo A and B from $7.76\pm 0.35\,\mathrm{L}_{\odot}$ and $4.69\pm
0.74\,\mathrm{L}_{\odot}$ to $7.531\pm 0.110\,\mathrm{L}_{\odot}$ and
$4.692\pm 0.095\,\mathrm{L}_{\odot}$. The new luminosities are about three and
seven times more precise.

The radius of 12 Boo A is further constrained by our measurements of the
asteroseismic parameters $\Delta\nu$ and $\nu_{\mathrm{max}}$. If we simply
use the scaling relation (Ulrich, 1986; Kjeldsen & Bedding, 1995)

$\Delta\nu\propto\sqrt{\bar{\rho}}\propto\sqrt{\frac{M}{R^{3}}}$ (1)

we obtain a radius $2.456\pm 0.039\,\mathrm{R}_{\odot}$, which is slightly
more precise than the SED fit. The stellar modelling result, which has
asymmetric uncertainties, is more precise still. The mean density $\bar{\rho}$
and, to a lesser extent, surface gravity $\log g$ of 12 Boo A are much better
constrained than 12 Boo B. This is a natural consequence of the seismic data
for 12 Boo A: $\Delta\nu$ tightly constrains $\bar{\rho}$;
$\nu_{\mathrm{max}}$ constrains $\log g$.

Our results thus significantly improve the properties of the system that have
been reported in the literature. A more sophisticated analysis of the light
curve and power spectrum might allow the identification of individual mode
frequencies. Furthermore, 12 Boo is scheduled to be re-observed during TESS’s
Sector 50 (2022-03-26 to 2022-04-22), and additional data at a cadence of 120
or 20 seconds might allow for a mode identification through which the Sector
23 data can be better exploited.

12 Boo might also be an interesting target for the Stellar Oscillation Network
Group (SONG; Grundahl et al., 2014; Fredslund Andersen et al., 2019), which
aims to become a worldwide network of telescopes with which to measure radial
velocity variations in bright solar-like oscillators. The signal will be
complicated by the stars’ orbital motions but the scientific value of
measuring the individual mode frequencies remains, for now, unexplored.

## Acknowledgements

WHB and WJC thank the UK Science and Technology Facilities Council (STFC) for
support under grant ST/R0023297/1. AM acknowledges support from the ERC
Consolidator Grant funding scheme (project ASTEROCHRONOMETRY,
https://www.asterochronometry.eu, G.A. n. 772293). RAG acknowledges the
support from PLATO and GOLF CNES grants. LGC thanks the support from grant
FPI-SO from the Spanish Ministry of Economy and Competitiveness (MINECO)
(research project SEV-2015-0548-17-2 and predoctoral contract
BES-2017-082610). SM acknowledges support from the Spanish Ministry of Science
and Innovation (MICINN) with the Ramón y Cajal fellowship no. RYC-2015-17697
and grant no. PID2019-107187GB-I00, and through AEI under the Severo Ochoa
Centres of Excellence Programme 2020–2023 (CEX2019-000920-S). DLB acknowledges
support from the TESS GI Program under NASA awards 80NSSC18K1585 and
80NSSC19K0385. JC is supported by a grant from the Max Planck Society to
prepare for the scientific exploitation of the PLATO mission. CK is supported
by Erciyes University Scientific Research Projects Coordination Unit under
grant number DOSAP MAP-2020-9749.

This paper includes data collected by the TESS mission, which are publicly
available from the Mikulski Archive for Space Telescopes (MAST). Funding for
the TESS mission is provided by the NASA Explorer Program.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

## Data availability

Original TESS light curves and pixel-level data are available from the
Mikulski Archive for Space Telescopes at http://mast.stsci.edu/. Other data
underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Abt & Levy (1976) Abt H. A., Levy S. G., 1976, ApJS, 30, 273
  * Addison et al. (2021) Addison B. C., et al., 2021, MNRAS, 502, 3704
  * Aerts (2021) Aerts C., 2021, Reviews of Modern Physics, 93, 015001
  * Antia & Basu (1994) Antia H. M., Basu S., 1994, A&AS, 107, 421
  * Appourchaux et al. (2015) Appourchaux T., et al., 2015, A&A, 582, A25
  * Auvergne et al. (2009) Auvergne M., et al., 2009, A&A, 506, 411
  * Bacharach (1979) Bacharach M., 1979, Journal of the American Statistical Association, 74, 837
  * Balachandran (1990) Balachandran S., 1990, ApJ, 354, 310
  * Ball et al. (2020) Ball W. H., et al., 2020, MNRAS, 499, 6084
  * Behr et al. (2011) Behr B. B., Cenko A. T., Hajian A. R., McMillan R. S., Murison M., Meade J., Hindsley R., 2011, AJ, 142, 6
  * Benomar et al. (2012) Benomar O., Baudin F., Chaplin W. J., Elsworth Y., Appourchaux T., 2012, MNRAS, 420, 2178
  * Boden et al. (2000) Boden A. F., Creech-Eakman M. J., Queloz D., 2000, ApJ, 536, 880
  * Boden et al. (2005) Boden A. F., Torres G., Hummel C. A., 2005, ApJ, 627, 464
  * Borucki et al. (2010) Borucki W. J., et al., 2010, Science, 327, 977
  * Bouchy & Carrier (2001) Bouchy F., Carrier F., 2001, A&A, 374, L5
  * Brown et al. (1991) Brown T. M., Gilliland R. L., Noyes R. W., Ramsey L. W., 1991, ApJ, 368, 599
  * Campbell & Wright (1900) Campbell W. W., Wright W. H., 1900, ApJ, 12, 254
  * Chaplin et al. (2020) Chaplin W. J., et al., 2020, Nature Astronomy, 4, 382
  * Chontos et al. (2021) Chontos A., et al., 2021, ApJ, 922, 229
  * Christensen-Dalsgaard (2008a) Christensen-Dalsgaard J., 2008a, Ap&SS, 316, 113
  * Christensen-Dalsgaard (2008b) Christensen-Dalsgaard J., 2008b, Ap&SS, 316, 13
  * CoRot Team (2016) CoRot Team 2016, The CoRoT Legacy Book: The adventure of the ultra high precision photometry from space, by the CoRot Team. EDP Sciences, doi:10.1051/978-2-7598-1876-1
  * Davies et al. (2015) Davies G. R., et al., 2015, MNRAS, 446, 2959
  * Demarque et al. (2008) Demarque P., Guenther D. B., Li L. H., Mazumdar A., Straka C. W., 2008, Ap&SS, 316, 31
  * Fausnaugh et al. (2021) Fausnaugh M. M., et al., 2021, TESS Data Release Notes: Sector 20, DR27, NASA/TM—2019––220500, %****␣12_boo.bbl␣Line␣150␣****https://archive.stsci.edu/missions/tess/doc/tess_drn/tess_sector_20_drn27_v03.pdf
  * Fredslund Andersen et al. (2019) Fredslund Andersen M., et al., 2019, A&A, 623, L9
  * Gaia Collaboration et al. (2016) Gaia Collaboration et al., 2016, A&A, 595, A1
  * Gaia Collaboration et al. (2021) Gaia Collaboration et al., 2021, A&A, 649, A1
  * García & Ballot (2019) García R. A., Ballot J., 2019, Living Reviews in Solar Physics, 16, 4
  * García et al. (2014) García R. A., et al., 2014, A&A, 568, A10
  * Grundahl et al. (2014) Grundahl F., et al., 2014, in Guzik J. A., Chaplin W. J., Handler G., Pigulski A., eds, IAU Symposium Vol. 301, IAU Symposium. pp 69–75, doi:10.1017/S1743921313014117
  * Halbwachs (1986) Halbwachs J. L., 1986, A&AS, 66, 131
  * Harper (1914) Harper W. E., 1914, Publications of the Dominion Observatory Ottawa, 1, 301
  * Høg et al. (2000a) Høg E., et al., 2000a, A&A, 355, L27
  * Høg et al. (2000b) Høg E., et al., 2000b, A&A, 357, 367
  * Hosking & Wallis (1997) Hosking J. R. M., Wallis J. R., 1997, Regional Frequency Analysis. Cambridge Univ. Press, Cambridge
  * Howell et al. (2014) Howell S. B., et al., 2014, PASP, 126, 398
  * Huber et al. (2019) Huber D., et al., 2019, AJ, 157, 245
  * Huber et al. (2022) Huber D., et al., 2022, AJ, 163, 79
  * Hut (1981) Hut P., 1981, A&A, 99, 126
  * Jiang & Gizon (2021) Jiang C., Gizon L., 2021, Research in Astronomy and Astrophysics, 21, 226
  * Jiang et al. (2011) Jiang C., et al., 2011, ApJ, 742, 120
  * Kervella et al. (2017) Kervella P., Bigot L., Gallenne A., Thévenin F., 2017, A&A, 597, A137
  * Kjeldsen & Bedding (1995) Kjeldsen H., Bedding T. R., 1995, A&A, 293, 87
  * Kjeldsen et al. (2005) Kjeldsen H., et al., 2005, ApJ, 635, 1281
  * Konacki et al. (2010) Konacki M., Muterspaugh M. W., Kulkarni S. R., Hełminiak K. G., 2010, ApJ, 719, 1293
  * LaPlace (1814) LaPlace P. S., 1814, Essai philosophique sur les probabilités. Courcier
  * Lèbre et al. (1999) Lèbre A., de Laverny P., de Medeiros J. R., Charbonnel C., da Silva L., 1999, A&A, 345, 936
  * Mathur et al. (2010) Mathur S., García R. A., Régulo C., et al. 2010, A&A, 511, A46
  * Maxted et al. (2020) Maxted P. F. L., et al., 2020, MNRAS, 498, 332
  * Mermilliod (2006) Mermilliod J. C., 2006, VizieR Online Data Catalog, p. II/168
  * Merrill (1922) Merrill P. W., 1922, ApJ, 56, 40
  * Metcalfe et al. (2012) Metcalfe T. S., et al., 2012, ApJ, 748, L10
  * Metcalfe et al. (2020) Metcalfe T. S., et al., 2020, ApJ, 900, 154
  * Metcalfe et al. (2021) Metcalfe T. S., et al., 2021, ApJ, 921, 122
  * Miglio et al. (2007) Miglio A., Montalbán J., Maceroni C., 2007, MNRAS, 377, 373
  * Miglio et al. (2014) Miglio A., Chaplin W. J., Farmer R., Kolb U., Girardi L., Elsworth Y., Appourchaux T., Handberg R., 2014, ApJ, 784, L3
  * Nielsen et al. (2020) Nielsen M. B., et al., 2020, A&A, 641, A25
  * Paunzen (2015) Paunzen E., 2015, A&A, 580, A23
  * Paxton et al. (2019) Paxton B., et al., 2019, ApJS, 243, 10
  * Pires et al. (2015) Pires S., Mathur S., García R. A., Ballot J., Stello D., Sato K., 2015, A&A, 574, A18
  * Ricker et al. (2015) Ricker G. R., et al., 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Riello et al. (2021) Riello M., et al., 2021, A&A, 649, A3
  * Silva Aguirre et al. (2017) Silva Aguirre V., et al., 2017, ApJ, 835, 173
  * Skrutskie et al. (2006) Skrutskie M. F., et al., 2006, AJ, 131, 1163
  * Stassun & Torres (2016) Stassun K. G., Torres G., 2016, AJ, 152, 180
  * Stassun et al. (2017) Stassun K. G., Collins K. A., Gaudi B. S., 2017, AJ, 153, 136
  * Stassun et al. (2018) Stassun K. G., Corsaro E., Pepper J. A., Gaudi B. S., 2018, AJ, 155, 22
  * Stone (1961) Stone M., 1961, Ann. Math. Statist., 32, 1339
  * Thompson et al. (1978) Thompson G. I., Nandy K., Jamar C., Monfils A., Houziaux L., Carnochan D. J., Wilson R., 1978, Catalogue of stellar ultraviolet fluxes : a compilation of absolute stellar fluxes measured by the Sky Survey Telescope (S2/68) aboard the ESRO satellite TD-1 /. Science Research Council
  * Tomkin & Fekel (2006) Tomkin J., Fekel F. C., 2006, AJ, 131, 2652
  * Townsend & Teitler (2013) Townsend R. H. D., Teitler S. A., 2013, MNRAS, 435, 3406
  * Townsend et al. (2018) Townsend R. H. D., Goldstein J., Zweibel E. G., 2018, MNRAS, 475, 879
  * Ulrich (1986) Ulrich R. K., 1986, ApJ, 306, L37
  * White et al. (2012) White T. R., et al., 2012, ApJ, 751, L36
  * White et al. (2017) White T. R., et al., 2017, A&A, 601, A82
  * Wright et al. (2010) Wright E. L., et al., 2010, AJ, 140, 1868
  * Çelik Orhan et al. (2021) Çelik Orhan Z., Yıldız M., Kayhan C., 2021, MNRAS, 503, 4529
  * de Medeiros & Udry (1999) de Medeiros J. R., Udry S., 1999, A&A, 346, 532