– Preprint 2023 Full Paper – MIDL 2023 submission Hanxue Gu 1
<EMAIL_ADDRESS>
Hongyu He 1<EMAIL_ADDRESS>
Roy Colglazier 2<EMAIL_ADDRESS>
Jordan Axelrod 3<EMAIL_ADDRESS>
Robert French2<EMAIL_ADDRESS>
Maciej A Mazurowski1,2,3,4<EMAIL_ADDRESS>
1 Department of Electrical and Engineering Department, Duke University, NC,
USA
2 Department of Radiology, Duke University, NC, USA
3 Department of Computer Science, Duke University, NC, USA
4 Department of Biostatistics and Bioinformatics, Duke University, NC, USA

# SuperMask: Generating High-resolution object masks from multi-view,
unaligned low-resolution MRIs

###### Abstract

Three-dimensional segmentation in magnetic resonance images (MRI), which
reflects the true shape of the objects, is challenging since high-resolution
isotropic MRIs are rare and typical MRIs are anisotropic, with the out-of-
plane dimension having a much lower resolution. A potential remedy to this
issue lies in the fact that often multiple sequences are acquired on different
planes. However, in practice, these sequences are not orthogonal to each
other, limiting the applicability of many previous solutions to reconstruct
higher-resolution images from multiple lower-resolution ones. We propose a
weakly-supervised deep learning-based solution to generating high-resolution
masks from multiple low-resolution images. Our method combines segmentation
and unsupervised registration networks by introducing two new regularizations
to make registration and segmentation reinforce each other. Finally, we
introduce a multi-view fusion method to generate high-resolution target object
masks. The experimental results on two datasets show the superiority of our
methods. Importantly, the advantage of not using high-resolution images in the
training process makes our method applicable to a wide variety of MRI
segmentation tasks.

###### keywords:

High-resolution shape generation, medical image segmentation

††editors: Preprint for MIDL 2023

##  1 Introduction

fig:target

Figure 1: The objective of our strategy is to generate realistic 3D masks for
non-isotropic MRIs (have low-resolution at through-plane directions). The left
is an illustration of the definition of in-plane slices and through-plane
slices. The 3D object directly from manual annotations has a coarse and grid-
like appearance (the first 3D object); our method is targeted to generate
higher-detailed and more realistic high-resolution masks (the middle 3D
object) when only low-resolution images and annotations are available. The
generated 3D object has more details and higher similarity to real objects
(the right 3D object).

As a non-invasive and low-radiation imaging technique, magnetic resonance
imaging (MRI) plays an important role in disease diagnosis and
characterization. In practical practice, MRI scans are done with relatively
few slices and a significant slice thickness, owing to the limits imposed by
the technique’s slow acquisition speed. In replace of an isotropic high-
resolution 3D volume, highly anisotropic images, which can be seen as a stack
of 2D slices [Erasmus et al. (2004)], are acquired with better resolution
inside the slices than in the slice-selection (or through-plane) direction
[Van Reeth et al. (2012)], see fig:target (green box). When directly
segmenting the target objects on the anisotropic volumes, the low resolution
(LR) in the slice-selection direction might result in extremely coarse
predictions (fig:target, red box) that are far from the real-object shape,
hence hindering the later diagnosis.

Machine learning-based super-resolution (SR) methods have been widely used for
the reconstruction of a high-resolution (HR) magnetic resonance (MR) 3D volume
from multi-planar LR 2D scans [Van Reeth et al. (2012), Plenge et al. (2012),
Gholipour et al. (2010), Jia et al. (2017), Sui et al. (2019)]. In recent
years, several studies have further investigated convolutional neural networks
(CNNs) based architectures for HR MRI reconstruction [Pham et al. (2019),
Jurek et al. (2020)]. Ebner et al. (2020) presented a fully automated
framework for fetal brain reconstruction that includes coarse fetal brain
localization, fine segmentation, and super-resolution reconstruction. Chai et
al. (2020) used a generative adversarial network (GAN) to restore the through-
plane slices. However, the majority of the previously proposed methods are
impractical because current MRI high-resolution recreation methods from multi-
planar views often require well-aligned and near-perfect orthogonal images
Zhang et al. (2021); Zhou et al. (2019). This is not commonly happening in the
real case because the views are defined in the anatomy coordination instead of
the world coordination, and patients may switch poses largely between views,
as seen in Appendix A. Images can be largely misaligned because of the
patient’s movement, or other views are not taken simultaneously.

When the resolution at the through-plane direction is too low to distinguish
the details, it is impossible to extract precise and realistic masks from a
single MRI volume, as seen in fig:target (left volume), and applying multi-
planar scans into segmentation tasks is also an effective strategy. In order
for the recreation of HR masks from multi-view MRIs, image registration and
alignment are essential. Some studies Askin Incebacak et al. (2022) mentioned
an initial registration procedure as the preprocessing. However, traditional
registration techniques Ferrante and Paragios (2017) are time-consuming,
isolated to downstream tasks, and considered different views are taken
simultaneously with only small variations in the patient’s mobility instead of
substantial posture switching. On the other side, our target images have
larger displacements, making the registration significantly harder.

fig:pipeline

Figure 2: This is the pipeline of our proposed method SuperMask. In training
phase 1, an image registration network (AlignNet) and a segmentation network
are trained separately. In training phase 2, the AlignNet and Segmentor are
intertwined to fine-tune with two newly introduced regularization losses. In
the test phase for our method, Coronal and Sagittal view images are aligned
into the axial view first, and a 1-D Gaussian fusion is applied to generate a
high-resolution mask.

In this paper, we introduce a weakly-supervised framework called SuperMask
that can automatically register low-resolution, unaligned 2D MRI scans
obtained from different orientations and produce HR and precise masks. Our
approach utilizes deep learning techniques to learn the deformations required
to register the images and the segmentations of target subjects on different
views of LR scans. The framework consists of three stages: in the first,
registration and segmentation networks are pre-trained independently; in the
second, segmentation and registration mutually promote one another (referred
to as intertwined learning); and in the third, segmentation results are fused.
By introducing two extra regularization terms, our innovative design of
intertwined learning can facilitate each part’s learning. This complementary
learning approach can achieve an optimal registration and target object
segmentation solution. In addition, our simple single-dimensional uncertainty
Gaussian fusion, inspired by Gaussian Mixture Models (GMM) Reynolds (2009),
can efficiently fuse information from multiple views, and we consider it a
good fit for MRIs’ through-plane uncertainty compared with other fusion
strategies.

We demonstrate the effectiveness of our framework on a variety of datasets
with varying distances between slices. The experiments show that our approach
is able to outperform previous methods, particularly in cases where the images
are low-resolution or poorly aligned. In contrast with previously generative-
based methods Chai et al. (2020); Yuan et al. (2020) and image-to-image
translation methods Masutani et al. (2020), our method does not require any
high-resolution images or masks to be involved in the training and design
processes. Since high-resolution MRI images are exceedingly rare for several
body parts 3D , our method is extensible and effective for a variety of use
cases.

##  2 Method

Our SuperMask is made to effectively auto-register low-resolution (LR) and
unaligned images, segment the targets, and generate HR masks that have more
accurate target objects’ shape representations. Section 2.2 describes a coarse
pre-training for image registration and segmentation, followed by an
intertwined training for fine-tuning in Section 2.3. Finally, we show the
generation of HR masks using our novel 1D-Gaussian fusion in Section 2.4.

###  2.1 Pre-processing

In the target task, each patient contains three low-resolution 2D MR scans
from different views (axial, coronal, and sagittal) of the same body part,
$I_{1}\in\mathbf{R}^{X\times Y\times Z_{l}}$, $I_{2}\in\mathbf{R}^{X\times
Y_{l}\times Z}$,$I_{3}\in\mathbf{R}^{X_{l}\times Y\times Z}$. $X_{l}$,
$Y_{l}$, $Z_{l}$ represent the number of slices on lower-sampled dimensions,
which are much smaller than the in-plane dimensions $X$, $Y$, and $Z$. The
ground-truth masks for these three images are $Y_{1},Y_{2},Y_{3}$ of the same
dimension. For 2D MRI, which is often not isotropic, its voxel size can be
expressed in terms of pixel spacing (p) and distance between slices (d). For
example, $I_{1}$ has a voxel size of $p\times p\times
d,d_{s}=\frac{p}{d}=\frac{Z_{l}}{Z}$, where we define $d_{s}$ as a relative
ratio of the slice distance vs. in-plane pixel spacing. In order to better
register $I_{1}$, $I_{2}$, and $I_{3}$, we first need to match their voxels to
the same unit, i.e., by up-sampling and fill the missing slices by nearest
interpolation on 2D scans and masks to get a voxel size of $p\times p\times
p$, seen as fig:pipeline (bottom left). We get 3D volumes
$I^{\prime}_{1},I^{\prime}_{2},I^{\prime}_{3},Y^{\prime}_{1},Y^{\prime}_{2},Y^{\prime}_{3}\in\mathbf{R}^{X\times
Y\times Z}$.

###  2.2 Coarse segmentation and registration

In phase 1 training, we train the coarse segmentation and registration
separately, as shown in the top left of figure fig:pipeline. For segmentation,
we assemble all of the available up-sampled images and masks into a training
set
$\mathbf{I^{\prime}}={\mathbf{I^{\prime}_{1}}\cup\mathbf{I^{\prime}_{2}}\cup\mathbf{I^{\prime}_{3}}}$,
$\mathbf{Y}={\mathbf{Y^{\prime}_{1}}\cup\mathbf{Y^{\prime}_{2}}\cup\mathbf{Y^{\prime}_{3}}}$
and feed them into a 3D U-net Çiçek et al. (2016) to train a coarse
segmentation network $\mathcal{S}_{1}$. Let $I^{\prime}_{i}$ as the input
volume, $M_{i}=\mathcal{S}_{1}(I^{\prime}_{i})$ as the predicted mask and
$Y_{i}$ as the corresponding ground truth mask. Dice loss Sudre et al. (2017)
(noted as $L_{seg_{1}}$) is applied for penalizing the segmentation objective.

For registration, we implement a coarse 3D spatial transformer network
$\mathcal{G}_{1}$ (Jaderberg et al. (2015) modified from a 2D version). For
each patient, we sample two views,
$I^{\prime}_{i},I^{\prime}_{j}\in{I^{\prime}_{1},I^{\prime}_{2},I^{\prime}_{3}}$
and apply a random affine transformation to each to augment the training set.
The localization network takes two 3D volumes, ${I^{\prime}_{i}}$ as the
moving image and ${I^{\prime}_{j}}$ as the fixed image. It generates 12 affine
parameters that reflect the transformation from $I^{\prime}_{1}$ to
$I^{\prime}_{2}$. Considering that MRI responds to real-life objects and does
not have the deformation and scaling of objects, we reduce the number of
affine parameters by setting the shearing and scaling parameters all to 1.
After sampling grid $G$ based on these affine parameters $\theta_{i\rightarrow
j}$ to get registration field $\phi_{i\rightarrow j}=\mathcal{T_{\theta}}(G)$,
we apply a differentiable image sampling on the input moving image
$I^{\prime}_{i}$ to generate a moved image $f_{i\rightarrow
j}=\mathcal{G}_{1}(I^{\prime}_{i},I^{\prime}_{j})=I^{\prime}_{i}\circ\phi_{i\rightarrow
j}$ that is registered into the pose of $I^{\prime}_{j}$.

The unsupervised registration loss combines two components:
$\mathcal{L}_{sim}$ that penalizes the difference between the moving and fixed
image, and $\mathcal{L}_{ID}$ that prevents the transformation between two
identical inputs,

$\mathcal{L}_{align_{1}}=\mathcal{L}_{sim}(I^{\prime}_{j},f_{i\rightarrow
j})+\lambda_{1}\mathcal{L}_{ID},$ (1)

where we apply $L_{sim}$ as local cross-correlation Boyd (2001), which is more
robust to intensity variations across scans and datasets, and
$L_{ID}=||I^{\prime}_{i},\mathcal{G}(I^{\prime}_{i},I^{\prime}_{i})||_{2}+||I^{\prime}_{j},\mathcal{G}(I^{\prime}_{j},I^{\prime}_{j})||_{2}$,
where $||\cdot||$ denotes L2-norm. Also, $\mathcal{L}_{align_{1}}$ can be
extended to a supervised version considering the LR masks are available, which
are finally
$\mathcal{L}_{align_{1}^{*}}=\mathcal{L}_{align_{1}}+\lambda_{2}\mathcal{L}_{sim}(Y^{\prime}_{j},Y_{i}\phi_{i\rightarrow
j})$. The results of these two versions of registration loss had no
significant differences based on our experiments.

###  2.3 Intertwined-tuning segmentation and registration

After pre-training the segmentation $\mathcal{S}_{1}$ and $\mathcal{G}_{1}$,
we fine-tune them by intertwining the segmentation and registration steps.
Similarly to the previous stage, we randomly select two different views from a
single patient at one time,
$I^{\prime}_{i},I^{\prime}_{j}\in{I^{\prime}_{1},I^{\prime}_{2},I^{\prime}_{3}}$.
We also introduce two new loss functions, cross-view supervision,
$L_{cons_{1}}$ and $L_{cons_{2}}$ as

$\aligned\mathcal{L}_{cons_{1}}&=||M_{Fj\rightarrow
i}-M_{i}||_{2}+||M_{Fi\rightarrow j}-M_{j}||_{2},\\\
\mathcal{L}_{cons_{2}}=||F_{Mi}-M_{i}||_{2}+||F_{Mj}-M_{j}||_{2},$ (2)

where $M_{Fj\rightarrow i}=\mathcal{S}(f_{j\rightarrow i})$ represents the
segmentation masks obtained from the aligned image $f_{j\rightarrow i}$, and
similarly for $M_{Fi\rightarrow j}=\mathcal{S}(f_{i\rightarrow j})$.
$F_{Mi}=\mathcal{S}(I^{\prime}_{i})\circ\phi_{i\rightarrow j}$ was obtained by
first segmenting the original image $I^{\prime}_{i}$ and then registering the
segmented mask into the pose of image $I^{\prime}_{j}$. Similarly for
$F_{Mj}=\mathcal{S}(I^{\prime}_{j})\circ\phi_{j\rightarrow i}$.
$\mathcal{L}_{cons_{1}}$ encourages the segmentation network $\mathcal{S}$ to
predict masks from a single view of the image that looks similar to those from
aligned images, and $\mathcal{L}_{cons_{2}}$ encourages the registration
network $\mathcal{G}$ to align the segmented masks to have more overlap. These
two intertwined regularizations are applied separately to $\mathcal{S}$ and
$\mathcal{G}$ to achieve more accurate and precise segmentation and
registration.

In training stage 2, the AlignNet and Segmentor are updated iteratively, and
the final objectives to update the segmentation network and registration
network are $\mathcal{L}_{seg_{2}}=L_{seg_{1}}+\alpha_{1}L_{cons_{1}}$,
$\mathcal{L}_{align_{2}}=L_{align_{1}}+\alpha_{2}L_{cons_{2}}$, respectively.

###  2.4 1-D Gaussian fusion to generate high-resolution mask

During inference, we take the three views of scans $I_{1}^{\prime}$,
$I_{2}^{\prime}$ and $I_{3}^{\prime}$, and align the last two views
($I_{2}^{\prime}$ $I_{3}^{\prime}$ into the axial scans $I^{\prime}_{1}$)
first; then these aligned scans are fed into the segmentation networks to get
their predicted masks. Without loss of generality, let us examine the sagittal
view as an example. We introduce a sampling matrix called $S_{3}$, that
reflects the sampling voxels in the up-sampled image matrix, where
$S_{3}[i,j,k]=1$ when this voxel point is taken from the original LR image
$I_{3}$ instead of an interpolated value. Thus we can see that the matrix
$S_{3}$ is like a matrix with the YZ-plane assigned to 1 at regular intervals
in the x-dimension, as shown in fig:pipeline (bottom branch). Inspired by
descriptions of uncertainties in MRIs Van Reeth et al. (2012), we found that
1D-Gaussian was a perfect match with the uncertainties between slices. Thus,
we apply a 1D-gaussian along the through-plane dimension (x dimension for
sagittal views) with a Gaussian kernel size of $d_{s}/2$, we could get a
probability matrix ($P_{3}$) that reflects the confidence of this scan at each
voxel, and this probability matrix can be aligned into the registered images
as $P_{3\rightarrow 1}=S_{3}\circ\phi_{3\rightarrow 1}$. The final mask is
fused by the following equation:

$M_{hr}=\frac{1}{\omega_{1}}M_{1}*P_{1}+\frac{1}{\omega_{2}}M_{F2\rightarrow
1}*P_{F2\rightarrow 1}+\frac{1}{\omega_{3}}M_{F3\rightarrow
1}*P_{F3\rightarrow 1};$ (3)

where $w_{1}$, $w_{2}$, $w_{3}$ are the normalization factors with
$\omega_{i}=\frac{P_{i}}{\sum_{1}^{3}P_{k}}$.

##  3 Experiments

fig:my_results

Figure 3: Qualitative results of the high-resolution mask generation methods.
The methods depicted here are Unet-LR, Unet-cc-nst, Unet-cc-vote, Ours, and
Unet-HR. Here are two examples from Heart 16 (upper two rows) and Brain 16
(bottom two rows), respectively. For each example, the first row is the
2D-slice view of the HR images with masks; the green curves are the ground
truths, and the pink curves are the predictions. The bottom row shows the 3D
view of the objects.

###  3.1 Datasets

As above-mentioned, most body parts are not imaged in high-resolution (HR)
MRI, so to prove our method’s effectiveness, we test our method by the body
parts with HR MRI available. By training the networks on the LR images, we can
evaluate our method’s performance on the HR masks. In this work, we obtain two
MRI datasets, Brats Menze et al. (2015) and Heart Simpson et al. (2019).

Brats contains 273 patient studies. We split them into 164 studies in the
training set and 42 in the test set. The original volumes are isotropic, with
a voxel spacing of 1 mm * 1 mm * 1 mm. Heart contains 20 patient studies We
split them into 15 studies in the training set and 5 in the test set. The
original images have a voxel spacing of 1.25mm*1.25mm*1.37mm, which has a
slightly lower but acceptable resolution in the third dimension. To make a
consistent 3D input size for networks, we first resample the volumes with a
1mm*1mm*1mm voxel spacing. For each of the datasets, we start by centering the
object and padding it to a cube of $256\times 256\times 256$ voxels. Then, we
apply a random 3D affine transformation, consisting of a rotation ranging from
-45 degrees to 45 degrees and a translation ranging from -30 voxels to 30
voxels, and auto-contrast to imitate the MRIs gotten at different times. To
get the LR images, we resample each dimension with a slice distance of 8 mm
(noted as datasets Brain 8 and Heart 8) and 16 mm (noted as datasets Brain 16
and Heart 16), where the resampled images have dimensions $32$ for voxel
spacing of 8 mm and $16$ for voxel spacing of 16 mm.

###  3.2 Baseline methods

We compare our methods with the following: The first baseline, Unet-LR, is a
segmentation algorithm (3D Unet) that uses all low-resolution images with all
available views, similar to our stage 1 segmentation ($\mathcal{S}_{1}$).
Second, Unet-cc-img-nst, is the Unet-LR plus image correlation, a traditional,
non-learning-based iterative registration method that uses image correlation
as a similarity metric and has a convergence minimum value of 1E-6 to register
three-view images. Then the segmented masks are fused by a 3D nearest neighbor
interpolation (Unet-cc-img-nst). Third, Unet-cc-img-vote, a similar version of
the second but using majority voting, commonly used in ensemble learning, as
the final fusion algorithm. The fourth and fifth, Unet-cc-msk-nst and Unet-cc-
msk-vote, are similar versions of the second and third but apply registration
directly to the segmented masks. Lastly, Unet-HR is trained on 1 mm* 1 mm* 1
mm voxel-spaced HR images. It is not appropriate to compare to the other
methods, but it can serve as an “upper bound” for all methods.

###  3.3 Metrics

We compare the predicted masks with the ground-truth HR masks $Y_{HR}$. To
make the comparison fair, we assume that the HR masks are perfectly aligned
with the axial view images, as the axial view is unchanged. By directly
calculating the metrics on the predicted axial view masks, we can compare the
predicted LR masks $M_{1}$ with HR ground truth masks $Y_{HR}$. We quantify
the generated HR masks with $M_{HR}$ by first the dice score (DSC) which
quantifies the overlap of the predicted masks and HR ground truths. Then, the
under segmentation (US), over segmentation (OS), root mean squared (RMS)
Monteiro and Campilho (2006) which quantify the precision of the predicted
masks. Lastly, in light of the fact that the DSC evaluates overlap but places
little emphasis on the shape information of segmented volumes, we also add a
boundary accuracy (mBA) Taha and Hanbury (2015) that focuses on the shape
details.

###  3.4 Implementation details

For our methods, during training stage 1, Segmentor and Alignet are trained
separately with a learning rate of 0.001 for 40 epochs. During training stage
2, the learning rate is set at 0.0001 with a step-wise learning rate decay for
every 200 iterations as gamma = 0.9, and stage 2 training is done for 60
epochs. Before feeding into the networks, rescale intensity after a 2%-98%
thresholding was applied to normalize image intensities into [0,1]. During all
the training, the batch size is set to 2, and 2 GPUs with the Geforce A6000
are used. For the hyper-parameters, $\alpha 1=\alpha 2=0.05$,
$\lambda_{1}=0.1$, and $\lambda_{2}=0.05$, and it was chosen by the
experiments on Heart 16 dataset among several hyper-parameters combinations.
All the baseline methods are set for training over 100 epochs with the same
learning rate decay and batch size as our method.

tab:results  Methods Heart 8 Brain 8 DSC US OS RMS mBA DSC US OS RMS mBA Unet-
LR 0.863$\pm.008$ 0.118$\pm.015$ 0.151$\pm.013$ 0.141$\pm.015$ 0.850$\pm.010$
0.839$\pm.011$ 0.127$\pm.013$ 0.173$\pm.010$ 0.166$\pm.013$ 0.819$\pm.012$
Unet-img-cc-nst 0.819$\pm.017$ 0.202$\pm.019$ 0.156$\pm.018$ 0.182$\pm.019$
0.803$\pm.021$ 0.634$\pm.024$ 0.194$\pm.022$ 0.397$\pm.041$ 0.195$\pm.039$
0.789$\pm.029$ Unet-img-cc-vote 0.879$\pm.014$ 0.063$\pm.012$ 0.174$\pm.015$
0.130$\pm.013$ 0.859$\pm.016$ 0.813$\pm.019$ 0.058$\pm.020$ 0.262$\pm.032$
0.196$\pm.033$ 0.807$\pm.019$ Unet-msk-cc-nst 0.799$\pm.026$ 0.121$\pm.028$
0.251$\pm.023$ 0.204$\pm.028$ 0.814$\pm.022$ 0.728$\pm.036$ 0.086$\pm.033$
0.359$\pm.039$ 0.279$\pm.039$ 0.770$\pm.035$ Unet-msk-cc-vote 0.863$\pm.022$
0.198$\pm.028$ 0.064$\pm.025$ 0.178$\pm.024$ 0.842$\pm.019$ 0.761$\pm.029$
0.053$\pm.032$ 0.330$\pm.033$ 0.216$\pm.021$ 0.776$\pm.027$ Ours-1seq
0.871$\pm.005$ 0.098$\pm.012$ 0.117$\pm.010$ 0.108$\pm.012$ 0.855$\pm.009$
0.849$\pm.007$ 0.142$\pm.008$ 0.151$\pm.011$ 0.157$\pm.009$ 0.821$\pm.003$
Ours-2seq 0.886$\pm.006$ 0.095$\pm.015$ 0.121$\pm.016$ 0.114$\pm.015$
0.857$\pm.011$ 0.855$\pm.011$ 0.136$\pm.009$ 0.131$\pm.015$ 0.134$\pm.011$
0.830$\pm.007$ Ours(SuperMask) 0.896$\pm.002$ 0.093$\pm.010$ 0.113$\pm.010$
0.105$\pm.010$ 0.870$\pm.009$ 0.883$\pm.012$ 0.109$\pm.014$ 0.116$\pm.015$
0.126$\pm.015$ 0.848$\pm.010$ Heart 16 Brain 16 DSC US OS RMS mBA DSC US OS
RMS mBA Unet-LR 0.812$\pm.009$ 0.201$\pm.018$ 0.173$\pm.016$ 0.189$\pm.018$
0.780$\pm.014$ 0.81$\pm.014$ 0.134$\pm.021$ 0.224$\pm.024$ 0.195$\pm.025$
0.797$\pm.019$ Unet-img-cc-nst 0.664$\pm.054$ 0.338$\pm.065$ 0.330$\pm.045$
0.336$\pm.063$ 0.687$\pm.035$ 0.638$\pm.099$ 0.441$\pm.097$ 0.162$\pm.056$
0.347$\pm.095$ 0.794$\pm.074$ Unet-img-cc-vote 0.683$\pm.035$ 0.319$\pm.066$
0.302$\pm.055$ 0.310$\pm.063$ 0.703$\pm.045$ 0.638$\pm.095$ 0.504$\pm.094$
0.042$\pm.007$ 0.359$\pm.094$ 0.715$\pm.029$ Unet-msk-cc-nst 0.683$\pm.043$
0.176$\pm.054$ 0.401$\pm.074$ 0.312$\pm.064$ 0.730$\pm.043$ 0.629$\pm.104$
0.464$\pm.122$ 0.125$\pm.067$ 0.353$\pm.069$ 0.719$\pm.054$ Unet-msk-cc-vote
0.763$\pm.024$ 0.059$\pm.007$ 0.357$\pm.045$ 0.256$\pm.043$ 0.766$\pm.016$
0.670$\pm.047$ 0.040$\pm.005$ 0.460$\pm.073$ 0.321$\pm.071$ 0.728$\pm.043$
our-1seq 0.819$\pm.008$ 0.200$\pm.016$ 0.157$\pm.012$ 0.182$\pm.015$
0.791$\pm.016$ 0.820$\pm.013$ 0.156$\pm.024$ 0.190$\pm.022$ 0.184$\pm.023$
0.801$\pm.017$ our-2seq 0.833$\pm.005$ 0.187$\pm.014$ 0.146$\pm.015$
0.177$\pm.014$ 0.798$\pm.015$ 0.836$\pm.013$ 0.140$\pm.027$ 0.185$\pm.015$
0.179$\pm.017$ 0.810 $\pm.011$ Ours(SuperMask) 0.875$\pm.004$ 0.139$\pm.012$
0.131$\pm.014$ 0.137$\pm.013$ 0.842$\pm.010$ 0.861$\pm.015$ 0.108$\pm.017$
0.165$\pm.020$ 0.152$\pm.017$ 0.823$\pm.014$ HR (upper bound) HR (upper bound)
Unet-HR 0.923$\pm.009$ 0.102$\pm.010$ 0.051$\pm.006$ 0.082$\pm.009$
0.908$\pm.013$ 0.900$\pm.014$ 0.068$\pm.013$ 0.117$\pm.017$ 0.110$\pm.015$
0.877$\pm.016$

Table 1: Quantitative results of the high-resolution mask generation methods.
Dataset 8 and 16 note the slice distance in the through-plane direction.
Results are reported based on 4 repeated training sections with average and
standard deviation ($\pm$). our-1seq denotes the evaluation situation when
only 1 view (axial-view) is available, and our-2seq denotes 2 views (axial-
view and sagittal-view) are available.

###  3.5 Results and discussion

tab:results shows the results for the HR mask generation on various datasets.
It is shown that our method outperforms all the baselines on all the datasets
when only LR images are available. Compared with the drop in performance from
Unet-HR to Unet-LR, there is a drop in DSC of 0.06 for slice distances of 8 mm
and a drop of 0.112 and 0.09 for 16 mm. We can see that only training a simple
3D segmentation network with LR images and masks is not enough to achieve
acceptable HR masks. The loss of information between through-plane slices is
detrimental. The more distant the slices are from each other, the lower the
resolution of the image is, and the more significant this performance
reduction is. This could also be seen in fig:my_results, where, despite up-
sampling to the real-world coordination, the segmented masks from Unet-LR have
a very distinct ladder-like appearance.

Even when only getting LR images, our method gets an increase in DSC of 0.03
and 0.043 for a slice thickness of 8 mm and a significant increase of 0.052
and 0.09 for a slice thickness of 16 mm, for the heart and brain datasets,
compared with Unet-LR. This confirms the ability and effectiveness of our
method to combine more details by extracting information from non-orthogonal
LR images. Comparing RMS, it can be reduced by about 28% on the Heart 16
dataset and 22.1% on the brain 16 dataset. Our approach can substantially
reduce the RMS due to the through-plane uncertainty. Using mBA, we observe an
obvious increase in mBA for our methods, especially with a slice thickness of
16 mm, where a large part of the boundary details is lost from the input
images. Although there are some smaller US for nearest or voting fusion
methods, they suffer from a much larger OS, indicating an over-segmentation
bias. Our method achieves a better balance.

The drop in performance for baselines Unet-img-cc-* and Unet-msk-cc-* implies
that generating HR masks from multiple views requires highly precise image
registration. The traditional image registration methods based on image
correlation can not handle the low through-plane resolution well at different
directions; see Appendix B for more details. Using failed image registration
to construct segmentations from multiple views is detrimental. Our learning-
based registration, combined with the target object’s segmentation, can
circumvent this problem and make connections between views, which contributes
to the fusion of the target object’s segmentation. Also, because the training
was a two-view setting, our work can be easily extended when only two views
are available (our-2seq), and also get an improvement. More details of the
effectiveness of each component are shown in appendix D tab:ablation. Also
seen from the illustration in fig:my_results, our final generated masks,
though free from any HR images or masks during the training and evaluation,
can preserve the details of the object boundaries and have a high consensus
with the HR masks.

##  4 Conclusion

In this work, we proposed a method SuperMask that generates high-resolution
target segmentation from multiple unaligned 2D MRIs. We experimentally
demonstrate the effectiveness of our methods for improving segmentation
performance when only low-resolution masks are available. Importantly, our
method does not require HR images in the training stage, which makes it
broadly applicable. Future work will consider extending the method from
generating HR masks to generating HR images.

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##  Appendix A Illustration of the orientation of three different volumes

fig:data shows that the ideal assumption for multi-view MRIs is that
axial/sagittal/coronal views are orthogonal to each other with no translation
offset. However, in reality, these three perspectives are based on an
anatomical term in which the objects may have large positional movements or
orientations. If we translate these three views into real-world units and
directly combine these three scans (fig:data (c), (d), (e)), the ulna/radius
bones cannot overlap due to the positional/orientation alterations.

fig:data

Figure 4: Illustration of input MRIs and the orientation of three different
volumes. (a)(ideal case) anatomical coordinate system aligned with image
coordinate system; (b)(real case) anatomical coordinate system not aligned
with image coordinate system; (c)axial-view scans; (d)coronal-view scans;
(e)sagittal-view scans; (f) integration of three views from the same patient
in real-clinical forearm MRIs.

##  Appendix B The results for image registration

fig:align shows that, when we are to align coronal views and sagittal views
which have the low resolution at different planes, traditional registration
can achieve to-some-extent the correction between images. But the accuracy of
its correction is not as high as that of our method. For example, if we look
at the bottom two rows in fig:align (aligned sagittal for trade-img), it has a
rotation error with the other two views after registration. These slight
imperfections are detrimental if we want to rely on the predicted mask on
these aligned images.

It can also be seen from tab:align, using our 2-stage intertwined learning can
guarantee better agreements between views, where the DSC between views is
higher compared with an isolated set of image segmentation and registration
(Unet-img-cc-*). Also, applying the registration directly to segmented masks
can achieve a large overlap, but these alignments might be far from the real
objects (i&HR are lower).

fig:align

Figure 5: Qualitative results for image registration from multi-views,

tab:align  DSC between views 1&2 1&3 2&3 2&HR 3&HR Unet-img-cc-* 0.71 0.661
0.697 0.765 0.717 Unet-msk-cc-* 0.750 0.725 0.696 0.749 0.715 Ours 0.784 0.75
0.755 0.816 0.783

Table 2:  Quantitative performance of image registration for multi-view MRIs.
Shown are the DSC between segmented masks on the aligned views of the Brat 16
dataset. 1&2 means DSC calculated on view 1 and view 2 masks.

##  Appendix C Details of hyperparameter selection

We selected our model’s final hyper-parameters based on the evaluated DSC on
the Heart 16 dataset, where we put extra 4 volumes in training set as the
evaluation set here. There are enormous combinations of parameters
$\alpha_{1}$, $\alpha_{2}$, $\lambda_{1}$, $\lambda_{2}$, and we did not
through all the potential parameter settings. Based on the five different
combinations, we found that $\lambda_{1}$, $\lambda_{2}$, which controls the
registration loss, were preferable under a scale of 0.5. $\alpha_{1}$,
$\alpha_{2}$, which controls the scales of regularization terms, are not too
influential on the final performance. We assumed that is because of the two-
stage training, the previous loss terms as $L_{align1}$ and $L_{seg1}$ were
already optimized in pre-training, and stage 2 would care more from the
gradients contributed from $L_{cons1}$ and $L_{cons2}$ no matter the scaling
is. We agreed that we are not thoroughly optimizing these parameters on each
individual dataset, but the general selection rule for the scales of
parameters found by these five combinations could give us a relatively good
performance on other datasets.

tab:param-selection  model version $\alpha_{1}$ $\alpha_{2}$ $\lambda_{1}$
$\lambda_{2}$ DSC model 1 0.1 0.1 0.1 0.1 0.87 model 2 0.2 0.2 0.5 0.5 0.845
model 3 0.05 0.05 0.1 0.05 0.871 model 4 0.5 0.5 0.1 0.05 0.867 model 5 0.05
0.05 0.1 0.05 0.872

Table 3: The details of hyper-parameter selection on Heart 16 dataset, and the
supporting metric is DSC.

##  Appendix D The results for ablation studies

This section shows the ablation studies for our method that demonstrate the
effectiveness of each component of our method. Our-1-stage is our methods’
one-stage training without an intertwined regularization. Our-voting is
replacing the 1D-fusion in our method with voting, and Our-voting is replacing
the 1D-fusion in our method with the nearest interpolation.

tab:ablation  Ablation methods componenets heart-16 two-stage Alignet fusion
DSC US OS RMS mBA Our-1-stage 0 1 our fusion 0.849 0.178 0.120 0.154 0.819 Our
- voting 1 1 voting 0.826 0.154 0.161 0.157 0.821 Our - nearest 1 1 nearest
0.763 0.286 0.227 0.237 0.750

Table 4: Ablation study of each component of our model.


# Revealing mysteries in gamma-ray bursts: the role of gluon condensation

Wei Zhua111Corresponding author, E-mail<EMAIL_ADDRESS>Xu-Rong Chenb
and Yu-Chen Tangc

aDepartment of Physics, East China Normal University, Shanghai 200241, P.R.
China
bInstitute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,P.R.
China
cKey Laboratory of Dark Matter and Space Astronomy, Purple Mountain
Observatory,
Chinese Academy of Sciences, Nanjing 210008, P.R. China

###### Abstract

We use a special gluon distribution in the nucleon, which was predicted by a
QCD evolution equation to consistently explain several intriguing phenomena
associated with gamma-ray bursts. They are the GeV-TeV spectra of GRB 221009A,
the remarkably symmetrical explosion cloud in kilonova AT2017gfo, and the
absence of a very high-energy gamma-ray signature in GRB 170817A. We find that
these occurrences can be attributed to the gluon condensation within nucleons,
i.e., a significant number of soft gluons within nucleons are condensed at a
critical momentum, resulting in the emergence of a steep and high peak in the
gluon distributions. Through this profound connection between microscopic and
macroscopic phenomena, we have not only expanded the applications of the
hadronic scenario in cosmic gamma-ray emissions but also presented new
evidence for the existence of gluon condensation.

keywords: Gamma-ray bursts; GRB 221009A; AT2017gfo/GRB 170817A/GW170817; Gluon
condensation

##  1 INTRODUCTION

Gamma-ray bursts (GRBs) stand as the most potent and radiant electromagnetic
events witnessed in distant galaxies since the inception of the universe.
Long-duration GRBs are theorized to stem from the collapse of massive stars,
whereas short-duration ones are believed to arise from the merger of two
compact celestial objects, such as neutron stars or black holes. Despite the
plethora of theoretical models aimed at elucidating both long and short GRBs,
the core question surrounding these phenomena persists as an enigma. To
exemplify, we present two recent instances as follows.

Recently, a highly energetic outburst, initially detected by Swift-XRT [1],
was promptly confirmed as GRB 221009A by Fermi-GBM [2,3] and Fermi-LAT [4,5].
Space-based detectors, constrained in their resolution capabilities to gamma-
ray signals up to the 10 GeV energy level, faced limitations. However, the
Large High Altitude Air Shower Observatory (LHAASO) detectors overcame this
hurdle, recording very high-energy (VHE) photons in the TeV energy range from
GRB 221009A within the next 30 minutes [6]. The extensive array of ground-
based detectors allowed for an unprecedented level of precision, revealing
that the gamma-ray brightness of GRB 221009A surpassed that of any previously
observed GRB by 50 times. Instances of such GRBs are exceptionally rare, with
an estimated probability of occurrence at one in a thousand years. This
extraordinary event has provided us with a more comprehensive VHE spectrum of
the brightest GRB ever detected. It is conceivable that this long-duration GRB
originated from the gravitational collapse of a massive star, leading to the
formation of either a neutron star or a black hole. Upon the creation of a
neutron star or black hole, the resultant formation of powerful particle jets
containing protons, nuclei, and electrons occurs. These particles undergo
acceleration, reaching velocities approaching the speed of light, primarily
driven by shock waves. Subsequently, they release high-energy and VHE photons
into space while traversing through remnants of the progenitor stars. The
spectra of GRB 221009A present a potential clue in unraveling the mystery of
GRBs, particularly due to its distinctive structure in the GeV-TeV band.
Despite the numerous theoretical models proposed to elucidate these high-
energy events, the true nature of the VHE radiation source remains elusive.

An illustrative example of a short-duration GRB is the kilonova AT2017gfo [7],
originating from the gravitational wave event GW 170817 [8], also referred to
as short-duration GRB 170817A due to the observation of electromagnetic (EM)
counterparts [9]. This event marked the first multi-messenger observation
related to the coalescence of two neutron stars. Upon scrutinizing the spectra
of AT2017gfo, researchers made a surprising discovery: during its early
stages, this kilonova exhibited a perfectly spherical shape [10]. This
contradicts the expected characteristics of ultra-dense matter, as
hydrodynamical models suggested that the resulting explosion cloud should have
a flat shape. This finding implies that the merger of two compact neutron
cores occurred with minimal resistance. The specific mechanism underlying this
spherical explosion remains a mystery.

There are generally two sources of gamma-rays in cosmic rays [11]: (i) the
leptonic scenario, involving curvature radiation or inverse Compton scattering
of high-energy electrons, accompanied by low-energy (KeV) synchrotron
radiation spectra, and (ii) the hadronic scenario, occurring in a two-step
process: the proton collisions initially produce a large number of secondary
particles, with the $\pi$ meson being chief among them, and the neutral
$\pi^{0}$ decays electromagnetically, resulting in a pair of photons. It is
reasonable to consider that the hadronic scenario should dominate as a
mechanism. This is because protons and nuclei carry most of the kinetic energy
in the primary particle jets and are also accelerated by shock waves.
Additionally, their radiation loss is minimal. However, in GRB studies, the
hadronic scenario is typically excluded from the GRB picture. The reason is as
follows.

A proton is composed of quarks and gluons. In a high-energy proton collision,
hundreds or even thousands of secondary particles, predominantly $\pi$ mesons,
can be generated. As the collision energy escalates, an increasing number of
gluons with small values of $x$ within the proton, (where $x$ denotes the
ratio of the longitudinal momentum of the gluon to that of the proton),
contribute to the production of secondary particles. Consequently, gluons play
a dominant role in $\pi$-meson production. The abundance of gluons rises
steadily as $x$ decreases, without forming a distinct peak in the meson energy
spectrum. This finding aligns with experimental observations at the Large
Hadron Collider (LHC), presently the largest high-energy collider. On the
other hand, the electromagnetic decay of the neutral $\pi^{0}$ results in a
peak in the gamma-ray spectrum at $m_{\pi}/2$, which is shifted to nearly 1
GeV due to the increased production of $\pi$ mesons. Consequently, the energy
spectra of the conventional hadronic scenario consistently cluster around
$\sim 1~{}GeV$ in $E_{\gamma}^{2}\Phi_{\gamma}$ (referred to as the
$\pi$-bump) [12]. Therefore, the hadronic scenario appears to be less
appealing for independently explaining TeV emission in GRBs. The case of GRB
221009A has prompted us to revisit the structure of the proton. It is
conceivable that the energy of the proton may surpass the acceleration
capabilities of the LHC, potentially leading to a new understanding of the
nature of gluons within the proton.

In this paper, we focus on the hadronic scenario, but taking into account a
novel understanding of nucleon structure. According to the evolution equations
in quantum chromodynamics (QCD), gluon distributions are predicted to
gradually approach an equilibrium state characterized by the splitting and
fusion of gluons, a phenomenon known as the color glass condensate (CGC) [13],
although this is not a true physical condensation. A further development of
the evolution equations is the Zhu-Shen-Ruan (ZSR) equation [14,15], which
posits that at high energy, the evolution of gluon distributions in nucleons
exhibits chaotic behavior, resulting in extremely strong shadowing and
antishadowing effects. This leads to the accumulating numerous gluons within a
narrow phase space defined by a critical momentum $(x_{c},k_{c})$. This
phenomenon is referred to as the gluon condensation (GC) (see Fig. 1) [16,17].

Figure 1: The evolution of gluon distributions in a QCD evolution equation
[16,17] from the CGC to the GC, where gluons at $x<x_{c}$ are condensed at a
critical momentum $(x_{c},k_{c})$. All coordinates are on the logarithmic
scale. There are two characteristics of this distribution: a sharp peak at the
critical momentum and no gluons present at $x<x_{c}$. We will show their
different performances in GRB 221009A and GRB 170817A, respectively.

A substantial distortion in the gluon distribution function, as suggested by
the GC effect, would likely have been noticeable in proton colliders. The
absence of its detection can be attributed to the limitation that the energy
levels at the LHC are insufficient for the gluon peaks at $(x_{c},k_{c})$ to
have reached the interaction zone. This realization prompted us to explore
cosmic rays, where protons can be accelerated to energy levels surpassing
those achievable at the LHC. The approach is straightforward and can be
computed by directly substituting the distribution illustrated in Fig. 1 into
the conventional hadronic framework. A concise derivation can be found in Sec.
2, with a more detailed explanation provided in [18-20]. In Section 3, we
demonstrate that the resulting gamma-ray spectrum conforms to a broken power-
law (BPL) distribution, featuring an attenuation factor in the tail. We refer
to the traditional hadronic scenario incorporating the gluon condensation
effect as the GC-model, which exhibits exceptional agreement with the GRB
221009A spectrum. The concept of GC not only involves the accumulation of
gluons in the vicinity of $(x_{c},k_{c})$ but also implies the disappearance
of gluons smaller than $x_{c}$. Consequently, when the proton-proton (pp)
collisions energy surpasses the threshold $s_{cut}$, its total scattering
cross-section is significantly suppressed, rendering the colliding protons
transparent. Given that the threshold $s_{cut}$ is very high for pp and pA
collisions, observing this phenomenon becomes challenging. An exception may be
binary neutron star mergers, as the neutron star nucleus, being infinitely
heavy, can have a sufficiently low $s_{cut}$ making the merger drag-free. We
will delve into the explanation of the kilonova AT2017gfo event in Sec. 4.
Further we attempt to offer a comprehensible explanation for the cause of the
GC and give a summary in Sec. 5.

##  2 THE GC-MODEL

The GC-model may provide a link between the traditional hadronic scenario and
the GC in the nucleon. To elucidate the underlying assumptions in this model,
we systematically reconstruct the derivation outlined in [18-20]. In the
context of high-energy radiation within the hadronic scenario, during a $pp$
collision, approximately half of the energy from parent protons is absorbed by
valence quarks, which subsequently form the leading particles. The residual
energy undergoes conversion into secondary hadrons, predominantly pions,
within the central region. Subsequently, $\pi_{0}\rightarrow 2\gamma$ takes
place. The spectrum of these photons in the Lab. system is computed by

$\Phi_{\gamma}(E_{\gamma})=C_{\gamma}\left(\frac{E_{\gamma}}{\mathrm{GeV}}\right)^{-\beta_{\gamma}}\int_{E_{\pi}^{min}}^{E_{\pi}^{cut}}dE_{\pi}\left(\frac{E_{p}}{\mathrm{GeV}}\right)^{-\beta_{p}}N_{\pi}(E_{p},E_{\pi})\frac{d\omega_{\pi-\gamma}(E_{\pi},E_{\gamma})}{dE_{\gamma}},$
$None$

where the spectral index $\beta_{\gamma}$ incorporates the energy loss due to
pion absorption by the medium, and $N_{\pi}$ is the multiplicity of pions,
$C_{\gamma}$ is a normalization constant that includes the kinematic factor
and the flux dimension. As is customary, the accelerated protons are assumed
to follow a simple $N_{p}\propto E_{p}^{-\beta_{p}}$ near the source.

Substituting the GC-distribution in Fig. 1 to the multiplicity $N_{g}$ of
gluon mini-jet [21,22]

$\frac{dN_{g}}{dk_{T}^{2}dy}=\frac{64N_{c}}{(N^{2}_{c}-1)k_{T}^{2}}\int_{0.1}^{100}q_{T}dq_{T}\int_{0}^{2\pi}d\phi\alpha_{s}(\Omega)\frac{F(x_{1},\frac{1}{4}(k_{T}+q_{T})^{2})F(x_{2},\frac{1}{4}(k_{T}-q_{T})^{2})}{(k_{T}+q_{T})^{2}(k_{T}-q_{T})^{2}},$
$None$

where $\Omega=Max\\{k_{T}^{2},(k_{T}+q_{T})^{2}/4,(k_{T}-q_{T})^{2}/4\\}$; The
longitudinal momentum fractions of interacting gluons are fixed by kinematics:
$x_{1,2}=k_{T}e^{\pm y}/\sqrt{s}$.

Figure 2:  A schematic diagram for the ratio of the multiplicity of gluon
mini-jets with and without the GC effect. The empty circles are the solutions
of Eq. (2) using the input in Fig. 1 and the solid curve is their smoothed
result. All scales are taken as logarithmic unit.

As anticipated, an evident surge in $N_{g}$ becomes apparent when $\sqrt{s}$
surpasses $\sqrt{s_{GC}}$, signifying the active involvement of the gluon
distribution peaks in the generation of secondary particles (see Fig. 2).
Nevertheless, this augmentation ceases beyond a certain higher energy
threshold, leading to a rapid decline in meson yield to zero. This decline
occurs due to the unavailability of gluons smaller than $x_{c}$. The resulting
high plateau effect in the meson yield distinctly surpasses the modest peak
associated with $\pi^{0}$ decay, contributing to a bulge in the gamma-ray
spectrum. The relationships among $\sqrt{s_{GC}}$, $\sqrt{s_{cut}}$, and
$x_{c}$ are

$x_{c}=\frac{k_{T}}{\sqrt{s_{GC}}}e^{-y_{max}}\simeq\frac{k^{2}_{c}}{s_{GC}},$
$None$

are

$x_{c}=\frac{k_{T}}{\sqrt{s_{cut}}}e^{0}.$ $None$

In the subsequent stage, we encounter $N_{\pi}$, a complex quantity
interrelated with $N_{g}$ and characterized by intricate nonperturbative QCD
effects. Conventionally, a multi-parameter empirical formulation is employed
to articulate $N_{\pi}$. However, when addressing cosmic rays, whose energies
may surpass the scale of the LHC, empirical formulas are non-existent.

Fortunately, the GC effect can be employed to simplify this relationship.
Experimental observations indicate that as collision energy increases,
$N_{\pi}$ also increases due to the greater participation of gluons in the
process of generating new particles. Recent research by Roberts et al.
suggests that the effective hadron mass might be dynamically generated as a
result of gluon nonlinearities [23]. We envisage that when a substantial
number of condensed gluons at the threshold $x_{c}$ suddenly participate in
the $pp$-collisions, it inevitably leads to a dramatic increase in the
production of secondary mesons. Since mesons possess mass, their yield
$N_{\pi}$ is inherently constrained. In accordance with quantum theory, newly
generated particles through collisions are initially excited from off-mass-
shell to on-mass-shell, acquiring relative kinetic energy from the remaining
interaction energy. In principle, the condensed gluons engaging in collisions
may simultaneously generate a considerable number of secondary off-mass-shell
mesons at a given interaction energy. If these mesons are excited to on-mass-
shell, they are capable of saturating all available energy, indicating that
nearly all kinetic energies in collisions at the center-of-mass (C.M.) frame
are utilized in creating the rest pions. This results in almost no relative
momentum for the newly-formed mesons, leading to the maximum value of
$N_{\pi}$. While the validity of this saturation approximation will be
scrutinized by subsequent observed data, adopting this limit allows us to
circumvent the complexities associated with the hadronization mechanism.

Figure 3: $pp\rightarrow N_{\pi}\pi$ in the different systems, where almost
all available kinetic energies of collisions at the C.M. system are used to
create pions due to the GC effect.

Using energy conservation in Fig. 3, we have

$E_{p}+m_{p}=\tilde{m}_{p}\gamma_{1}+\tilde{m}_{p}\gamma_{2}+N_{\pi}m_{\pi},$
$None$

where $\tilde{m}_{p}$ marks the leading particle and $\gamma_{i}$ is the
Lorentz factor. On the other hand, the square of relativistic invariant total
energy $s=(p_{1}+p_{2})^{2}$ in the two systems has

$2m_{p}^{2}+2E_{p}m_{p}=(2\tilde{E}^{*}_{p}+N_{\pi}m_{p})^{2}.$ $None$

Using

$2E_{p}\equiv\left(\frac{1}{k}-1\right)N_{\pi}m_{\pi},$ $None$

$k\simeq 1/2$ is the inelasticity, we have

$\tilde{m}_{p}\gamma_{1}+\tilde{m}_{p}\gamma_{2}=\left(\frac{1}{k}-1\right)N_{\pi}m_{\pi}\gamma.$
$None$

One can straightforwardly obtain

$\ln N_{\pi}=0.5\ln(E_{p}/1~{}\mathrm{GeV})+a,~{}~{}\ln
N_{\pi}=\ln(E_{\pi}/1~{}\mathrm{GeV})+b,$ $None$

where $a\equiv
0.5\ln(2m_{p}/1~{}\mathrm{GeV})-\ln(m_{\pi}/1~{}\mathrm{GeV})+\ln k$ and
$b\equiv\ln(2m_{p}/1~{}\mathrm{GeV})-2\ln(m_{\pi}/1~{}\mathrm{GeV})+\ln k$.
Thus, under the saturation condition in Eq. (5), only two universal conditions
(relativistic covariance and conservation of energy) can obtain the $\pi$
meson yield, which is a straight line in double logarithmic coordinates, i.e.,
the PL in $E_{\gamma}\in[E_{\pi}^{GC},E_{\pi}^{cut}]$ at Fig. 4.

Figure 4:  A schematic diagram for the ratio of the multiplicity of pion with
and without the GC effect. Note that the fluctuations in Fig. 2 have
disappeared due to Eq. (9). All scales are taken as logarithmic units.

Substituting Eq. (9) and a standard spectrum of $\pi^{0}\rightarrow 2\gamma$
into Eq. (1), we can analytically obtain the GC-spectrum in Fig. 5, i.e.,

$E_{\gamma}^{2}\Phi^{GC}_{\gamma}(E_{\gamma})\simeq\left\\{\begin{array}[]{ll}\frac{2e^{b}C_{\gamma}}{2\beta_{p}-1}(E_{\pi}^{GC})^{3}\left(\frac{E_{\gamma}}{E_{\pi}^{GC}}\right)^{-\beta_{\gamma}+2}\\\
{\rm~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}if~{}}E_{\gamma}\leq
E_{\pi}^{GC},\\\ \\\
\frac{2e^{b}C_{\gamma}}{2\beta_{p}-1}(E_{\pi}^{GC})^{3}\left(\frac{E_{\gamma}}{E_{\pi}^{GC}}\right)^{-\beta_{\gamma}-2\beta_{p}+3}\\\
{\rm~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}if~{}}E_{\pi}^{GC}<E_{\gamma}<E_{\pi}^{cut},\\\
\\\
\frac{2e^{b}C_{\gamma}}{2\beta_{p}-1}(E_{\pi}^{GC})^{3}\left(\frac{E_{\gamma}}{E_{\pi}^{GC}}\right)^{-\beta_{\gamma}-2\beta_{p}+3}\exp\left(-\frac{E_{\gamma}}{E_{\pi}^{cut}}+1\right).\\\
{\rm~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}if~{}}E_{\gamma}\geq
E_{\pi}^{cut},\end{array}\right.$ $None$

where $E_{\pi}^{GC}$ is the break energy at the GC-threshold determined by the
critical momentum $(x_{c},k_{c})$. $E_{\pi}^{cut}$ is the cutoff energy. A
phenomenological exponential cut factor in Eq. (10) describes the fast
suppression of the energy spectrum at $E_{\gamma}>E_{\pi}^{cut}$ if the
protons are accelerated to a enough high energy range. The reason is that the
gluons at $x<x_{c}$ have been condensed at $x_{c}$ and where no gluons to
participate in the $pp$ interaction. Note that

$E_{p}+m_{p}=\frac{2m_{p}}{m^{2}_{\pi}}E_{\pi}^{2},$ $None$

which gives the relation between the energy of the incident proton and
$E_{\pi}$ in the Lab. system. Additionally, the two parameters of
$E_{\pi}^{GC}$ and $E_{\pi}^{cut}$ are connected with each other through

$E_{\pi}^{cut}=e^{b-a}\sqrt{\frac{2m_{p}}{k_{c}^{2}}}\left(E_{\pi}^{GC}\right)^{2},~{}~{}~{}if~{}E_{p}\geq
E_{p}^{cut},$ $None$

where all energies take the GeV unit. Note that if the protons cannot be
accelerated to $E_{p}^{cut}$, the value of $E_{\pi}^{cut}$ needs to be
reduced.

Figure 5: A schematic GC-spectrum predicted by the hadronic model with the GC
effect.

The GC-spectrum has the following characteristics. (i) The power law at
$E_{\gamma}<E_{\pi}^{GC}$ origins from a steep increase in the pion number
(see Fig. 4), which makes the lower limit of integration in Eq. (1) is
independent of $E_{\gamma}$, while the power law at $E_{\gamma}>E_{\pi}^{GC}$
is a result of Eq. (9); (ii) The gamma signal disappears at
$E_{\gamma}>E_{\pi}^{cut}$ if the protons are accelerated above $E_{p}^{cut}$.
We will show these two examples in the following sections.

##  3 THE BROKEN POWER-LAW IN THE GRB 221009A

We employ the GC-model to investigate the GRB 221009A event. When analyzing
events of this nature, corrections for the influence of extragalactic
background light (EBL) become imperative, particularly as high-energy photons
traverse vast interstellar distances. Notably, LHAASO has recently provided
new intrinsic spectral energy distributions that have undergone these
corrections, covering the time span from 230 to 900 seconds after the trigger
[24]. Regrettably, the available LHAASO data are constrained to energies above
200 GeV. However, gamma-ray spectra at lower energies for GRB 221009A have
been captured by Fermi-LAT over the initial 2000 seconds following the Fermi-
GBM trigger [25]. It is worth noting that data within the first 300 seconds
from Fermi-LAT may potentially saturate the detectors [26]. Consequently, we
exclusively utilize the LHAASO data during the interval of 300-900 seconds to
seamlessly connect with the Fermi-LAT data, as illustrated in Fig. 6.

Figure 6:  An intrinsic spectral energy distribution of GRB 221009A compares
with the GC-spectrum for the p-heavy nuclei, pp collisions (the solid curve)
and sum of the of them (point-broken curve), where the contributions of the
EBL attenuation are deducted from the observed spectra (the dashed curve)
using the model of Saldana-Lopez et al [27]. Data are taken from LHAASO in
[300-900]s [24] and Fermi-LAT in [300,900]s [25]. The parameters: For the
p-heavy nuclei collisions, $E_{\pi}^{GC}=100~{}GeV$, $C_{\gamma}=3.09\times
10^{-12}(GeV^{-2}cm^{-2}s^{-1})$, $\beta_{\gamma}=1.53$, $\beta_{p}=0.867$ and
$E_{\pi}^{cut}=14~{}TeV$ if $E_{p}^{max}=2~{}EeV\ll E_{p}^{cut}$; For the
$pp$-collisions, $E_{\pi}^{GC}=20~{}TeV$, $C_{\gamma}=9\times
10^{-19}(GeV^{-2}cm^{-2}s^{-1})$, $\beta_{\gamma}=0$, $\beta_{p}=0.867$. Note
that although both the LHAASO recorded points and the contributions of the EBL
are described by two different irregular curves (see [24]), their synthesized
intrinsic spectrum presents a simple power-law as predicted by the GC-model.

Figure 6 presents the spectral energy distribution of GRB 221009A and its
comparison with the GC-spectrum. The value of $E_{\pi}^{GC}$ relies on the
baryon number $A$ of the target nucleus in the $pA$ (or $AA$) collisions. For
instance, $E_{\pi}^{GC}$ is ${\sim}20~{}TeV$ in the $pp$-collisions, and it is
$100~{}GeV$ for the $p$-heavy nucleus collisions [17]. Note that the broken
point at $\sim 100~{}GeV$ in Fig. 6 is also consistent with the previous
fitting GRB 180720B and GRB 190114C in [20]. It is worth noting that this
value of $\beta_{p}$ is harder than the index ($\sim 2.7$) of power-law in the
cosmic rays since the former is near the source while the later is observed on
the earth.

The observational precision of LHAASO has seen significant improvement owing
to the extensive signals collected by its expansive detector array. The
notable alignment between the linear segment of the GC-spectrum and the LHAASO
data is particularly striking, even accounting for the exceptionally small
statistical errors. The power-law behavior of the meson yield in Eq. (9) is a
consequence of the saturation approximation and the two universal principles
mentioned earlier. This outcome serves to reinforce the credibility of the
saturation assumption.

The observed BPL form in gamma-ray spectra is not exclusive to GRB 221009A, it
has also been documented in other GRBs [20], sub-TeV gamma-ray spectra from
AGNs [20], and the GeV gamma-ray spectra of pulsars [28,29]. These
observations imply that the hadronic scenario involving the GC effect is a
prevalent phenomenon in nature, albeit with varied manifestations in different
environments. For instance, in a GRB event, gamma-ray emissions occur when
particle jets collide with the remnants of precursor stars, leading to prompt
emissions over a short duration. In contrast, in AGNs, similar collisions may
persist for an extended period due to the continuous production of a large
number of PeV protons through potent acceleration mechanisms, although our
understanding of these mechanisms remains incomplete. This diversity in
manifestations underscores the complexity and richness of the hadronic
scenario associated with the GC effect across different astrophysical
contexts.

The LHAASO data reveal that gamma-ray events beyond $10$ TeV in the intrinsic
spectrum of GRB221009A do not exhibit softening. This phenomenon is also
observed in the VHE gamma-ray spectra of some AGNs [20]. The observation of
the hardening phenomenon in TeV gamma-rays from cosmological distances carries
significant implications for potential new physics beyond the standard model.
For instance, it has been interpreted as originating from axions [30] or
Lorentz invariance violation [31]. Axions are hypothetical particles
introduced by theoretical models beyond the Standard Model (SM). These
unobserved particles couple with two photons, leading to a suppression of EBL
absorption of high-energy gamma-rays and increased transparency of space.
Alternatively, if the threshold for interacting with EBL is slightly increased
due to Lorentz invariance violation, the process $\gamma\gamma\rightarrow
e^{+}e^{-}$ is suppressed, resulting in a more transparent universe. These
scenarios highlight the potential for TeV gamma-ray observations to provide
insights into novel physics beyond our current understanding.

The GC-model may provide a realistic explanation for the observed hardening
VHE gamma-ray spectra. In Fig. 6, we present a GC-spectrum represented by the
dashed lines, derived from the $pp$ collisions, with $\beta_{\gamma}\sim 0$
indicating negligible energy loss of gamma-rays in the $pp$-collision
environment [20]. Additionally, we use $\beta_{p}=0.867$. We emphasize that
the hardening gamma spectrum caused by the GC can occur in different types of
processes, as long as there are high-energy pp collisions there. In fact,
similar spectra have been observed in the energy spectra of various celestial
objects, including 1ES 0229+200, 1ES 1101-232, 1ES 0347-121, PKS 2005-489,
H1426+482, and MKn 501 (see Fig. 8 in [20]). The GC-spectra near
$E_{\pi}^{GC}\sim 20TeV$ imply a necessity for proton acceleration beyond
$E_{p}^{GC}\sim 40~{}EeV$. It is noteworthy that a recent detection of an
extremely energetic cosmic ray at 244 EeV by the surface detector array of the
Telescope Array experiment [32] is of particular relevance. Taking into
account the Greisen-Zatsepin-Kuzmin (GZK) suppression, this observation
suggests that EeVatrons in extragalactic sources could potentially serve as
the sources of the GC-spectrum beyond the TeV energies.

We make a comparison of the radiation powers between the hadronic scenario
without the GC effect and with the GC effect. In the former case, a
significant portion of the kinetic energy in hadronic collisions typically
transforms into heat for secondary particles in the central region of the
collisions. Consequently, the meson yield follows a logarithmic dependence,
i.e., $N_{\pi}\propto\ln(\sqrt{s}/m_{\pi})$, where $\sqrt{s}$ is the
interaction energy in the center-of-mass (C.M.) system. On the other hand, the
GC effect maximizes the utilization of available kinetic energy to generate
new particles, resulting in a meson yield proportional to $\sqrt{s}/m_{\pi}$.
As an illustrative example, taking $E_{\pi}^{GC}=100~{}GeV$, the corresponding
proton energy is $E_{p}^{GC}=10^{5}~{}GeV$ due to Eq. (11). Consequently, the
ratio of $N_{\pi}$ with the GC effect to $N_{\pi}$ without the GC effect is
approximately $10^{3}$. This outcome underscores that the GC effect
facilitates the efficient conversion of collision kinetic energy into
radiation energy.

The emissions of GRB 221009A are most likely generated by different physical
mechanisms occurring at different times and locations. The GC model dominates
the brightest radiation being emitted between 200 and 300 seconds after the
initial trigger Because the GC-effect has to be a very energetic proton flow,
this effect, especially the hardening tail of the gamma spectrum, may
disappear during the afterglow radiation ($T\gg 10^{3}s$) due to the weakening
of the proton flow energy, and the other radiation mechanisms (for example the
photohadronic model) becomes important. The main difference between them with
the GC-model is that the formers derive from the power law. This is important
to note in the study of the GRB 221009A spectra at different time periods
[33].

##  4 THE COLOR PENETRATION IN GRB 170817A

The distribution of condensed gluons within a nucleon not only reveals a
pronounced peak at $(x_{c},k_{c})$ but also indicates the absence of gluons at
$x<x_{c}$. Let’s consider a neutron (n) incident on a nucleus (A). The average
penetration distance ($L$) of the neutron is defined as $L=1/(\sigma N_{A})$,
where $N_{A}$ represents the nuclear density. The cross section
$\sigma(\sqrt{s})$ encompasses contributions from both elastic and inelastic
scattering, as well as resonance excitation. In the parton picture [34-36], at
high energy, valence quarks tend to traverse with a loss of only approximately
$2\%$ of their initial momentum. These valence quarks give rise to outgoing
hadrons in the fragmentation region. Interactions between small partons from
the two incident hadrons in the rapidity central range dominate the cross
section. The resistance between two colliding nucleons is contingent upon the
multiplicity and strength of interactions among their partons. Consequently,
the predominant contribution to the cross section $\sigma$ arises from the
exchange of small partons between the two colliding nucleons, particularly
since these small partons, mainly gluons, exhibit an abundant quantity and low
momentum transfer.

Specifically, the cross section $\sigma(pp\rightarrow X)$ at high energy is
dominated by the production of gluon mini-jet $\sigma(\sqrt{s})\propto
N_{g}(\sqrt{s})$. Note that $\sigma(\sqrt{s}>\sqrt{s_{cut}})\rightarrow 0$ due
to Fig. 4. It means that $L\rightarrow\infty$ at $\sqrt{s}>\sqrt{s_{cut}}$ and
the nucleus becomes traversable. We call it the color penetration.
$\sqrt{s_{cut}}$ is its threshold, which relates to the value of $x_{c}$. For
example, for a heavy nucleus, its $E_{\pi}^{GC}\sim 100~{}GeV$ and
$\sqrt{s_{cut}}\sim 1~{}PeV$. Thus, the color penetration exists in the extra
high energy hadronic collisions and it is an extremely rare event. In
reference [15] it was predicted that this effect may impede the increase of
new particle events in future ultra high energy hadron colliders.

However, the binary neutron star merger scenario may change this situation.
The core $A^{*}$ of a neutron star is a huge nucleus composed of nearly an
infinite number of neutrons, making the GC critical value $x_{c}$ significant
increases in such a large nucleus [17]. In principle, $x_{c}$ approaches 1 as
$A$ approaches infinity. However, the kinematical restriction leads to a
maximum $x_{c}<1$ at $A\rightarrow\infty$. Let us detail it. Since pions
dominate the secondary particles in $pp\rightarrow X$, we consider
$\sigma(pp\rightarrow\pi)$. The energy conservation demands $E_{p}\geq m_{p}$
and $E_{\pi}\geq m_{\pi}$ in Eq. (11). Therefore,
$E_{\pi,min}^{GC}=0.135~{}GeV,~{}~{}E_{p,min}^{GC}=0.938~{}GeV$, which
corresponds to $E_{\pi,min}^{cut}=0.255~{}GeV$, or
$\sqrt{s_{cut,min}}=3.7~{}GeV$ and $x_{c,max}=0.27$ if $k_{c}\sim 1~{}GeV$. It
implies that two neutrons from two star cores at the C.M. system have energies
$E_{c.m.}+E_{c.m.}=3.7~{}GeV$ and the Lorentz factor $\gamma=1.85/0.938=2$ or
$\beta=v/c=0.87$.

If we consider the neutron star in GRB 170817A with a radius of approximately
$R\sim 20~{}km$ and a spin frequency of $f\sim 2000/s$, each neutron during
the merging process exhibits a $\beta\sim 0.67$. Taking into account the
additional contribution of the star orbit rotation, the phenomenon of color
penetration becomes plausible. When a collision manages to breach the shell
barrier of neutron stars and penetrate the core area from both sides, a
notable reduction in the cross-sectional area ($\sigma$) is observed. This
suggests that binary neutron stars can merge with minimal resistance until two
neutrons reach each other’s boundaries. This scenario has the potential to
result in a spherically symmetric explosion source. However, it’s important to
note that not all binary neutron star mergers exhibit the color penetration
effect unless they possess a sufficiently high collision energy.

The color penetration observed in GRB 170817A is accompanied by a
characteristic gamma-ray spectrum. Cosmic gamma-rays typically arise from the
$pp\rightarrow\pi^{0}\rightarrow 2\gamma$ process within the hadronic
scenario. This process is highly affected by the gluon distributions in
nucleons. The GC effect may give rise to the typical BPL with an exponential
suppression factor in gamma-ray energy spectra. Besides, in a neutron-neutron
collisions,
$\Phi^{GC}_{\gamma,max}(pA)/\Phi^{GC}_{\gamma,max}(A^{*}A^{*})\sim\sqrt{s_{cut}(pA)}/\sqrt{s_{cut}(A^{*}A^{*})}\gg
1,$ since $E_{\pi}^{cut}(pA)\gg E_{\pi}^{cut}(A^{*}A^{*})$. Therefore, the GC-
peak at the MeV band in the merger of binary neutron stars is significantly
reduced.

It is noteworthy that no hard gamma-rays were detected by Fermi-GBM [37] and
INTEGRAL [38] during the merger time of GW 170817. Despite subsequent analyses
of data recorded by high-energy ($>100MeV$) and VHE ($>100GeV$) gamma-ray
detectors, such as Fermi-LAT, no instances of high-energy counterparts
conclusively associated with the event were identified [39,40]. An
interruption in data collection by Fermi-LAT during the trigger moment of GW
170817, caused by its passage through the South Atlantic Anomaly, prevented an
assessment of the presence or absence of high-energy emission. Nevertheless,
negative observations were recorded with H.E.S.S., a detector that commenced
observations merely five hours after GW 170817. The GC-model predicts that the
gamma-ray spectrum of GRB 170817A exhibits an isolated, narrow, and weak
spectrum in the MeV range.

An energy range approximately centered around $\sim 200~{}MeV$ represents an
intermediate zone, positioned between the KeV and GeV focal ranges, which has
received limited attention in investigations of GRB emissions. This is
attributed to both the characteristics of scintillation detectors used for GRB
observations, where the efficiency of photon detection in this range is
insufficient, and the steeper and narrower BPL shape of the GC-spectrum.
Fortunately, a new space-based gamma-ray telescope, GAMMA-400, has been
designed to continuously measure within the 20 MeV to several TeV energy range
[41]. The exceptional characteristics of this instrument make it a potent tool
for identifying GC-spectral signatures at MeV in short GRB events.

##  5 DISCUSSIONS AND CONCLUSION

Comparing the GC with the Bose-Einstein condensation (BEC), we observe that,
despite both involving multi-bosons sharing a single wave function, they
represent distinct physical phenomena. Thus, the GC opens a new window into
nature. For example, the GC-model efficiently generates electromagnetic
radiation in the universe. While positive and negative electron annihilation
undoubtedly maximizes photon production efficiency, such events are rare in
nature. Inverse Compton scattering is a common mechanism for explaining VHE
gamma-ray spectra, but it requires an ample supply of soft photons to serve as
targets for scattering electrons. Furthermore, the GC represents the first
instance of the butterfly effect occurring in elementary particles.
Consequently, concerns about the efficacy of perturbation treatment are
unnecessary, as once the CGC exhibits a strong chaotic phenomenon, it proves
challenging to suppress. Additionally, the $\pi$-cluster, as an intermediate
state in the GC-model, closely resembles $\pi$-condensation, a phenomenon
anticipated in nuclear physics. This warrants further investigation.

We have noted that the maximum energies in $p-Pb$ and $Pb-Pb$ collisions at
the LHC are 8.16 TeV and 5.02 TeV, respectively. Additionally, the Auger
collaboration indirectly used cosmic ray data at the top of the atmosphere and
found that the $pA$ cross section at $\sqrt{s}\sim 100TeV$ is a normal value
of $\sim 567mb$ with no significant increment. An important question arises:
why have they not recorded the GC effect? We consider $E_{\pi}^{GC}(Pb-
Pb)=100GeV$ and $E_{\pi}^{GC}(p-light~{}nuclei)=1TeV$ according to Fig. 11 in
[17]. The results are $\sqrt{s_{Pb-Pb}^{GC}}=1TeV$, $\sqrt{s_{Pb-
Pb}^{cut}}=100TeV$, $\sqrt{s_{p-light~{}nuclei}^{GC}}=10TeV$, and
$\sqrt{s_{p-light~{}nuclei}^{cut}}=10~{}PeV$. Since these estimations are
based on central collisions, the actual collision threshold should be larger
than these estimations. Therefore, we caution that further increases in hadron
collision energies in the upcoming LHC plans may lead to unexpectedly intense
gamma-rays in the accelerator, resembling artificial mini-GRBs, and
potentially causing damage to the detectors [17]. Moreover, the GC raises the
possibility of a sudden disappearance of gluon distributions, potentially
halting the increase of new particle events in an ultra-high-energy heavy-ion
collider [15].

To summarize, we illustrate a simple logical relationship between the GC in
the proton and the GRB spectra during the burst phase. When the CGC gluons
undergo evolution through the singularity of a nonlinear QCD evolution
equation, these gluons will enter a chaotic state. The resulting intense
oscillations in the gluon distribution may lead to strong antishadowing,
causing a large number of soft gluons to condense at a critical momentum. On
the other hand, during the burst phase of a GRB event, protons are accelerated
above the GC threshold, leading to a sudden influx of condensed gluons into
the proton-nuclei collisions, giving rise to the observed bump in the gamma-
ray spectrum. Moreover, as long as these pion yield saturates, energy
conservation and relativistic covariance directly result in the BPL form the
gamma spectrum, as observed by Fermi-LAST and LHAASO. Besides, the remarkably
symmetrical explosion cloud in kilonova AT2017gfo and the absence of a very
high-energy gamma-ray signature in GRB 170817A also can explained by the above
GC model.

### Acknowledgments

We thank Zi-Qing Xia for the useful comments. We also thank Qi-hui Chen and
Wei Kou for the help in the calculations.

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# Code Generation Tools (Almost) for Free?
A Study of Few-Shot, Pre-Trained Language Models on Code

Patrick Bareiß  University of StuttgartGermany ,  Beatriz Souza  Federal
University of PernambucoBrasil ,  Marcelo d’Amorim  Federal University of
PernambucoBrasil and  Michael Pradel  University of StuttgartGermany

###### Abstract.

Few-shot learning with large-scale, pre-trained language models is a powerful
way to answer questions about code, e.g., how to complete a given code
example, or even generate code snippets from scratch. The success of these
models raises the question whether they could serve as a basis for building a
wide range code generation tools. Traditionally, such tools are built manually
and separately for each task. Instead, few-shot learning may allow to obtain
different tools from a single pre-trained language model by simply providing a
few examples or a natural language description of the expected tool behavior.
This paper studies to what extent a state-of-the-art, pre-trained language
model of code, Codex, may serve this purpose. We consider three code
manipulation and code generation tasks targeted by a range of traditional
tools: (i) code mutation; (ii) test oracle generation from natural language
documentation; and (iii) test case generation. For each task, we compare few-
shot learning to a manually built tool. Our results show that the model-based
tools complement (code mutation), are on par (test oracle generation), or even
outperform their respective traditionally built tool (test case generation),
while imposing far less effort to develop them. By comparing the effectiveness
of different variants of the model-based tools, we provide insights on how to
design an appropriate input (“prompt”) to the model and what influence the
size of the model has. For example, we find that providing a small natural
language description of the code generation task is an easy way to improve
predictions. Overall, we conclude that few-shot language models are
surprisingly effective, yet there is still more work to be done, such as
exploring more diverse ways of prompting and tackling even more involved
tasks.

††copyright: none

##  1\. Introduction

Various software engineering tools assist developers by generating source
code. One group of approaches reasons about existing code and modifies it in a
way suitable to achieve some goal. For example, code mutation tools
(jia2010analysis; papadakis2019mutation) introduce mistakes to measure the
effectiveness of test suites, and automated program repair tools
(cacm2019-program-repair; monperrus2018automatic) suggest how to fix
programming mistakes. Another group of approaches generates new code from
scratch, given some existing code that the new code is supposed to relate to.
For example, test case generators (Csallner2004; Pacheco2007; Fraser2011a)
automatically create tests that exercise a given method under test, and code
completion tools (Bruch2009; Raychev2014; Hellendoorn2019) generate code that
completes an existing code snippet in a suitable way. Finally, a third group
of code generation tools does not require any existing code as an input, but
instead generates new code given some natural language artifact. For example,
some approaches generate test oracles based on informal API documentation
(Goffi2016; Blasi2018; Blasi2021), infer API usage protocols (Zhong2009a), or
suggest missing type annotations (icse2019).

The traditional way of creating such code manipulation tools is based on
program analysis combined with various rules and heuristics. Program analysis
can, at least in principle, ensure that the generated code is guaranteed to
have certain properties, e.g., to be type-correct or to pass a given set of
test cases. Hand-coded rules and heuristics are typically required to enable a
technique to be effective and efficient on real-world software. More recently,
learning-based approaches have started to complement traditional program
analysis-based code generation tools (NeuralSoftwareAnalysis). Typically,
these approaches formulate the specific code generation task as a supervised
learning problem, and require large amounts of training data to obtain an
effective machine learning model. A commonality of both traditional program
analyses and learning-based approaches is that creating a new code generation
tool involves significant human effort. Even worse, this effort often must be
repeated for each new combination of a task to achieve and a programming
language to target.

A recent trend in the natural language processing (NLP) community promises a
form of “general intelligence” that remedies many of the problems of building
task-specific techniques: _few-shot learning with large-scale, pre-trained
language models_ (Brown2020), henceforth abbreviated with _FSLMs_. These
models are trained on huge amounts of data without focusing on a specific
downstream task. Instead, the training is based on generic pseudo-tasks for
which it is trivial to obtain sufficient training data, e.g., predicting
masked words or whether two sentences belong together. Once trained, FSLMs are
effective at various question answering and text generation tasks, e.g.,
reading comprehension, trivia quizzes, translation between languages, and text
completion (Brown2020).

Applying FSLMs to code is still a relatively sparsely explored area. While
recent work employs pre-training of models of code as a means to reduce the
amount of required training examples (Ahmad2021; Guo2021; Liu2020; Feng2020),
these approaches still fine-tune a model for a specific purpose and hence
require moderately large amounts of labeled training examples. Noteworthy
exceptions include GitHub’s Copilot code completion
system111https://copilot.github.com/, which is based on the Codex FSLM
(Chen2021), and the recently released, open-source PolyCoder model family
(Xu2022). While the results of these models are impressive, code completion is
only one of many code generation tasks. Do the abilities of FSLMs generalize
to other software engineering tasks that traditionally have been addressed by
special-purpose code generation techniques? In case of a positive answer,
FSLMs offer the potential to obtain code generation tools (almost) for free,
as an FSLM gets trained once and can then be applied to many different tasks.
Despite this potential and the strong interest of the software engineering
community in automated code generation techniques, there currently is no
systematic study of the abilities of FSLMs on such tasks.

Table 1. Representative examples of results obtained with FSLM-based code
generation tools.