# Practical Edge Detection via Robust Collaborative Learning

Yuanbin Fu  Tianjin UniversityTianjin, China<EMAIL_ADDRESS>and  Xiaojie
Guo  Tianjin UniversityTianjin, China<EMAIL_ADDRESS>
(2023)

###### Abstract.

Edge detection, as a core component in a wide range of vision-oriented tasks,
is to identify object boundaries and prominent edges in natural images. An
edge detector is desired to be both efficient and accurate for practical use.
To achieve the goal, two key issues should be concerned: 1) How to liberate
deep edge models from inefficient pre-trained backbones that are leveraged by
most existing deep learning methods, for saving the computational cost and
cutting the model size; and 2) How to mitigate the negative influence from
noisy or even wrong labels in training data, which widely exist in edge
detection due to the subjectivity and ambiguity of annotators, for the
robustness and accuracy. In this paper, we attempt to simultaneously address
the above problems via developing a collaborative learning based model, termed
PEdger. The principle behind our PEdger is that, the information learned from
different training moments and heterogeneous (recurrent and non recurrent in
this work) architectures, can be assembled to explore robust knowledge against
noisy annotations, even without the help of pre-training on extra data.
Extensive ablation studies together with quantitative and qualitative
experimental comparisons on the BSDS500 and NYUD datasets are conducted to
verify the effectiveness of our design, and demonstrate its superiority over
other competitors in terms of accuracy, speed, and model size. Codes can be
found at https://github.com/ForawardStar/PEdger.

Edge detection; Neural networks; Label denoising

††journalyear: 2023††copyright: acmlicensed††conference: Proceedings of the
31st ACM International Conference on Multimedia; October 29-November 3, 2023;
Ottawa, ON, Canada††booktitle: Proceedings of the 31st ACM International
Conference on Multimedia (MM ’23), October 29-November 3, 2023, Ottawa, ON,
Canada††price: 15.00††doi: 10.1145/3581783.3612099††isbn:
979-8-4007-0108-5/23/10††ccs: Computing methodologies Image manipulation

(a) Input & GT

(b) RCF

(c) PiDiNet

(d) BDCN

(e) Ours

Figure 1. Visual comparison with other methods. Our detected edges are thinner
and clearer than the others.

##  1\. Introduction

Edge detection aims to extract object boundaries and salient edges from given
images, please see Fig. 1 for an example. As an indispensable component to a
wide spectrum of applications, such as object detection (Ren et al., 2017),
semantic segmentation (Hu et al., 2020), and image editing (Elder and
Goldberg, 2001), to just name a few, this task has been drawing intensive
attention from the community with a great progress made over last decades,
especially with the emergence of deep learning. Even though, by simultaneously
considering the detection accuracy, processing speed, and model size,
developing practical edge detectors for real-time applications remains highly
desired.

Early edge detectors concentrated on local features, like image gradients
(Canny, 1986) and manually-designed features (Martin et al., 2004; Arbelaez et
al., 2011; Dollár and Zitnick, 2013). These methods determine edges solely
based on shallow/low-level cues, the accuracy of which is often unsatisfactory
in practice. To exploit semantic information, numerous works (Xie and Tu,
2017; He et al., 2019; Liu et al., 2019; Pu et al., 2022) have been recently
built on top of deep learning techniques and have brought tremendous
progresses, benefiting from the strong representative ability of deep
networks. These approaches leveraged the models like VGG16 (Simonyan and
Zisserman, 2015), ResNet50 (He et al., 2016), and ViT-B/16 (Dosovitskiy et
al., 2021) pre-trained on the large-scale ImageNet (Deng et al., 2009), as
their backbones, to embody abundant knowledge. In spite of the remarkable
improvement in terms of accuracy, they inevitably pay high prices at
computation and memory, due to the (large) pre-trained backbones. The
inefficiency limits their applicability in real-time scenarios. In the
literature, a good trade-off technique goes to the recent PiDiNet (Su et al.,
2021), which liberates from heavy pre-trained backbones, and achieves fast and
reasonably accurate performance. Even though, there still has a large room for
improving the accuracy, as illustrated in Fig. 2. An important issue rarely
considered by these mentioned methods is the negative impact from noisy or
even wrong annotations in training data on the quality of learning.
Particularly in edge detection datasets, such problematic labels widely exist
due to the subjectivity and ambiguity of annotators. Figure 3 depicts noisy
annotations.

Figure 2. Performance comparison in Optimal Dataset Scale F-score (ODS-F). For
all the methods, the running speed is tested on an RTX 3090 GPU with
$200\times 200$ images as input.

To take a step forward in deep learning era for accurate, fast, and
lightweight edge detection, as previously analyzed, we have to face two main
challenges. On the one hand, the high accuracy of most deep models is
primarily attributed to the abundant knowledge implicated in (large) pre-
trained parameters, but these pre-trained parameters significantly increase
the computational complexity and model size. A question naturally arises,
i.e., how to liberate deep edge models from pre-trained backbones for saving
the computational cost and cutting the model size? On the other hand, for a
given image, different annotators may provide nonidentical edge labels. In
other words, the inconsistent/noisy labels by ambiguous and subjective
annotators can hardly be avoided. Hence, another challenge, i.e., how to
mitigate the negative influence from noisy or even wrong labels in training
data for the robustness and accuracy, needs to be solved. It seems that pre-
trained models can reduce the influence of noisy labels thanks to the
regularization ability from extra data, but it again falls into the first
dilemma.

For jointly conquering the above two challenges, we alternatively investigate
to integrate useful knowledge, probably weak, from different sources (except
for large pre-trained models) as a strong agent. By this means, the
bias/inconsistency of the training data can be relieved from the perspective
of ensemble learning (Cho et al., 2022; Lu and He, 2022), while dispensing
with the pre-training. Although the knowledge learned either at a particular
training moment or by a single model is easily disturbed by noisy/outlier
samples, especially when the batch size is small, it shall be qualified to act
as a source for ensemble. To verify the primary claim of this work, the
information from both different training moments and heterogeneous network
architectures, is taken into consideration.

Image1 Annotation1 Image2 Annotation2

Figure 3. Some examples of noisy annotations in BSDS. The labeling of eyes and
mouth indicated by the blue squares in the first case is contradictory. The
red squares in the second case show the wrong annotations that some pixels
within the grass are labelled as edges, which actually are textures.

#### Contribution.

With regard to heterogeneous architectures, two networks with recurrent
(slower) and non recurrent (faster) structures are incorporated during
training. The knowledge learned by these two structures will be fused using a
simple uncertainty-aware weighting method. For integrating the information
across different training moments, we ensemble the network parameters
corresponding to different epochs in a momentum way. During training, both the
networks are updated collaboratively, that is, they are trained in one stage
without waiting for the training of another to be finished beforehand. Please
notice that in the testing phase, only one network needs to be executed. Our
major contributions are summarized as:

  * •

We customize a collaborative learning scheme for practical edge detection via
assembling the information across different network architectures and training
moments. It is able to prevent the network from over-fitting to noises without
the help of any pre-trained model.

  * •

We design two architectures to capture diversified knowledge for robustness,
which will be fused in an uncertainty-aware fashion. Particularly, a recurrent
structure with shared parameters across scales and a non-recurrent one having
different amount of parameters for different scales, are maintained in the
collaborative learning process.

  * •

We conduct extensive experiments to verify the effectiveness of our design on
several benchmark datasets. The results indicate that our method can achieve
superior performance over other state-of-the-art competitors by
comprehensively considering the accuracy, speed, and model size.

##  2\. Related Work

#### Edge Detection.

Early attempts on edge detection may trace back to gradient based edge
operators like Canny (Canny, 1986). Though being efficient, the detection
quality by these operators hardly reaches the requirement in real-world
applications. Another category refers to machine learning approaches. As a
representative, gpb-UCM (Arbelaez et al., 2011) feeds the oriented gradient of
three brightness channels and one texture channel as the features into a
logistic regression classifier. Dollar et al. (Dollár and Zitnick, 2013)
proposed to employ the random forest for edge detection, which takes full
advantage of the inherent structure in local patches. With the emergence of
deep learning, various approaches have been proposed, such as HED (Xie and Tu,
2017) and RCF (Liu et al., 2019). They output multi-scale side-outputs based
on the pre-trained features from VGG16. But, all the side-outputs are trained
under identical supervision, ignoring the diversity. To address this issue,
several works (Liu and Lew, 2016; He et al., 2019) advocated to impose
different supervisions on these side-outputs. More recently, EDTER (Pu et al.,
2022) accomplishes the edge detection task using ViT (Dosovitskiy et al.,
2021). Although considerably improving the accuracy, these mentioned deep
learning based methods all employ external networks (e.g., VGG16, ViT-B/16)
with substantial parameters pre-trained on large-scale datasets (e.g. ImageNet
(Deng et al., 2009)), as assistant. Consequently, their computational costs
also increase. To save the computational expense, PiDiNet (Su et al., 2021)
introduces the pixel difference convolution to extract edge-related features
without employing pre-trained backbones. Compared to RCF (Liu et al., 2019),
it considerably reduces the model size and accelerates the speed with
competitive accuracy.

#### Learning from Noisy Labels.

Recently, numerous approaches have been raised to train deep networks in the
presence of label noises (Lukov et al., 2022; Fu et al., 2022; Wang et al.,
2019; Park et al., 2020). Among them, rectifying the original noisy targets
based on the useful information/knowledge investigated in training, is
demonstrated to be effective. For example, PENCIL (Yi and Wu, 2019) adopts the
information in back-propagation to probabilistically correct noisy labels,
which does not need any prior information about noises. Similarly, Ren et al.
(Ren et al., 2018) estimated the importance score of samples according to
gradient directions in back-propagation.

Besides the information from back-propagation, softening the noisy targets
using the knowledge learned from previous iterations, i.e., online knowledge
distillation, is also revered by many works. As representatives, Kim et al.
(Kim et al., 2021) produced the soft target based on the predictions of the
network itself, using the parameters updated at the last iteration. (Zhang et
al., 2018) utilized peer-teaching strategy that trains collaboratively
multiple networks to learn from each other. But, one independent network may
not provide reasonable guidance. Hence, (Lan et al., 2018; Guo et al., 2020)
established the supervision information on-the-fly through ensembling the
predictions of multiple collaborative networks, which achieve better
performance. The methods in this category suffer from two main limitations: 1)
The network architectures are identical or similar, leading to the poor
diversity; 2) They utilize the parameters updated by the last mini-batch to
generate soft targets, resulting in the inconsistent optimization directions
since the parameters change dramatically for each iteration.

All the above mentioned researches point at image-level noisy annotations for
classification, which may not be suitable for pixel-level noises. For dealing
with dense prediction tasks, (Feng et al., 2022; Zhang et al., 2021) assigned
a pixel-wise soft weight to each sample, which is determined by the variance
of predictions at different iterations, for the sake of encouraging noisy
pixels to contribute less than other pixels. Unfortunately, besides involving
only one homogeneous network structure during training, their predictions are
produced only based on the snapshot knowledge at each iteration, without the
consideration of knowledge across different training moments. To mitigate the
pixel-level noises in edge detection datasets, we take advantage of the robust
knowledge from different training moments and heterogeneous models, where the
poor diversity of models and inconsistent label constraints during training,
can be avoided.

##  3\. Methodology

This work aims to devise a practical edge detector by simultaneously
addressing 1) the dependence on large pre-trained models, and 2) the negative
influence of noisy labels in training data. To achieve the goal, we propose to
integrate the knowledge across different training moments and network
architectures (as schematically shown in Fig. 4). We denote the training data
by $\textbf{D}\mathrel{\mathop{:}}=\\{(X_{n},Y_{n}),n=1,2,...,N\\}$, where $N$
is the total number of training samples, each sample $X_{n}$ is the raw input
image, and $Y_{n}$ is the ground truth edge map for $X_{n}$.

###  3.1. Network Architecture

Our framework comprises recurrent and non recurrent networks with different
randomly initialized parameters, to model diversified features for robustness,
which are trained collaboratively in one stage. It is rational since
integrating the predictions from discrepant models, helps to prevent from
over-fitting to noises/outliers, as verified by the literature of model
ensemble (Cho et al., 2022; Lu and He, 2022). During inference, only the non-
recurrent structure is executed.

#### Recurrent Architecture.

The blueprint of our recurrent network is shown in Fig. 4 (a). For each data
sample $X_{n}$, it is firstly taken as the input of the encoding module
consisting of one plain convolutional layer and two residual layers. The
features extracted by the encoding module will be fed into the recurrent
module whose parameters are shared across every recurrent steps. Specifically,
at $t$-th step ($t>1$), the recurrent module takes the features produced by
the $(t-1)$-th step as input (for the first recurrent step, that is $t=1$, the
features generated by the encoding module are inputted), and then outputs the
features of this step that will be taken as the input of next step again. This
process is executed iteratively until reaching the total recurrent step $T$.

(a) Recurrent Architecture.

(b) Non Recurrent Architecture.

Figure 4. Recurrent & non recurrent architecture. $K$, $S$, and $D$ are the
kernel size, stride, and dilation rate, respectively. The number of output
channels is shown below each block.

Notice that the recurrent module is connected with a max-pooling operation at
every steps except for the first step. As demonstrated by (Xie and Tu, 2017;
Liu et al., 2019; He et al., 2019), the usage of max-pooling operation can
increase the receptive fields, which is beneficial to generate multi-scale
side-output edges. The useful information captured by each step becomes
coarser as its receptive field size increasing, and the responses at larger
objects are stronger. In addition, inspired by the works (Deng et al., 2018;
Wang et al., 2017; He et al., 2019) that aggregate the information from fine-
to-coarse and coarse-to-fine for edge detection, we introduce two branches in
the decoding module as shown in Fig. 4, to aggregate the information from the
early recurrent steps (fine) to late steps (coarse), and from the late steps
to early, respectively. Concretely, for each step $t$, the side-output
features $R_{n}^{f2c(t)}$ at $t$-th step are obtained through adding the
outputs by the fine-to-coarse branch of this step, to the down-sampled side-
output features of previous step, say $R_{n}^{f2c(t-1)}$. Similarly,
$R_{n}^{c2f(t)}$ is obtained through adding the outputs by the coarse-to-fine
branch of this step, to the up-sampled features $R_{n}^{c2f(t+1)}$ of the late
step. Finally, all the side-output features are fused by a $1\times 1$
convolution, and processed by the Sigmoid activation to output the edge
detection results $E_{n}^{R}$. Due to the shared parameters, the recurrent
network is compact and converges fast. Also, the number of channels in
recurrent model is larger than the non recurrent one, thanks to that the
shared parameters can save parameter space for expanding channels.

#### Non-recurrent Architecture.

Objectively, recurrent models are typically slow in speed, because the
recurrent module (shown in blue part of Fig.4 (a)) needs to be inferred for
many times, where the network configurations are fixed. To construct a faster
network, we propose to train different amount of parameters specific to
different scales (shown in blue part of Fig.4 (b)). Specifically, the
parameter amount of the module at the first stage is the least, and gradually
increases as the feature size decreases. The reason for this design is that,
the earlier stages mainly focus on extracting lower-level features with larger
resolutions, while the later stages produce higher-level semantic features
with smaller resolutions. In short, the higher-level semantic structures at
later stages require more parameters to learn than lower-level details, and it
takes more time to process the feature maps at later stages with larger
resolutions than feature maps with smaller resolutions. Thus, using lesser
(more) parameters for earlier (later) stages can achieve minimal performance
degradation and maximal speed acceleration. It can be seen from Fig. 4 that
the difference between the recurrent and non recurrent structure is whether
the parameters of the modules indicated by the blue and black (the decoding
module) are shared across multiple scales. Notice that the non recurrent
architecture also adopts the bidirectional aggregation, the operation of which
is exactly the same as the recurrent one.

Algorithm 1 Collaborative Learning for Edge Detection

Input: Training dataset D; the total number of samples $N$; the total number
of training epochs $EP$; two network $G^{R}$ and $G^{NonR}$ trained by back-
propagation; two momentum networks $G_{m}^{R}$ and $G_{m}^{NonR}$; a hyper-
parameter $\eta_{EP}$
Output: Updated target network $G_{m}^{NonR}$
Initialize: $ep\mathrel{\mathop{:}}=0$; $\eta_{EP}\mathrel{\mathop{:}}=0.8$;
Randomly initialize $G^{R}$ and $G^{NonR}$

1: while $ep<EP$ do

2: Compute $\eta_{ep}$ by Eq. (5);

3: while $n<N$ do

4: Sample data $X_{n}$, $Y_{n}$ from D;

5: if $ep>0$ then

6: $M_{n}^{R}\mathrel{\mathop{:}}=G_{m}^{R}(X_{n})$,
$M_{n}^{NonR}\mathrel{\mathop{:}}=G_{m}^{NonR}(X_{n})$;

7: Compute $M_{n}$ by Eq. (LABEL:fusion) and $\widetilde{Y}_{n}$ by Eq. (4);

8: else

9: $\widetilde{Y}_{n}\mathrel{\mathop{:}}=Y_{n}$

10: end if

11: $E_{n}^{R}\mathrel{\mathop{:}}=G^{R}(X_{n})$,
$E_{n}^{NonR}\mathrel{\mathop{:}}=G^{NonR}(X_{n})$;

12: Compute $L^{R}$ by Eq. (7) and $L^{NonR}$ by Eq. (8);

13: Update $G^{R}$ and $G^{NonR}$ through back-propagating from $L^{R}$ and
$L^{NonR}$, respectively;

14: $n\mathrel{\mathop{:}}=n+1$

15: end while

16: if $ep>0$ then

17: Update $G_{m}^{R}$ and $G_{m}^{NonR}$ by Eq. (1);

18: else

19: Copy the parameters of $G^{R}$ and $G^{NonR}$, to $G_{m}^{R}$ and
$G_{m}^{NonR}$, respectively;

20: end if

21: $ep\mathrel{\mathop{:}}=ep+1$;

22: end while

###  3.2. Robust Collaborative Learning

As illustrated before, the original target $Y_{n}$ may contain ambiguous or
wrong annotations, which provide mislead optimization direction in training.
To avoid the disturbance of noisy labels, we design a collaborative learning
algorithm, to explore the robust knowledge across training moments and
heterogeneous architectures. Doing so helps to break free from the dependence
of the knowledge in pre-trained models, and therefore enables the network to
have both high accuracy and fast inference speed.

Concretely, the knowledge at different training moments throughout the whole
training process, can be aggregated by updating the parameters of a momentum
network:

(1)
$\displaystyle\textbf{W}_{m}^{(ep)}\mathrel{\mathop{:}}=(\textbf{W}_{bp}^{(ep)}+\textbf{W}_{m}^{(ep-1)})/2,$

where $\textbf{W}_{m}^{(ep)}$ and $\textbf{W}_{bp}^{(ep)}$ are the parameters
of a momentum network, and its corresponding network trained by back-
propagation, after the $ep$-th epoch, respectively.

Input & GT RCF PiDiNet BDCN EDTER PEdger

Figure 5. Visual comparison with other competitors.

To combine the knowledge from networks with different structures, we maintain
two momentum networks $G_{m}^{R}$ and $G_{m}^{NonR}$ corresponding to the
recurrent and non recurrent network $G^{R}$ and $G^{NonR}$ that trained by
back-propagation, respectively. We propose to fuse the predictions of two
momentum networks based on the pixel-wise uncertainty of these two models:

(2) $\displaystyle M_{n}\mathrel{\mathop{:}}=M^{R}_{n}\circ
U^{R}_{n}+M^{NonR}_{n}\circ U^{NonR}_{n},$

in which

(3) $\displaystyle U^{R}_{n}\mathrel{\mathop{:}}=$
$\displaystyle\frac{|M^{R}_{n}-0.5|}{|M^{R}_{n}-0.5|+|M^{NonR}_{n}-0.5|},$
$\displaystyle U^{NonR}_{n}\mathrel{\mathop{:}}=$
$\displaystyle\frac{|M^{NonR}_{n}-0.5|}{|M^{R}_{n}-0.5|+|M^{NonR}_{n}-0.5|},$

where $\circ$ is the Hadamard product,
$M^{R}_{n}\mathrel{\mathop{:}}=G_{m}^{R}(X_{n})$ and
$M^{NonR}_{n}\mathrel{\mathop{:}}=G_{m}^{NonR}(X_{n})$, $|\cdot|$ is pixel-
wise absolute operation, $U^{R}_{n}$ and $U^{NonR}_{n}$ are the uncertainty
scores of the predictions. The closer (farther) the predictions by a network
are to 0.5 compared to the other, the more (less) uncertain about the
predictions by this network, and thus the less (more) contributions it should
make to fusion. It can be seen from Table 2 that, the uncertainty-aware
weighting method achieves higher edge detection accuracy, compared to simply
averaging the predictions with the same weight.

Having the robust $M_{n}$ that integrates the knowledge across training
moments and models, the original noisy target $Y_{n}$ can be replaced with the
corrected $\widetilde{Y}_{n}$, which is calculated through:

(4) $\widetilde{Y}_{n}\mathrel{\mathop{:}}=\eta_{ep}\cdot
M_{n}+(1-\eta_{ep})\cdot Y_{n},$

where $\eta_{ep}$ is the weight that controls how much we are going to trust
the knowledge from $M_{n}$, at $ep$-th epoch. It is worth to mention that,
$\widetilde{Y}_{n}$ can also be regarded as the soft target (Diaz and Marathe,
2019; Zi et al., 2021; Lukov et al., 2022; Zhou et al., 2021; Wang et al.,
2022; Lienen and Hüllermeier, 2021), which suppresses over-confident class
probabilities in the original hard target. In our collaborative setting, since
all the models are optimized collaboratively from scratch, they generally do
not have enough knowledge about data distributions at the early stage of
training. Hence, inspired by the self distillation technique (Kim et al.,
2021), we gradually increase $\eta_{ep}$ in training, which is computed as:

(5) $\eta_{ep}\mathrel{\mathop{:}}=\eta_{EP}\times\frac{ep}{EP},$

where $EP$ is the total number of epoch for training and $\eta_{EP}$ is the
$\eta_{ep}$ at the last epoch. In this paper, we set
$\eta_{EP}\mathrel{\mathop{:}}=0.8$. The whole process can be summarized in
Algorithm 1.

###  3.3. Loss Function

As for the losses in training, we adopt the widely-used cross-entropy loss on
all the side-outputs and final edge maps. The formulation is:

(6) $\displaystyle
l(E_{n},\widetilde{Y}_{n})\mathrel{\mathop{:}}=-\alpha\cdot$
$\displaystyle\sum_{i=1}^{|X_{n}|}\widetilde{Y}_{n}^{(i)}\text{
Log}(E_{n}^{(i)})-$ $\displaystyle-\beta\cdot$
$\displaystyle\sum_{i=1}^{|X_{n}|}(1-\widetilde{Y}_{n}^{(i)})\text{
Log}(1-E_{n}^{(i)}),$

where
$\alpha\mathrel{\mathop{:}}=\lambda\cdot\frac{|Y_{n}^{+}|}{|Y_{n}^{+}|+|Y_{n}^{-}|}$
and $\beta\mathrel{\mathop{:}}=\frac{|Y_{n}^{-}|}{|Y_{n}^{+}|+|Y_{n}^{-}|}$.
Moreover,
$|Y_{n}^{+}|\mathrel{\mathop{:}}=||Y_{n}\circ\widetilde{Y}_{n}||_{1}$ and
$|Y_{n}^{-}|\mathrel{\mathop{:}}=||(1-Y_{n})\circ(1-\widetilde{Y}_{n})||_{1}$
denote the positive and negative sets of the corrected ground truth
corresponding to the $n$-th sample, respectively. $\lambda$ is used to balance
the positive and negative pixels. $|X_{n}|$ denotes the number of pixels in
the image $X_{n}$. $\widetilde{Y}_{n}^{(i)}$ and $E_{n}^{(i)}$ refer to the
$i$-th pixel in $\widetilde{Y}_{n}$ and $E_{n}$, respectively. The final loss
functions $L^{R}$ and $L^{NonR}$ respectively for $G^{R}$ and $G^{NonR}$, are
formulated as:

(7) $\displaystyle L^{R}\mathrel{\mathop{:}}=$ $\displaystyle
l(E_{n}^{R},\widetilde{Y}_{n})+$
$\displaystyle\sum_{t=1}^{T}[l(\sigma(R_{n}^{f2c(t)}),$
$\displaystyle\widetilde{Y}_{n})+l(\sigma(R_{n}^{c2f(t)}),\widetilde{Y}_{n})],$
(8) $\displaystyle L^{NonR}\mathrel{\mathop{:}}=$ $\displaystyle
l(E_{n}^{NonR},\widetilde{Y}_{n})+$
$\displaystyle\sum_{t=1}^{4}[l(\sigma(O_{n}^{f2c(t)}),$
$\displaystyle\widetilde{Y}_{n})+l(\sigma(O_{n}^{c2f(t)}),\widetilde{Y}_{n})],$

where $\sigma(\cdot)$ is the Sigmoid function,
$E_{n}^{R}\mathrel{\mathop{:}}=G^{R}(X_{n})$ and
$E_{N}^{NonR}\mathrel{\mathop{:}}=G^{NonR}(X_{n})$.

Table 1. Comparison on the BSDS dataset. The running speeds are tested on an RTX 3090 GPU with 200 $\times$ 200 images as input if not specified, while $\dagger$ indicates cited GPU speeds. Our results are highlighted in bold. Method | Pub.’Year | ODS | OIS | Throughput | Params | Pre-training
---|---|---|---|---|---|---
Human | - | - | .803 | .803 | - | - | -
Canny (Canny, 1986) |  | PAMI’86 | .600 | .640 | 31FPS | - | No
Pb (Martin et al., 2004) |  | PAMI’04 | .672 | .695 | - | - | No
gPb-UCM (Arbelaez et al., 2011) | Traditional | PAMI’11 | .726 | .757 | - | - | No
ISCRA (Ren and Shakhnarovich, 2013) | Methods | CVPR’13 | .717 | .752 | - | - | No
SE (Dollár and Zitnick, 2013) |  | ICCV’13 | .743 | .763 | 12.5FPS$\dagger$ | - | No
OEF (Hallman and Fowlkes, 2015) |  | CVPR’15 | .746 | .770 | 2/3FPS$\dagger$ | - | No
DeepEdge (Bertasius et al., 2015a) |  | CVPR’15 | .753 | .772 | 1/1000FPS$\dagger$ | - | ImageNet
DeepContour (Shen et al., 2015) |  | CVPR’15 | .757 | .776 | 1/30FPS$\dagger$ | 27.5MB | ImageNet
HFL(Bertasius et al., 2015b) |  | ICCV’15 | .767 | .788 | 5/6FPS$\dagger$ | - | ImageNet
CEDN (Yang et al., 2016) |  | CVPR’16 | .788 | .804 | 10FPS$\dagger$ | - | ImageNet
COB (Maninis et al., 2017) |  | PAMI’17 | .793 | .820 | - | - | ImageNet
HED (Xie and Tu, 2017) | CNN | IJCV’17 | .788 | .808 | 73FPS | 14.7MB | ImageNet
CED (Wang et al., 2017) | based | CVPR’17 | .794 | .811 | - | 14.9MB | ImageNet
AMH-Net (Xu et al., 2017) | Methods | NeurIPS’17 | .798 | .829 | - | - | ImageNet
LPCB (Deng et al., 2018) |  | ECCV’18 | .808 | .824 | 30FPS$\dagger$ | - | ImageNet
RCF (Liu et al., 2019) |  | PAMI’19 | .806 | .823 | 61FPS | 14.8MB | ImageNet
BDCN (He et al., 2019) |  | CVPR’19 | .820 | .838 | 44FPS | 16.3MB | ImageNet
DSCD (Deng and Liu, 2020) |  | ACMMM’20 | .813 | .836 | - | - | ImageNet
LDC (Deng et al., 2021) |  | ACMMM’21 | .812 | .826 | - | - | ImageNet
FCL (Xuan et al., 2022) |  | NN’22 | .815 | .834 | - | - | ImageNet
EDTER (Pu et al., 2022) | ViT based | CVPR’22 | .832 | .847 | 2.1FPS | 468.84MB | ImageNet
PiDiNet (Su et al., 2021) | CNN | ICCV’21 | .807 | .823 | 93FPS | 710KB | No
PEdger | based | - | .821 | .840 | 92FPS | 734KB | No
PEdger-large | Methods | - | .823 | .841 | 71FPS | 979KB | No

##  4\. Experimental Validation

###  4.1. Implementation Details

We implement our network using PyTorch library on RTX 3090 GPU. In detail,
PEdger is randomly initialized and trained for 30 epochs using AdamW
optimizer. The momentum and weight decay of the optimizer are set to 0.9 and
0.001, respectively. The warm-up strategy is adopted for the training of the
first 4 epochs whose learning rates are gradually increased to 0.001, and then
decayed in a linear way. The recurrent step $T$ is set to 5 for recurrent
models. $\lambda$ is set to 1.1 for BSDS, and 1.3 for NYUD. Given a predicted
edge confidence, a threshold is needed to produce the binary edge map. The
threshold for binarizing the ground truth edge maps is set to 0.2 for both
BSDS500. No threshold is needed for NYUD since the images are singly
annotated. The batch size is set to be 16, and is implemented by gradient
accumulation since the sizes of images are varied in training. During
training, we randomly perturb the brightness, contrast, saturation, and hue of
input images, from 50% to 150% of their original values, and randomly convert
input images to gray-scale images in a probability of 0.2, for data
augmentation. When evaluating, standard non-maximum suppression (NMS) (Dollár
and Zitnick, 2013) is applied to thin detected edges, and both Optimal Dataset
Scale (ODS) and Optimal Image Scale (OIS) F-score are reported. The Sigmoid
activation is also replaced with $e^{x-0.5}/(e^{x-0.5}+e^{-x+0.5})$ in
testing.

###  4.2. Datasets & Competitors

We compare our method with some non-deep learning algorithms, including Canny
(Canny, 1986), Pb (Martin et al., 2004), gPb-UCM (Arbelaez et al., 2011),
ISCRA (Ren and Shakhnarovich, 2013), SE (Dollár and Zitnick, 2013), and OEF
(Hallman and Fowlkes, 2015), and some recent deep learning based approaches,
including DeepEdge (Bertasius et al., 2015a), DeepContour (Shen et al., 2015),
HFL (Bertasius et al., 2015b), CEDN (Yang et al., 2016), HED (Xie and Tu,
2017), COB (Maninis et al., 2017), CED (Wang et al., 2017), AMH-Net (Xu et
al., 2017), LPCB (Deng et al., 2018), RCF (Liu et al., 2019), BDCN (He et al.,
2019), DSCD (Deng and Liu, 2020), PiDiNet (Su et al., 2021), LDC (Deng et al.,
2021), FCL (Xuan et al., 2022), and EDTER (ViT-based) (Pu et al., 2022).

Two datasets employed for evaluation include: 1) The BSDS500 (Arbelaez et al.,
2011) has 200 training, 100 validation and 200 test images. The label is
annotated by 4 to 9 participants. Following previous works, we train our model
on the data consisting of augmented BSDS and VOC Context dataset; and 2) The
NYU Depth (NYUD) dataset (Gupta et al., 2013) contains 1449 RGB and HHA image
pairs, which is split into train (381 images), validation (414 images), and
test sets (654 images). Following previous works, we increase the maximum
tolerance allowed for correct matches of edge predictions to ground truth to
.011, when evaluating.

###  4.3. Ablation Study

We conduct our ablation study on the mixture of augmented BSDS and PASCAL VOC
(Mottaghi et al., 2014) dataset.

#### Effectiveness of integrating across training moments and models.

We report the results of four alternatives: 1) Baseline. Training a model with
non recurrent structure supervised by the original noisy targets $Y_{n}$; 2)
No Information from neither different Moments nor heterogeneous Structures
(NIMS). Generating the targets $\widetilde{Y}_{n}$ neither based on knowledge
from different training moments nor heterogeneous architectures. The knowledge
learned by a single model trained after the last epoch, are used to produce
$\widetilde{Y}_{n}$; 3) Ensemble only Across Different Training Moments
(EADM). Generating the targets $\widetilde{Y}_{n}$ only based on the knowledge
across different training moments. In this setting, only the momentum and
back-propagation network with non recurrent structure, is maintained in
training; 4) Ensemble only Across Different Structures (EADS). Generating the
targets $\widetilde{Y}_{n}$ only based on the knowledge across different
architectures without maintaining momentum networks, where the knowledge
learned from the last epoch, are fused to obtain $M_{n}$. 5) Mutual Learning
of Heterogeneous Structures (MLHS). Similar to (Zhang et al., 2018), the
recurrent and non recurrent network will mutually supervise each other. The
recurrent model is supervised by the non recurrent model and $Y_{n}$, while
the non recurrent model is supervised by the recurrent one and $Y_{n}$. In
this alternative, the knowledge across training moments is incorporated.

Table 2. Ablation study on the effectiveness of the proposed collaborative learning strategy on BSDS dataset in terms of ODS-F. All the results are computed with single-scale inputs.  Baseline | NIMS | EADS | EADM
---|---|---|---
.809 | .811 | .814 | .818
MLHS | Average | Synchron. | Ours
.818 | .817 | .819 | .821
Table 3. Ablation study on the effectiveness of the two network architectures involved in training, on BSDS dataset in terms of ODS-F. All results are computed with single-scale inputs, and our results are highlighted in bold. $\dagger$ denotes the non recurrent structure whose number of parameters decreases as the resolutions of features decrease. Network 1 | Network 2
---|---
Architecture | ODS | Architecture | ODS
Recurrent | .815 | Recurrent | .816
Non-Recurrent | .818 | Non-Recurrent | .819
Recurrent | .816 | Non-Recurrent$\dagger$ | .818
Recurrent | .817 | Non-Recurrent | .821

As can be seen from Table 2, the accuracy of Baseline is the worst one among
all the alternatives, due to the mislead noises/outliers in manually annotated
labels. Though slightly better than the baseline when using
$\widetilde{Y}_{n}$ instead of noisy $Y_{n}$, NIMS still lags behind other
alternatives. In comparison, using the targets $\widetilde{Y}_{n}$ produced by
either EADM or EADS can evidently improve the performance, proving the
necessity of either training moments or heterogeneous architectures, in
solving the problem of noisy annotations. When combining the knowledge from
both of these two aspects (_Ours_), the best result is achieved, benefiting
from the robust information that resists the disturbance of noises. It is
worth mentioning that, MLHM is inferior to our method, which shows the
effectiveness of fusing the predictions of recurrent and non recurrent
networks using Eq. LABEL:fusion.

Table 4. Comparison on the NYUD dataset. All results are computed with single-scale RGB inputs.  Methods | Pub.’Year | ODS | OIS | Throughput
---|---|---|---|---
gPb-UCM (Arbelaez et al., 2011) | PAMI’10 | .632 | .661 | 1/360FPS$\dagger$
SE (Dollár and Zitnick, 2013) | ICCV’13 | .695 | .708 | 5FPS$\dagger$
OEF (Hallman and Fowlkes, 2015) | CVPR’15 | .651 | .667 | -
HED (Xie and Tu, 2017) | IJCV’17 | .720 | .734 | 67FPS
LPCB (Deng et al., 2018) | ECCV’18 | .739 | .754 | -
RCF (Liu et al., 2019) | PAMI’19 | .743 | .757 | 58FPS
AMH-Net (Xu et al., 2017) | NeurIPS’17 | .744 | .758 | -
BDCN (He et al., 2019) | CVPR’19 | .748 | .763 | 35FPS
EDTER (Pu et al., 2022) | CVPR’22 | .774 | .789 | 2.2FPS
PiDiNet (Su et al., 2021) | ICCV’21 | .733 | .747 | 62FPS$\dagger$
PEdger | - | .742 | .757 | 92FPS

#### Two-Training stage.

To compare the performance of two-stage training, we maintain a momentum
network with recurrent structure and its corresponding back-propagation
network in the first training stage, where $\widetilde{Y}_{n}$ is produced
solely based on the predictions of the momentum recurrent network. In the
second stage, the momentum and back-propagation network with non recurrent
structure is updated, while the networks with recurrent structure are frozen.
The target $\widetilde{Y}_{n}$ for supervising the training of second stage,
is produced according to Eq LABEL:fusion and Eq. 4. The only difference
between the second training stage and our original collaborative learning, is
whether the networks with recurrent structure are updated. The quantitative
result of two-stage learning (_Synchron._ in Table 2) is 0.819, being slightly
worse than ours. The reason may be that the performance of recurrent networks
is fixed during the second stage, which limits the improvement of non
recurrent networks.

#### Effectiveness of network architecture design.

We conduct experiments to see how the choice of two network architectures in
training affect performance. As can be seen from the first two rows in Table
3, involving the same network architectures in training, e.g., all networks
are with recurrent or non recurrent structure, leads to the sub-optimal
performance. It shows that using two discrepant network structures (recurrent
& non recurrent in this work) is critical to capture the diversified knowledge
for improving the robustness and accuracy.

In addition, as we have mentioned in Section 3.1, the amount of parameters for
the non recurrent network, should gradually increases as the features size
decreases. To verify the effectiveness of this design, we allocate more (less)
parameters for earlier (later) stages with larger (smaller) feature sizes for
comparison. Besides slower than our original design, it can be seen that the
accuracy of both recurrent and non recurrent network is inferior to ours
(recurrent: 0.817 vs. 0.816, non recurrent: 0.821 vs. 0.818) when the
parameter amount decreases for later stages as denoted by $\dagger$. The
reason lies in that later stages for learning semantic higher-level
information, require more parameters than eariler stages for lower-level
details.

#### Effectiveness of uncertainty-aware fusion.

In this section, we conduct the experiment to validate the effectiveness of
our proposed uncertainty-aware combination strategy, as formulated in Eq.
LABEL:fusion. To compare between with and without considering the
uncertainties, we alternatively average the predictions of recurrent and non
recurrent networks using the same weight, that is to say, $M_{n}$ is produced
through $(M^{R}_{n}+M^{NonR}_{n})/2$. The quantitative result of simply
averaging the predictions is 0.817 (_Average_ in Table 2), which is lower than
our method in a large margin. It verifies that the larger the confidence of
the model in its prediction, the more contribution it should make when
emsembling. Otherwise, the low-confident predictions, which are highly likely
to be wrong, will cause harmful impacts in fusion.

Figure 6. Precision-Recall curves of our method and other competitors on the
BSDS dataset.

###  4.4. Comparison with State-of-the-arts

It can be seen from Fig. 6 and Tab. 1 that our PEdger can achieve high
accuracy on BSDS, i.e., with ODS-F of 0.821 and OIS-F of 0.840. Though
inferior to EDTER, our PEdger runs about 49$\times$ faster than it with merely
734KB parameters. Notice that the two-stage training is required for the
global and local transformers of EDTER, making its training procedure
inefficient. For the CNN based approaches, our PEdger outperforms the recently
proposed BDCN while still achieving nearly 100 FPS. Compared with the only
work for efficient edge detection, i.e., PiDiNet, we have a significant
advantage in terms of ODS/OIS (ODS: 0.821 vs. 0.807, OIS: 0.840 vs. 0.815). We
also show the visual results in Fig. 5. It can be seen that our results are
much more clearer than those of other competitors, with less superfluous
aesthetic predictions in the non edge regions.

The results on NYUD are shown in Tab. 4. We can see that our method is able to
achieve highly comparable results among the state-of-the-art methods while
being efficient. Notice that we adopt the identical network architectures and
hyper-parameters for all datasets if not otherwise specified.

##  5\. Conclusion

We proposed a novel collaborative learning framework, namely PEdger, to
simultaneously address the 1) dependence on a large amount of pre-trained
parameters, and 2) the performance degradation caused by noisy labels. It is
able to integrate the knowledge from different training moments in the whole
training procedure, and heterogeneous network architectures. Moreover, we
customized an efficient and compact network, contributing to fast and accurate
edge detection. Our PEdger can be trained from scratch and surpass human-level
performance, breaking the dependence of backbone pre-trained on large scale
datasets. Experimental results have revealed the advances of our proposed
PEdger over the state-of-the-arts, making it attractive for real-time
applications.

## Acknowledgements

This work was supported by National Natural Science Foundation of China under
Grant 62072327.

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[a]João R. T. de Mello Neto

# The Giant Radio Array for Neutrino Detection

###### Abstract

Ultra-high-energy cosmic neutrinos (UHE), with energies above 100 PeV, are
unparalleled probes of the most energetic astrophysical sources and weak
interactions at energies beyond the reach of accelerators. GRAND is an
envisioned observatory of UHE particles - neutrinos, cosmic rays, and gamma
rays - consisting of 200,000 radio antennas deployed in sub-arrays at
different locations worldwide. GRAND aims to detect the radio emission from
air showers induced by UHE particle interactions in the atmosphere and
underground. For neutrinos, it aims to reach a flux sensitivity of $\sim
10^{-10}$ GeV cm-2 s-1 sr-1, with a sub-degree angular resolution, which would
allow it to test the smallest predicted diffuse fluxes of UHE neutrinos and to
discover point sources. The GRAND Collaboration operates three prototype
detector arrays simultaneously: GRAND@Nançay in France, GRANDProto300 in
China, and GRAND@Auger in Argentina. The primary purpose of GRAND@Nançay is to
serve as a testbench for hardware and triggering systems. On the other hand,
GRANDProto300 and GRAND@Auger are exploratory projects that pave the way for
future stages of GRAND. GRANDProto300 is being built to demonstrate autonomous
radio-detection of inclined air showers and study cosmic rays near the
proposed transition between galactic and extragalactic sources. All three
arrays are in the commissioning stages. It is expected that by 2028, the
detector units of the final design could be produced and deployed, marking the
establishment of two GRAND10k arrays in the Northern and Southern hemispheres.
We will survey preliminary designs, simulation results, construction plans,
and the extensive research program made possible by GRAND.

##  1 Introduction

Ultra-high-energy cosmic rays (UHECRs) are atomic nuclei with energies
exceeding approximately $10^{18}$ electron volts (eV), and their origins
remain a puzzle [1]. The mechanisms by which they attain such extreme energies
are still poorly understood. Throughout their journey from the point of
acceleration to their arrival at Earth, cosmic rays interact with matter and
radiation fields along their trajectory, producing numerous secondary
particles, including neutrinos and photons. This production establishes a
significant multi-messenger connection. UHECRs are deflected from their
original paths by intervening magnetic fields and are attenuated by cosmic
photon backgrounds during their propagation. UHE photons are similarly
attenuated. In contrast, neutrinos travel to Earth largely unaffected by
intervening obstacles However, neutrinos, being largely unaffected by
obstacles, making them a valuable messenger for exploring the vast reaches of
the high-energy, large-redshift non-thermal Universe.

The Giant Radio Array for Neutrino Detection (GRAND) [2] is a proposed large-
scale observatory specifically designed to unravel and investigate the sources
of UHECRs. One of its primary objectives is to discover and study UHE
neutrinos. GRAND will detect the radio signals emitted when UHE cosmic rays,
gamma rays, and neutrinos produce extensive air showers (EAS) in the Earth’s
atmosphere. Its configuration also enables comprehensive fundamental particle
physics, cosmology, and radioastronomy studies. GRAND will also play a
significant role in detecting neutrino emissions from transient astrophysical
sources [3]. Below, we will delve into the concept of GRAND, its physics
topics, simulated performance, the ongoing development of prototyping arrays,
and the proposed phased implementation.

##  2 The GRAND project

###  2.1 Radio-detection of UHE neutrinos

When a cosmic particle interacts with the Earth’s atmosphere, it initiates an
EAS. This cascade of particles, in turn, produces electromagnetic radiation
primarily due to the deflection of charged particles within the shower by the
Earth’s magnetic field. This phenomenon, known as geomagnetic emission,
exhibits coherence in the tens of MHz frequency range. Consequently, it
generates short-duration ($\sim 100$ ns), transient electromagnetic pulses
with amplitudes significant enough to enable the detection of the EAS, given
that the energy of the shower exceeds approximately $10^{16.5}$ eV [4, 5].
Radio-detection of EAS is a mature technique that benefits from the valuable
experience gained through numerous previous experiments, such as AERA, LOFAR,
CODALEMA, Tunka-Rex, and TREND, that presented the proof of principle that an
antenna array can detect EAS in stand-alone mode [6].

Cosmic neutrinos are less likely to be detected through interactions with the
atmosphere due to their extremely small interaction cross-section with matter.
Nevertheless, $\nu_{\tau}$ neutrinos can produce $\tau$ leptons beneath the
Earth’s surface via charged-current interactions with rock. Thanks to their
considerable range in rock (50 m per PeV of energy before decaying) and short
lifetime (0.29 ps), $\tau$ leptons can emerge into the atmosphere and decay,
initiating a detectable EAS [7]. Only Earth-skimming trajectories allow for
such a scenario, since the Earth acts as a barrier to neutrinos with energies
surpassing $10^{17}$ eV.

This characteristic proves to be advantageous for radio-detection purposes.
Due to relativistic effects, the radio emission becomes highly focused in a
forward-directed cone, with its opening defined by the Cherenkov angle
$\theta_{\textrm{C}}\leq 1^{\circ}$. For vertically incoming showers, this
results in a radio footprint on the ground that spans only a few hundred
meters in diameter. Consequently, a dense array of antennas is required to
sample the signal in this scenario adequately. However, for an air shower with
highly inclined trajectory, the increased distance between the antennas and
the emission zone, coupled with the projection effect of the signal on the
ground, leads to a few kilometers-long footprint [4, 5]. By targeting air
showers with such inclined trajectories, it becomes feasible to detect them
using a sparser and larger array, typically employing one antenna per square
kilometer. This capability serves as a crucial feature of the GRAND detector.

GRAND also incorporates the strategy of selecting mountainous regions with
advantageous topographies as deployment sites. An optimal topography involves
two parallel mountain ranges spaced several tens of kilometers apart. One
range serves as the target for neutrino interactions, while the other
functions as a screen onto which the subsequent radio signal is projected.
Simulations reveal that such configurations lead to an enhanced detection
efficiency, approximately four times greater than that of a flat site [2].

###  2.2 Simulated performance

The end-to-end simulation pipeline used to determine the sensitivity of GRAND
incorporates the intricate topography of the deployment site and the extensive
instrumented area. Given the complexity of the task at hand, we ensure the
inclusion of all relevant physics while striving to optimize computational
performance. We individually validate each simulation stage by comparing it
with existing codes. The complete simulation chain is described in detail in
[2].

The approach described in [8] has been applied in a preliminary analysis to
reconstruct the depth of maximum development of cosmic ray-induced showers
($X_{\textrm{max}}$) using a GRAND-like array. This method achieves
$X_{\textrm{max}}$ resolution smaller than 40 g cm-2, assuming knowledge of
the shower energy and core position [9]. Another study, based on a spherical
fit of the wavefront, even though it does not measure $X_{\textrm{max}}$
directly, uses a figure of merit to estimate that the resolution will be
slightly worse than 17 g cm-2 [10]. See figure 1, upper right.

Innovative reconstruction techniques performing fits to the strength of the
radio signal as a function of the angle from the shower axis (angular
distribution function, ADF) have showcased the potential to achieve angular
resolutions of approximately $0.1^{\circ}$ in determining the arrival
direction of particles, as shown in figure 1, lower right [11]. Even though
this result was developed and tested using simulated data only, this level of
precision opens up the possibility of conducting neutrino and gamma-ray
astronomy with the GRAND observatory.

For neutrino searches, the shower energy of the EAS only provides a lower
bound on the initial neutrino energy. The initial findings regarding energy
resolution are promising. Using a preliminary reconstruction based on deep
learning methods, an energy resolution of 15% for incoming protons and iron
nuclei was achieved without implementing an antenna response and in an
idealized scenario for radio-detection [12]. Another preliminary global
reconstruction method that uses the angular distribution function also yields
a 20% energy resolution [11]. Other machine learning and analytical methods
are under development within the Collaboration. Consequently, a final energy
resolution of 10% can likely be attained.

Figure 1: Left: Predicted cosmogenic neutrino flux, compared to experimental
upper limits and sensitivities. Gray-shaded regions are generated by fitting
UHECR simulations to Auger spectral and mass-composition data [2]. Upper
right: Mean value of the radio grammage ${\\{X}_{\textrm{e}}$ distribution per
energy slices for proton and iron. ${{X}_{\textrm{e}}}$ can be considered in
this model as the (static) point-like source of the radio emission and it is a
free parameter of the wavefront model. The error bars show the statistical
fluctuations, taken equal to $\sigma_{{X}_{\textrm{e}}}/\sqrt{N}$ where
$\sigma_{{X}_{\textrm{e}}}$ is the standard deviation of the
${X}_{\textrm{e}}$ values for each simulation and $N$ is the number of
simulated showers per energy slice. The shaded areas correspond to the
additional uncertainties associated to error on the direction of origin of the
showers for $1-\sigma$ values of $0.1^{\circ}$ and $0.5^{\circ}$, where
$\sigma$ is the standard deviation of the gaussian distribution around the
true direction to account for reconstruction errors. [10]. Lower right:
Distributions of the angular distances $\Psi$ for the GRANDProto300-like
layout (see section 4) [11].

##  3 GRAND science case

###  3.1 Ultra-high-energy messengers

The interaction between UHECRs and the cosmic microwave background (CMB) and
extragalactic background light (EBL) generates cosmogenic UHE neutrinos (and
photons), with energies in excess of $10^{17}$ eV. Despite our limited
understanding of the sources of UHECRs, their existence is assured. Even with
pessimistic assumptions, GRAND has the potential to discover UHE neutrinos -
even if their flux is low. The sensitivity of GRAND to UHE neutrino reaches
$4\times 10^{-10}$ GeV cm-2 s-1 sr-1 as shown in figure 1, left. Cosmogenic
neutrino studies indicate that the outcomes of measurements conducted by GRAND
will have significant implications for constraining the sources of UHECRs [13,
14]. Additionally, they will provide constraints on the proton fraction at
ultra-high energies [15]. The remarkable sensitivity of GRAND, coupled with
its sub-degree angular resolution, will unlock the potential for conducting
UHE neutrino astronomy, enabling the identification of point sources.

Figure 2, left, shows the sensitivity limit of GRAND for point sources. The
sources of UHECRs and UHE neutrinos could be distinct. Therefore, even if a
heavy composition is observed in UHECRs, it does not necessarily imply a
suppression in the flux of neutrinos at EeV energies [16].

Similarly to UHE neutrinos, the cosmogenic flux of UHE photons is also
guaranteed. They may be emitted by astrophysical sources, depending on their
opacity. However, distant objects cannot be directly observed as they are
attenuated by the CMB and EBL and reprocessed to lower energies. The most
stringent upper limits on UHE photons can be improved by two orders of
magnitude after three years of data-taking by GRAND. See figure 2, right [2].

###  3.2 Multimessenger astronomy

With its excellent angular resolution and extensive sky coverage, GRAND has
the potential to detect UHE neutrinos associated with transient events in
conjunction with electromagnetic emissions. Using a single array site
containing 10,000 antennas (GRAND10k), the instantaneous field of view is
about 5% of the sky; the daily field of view, however, reaches about 80%.
Using multiple such sites deployed at different locations, as is intended for
final configuration of GRAND, the field of view is even larger.

GRAND will be a crucial triggering partner for multimessenger observations,
enabling the precise reconstruction of the arrival direction of neutrino-
induced air showers near the horizon with sub-degree accuracy and minimal
latency. This capability allows GRAND to issue alerts to other experiments or
coordinated systems promptly. Additionally, as a follow-up partner, GRAND can
swiftly validate alerts generated by other experiments as well as
gravitational-wave detectors. If the target directions fall within the
instantaneous field of view of GRAND, it becomes possible to establish
constraints on UHE neutrino emissions originating from a transient.

Figure 2:  Left: The sensitivity limits of GRAND for point sources [2]. It is
important to note that these GRAND limits assume the deployment of 200k
antennas at a single location. Right: The projected upper limits of GRAND on
the sensitivity to UHE photons after three years of operation are presented.
For comparison, we also include the current upper limits from Auger and TA and
the projected capabilities of Auger by 2025. Additionally, we overlay the
predicted cosmogenic UHE photon flux resulting from pure-proton and pure-iron
UHECRs, as estimated in [17].

##  4 Experimental setups for prototyping

A dedicated design was formulated for the antenna used in the GRAND project,
known as the HorizonAntenna [2]. Its design is optimized to select very
inclined EAS. This antenna features three perpendicular arms, enabling
comprehensive polarization measurements of the signal. Positioned at a height
of 3.5 m above the ground and optimized for the frequency range of 50–200 MHz,
it exhibits optimal sensitivity to near-horizontal signals. Currently, the
GRAND Collaboration has three ongoing prototype arrays taking data.

Thirteen detection units (GRANDProto13), consisting of antennas and associated
electronics, were deployed in the Gobi desert, Gansu Province, China, in
February 2023. Data is being collected and analyzed from this initial setup,
which serves as the foundation for the GRANDProto300 array [18]. One transient
event is shown in figure 3, right. We are validating the detector unit design
with the ones already deployed, and once this is done, we will deploy the
remaining 70 units already built. This array should already be enough to
detect cosmic rays. In the near future, we will build and deploy the remaining
200 units to form GRANDProto300.

The addition of particle detectors to the prototype array is still under
consideration. By utilizing the GRANDProto300 array, we will have the
capability to investigate cosmic rays within the energy range of $10^{16.5}$
to $10^{18}$ eV, which encompasses the transitional region between between
Galactic and extragalactic UHECRs. The array will also allow for the detection
of radio transients. Moreover, if particle detectors are used, they may help
to address the discrepancies observed between simulations and measurements of
muons [19].

Four detection units were deployed at the Nançay Radio Observatory in France
in the autumn of 2022 [20]. The primary objective of the GRAND@Nançay test
array is to conduct hardware and trigger testing. A preliminary spectrum
obtained on-site is shown in figure 3, upper left.

Additionally, ten detection units are being deployed at the Pierre Auger
Observatory site from March to August 2023. The main goal is to perform cross-
calibration and validation of reconstruction using coincident events with
Auger. By taking an average spectrum we could verify the presence of FM radio
and TV stations in Malargue, as shown in the preliminary spectrum of figure 3,
lower left.

The GRAND Collaboration will complete the deployment and continue the
operation of prototype arrays: GRAND@Nançay, GRANDProto300, and GRAND@Auger.
Meanwhile, it will also focus on characterizing the radio background and
exploring the features of autonomous detection of inclined extensive air
showers.

Furthermore the GRAND Collaboration is committed to minimizing its carbon
footprint throughout its operations. By implementing sustainable practices and
using energy-efficient technologies, we aim to reduce our environmental
impact. Efforts are made to optimize the use of resources, reduce waste
generation, and promote recycling and reuse. The Collaboration strives to
minimize travel-related emissions by employing remote collaboration tools and
favoring virtual meetings whenever possible [21].

##  5 Future GRAND timeline

We expect that, by 2028, the design of the detector units will be finalized,
leading to the construction of two GRAND10k arrays (with 10000 units each).
Candidates for the bases of GRAND-North and GRAND-South are China and
Argentina, respectively, which will assure a full sky coverage. Subsequently,
in the 2030s, the replication of GRAND10k is expected to commence, resulting
in twenty subarrays comprising the entire GRAND project. By scaling up
production to an industrial level, the front-end electronics can be
transitioned to a fully integrated ASIC design, leading to cost reduction,
improved reliability, and greater reproducibility of individual units.
Furthermore, the design of each subarray can be customized based on location,
topography, or specific scientific objectives.

Figure 3: Upper left: Mean power spectrum (PSD) measured at GRAND@Nançay. A
filter is added in the RF acquisition chain of the setup to cut out FM
signals, very strong at that location. Lower left: Average FFT of one detector
unit at GRAND@Auger. Right: Transient event measured in GRANDProto13. Channel
1=X, 2=Y, 3=Z and channel 0 is free floating.

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