# Noise Perturbation for Saliency Prediction with
Psychophysical Synthetic Images

###### Abstract

Convolutional neural networks (CNNs) have achieved great success in natural
image saliency prediction. The primary goal of this study is to investigate
the performance of saliency prediction in CNN and classic models with
psychophysical synthetic images under noise perturbation. Is it still as
decent as natural images in terms of performance? In the meantime, it can be
used to investigate the relationship between CNNs and human vision, mainly
low-level vision functions. On the other hand, are CNNs exact replicas of
human visual function? This study used CNNs, Fourier, and spectral models
inspired by low-level vision systems to investigate saliency prediction on
psychophysical synthetic images rather than natural images. According to our
findings, saliency prediction models inspired by Fourier and spectral theory
outperformed current pre-trained deep neural networks on psychophysical images
with noise perturbation. However, psychophysical models were more unstable in
noise than pre-trained deep neural networks. Meanwhile, we suggested that
investigating CNNs with psychophysical methods could benefit visual
neuroscience and artificial neural network studies.

Index Terms—  Convolutional Neural Networks, Noise Perturbation, Saliency
Prediction, Psychophysical Synthetic Images

##  1 Introduction

The attention function was first introduced into artificial neural networks,
and it has achieved great success in computer vision tasks such as image
classification [1], few-show learning [2], and so on. There is no denying
CNN’s victory in both academia and industry. Moreover, Vision Transformer
(ViT) attains excellent results compared to state-of-the-art convolutional
networks while requiring substantially fewer computational resources to train
[1]. The vital message transmitted from all of these is Attention is All you
Need. Although there are some arguments about attention in artificial neural
networks, it’s beyond the scope of this paper.

The natural world includes heavy redundant information, and the human visual
system can’t process all this information due to the limited information
bottleneck of the vision system. However, related research about attention
using physiology methods has a long history, with a specific focus on the
vision system [3]. According to [4] three-stage vision information processing
models, the input information is encoded in the retina at 20MB/second [5] and
transmitted into the selection phase. Here is where attention happens, and it
can remove irrelevant messages. After the selection step, information is
compressed into 1MB/second and then conveyed to the primary visual cortex
(V1), leaving only 40 bits/second of information in V1 [6, 4]. A more critical
and classical principle that connects to the visual attention mechanism is the
Efficient Coding Principle [7]. After that, a larger body of psychophysical
experiments investigated visual attention. Meanwhile, different specifically
designed computational attention models were proposed based on physiology
experiments, such as the earliest Itti models for visual saliency prediction
[8].

In the age of deep learning, natural image saliency prediction has done very
well [9], as shown by ML_Net [10], and DeepGaze II [11]. However, there are
fewer studies that examine saliency prediction on psychophysical images under
noise perturbation using CNNs and custom-designed psychophysical models and
the relationship between CNNs and psychophysical saliency prediction models.
This study is limited to predicting saliency for psychophysically oriented
images with consideration of noise rather than natural images. Meanwhile,
investigating artificial deep networks and human vision systems, specifically
low-level vision functions, is central to this study. The connections between
CNNs and low-level vision functions in humans are not fully understood at the
moment. We find it puzzling that CNNs, on the other hand, rely entirely on
low-level human visual processes. Additionally, we want to comprehend and test
CNNs using psychophysical approaches and then improve their general and
transferable learning abilities across various tasks.

##  2 Related Work

In the deep learning era, general saliency prediction on natural images has
shown tremendous success [9]. Predicting visual saliency has become a hot
topic in machine vision and visual neuroscience. One way of understanding
black-box CNNs is visualization, and attention maps become a way of
understanding the features of information processing in CNNs. The classical
pre-trained with different depths of general CNNs were employed to investigate
their attention maps, such as Alexnet [12], GoogLeNet [13], Inceptionv3 [14],
Densenet201 [15], respectively. There are several top-ranking CNNs dedicated
to natural image saliency prediction, for example, DeepGazeII, based on the
MIT/Tuebingen Saliency Benchmark evaluation111https://saliency.tuebingen.ai/.
Furthermore, three of the highest-ranked psychophysical saliency prediction
models were introduced in this study, and their design is entirely based on
the functions of the human low-level vision system [16]. The first one is the
HTF model, which is a bottom-up paradigm for visual saliency prediction based
on frequency domain [17]. The second model is Incremental Coding Length (ICL),
which is designed based on information-theoretic principles [18]. The model
assumes that attention regions in images may elicit entropy gain in the
perception state and then attract high energy. The last one is the spectral-
oriented saliency prediction model, DCTS, which predicts image saliency based
on image signature [19]. The following section will examine saliency
prediction using all of the aforementioned models on psychophysical images.

##  3 Methods

###  3.1 Dataset: Psychophysical images

A SID4VAM dataset was used in this study, composed of 230 synthetic images
with known salient regions (see Fig. 1). Images with 15 distinct low-level
features (e.g., orientation, brightness, color, and size) were generated using
a target-distractor pop-out type of synthetic pattern. They have used free-
viewing and visual search task instructions and 7 feature contrasts for each
feature category [20, 21].

Fig. 1: Selected part of images from SID4VAM dataset. The visual conspicuity
targets are marked by a red rectangle in selected synthetic images.

###  3.2 Pre-trained CNNs and psychophysical models on psychophysical images

As we mentioned before, here we investigated pre-trained neural networks, such
as AlexNet, GoogLeNet, Inceptionv3, Densenet201, and DeepGazeII, to visualize
saliency maps with psychophysical synthetic images. Meanwhile, we also used
some classical Fourier and spectral saliency prediction models, which were
inspired by human low-level vision functions such as HFT, ICL, and DCTS. In
pre-trained neural networks, saliency maps are estimated with gradient-
weighted class activation mapping (Grad-CAM), which is a simple technique for
understanding which parts of an image are most important for a deep network’s
recognition [22]. Use Grad-CAM to gain a high-level understanding of what
image features a network uses to make a particular classification or perform
other tasks. The Grad-CAM map for a convolutional layer with $k$ feature maps
(channels), $A_{i,j}^{k}$, could be estimated with
$S=\operatorname{ReLU}\left(\sum_{k}\frac{1}{N}\sum_{i}\sum_{j}\frac{\partial
y^{c}}{\partial A_{i,j}^{k}}A^{k}\right)$, where $y^{c}$ is output,
representing the score for class $c$, $i,j$ indexes the pixels, $N$ is the
total number of pixels in the feature map. The rectified linear unit (ReLU)
activation ensures you get only the features that have a positive contribution
to the class of interest. The output is therefore a heatmap for the specified
class. The HFT, ICL, and DCTS were designed based on Fourier-spectral and
information-theoretic principles, respectively. The fundamental information
about models refer to Table. 1.

Network | Depth | Parameters (Millions) | Image Input Size
---|---|---|---
Alexnet | 8 | 61.0 | 227-by-227
Googlenet | 22 | 7.0 | 224-by-224
Inceptionv3 | 48 | 23.9 | 299-by-299
Densenet201 | 201 | 20.0 | 224-by-224
DeepGazeII | 19+ | - | -
HFT | - | - | -
ICL | - | - | -
DCTS | - | - | -

Table 1: Models information. The pretrained networks and psychophysical models
and some of their properties (e.g., depth, parameters and image input size).

###  3.3 Pre-trained CNNs and psychophysical models on degraded psychophysical
images

Here, we want to investigate the stability of various saliency prediction
models under noise interference (See Fig. 2). We added salt-and-pepper noise
with a density of 0.3 to each psychophysical image. Following that, we fed
degraded images into various models to see how noise affects the accuracy of
CNNs and psychophysical models in saliency prediction.

Fig. 2: Selected part of degraded psychophysical images The top row shows
degraded psychophysical stimuli images, and the second row shows saliency
prediction maps from humans.

###  3.4 Evaluation metrics

The accuracy (ACC) was adopted as statistical measures of the saliency
prediction performance. The ACC is defined as
$\mathrm{ACC}=\frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{P}+\mathrm{N}}$, where
TP, TN, P, and N refer to true positive, true negative, condition positive,
and condition negative, respectively. In this study, TP is the number of
positive samples correctly predicted as positive, TN the number of correctly
predicted negative samples, P the total number of positive samples, and N the
total number of negative samples.

##  4 Results

The visualization of saliency prediction with the above models is presented in
Fig. 3. Here we plotted a saliency prediction map with different models. We
observed that DCTS and HFT have prediction saliency that closely matches human
levels (baseline, second row). However, when compared to psychophysical
models, pre-trained CNNs perform worse. To statistically analyze the
performance of saliency prediction with different models, we measured the
receiver operator characteristic (ROC) curve, referring to Fig. 5. The
statistics result also shows psychophysical models have better prediction
accuracy than pre-trained CNNs. The reasons will be explained in the next
section. Further, we investigated the saliency prediction’s interference with
various models while operating in a noisy environment. In Fig. 4, we found
that for pre-trained (Alexnet, Googlenet, Inceptionv3, and Densenet201), and
even when using neural networks that were purpose-built (DeepGazeII), the
addition of noise had little effect. However, psychophysical models (HFT, ICL,
and DCTS) have a large effect by adding noise. The above result makes sense
because pre-trained neural networks already have fixed parameters and weights
for natural images. Still, psychophysical models need to be fed degraded
images without learning to produce saliency maps. However, we can’t evaluate
degraded images’ saliency performance because we don’t have human information
on degraded images. Therefore, we only report the clean image saliency
prediction metric in Fig. 5. Therefore, future work could collect human
saliency prediction data on the degraded image, and it would be fascinating.

Fig. 3: Predicting saliency using pre-trained convolutional neural networks
and biologically inspired saliency prediction models on selected images. The
top row shows psychophysical stimuli images, followed by the second row, which
shows the saliency prediction map from humans. The remaining rows show
saliency prediction maps from different models.

Fig. 4: Saliency predictions were made on selected degraded images using pre-
trained convolutional neural networks and psychophysical models. The top four
rows show degraded saliency predictions from pre-trained CNNs. The remaining
rows show saliency prediction maps from psychophysical models. From top to
bottom row, Alexnet, Googlenet, Inceptionv3, Densenet201, HFT, ICL, and DCTS
saliency prediction results are represented.

Fig. 5: AUC - ROC curve. The performance of saliency prediction with various
depths of pre-trained convolution neural networks and biologically inspired
saliency prediction models.

##  5 Conclusion and Future work

This study investigated saliency prediction accuracy using CNN, Fourier, and
spectral models inspired by low-level vision systems. On psychophysical
synthetic images, the saliency prediction models inspired by low-level vision
functions of humans outperformed pre-trained deep neural networks. As a
result, we discovered that CNNs do not accurately replicate the operation of
the human eye’s low-level vision system. However, we should be cautious of
deep CNNs that have been trained on natural images rather than psychophysical
synthetic images, as this will result in the current situation. Furthermore,
we found psychophysical models are more easily affected by noise compared to
pre-trained neural networks.

As mentioned above, some difficulties arise when psychophysical images are
applied to CNNs. On the one hand, there are not enough psychophysical images
used to train deep CNNs. On the other hand, CNN’s architecture models high-
level human visual systems, but do they model low-level visual functions, such
as color, orientation, brightness reduction, and adaptation? Finally, saliency
prediction on degradation distract tasks could be expanded with more diverse
degradation on both clean natural images and psychophysical synthetic images
to test saliency prediction stability. As previously stated, we must also
collect human saliency prediction data on degraded images as a baseline to
evaluate the model’s performance, and it would also be an exciting direction
to take in the future.

Last but not least, we should consider psychophysical methods when examining
the properties of artificial neural networks. Numerous psychophysical studies
on visual attention mechanisms have been conducted by visual neuroscientists,
and they have significantly increased our knowledge in those fields. It can
improve the generality and stability of artificial neural networks, but there
is insufficient research in this area. Nonetheless, it will be more
interesting to incorporate additional psychophysical studies into the
artificial neural network because they will deepen our understanding of the
human visual system and artificial neural network mechanisms.

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yields

$\overline{f}\left(\mathcal{B},\overline{\mathcal{B}}\right)>\begin{cases}\alpha_{s,n}^{c}-\frac{1}{R_{n}}-\frac{3}{R_{n+1}}-\frac{1}{R_{n}^{c}}-\frac{2}{\sqrt{l_{n-1}}}-\frac{3}{\sqrt{l_{n}}},&\text{if
}[w]_{s}\neq[\overline{w}]_{s},\\\
\beta_{n}^{c}-\frac{1}{R_{n}}-\frac{3}{R_{n+1}}-\frac{1}{R_{n}^{c}}-\frac{2}{\sqrt{l_{n-1}}}-\frac{3}{\sqrt{l_{n}}},&\text{if
}w\neq\overline{w}.\end{cases}$

We use the conditions on the sequence $(l_{n})_{n\in\mathbb{N}}$ from (10.2)
to estimate

$\frac{1}{R_{n}^{c}}+\frac{2}{\sqrt{l_{n-1}}}+\frac{3}{\sqrt{l_{n}}}\leq\frac{1}{\sqrt{l_{n-2}}}+\frac{3}{\sqrt{l_{n-1}}}\leq\frac{2}{\sqrt{l_{n-2}}}\leq\frac{1}{R_{n}},$

which implies (10.15). Similarly, we obtain (10.16). ∎

This accomplishes the induction step. In particular, we note with the aid of
(10.9) that

$\beta_{n+1}^{c}>\frac{\beta_{n}^{c}}{4p_{n+1}}-\frac{3}{R_{n+1}}\geq\frac{7\beta_{n}^{c}}{40p_{n+1}}\geq\frac{7}{R_{n+1}},$

that is, $R_{n+1}>\frac{7}{\beta_{n+1}^{c}}$ and the induction assumption
(10.6) for the next step is satisfied. Since we have

$\alpha_{s,M(s)}^{c}=\beta_{M(s)-1}^{c}-\frac{2}{R_{M(s)-1}}-\frac{4}{R_{M(s)}}\geq\beta_{M(s)-1}^{c}-\frac{2}{7}\beta_{M(s)-1}^{c}-\frac{\beta_{M(s)-1}^{c}}{10p_{M(s)}}>\frac{\beta_{M(s)-1}^{c}}{2}$

by (10.6) and (10.9), we also obtain

$\alpha_{s,n}^{c}\geq\alpha_{s,M(s)}^{c}-\frac{2}{R_{M(s)}}-\sum_{j>M(s)}\frac{6}{R_{j}}\geq\alpha_{s,M(s)}^{c}-\frac{\beta_{M(s)-1}^{c}}{20p_{M(s)}}-\frac{6\beta_{M(s)-1}^{c}}{40}>\frac{\alpha_{s,M(s)}^{c}}{2}$

for any $n>M(s)$ by conditions (10.6) and (10.9). Hereby, we see that for
every $s\in\mathbb{N}$ the decreasing sequence
$\left(\alpha_{s,n}^{c}\right)_{n\geq M(s)}$ is bounded from below by
$\frac{\alpha_{s,M(s)}^{c}}{2}$, and we denote its limit by $\alpha_{s}^{c}$.
Altogether, this proves the following proposition.

######  Proposition 89.

For every $n\geq M(s)$, we have

(10.17)
$\overline{f}\left(\mathcal{A},\overline{\mathcal{A}}\right)>\alpha_{s}^{c}$

on any substrings $\mathcal{A}$, $\overline{\mathcal{A}}$ of at least
$q_{n}/R_{n}^{c}$ consecutive symbols in $c_{n}(w)$ and $c_{n}(\overline{w})$,
respectively, for $w,\overline{w}\in\mathcal{W}_{n}$ with
$[w]_{s}\neq[\overline{w}]_{s}$.

###  10.5. Proof of Non-Kakutani Equivalence

We follow the strategy in Section 9 to show that for a tree
$\mathcal{T}\in\mathcal{T}\kern-1.42262ptrees$ without an infinite branch
$T_{c}=\mathcal{F}(\Psi(\mathcal{T}))$ and
$T_{c}^{-1}=\mathcal{F}(\Psi(\mathcal{T}))^{-1}$ are not Kakutani equivalent.

######  Lemma 90.

Let $\mathcal{T}\in\mathcal{T}\kern-1.42262ptrees$,
$T_{c}=\mathcal{F}(\Psi(\mathcal{T}))$, and $f:X\to\mathbb{Z}^{+}$ be
integrable. If $T_{c}^{f}\cong T_{c}$, then $\int f\mathrm{d}\mu=1$.

###### Proof.

The proof follows along the lines of Proposition 56. This time, one decomposes
the name $a(T_{c},P_{c},x)$ in the circular system into strings of the form
$\mathcal{C}_{n,j}\left(F_{n,i}\right)$, where $0\leq j<q_{n}$ and $F_{n,i}$
is a Feldman pattern of $1$-equivalence classes of $n$-words in the odometer-
based system (as considered in the proof of Proposition 56). ∎

Hereby, we can show as in Proposition 67 that if
$\mathbb{K}^{c}=\mathcal{F}(\Psi(\mathcal{T}))$ and
$(\mathbb{K}^{c})^{-1}=\mathcal{F}(\Psi(\mathcal{T}))^{-1}$ are Kakutani
equivalent, then it can only be by an even equivalence. We show that
$\mathbb{K}^{c}$ and $(\mathbb{K}^{c})^{-1}$ are not evenly equivalent if
$\mathcal{T}\in\mathcal{T}\kern-1.42262ptrees$ does not have an infinite
branch. For this purpose, we prove the following analogues of Lemmas 71 and
72.

######  Lemma 91.

Let $s\in\mathbb{N}$ and $\phi$ be a finite code from $\mathbb{K}^{c}$ to
$rev\left(\mathbb{K}^{c}\right)$. Then for sufficiently large
$n\in\mathbb{N}$, $0\leq i_{1},i_{2}<q_{n}$, and any pair of $s$-Feldman
patterns $P_{n,\ell}$ in $\mathbb{K}$ and $\overline{P}_{n,\overline{\ell}}$
in $rev(\mathbb{K})$ of different pattern type we have

$\overline{f}\left(\phi(\mathcal{G}),\overline{\mathcal{G}}\right)>\frac{\alpha^{c}_{s}}{10}$

for any substrings $\mathcal{G}$, $\overline{\mathcal{G}}$ of at least
$\frac{|P_{n,\ell}|\cdot
l_{n}q_{n}}{2^{2e(n)}}=\frac{|\overline{P}_{n,\overline{\ell}}|\cdot
l_{n}q_{n}}{2^{2e(n)}}$ consecutive symbols in
$\mathcal{C}_{n,i_{1}}\left(P_{n,\ell}\right)$ and
$\mathcal{C}_{n,i_{2}}^{r}\left(\overline{P}_{n,\overline{\ell}}\right)$,
respectively.

###### Proof.

The proof follows along the lines of Lemma 71. ∎

######  Lemma 92.

Let $s\in\mathbb{N}$ and $\phi$ be a finite code from $\mathbb{K}^{c}$ to
$rev\left(\mathbb{K}^{c}\right)$. Moreover, let $n\in\mathbb{N}$ be
sufficiently large, $0\leq i_{1},i_{2}<q_{n}$, and
$w,\overline{w}\in\mathcal{W}_{n+1}$ be any pair such that $w$ and
$rev(\overline{w})$ are of different $s$-pattern type. Then we have

$\overline{f}\left(\phi\left(\mathcal{A}\right),\overline{\mathcal{A}}\right)>\frac{\alpha_{s}^{c}}{16}$

on any substrings $\mathcal{A}$, $\overline{\mathcal{A}}$ of at least
$\frac{k_{n}l_{n}q_{n}}{2^{e(n)}}$ consecutive symbols in
$\mathcal{C}_{n,i_{1}}(w)$ and
$\mathcal{C}_{n,i_{2}}^{r}\left(\overline{w}\right)$, respectively.

###### Proof.

Analogously to the proof of Lemma 72, we divide $\mathcal{A}$ and
$\overline{\mathcal{A}}$ into circular images
$\mathcal{P}_{n,j}=\mathcal{C}_{n,i_{1}}\left(P_{n,j}\right)$ and
$\overline{\mathcal{P}}_{n,j}=\mathcal{C}_{n,i_{2}}^{r}\left(\overline{P}_{n,j}\right)$,
where $P_{n,j}$ and $\overline{P}_{n,j}$ are $s$-Feldman patterns in $w$ and
$\overline{w}$, respectively. Then the proof follows along the lines of Lemma
72. ∎

We can use Lemmas 91 and 92 to identify a well-approximating finite code with
an element of the group action in a similar way that we used Lemmas 71 and 72
in the proof of Lemma 73.

######  Lemma 93.

Let $s\in\mathbb{N}$, $0<\delta<\frac{1}{4}$,
$0<\varepsilon<\frac{\alpha_{s}^{c}}{200}\delta$, and $\phi$ be a finite code
from $\mathbb{K}^{c}$ to $rev(\mathbb{K}^{c})$. Then for $n$ sufficiently
large, any $0\leq i_{1}<q_{n}$, any $w\in\mathcal{W}_{n+1}$, and any string
$\overline{\mathcal{A}}$ in $rev(\mathbb{K}^{c})$ with

$\overline{f}\left(\phi\left(\mathcal{C}_{n,i_{1}}(w)\right),\overline{\mathcal{A}}\right)<\varepsilon,$

there must be exactly one string of the form
$\mathcal{C}_{n,i_{2}}^{r}(\overline{w})$ with
$|\overline{\mathcal{A}}\cap\mathcal{C}_{n,i_{2}}^{r}(\overline{w})|\geq(1-\delta)k_{n}l_{n}q_{n}$
for some $0\leq i_{2}<q_{n}$, $\overline{w}\in\mathcal{W}_{n+1}$. Moreover,
$rev(\overline{w})$ must be of the form
$\left[rev(\overline{w})\right]_{s}=\eta_{g}\left[w\right]_{s}$ for a unique
$g\in G_{s}$.

###### Proof.

As in the proof of Lemma 73 we neglect end effects and conclude that there can
be at most one string of the form $\mathcal{C}_{n,i_{2}}^{r}(\overline{w})$
with
$|\overline{\mathcal{A}}\cap\mathcal{C}_{n,i_{2}}^{r}(\overline{w})|\geq(1-\delta)k_{n}l_{n}q_{n}$.
Suppose that there is no such $\mathcal{C}_{n,i_{2}}^{r}(\overline{w})$, but
two strings $\mathcal{C}_{n,i_{2,1}}^{r}(\overline{w}_{1})$,
$\mathcal{C}_{n,i_{2,2}}^{r}(\overline{w}_{2})$ with
$\overline{w}_{1},\overline{w}_{2}\in\mathcal{W}_{n}$ and $0\leq
i_{2,1},i_{2,2}<q_{n}$ such that $\overline{\mathcal{A}}$ is a substring of
$\mathcal{C}_{n,i_{2,1}}^{r}(\overline{w}_{1})\mathcal{C}_{n,i_{2,2}}^{r}(\overline{w}_{2})$
and

$|\overline{\mathcal{A}}\cap\mathcal{C}_{n,i_{2,j}}^{r}(\overline{w}_{j})|\geq\left(\delta-3\varepsilon\right)\cdot
k_{n}l_{n}q_{n}>\frac{180}{\alpha_{s}^{c}}\varepsilon k_{n}l_{n}q_{n}\;\text{
for }j=1,2.$

As before there are two possible cases: Either at least one of
$rev(\overline{w}_{1}),rev(\overline{w}_{2})$ has a different $s$-pattern type
from that of $w$ or $rev(\overline{w}_{1}),rev(\overline{w}_{2})$ have the
same $s$-pattern type as $w$. We recall that the $s$-pattern structure of the
beginning and end of $w$ are different from each other by construction. To
investigate both cases we denote
$\overline{\mathcal{A}}_{j}\coloneqq\overline{\mathcal{A}}\cap\mathcal{C}_{n,i_{2,j}}^{r}(\overline{w}_{j})$.
Let $\pi$ be a best possible match in $\overline{f}$ between
$\phi\left(\mathcal{C}_{n,i_{1}}(w)\right)$ and $\overline{\mathcal{A}}$. Then
we denote the parts in $\phi\left(\mathcal{C}_{n,i_{1}}(w)\right)$
corresponding to $\overline{\mathcal{A}}_{j}$ under $\pi$ by
$\mathcal{A}_{j}$. If
$\overline{f}\left(\mathcal{A}_{j},\overline{\mathcal{A}}_{j}\right)>\alpha_{s}^{c}$,
we stop. Otherwise, we have
$\frac{8}{9}\leq\frac{|\mathcal{A}_{j}|}{|\overline{\mathcal{A}}_{j}|}\leq\frac{9}{8}$,
that is,
$|\mathcal{A}_{j}|\geq\frac{8}{9}|\overline{\mathcal{A}}_{j}|\geq\frac{160}{\alpha_{s}^{c}}\varepsilon
k_{n}l_{n}q_{n}$. Hence, for sufficiently large $n$ the strings are long
enough such that we can apply Lemmas 91 and 92. In both aforementioned cases
this yields with the aid of Fact 10 that

$\overline{f}\left(\phi(w),\overline{A}\right)\geq\frac{80}{\alpha_{s}^{c}}\varepsilon\cdot\frac{\alpha_{s}^{c}}{16}=5\varepsilon,$

which contradicts the assumption
$\overline{f}\left(\phi\left(\mathcal{C}_{n,i_{1}}(w)\right),\overline{\mathcal{A}}\right)<\varepsilon$.
Hence, there must be exactly one $\mathcal{C}_{n,i_{2}}^{r}(\overline{w})$
with
$|\overline{\mathcal{A}}\cap\mathcal{C}_{n,i_{2}}^{r}(\overline{w})|\geq(1-\delta)k_{n}l_{n}q_{n}$.

Lemma 92 also implies that $w$ and $rev(\overline{w})$ need to have the same
$s$-pattern structure. As before this only happens if
$\left[rev(\overline{w})\right]_{s}=\eta_{g}\left[w\right]_{s}$ for $g\in
G_{s}$. ∎

Assume $\mathbb{K}^{c}$ and $(\mathbb{K}^{c})^{-1}$ are evenly equivalent. Let
$(\varepsilon_{\ell})_{\ell\in\mathbb{N}}$ be a sequence of positive reals
satisfying $\sum^{\infty}_{i=\ell+1}\varepsilon_{i}<\varepsilon_{\ell}$. Then
we take a sequence of $(\varepsilon_{\ell},K_{\ell})$-finite $\overline{f}$
codes $\phi^{c}_{\ell}$ from $\mathbb{K}^{c}$ to $(\mathbb{K}^{c})^{-1}$ as
produced by Lemma 68. In the following we use
$\gamma^{c}_{s}=\frac{(\alpha_{s}^{c})^{4}}{5\cdot 10^{9}}$ for each
$s\in\mathbb{N}$. We apply Corollary 70 with $\gamma^{c}_{s}$ on our sequence
of codes and obtain $N(s)\in\mathbb{Z}^{+}$ such that for every $N\geq N(s)$
and $k\in\mathbb{Z}^{+}$ there is $n(s,N,k)\in\mathbb{Z}^{+}$ sufficiently
large such that for all $n\geq n(s,N,k)$ Lemma 93 with
$\delta^{c}_{s}=\frac{(\alpha^{c}_{s})^{3}}{2\cdot 10^{7}}$ and $\phi_{N}$
holds and there exists a collection of words
$\mathcal{W}^{\prime}_{n}\subset\mathcal{W}_{n}^{c}$ of circular $n$-words
satisfying the analogues of (C1) and (C2).

######  Lemma 94.

Suppose $\mathbb{K}^{c}$ and $(\mathbb{K}^{c})^{-1}$ are evenly equivalent and
let $s\in\mathbb{N}$. There is a unique $g_{s}\in G_{s}$ such that for every
$N\geq N(s)$ and sufficiently large $n\in\mathbb{N}$ we have for every
$w^{c}\in\mathcal{W}_{n}^{\prime}$ that there is $\overline{w}^{c}\in
rev(\mathcal{W}_{n}^{c})$ with
$\left[\overline{w}^{c}\right]_{s}=\eta_{g_{s}}\left[w^{c}\right]_{s}$ such
that

$\overline{f}\left(\phi_{N}^{c}(w^{c}),\overline{w}^{c}\right)<\frac{\alpha_{s}^{c}}{4},$

where $(\phi^{c}_{\ell})_{\ell}$ is a sequence of codes as described above.
Moreover, $g_{s}$ is of odd parity and the sequence $(g_{s})_{s\in\mathbb{N}}$
satisfies $g_{s}=\rho_{s+1,s}(g_{s+1})$ for all $s\in\mathbb{N}$.

###### Proof.

Let $w^{c}\in\mathcal{W}^{\prime}_{n+1}\subset\mathcal{W}_{n+1}^{c}$ and we
recall that $w^{c}=\prod_{i=0}^{q_{n}-1}\mathcal{C}_{n,i}(w)$ with
$w\in\mathcal{W}_{n+1}$. As in the proof of Lemma 74 we use property (C1) to
choose $n$ sufficiently large such that there exists $z\in
rev(\mathbb{K}^{c})$ with

$\overline{f}\left(\phi_{N}^{c}\left(\mathcal{C}_{n,i}(w)\right),z\upharpoonright[0,k_{n}l_{n}q_{n}-1]\right)<\gamma^{c}_{s}.$

We denote $\overline{\mathcal{A}}_{i}\coloneqq
z\upharpoonright[0,k_{n}l_{n}q_{n}-1]$ in $rev(\mathbb{K}^{c})$. By Lemma 93
there is a string of the form $\mathcal{C}_{n,j_{i}}^{r}(\overline{w})$ with
$|\overline{\mathcal{A}}_{i}\cap\mathcal{C}_{n,j_{i}}^{r}(\overline{w})|\geq(1-\delta^{c}_{s})k_{n}l_{n}q_{n}$
for some $0\leq j_{i}<q_{n}$, $\overline{w}\in\mathcal{W}_{n+1}$, and
$rev(\overline{w})$ must be of the form
$\left[rev(\overline{w})\right]_{s}=\eta_{g}\left[w\right]_{s}$ for a unique
$g\in G_{s}$. By Fact 9 we conclude
$\overline{f}\left(\phi_{N}^{c}\left(\mathcal{C}_{n,i}(w)\right),\mathcal{C}_{n,j_{i}}^{r}(\overline{w})\right)<\gamma^{c}_{s}+\delta^{c}_{s}$.
Since this holds true for every $i\in\left\\{0,\dots,q_{n}-1\right\\}$ and the
newly introduced spacers occupy a proportion of at most $\frac{1}{l_{n}}$, we
obtain for $n$ sufficiently large that
$\overline{f}\left(\phi_{N}^{c}(w^{c}),\mathcal{C}_{n}^{r}(\overline{w})\right)<\gamma^{c}_{s}+\delta^{c}_{s}+\frac{2}{l_{n}}<10^{-7}(\alpha_{s}^{c})^{3}$.
Hence, we choose $\overline{w}^{c}=\mathcal{C}_{n}^{r}(\overline{w})\in
rev(\mathcal{W}_{n+1}^{c})$ which satisfies
$\left[\overline{w}^{c}\right]_{s}=\eta_{g}\left[w^{c}\right]_{s}$ by
$\left[rev(\overline{w})\right]_{s}=\eta_{g}\left[w\right]_{s}$ and Remark 84.
To see that the same $g\in G_{s}$ is supposed to work for all
$w\in\mathcal{W}_{n+1}$ we repeat the argument as in Lemma 74 for a word in
$\mathcal{W}_{n+2}^{c}$ which is a concatenation of $(n+1)$-words by
construction. By the same arguments as in Lemma 74 one can also show that
$g_{s}=\rho_{s+1,s}(g_{s+1})$ for all $s\in\mathbb{N}$. ∎

As in Subsection 9.3 we conclude that $\mathbb{K}^{c}$ and
$(\mathbb{K}^{c})^{-1}$ cannot be Kakutani equivalent.

###  10.6. Proofs of Theorems 24 and 25

Now we can complete the transfer to the setting of diffeomorphisms.

###### Proof of Theorem 24.

In the proof of Theorem 21 we constructed a continuous reduction
$\Psi:\mathcal{T}\kern-1.42262ptrees\to\mathcal{E}$ that took values in the
strongly uniform odometer-based transformations. Under the functor
$\mathcal{F}$ from Subsection 10.3 these are transferred to strongly uniform
circular systems. During the course of the construction several parameters
with numerical conditions about growth and decay rates appear. In particular,
at stage $n$ the parameter $l_{n}$ is chosen last. Hence, we can choose it
sufficiently large to guarantee condition (10.2), as well as the convergence
to a $C^{\infty}$ diffeomorphism that is measure-theoretically isomorphic to
the circular system asserted in Lemma 77. We denote this realization map with
our choice of the sequence $(l_{n})_{n\in\mathbb{N}}$ by $R$. Hereby, we
define the map
$F^{s}:\mathcal{T}\kern-1.42262ptrees\to\text{Diff}^{\infty}(M,\lambda)$ by
$F^{s}=R\circ\mathcal{F}\circ\Psi$. We want to show that $F^{s}$ is a
continuous reduction.

######  Lemma.

$F^{s}:\mathcal{T}\kern-1.42262ptrees\to\text{Diff}^{\infty}(M,\lambda)$ is
continuous.

###### Proof.

Let $T=F^{s}(\mathcal{T})$ for $\mathcal{T}\in\mathcal{T}\kern-1.42262ptrees$
and $U$ be an open neighborhood of $T$ in $\text{Diff}^{\infty}(M,\lambda)$.
Let $T^{c}=\mathcal{F}\circ\Psi(\mathcal{T})$ be the circular system such that
$R(T^{c})=T$. By [FW19a, Proposition 61] there is $M\in\mathbb{N}$
sufficiently large such that for all $S\in\text{Diff}^{\infty}(M,\lambda)$ in
the range of $R$ we have the following property: If
$\left(\mathcal{W}_{n}^{c}(T^{c})\right)_{n\leq
M}=\left(\mathcal{W}_{n}^{c}(S^{c})\right)_{n\leq M}$, then $S\in U$. Here,
$S^{c}$ denotes the circular system such that $S=R(S^{c})$. Moreover,
$\left(\mathcal{W}_{n}(T^{c})\right)_{n\in\mathbb{N}}$ and
$\left(\mathcal{W}_{n}(S^{c})\right)_{n\in\mathbb{N}}$ denote the construction
sequences of $T^{c}$ and $S^{c}$, respectively. We recall from Subsection 10.3
that $\left(\mathcal{W}_{n}(T^{c})\right)_{n\leq M}$ is determined by the
first $M+1$ members in the construction sequence of the odometer based system
$F(\mathcal{T})$, that is, it is determined by
$\left(\mathcal{W}_{n}(\mathcal{T})\right)_{n\leq M}$. By (8.1) there is a
basic open set $V\subseteq\mathcal{T}\kern-1.42262ptrees$ containing
$\mathcal{T}$ such that for all $\mathcal{S}\in V$ the first $M+1$ members of
the construction sequences of $\Psi(\mathcal{T})$ and $\Psi(\mathcal{S})$ are
the same, that is, $\left(\mathcal{W}_{n}(\mathcal{T})\right)_{n\leq
M}=\left(\mathcal{W}_{n}(\mathcal{S})\right)_{n\leq M}$. Then it follows that
$F^{s}(\mathcal{S})\in U$ for all $\mathcal{S}\in V$ which yields the
continuity of $F^{s}$ by Fact 17. ∎

######  Lemma.

$F^{s}:\mathcal{T}\kern-1.42262ptrees\to\text{Diff}^{\infty}(M,\lambda)$
reduces the collection of ill-founded trees to the collection of ergodic
$C^{\infty}$ diffeomorphisms $T$ such that $T$ and $T^{-1}$ are Kakutani
equivalent.

###### Proof.

Since the realization map $R$ maps the circular system to a measure-
theoretically isomorphic diffeomorphism, it preserves Kakutani equivalence.
Hence, to see that $F^{s}$ is a reduction, it suffices to check that
$\mathcal{F}\circ\Psi$ is a reduction. Here, we let
$\mathbb{K}=\Psi(\mathcal{T})$ and
$\mathbb{K}^{c}=\mathcal{F}\circ\Psi(\mathcal{T})$ for
$\mathcal{T}\in\mathcal{T}\kern-1.42262ptrees$.

Suppose $\mathcal{T}\in\mathcal{T}\kern-1.42262ptrees$ has an infinite branch.
By the proof of part (1) of Proposition 23 in Section 5 there is an anti-
synchronous isomorphism $\phi:\mathbb{K}\to\mathbb{K}^{-1}$. Then we can apply
Lemma 82 on the functor $\mathcal{F}$ and obtain the existence of an anti-
synchronous isomorphism $\phi^{c}:\mathbb{K}^{c}\to(\mathbb{K}^{c})^{-1}$. The
converse direction follows from Subsection 10.5, where we have shown that if
$\mathcal{T}\in\mathcal{T}\kern-1.42262ptrees$ does not have an infinite
branch, then $\mathbb{K}^{c}$ and $(\mathbb{K}^{c})^{-1}$ are not Kakutani
equivalent. ∎

Both statements together yield that $F^{s}$ is a continuous reduction and we
conclude the statement as in the proof of Theorem 20. ∎

In an analogous manner we can also prove Theorem 25.

###### Proof of Theorem 25.

This time we let $R$ be the realization map from circular systems with fast
growing coefficients $(l_{n})_{n\in\mathbb{N}}$ to
$\text{Diff}_{\rho}^{\,\omega}(\mathbb{T}^{2},\lambda)$ obtained by Lemma 78.
Then the proof follows along the lines of the previous one, using [BKpp,
Proposition 7.10] instead of [FW19a, Proposition 61]. ∎

### Acknowledgements

We thank M. Foreman and B. Weiss for helpful discussions. We especially thank
Foreman for his encouragement and his detailed answers to our questions
regarding his work with D. Rudolph and B. Weiss.

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# Imaging Preflare Broadband Pulsations in the Decimetric-metric Wavelengths

Maoshui Lv  Institute of Space Sciences, Shandong University, Weihai,
Shandong, 264209, People’s Republic of China  Baolin Tan  CAS Key Laboratory
of Solar Activity, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100101, People’s Republic of China  School of Astronomy and
Space Science, University of Chinese Academy of Sciences, Beijing, 100049,
People’s Republic of China  Ruisheng Zheng  Institute of Space Sciences,
Shandong University, Weihai, Shandong, 264209, People’s Republic of China
Zhao Wu  Institute of Space Sciences, Shandong University, Weihai, Shandong,
264209, People’s Republic of China  Bing Wang  Institute of Space Sciences,
Shandong University, Weihai, Shandong, 264209, People’s Republic of China
Xiangliang Kong  Institute of Space Sciences, Shandong University, Weihai,
Shandong, 264209, People’s Republic of China  Yao Chen  Institute of Space
Sciences, Shandong University, Weihai, Shandong, 264209, People’s Republic of
China  Institute of Frontier and Interdisciplinary Science, Shandong
University, Qingdao, Shandong, 266237, People’s Republic of China

###### Abstract

Preflare activities contain critical information about the pre-cursors and
causes of solar eruptions. Here we investigate the characteristics and origin
of a group of broadband pulsations (BBPs) in the decimetric-metric
wavelengths, taking place during the preflare stage of the M7.1 flare dated on
2011 September 24. The event was recorded by multiple solar instruments
including the Nançay Radioheliograh that measure the properties of the radio
source. The BBPs start $\sim$24 min before the flare onset, extending from $<$
360 to above 800 MHz with no discernible spectral drift. The BBPs consist of
two stages, during the first stage the main source remains stationary, during
the second stage it moves outward along with a steepening extreme-ultraviolet
(EUV) wave driven by the eruption of a high-temperature structure. In both
stages, we observe frequent EUV brightenings and jets originating from the
flare region. During the second stage, the BBPs become denser in number and
stronger in general, with the level of the polarization increasing gradually
from $<$ 20% to $>$ 60% in the right-handed sense. These observations indicate
the steepening EUV wave is important to the BBPs during the second stage,
while the preflare reconnections causing the jets and EUV brightenings are
important in both stages. This is the first time such a strong association of
an EUV wave with BBPs is reported. We suggest a reconnection plus shock-
sweeping-across-loop scenario for the cause of the BBPs.

Solar coronal mass ejections (310), Solar activity (1475), Solar corona
(1483), Solar flares (1496), Solar radio emission (1522)

##  1 Introduction

Solar radio pulsations represent (quasi-)periodic or irregular short
fluctuations observed from the radio flux curves or the dynamic spectra, in
almost all wavelength ranges from metric to microwave. In the decimetric-
metric wavelengths, pulsations often appear as fine structures being
superposed on the wideband continuum of the type-IV solar radio burst (Young
et al. 1961; Kai & Takayanagi 1973; Droege 1977; Trottet et al. 1981; Khan et
al. 2002. See the review by Nindos & Aurass 2007). According to the bandwidths
of pulsations, they can be classified as broadband pulsations (BBPs) and
narrowband pulsations (NBPs).

At least two aspects of the mechanisms of the radio pulsations should be
considered. One is the radiation mechanism, the other is the cause of the
pulsations/modulations. For NBPs a coherent mechanism of plasma radiation may
be involved while for BBPs the gyrosynchrotron mechanism may be important. In
addition, the Type-III like coherent radiation mechanism excited by fast-
moving beam-type energetic electrons within large magnetic loops, as well as
the loss-cone maser instability driven by trapped energetic electrons (Aurass
et al. 2003; Zlotnik et al. 2003. See Ni et al. 2020, 2021 and Li et al. 2021
for latest theorical studies) have also been proposed as likely radiation
mechanisms for BBPs. On the other hand, the pulsations/modulations are often
explained with the following scenarios (Kuijpers, 1980; Aschwanden, 1987): (1)
modulations by the magnetohydrodynamic (MHD) oscillations in terms of, e.g.,
the fast sausage mode of the magnetic loop structure; (2) intrinsic
oscillations of the kinetic radiation process involving the nonlinear wave-
wave and wave-particle coupling; (3) modulations by the transient acceleration
of energetic electrons, by, e.g., intermittent or bursty magnetic
reconnections.

Most studies have focused on BBPs taking place during the impulsive or decay
phases of solar flares. During the preflare stage, the energy releases, if
exist, are much weaker than that during the main phase of the flares, thus the
resultant radio signatures can be easily missed. Zhang et al. (2015) reported
four flares with preflare microwave activities which contain fine structures
such as quasi-periodic pulsations (QPPs) and millisecond dots, using the high-
sensitivity data recorded by the Ond rejov radio spectrograph in the frequency
range of 0.8-2.0 GHz. This reveals novel signatures of preflare activities. In
a study on flare precursors in the low solar atmosphere, Wang et al. (2017)
found two episodes of weak emissions existing 10-30 min before the impulsive
phase of the flare. The accompanying microwave spectra were recorded by the
new Expanded Owens Valley Solar Array (EOVSA) in the frequency range of 2-18
GHz. They show that these precursor microwave emissions can be well modelled
as quasi-thermal, gyrosynchrotron emission sources, and the spectral fittings
have been used to deduce the magnetic field strengths and their temporal
variation. This provides important constraint on the location and
characteristics of the energy release from the radio perspective. Another
study identified preflare microwave QPPs with the 17 GHz data recorded by the
Nobeyama Radioheliograph (Li et al., 2020), in which the QPPs appear within
the flare source region according to the microwave imaging data. Their periods
increase from $\sim$ 300 to 500s. This has been used to infer the property of
electric currents of the pre-flare source region.

Here we report a rare decimetric-metric event of BBPs observed during the
preflare stage of an M7.1 limb flare. Multi-wavelength data, including both
EUV and radio imaging/spectral data, are available. The close-to-limb
perspective assures minor projection effect. The event provides a good
opportunity to investigate the origin of BBPs. The observational data and
results are presented in Sections 2 and 3. Conclusions and discussion are
presented in Section 4.

##  2 Data and Event Overview

The flare occurred on 2011 September 24, starting at 12:33 UT, peaking at
13:17 UT, and ending at 14:10 UT, according to the _Geostationary Operational
Environmental Satellite_ (_GOES_) flare list. It originated from the NOAA AR
11302 close to the northeastern limb of the solar disk. Figure 1(a) shows the
171 Å image observed by the Atmospheric Imaging Assembly (AIA; Lemen et al.
2012) on board the _Solar Dynamics Observatory_ (_SDO_ ; Pesnell et al. 2012)
at the flare peak time, from which overlying loops are clearly visible. Mainly
the EUV data at 171 Å (Fe IX, $\sim$0.6 MK), 193 Å (Fe XII, $\sim$1.6 MK), and
94 Å (Fe XVIII, $\sim$6 MK) were analyzed here. The EUV data have a pixel size
of 0.6″ and a cadence of 12s. An accompanying halo CME was observed by the
Large Angle and Spectrometric Coronagraph (LASCO; Brueckner et al. 1995) C2
onboard the _Solar and Heliospheric Observatory_ (_SOHO_ ; Domingo et al.
1995) first at 12:48 UT (see Figure 1(b)). According to the second-order
polynomial fit of the height measurements by C2, the CME starts at $\sim$12:33
UT.

The radio bursts were recorded by the spectrographs of e-Callisto Bleien
(175-870 MHz) with a time resolution of 0.25s (Figure 1(c)). The bursts took
place in all stages of the flare, from the preflare and early-rising stages to
the impulsive and decay stages. Liu et al. (2018) have analyzed the stationary
type IV continuum burst (12:40-13:00 UT) during the early-rising stage. Here
we focus on the radio bursts observed from $\sim$12:09-12:31 UT before the
onset of the flare.

The Nançay Radioheliograh (NRH; Kerdraon & Delouis 1997) provides imaging data
with polarization measurement at 10 frequencies from 150 to 445 MHz. The
spatial resolution depends on frequency and time of observation, being
$\sim$2′ at 445 MHz and $\sim$6′ at 150 MHz in summer and up to three times
larger along the NS direction during winter. Mainly the NRH data at 4
frequencies (360, 408, 432, and 445 MHz) were analyzed here.

The magnetic-field data on the photosphere are from the Helioseismic and
Magnetic Imager (HMI; Scherrer et al. 2012) on board _SDO_ , with a pixel
scale of 0.6″ and a cadence of 45s. The coronal magnetic field configuration
is extrapolated with the Potential Field Source Surface (PFSS; Schrijver & De
Rosa 2003) model.

##  3 Preflare BBPs and Coronal Activities

###  3.1 Spectral Characteristics and Source Properties of the BBPs

From Figure 2(a), the preflare BBPs started from $\sim$12:09 and ended at
$\sim$12:31 UT, being $\sim$24 min before the flare onset. They extend from
$<$ 360 to above 800 MHz with no discernible spectral drift. Hundreds of
decimetric-metric BBPs can be identified. They are rather intermittent from
$\sim$12:09 to 12:17 UT, and become denser in number and stronger in intensity
later. There exist several minutes of weak emission at $\sim$12:17 UT. This
has been used to split the whole BBPs into two stages, with stage I for
$\sim$12:09 to 12:17 UT and stage II for $\sim$12:17-12:31 UT. There exists
some background continuum that also becomes stronger during stage II. In
addition, the overall bandwidth of relatively strong emission becomes wider in
the later stage. During most time of this stage, the bandwidth is $>$500 MHz
that is close to the emission frequency. This is why we classify the emission
as BBPs.

Figure 2(b) presents the temporal curves of the brightness temperature
($T_{b}$) at four NRH frequencies (360, 408, 432, and 445 MHz). In accordance
with the spectral data, there exist plenty of local peaks of $T_{b}$. Five
blue vertical and dashed lines are plotted to show the correspondence between
the spectral data and temporal curves, so to confirm the absence of any
observable spectral drift of each BBP. The $T_{b}$ curves at the four
frequencies are similar to each other, with the values of $T_{b}$ being close
for the three larger frequencies while they are much smaller at 360 MHz. This
is consistent with the cutoff of the spectral data that is around 360 MHz. The
values of $T_{b}$ range from $\sim$107 K to $4\times 10^{8}$ K in stage I and
ranges from $\sim$107 K to $>$109 K in stage II.

Figure 2(c) presents the temporal variations of the levels of polarization at
the four NRH frequencies. The levels are close to each other, agreeing with
the fact they belong to the same BBPs. Another significant observation is that
the polarization remains at a weak level during stage I while increasing
consistently up to strong levels ($\sim$60%) at the end of stage II, which
continues to increase to $\sim 100{\%}$ after 12:31 UT.

Figure 2(d) shows the results of the wavelet analysis of the trend-subtracted
$T_{b}$ profile at the NRH 445 MHz, from which one can identify several
periodic components. The $\sim$ 5 min signal lasts for only about 5 min, thus
it is not convincing; the $\sim$ 3 min signal lasts for about 9 min
($\sim$12:14-12:23 UT). Their origin can be inferred from the three
intermittent enhancements of the spectral data (Figure 2(a)). There also exist
several periodic components at $\sim$1 min and $\sim$2 min. Comprehensive
studies exist regarding the origin of QPPs (reviewed briefly in the
introduction section), therefore they are not considered further here.

Figure 3 presents the NRH sources at the four frequencies (panels (a)-(d)).
Panel (e) shows the corresponding time line overplotted on the spectra. From
this figure and the accompanying movie, we observe a major source located
around the equatorial part of the limb. The source persists at the four NRH
frequencies during the pre-flare BBPs, indicating the existence of the
background continuum radiation. In addition to this major source, another two
sources located at the northern part of the images (out of the limb) appear
intermittently during the later period of stage II . Linking the three
sources, one can obtain a large-scale arcade-like structure.

The plus sign in each panel represents the centroid of the major source at 445
MHz, plotted for the convenience of comparison. The size of the source is
about 150-200″ according to the outmost contours which represent the 30% level
of the maximum $T_{b}$ (the 30% contours for short), close to the
corresponding spatial resolution of NRH. The source centroids of the four
frequencies are close to each other. During stage I the sources remain
basically stationary (being $\sim$ 40″ above the solar limb) while during
stage II the sources move outward systematically. This can be seen from Figure
4(a) which presents the source centroid locations with the 85% contours at
different times. The dashed line that connects the centroids of the three
frequencies (see Figure 4(a)) has been used as the slit to obtain the
distance-time ($d-t$) measurement of the radio sources with the NRH data. The
obtained $d-t$ data at 445 MHz are plotted in Figure 4(b), the error bars are
given by the dimension of the 85% contours of the source. They are overplotted
onto the $d-t$ map of the AIA data at 193 Å along the same slit that will be
analyzed later. From the radio data, the speed of the radio source during
stage I is not significant as expected, while the average speed increases to
$\sim$120 km s-1 around the end of stage II (from 12:28 to 12:31 UT).

###  3.2 Coronal Activities and their relation with the BBPs

Figure 5 and the accompanying movie present the AIA images at 171, 193, and 94
Å. The left two columns present the AIA images at the start of stage I and
around the end of stage II, respectively, the rightmost column presents the
difference images. The three passbands are selected to show (1) the system of
the coronal loops overlying the active region and the sporatic jets and EUV
brightenings within/around the flare source region with 171 Å, (2) the rise
and steepening of the EUV wave structure with 193 Å, and (3) the hot eruptive
structure that drives the EUV wave with 94 Å.

All these three aspects of the event are self-evident according to the AIA
data. We plot arrows in Figure 5 to indicate their occurrence at the specific
moment. What is important here is their physical connection to the BBP
sources.

The jets move outward from the flare source toward the main radio source. The
EUV brightenings take place frequently during the BBPs. The two phenomena
indicate the occurrence of intermittent magnetic reconnection during the pre-
flare stage. This can explain the similar intermittent behaviour of BBPs if
attributing the radio-emitting energetic electrons to the same reconnection
process.

The hot structure starts to rise around 12:10 UT according to the 94 Å data.
As mentioned, in Figure 4(b) we have plotted the $d-t$ map with the difference
data at 193 Å. Figure 4(c) presents the $d-t$ map at 94 Å along the same slit.
The rise of the ejecta consists of the gradual and the impulsive stage. The
turning point of the two stages is at $\sim$12:25 UT. The steepening of the
EUV wave also takes place around this moment (see Figure4(b)). According to
the $d-t$ maps and the AIA images, the EUV wave emerges at $\sim$12:16 UT,
which is always ahead of and thus is driven by the hot eruptive structure.
They have the similar gradual-impulsive two-stage dynamics, with the speed of
the EUV wave (hot structure) along the slit being $\sim$34 km s-1 ($\sim$36 km
s-1) during the first stage and being $\sim$121 km s-1 ($\sim$151 km s-1) on
during the second stage. Later, the EUV wave accelerates to $\sim$473 km s-1
(see Figure 4(b)), indicating its steepening into a shock structure.

In Figure 6 and the accompanying movie we present the BBP sources at six NRH
frequencies with the running-difference images at 193 Å. The major source
remains stationary first, after the appearance of the EUV wave the source is
located at the EUV wave front and moves together with the EUV wave. This
agrees with what has been observed from the $d-t$ map in Figure 4(b). Again,
the arcade-like structure linking the three sources correlates well with the
large-scale EUV wave front (see the white circles in Figure 6(g)).

##  4 Conclusions and Discussion

Here we presented detailed observations of the BBPs in the decimetric-metric
wavelengths. The purpose is to understand the process causing the pulsations
using multi-wavelength data including the radio-imaging data at several
frequencies from NRH and the EUV imaging data from AIA/SDO. Such complete data
set were reported for the first time ever for preflare decimetric-metric BBPs.
The BBPs start $\sim$24 min before the major eruption, during which frequent
EUV brightenings occur and several jets eject from the flare region towards
the main radio source. During the early stage (stage I) the BBP sources remain
stationary while during the later stage (stage II) the sources move outward
with a steepening/acclerating EUV wave that is driven by an eruptive hot
structure. The major BBP source, together with the other two sources, are
located at the EUV wave front. In addition, the BBPs get intensified overall,
and the level of polarization increases gradually upon the emergence of the
EUV wave. These observations strongly indicate that the EUV wave contributes
to the acceleration of the BBP-emitting energetic electrons during the later
stage, in addition to the low-lying reconnection process that may provide seed
particles to the EUV wave.

The combined action of the EUV wave and magnetic reconnection in releasing
BBPs is demonstrated here for the first time. To see how this happens, in
Figure 7(a) we show the HMI magnetogram superposed by (1) the magnetic field
lines extrapolated with the usual PFSS method, (2) the EUV profile delineated
with the 193 Å data at 12:31:55 UT (also see the white circles in Figure
6(g)), and (3) the 50$\%$ contour of the major source observed by NRH at
12:31:57 UT. Note that the PFSS result suffers from the potential-field
assumption and the usage of synoptic magnetogram, thus the extrapolated field
lines should be treated with caution. Nevertheless, the overall morphology of
the large-scale loop system agrees with the bright loops according to the 171
Å images (see Figure 5). The location of the major BBP source correlates with
the top of the loop system and the front of the EUV wave.

These observations are in line with the schematic of Figure 7(b). We suggest
that during stage I the BBPs are radiated by energetic electrons that are
released by the magnetic reconnection process taking place within the flare
source region as evidenced by jets and EUV brightenings, during stage II after
the EUV wave appearance these energetic electrons are further processed by the
steepening wave and the resultant BBP source are carried outward by the EUV
wave. In the meantime, the EUV wave sweeps a series of loop tops during its
outward propagation. According to Kong et al. (2015, 2016), such EUV (or
shock)-wave-sweeping-loop-top process favors the acceleration of energetic
particles since the loop tops serve as an efficient trapping agency within
which particles can be processed by the steepening wave/shock-like structure
for multiple times. The intensified pulsations during stage II are a natural
result of this further processing, and the gradual increase of polarization
level can be caused by the gradual change of the overall magnetic field
orientation within the loop tops that are swept by the EUV wave front.

We tried to derive some key parameters (e.g., magnetic field, plasma density)
in the radio sources through fitting the flux density spectra assuming the
gyrosynchrotron emission. Such fittings require one to prescribe a group of
free parameters, including the density of the background and energetic
electrons, the field strength, the viewing angle, the column depth, and the
size, etc. This means the obtained results are not unique and have very large
uncertainty. In addition, the available number/range of frequencies with
imaging data is quite limited here. For a proper fitting to deduce the source
conditions, more observational constraints, such as measurement of the flux
density at more frequencies and independent measurement of the magnetic field
or plasma density, are required.

This study is supported by the National Natural Science Foundation of China
(11973031, 11790303, and 11873036). The authors acknowledge the team of NRH
for making their data available to us. We thank the Institute for Data Science
FHNW Brugg/Windisch, Switzerland for providing the e-Callisto data.

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Figure 1: Overview of the Event: (a) the AIA 171 Å data observed around the
flare peak time, (b) the white light CME first observed by LASCO C2, (c) the
dynamic spectra given by the e-Callisto Bleien observatory with the GOES 1-8 Å
and 0.5-4 Å light curves. The two vertical dashed lines in panel (c) give the
period with the preflare BBPs. Figure 2: Overview of the preflare BBPs: (a)
the dynamic spectra observed from 12:06 to 12:36 UT for the frequency range of
[250, 870] MHz, (b) the temporal profiles of _T_ b at 1s cadence and (c) the
degree of polarization with 10s cadence at 360, 408, 432, and 445 MHz, (d) the
power spectrum given by the wavelet analysis using the _T_ b data at 445 MHz,
starting from 12:09 UT. The _T_ b and the polarization level are obtained by
averaging the corresponding data within the contour of 85% of the maximum
$T_{b}$. The two outmost vertical lines in panels (a)-(c) show the preflare
period to be analyzed, the line with short horizontal lines (representative of
the NRH frequencies) separate the BBPs into two stages (I and II), the blue
vertical dashed lines indicate five local _T_ b peaks to show the absence of
observable spectral drift. The dashed line in panel (d) gives the 95%
confidence level, and the cross-hatched region presents the cone of influence.
Figure 3: The radio sources observed by NRH at 360, 408, 432, and 445 MHz
(a-d), and the corresponding dynamic spectra showing the time line (the
vertical line) of the radio images (e). The contours in panels (a)-(d) are
given by the 30, 50, 70, and 90% levels of the corresponding maximum $T_{b}$
shown in each panel, and the plus sign represents the centroid of the source
at 445 MHz for comparison. The short horizontal lines in panel (e) indicate
the four NRH frequencies. An animation from 12:08 to 12:31 UT is available
online, which shows the evolution of the BBP sources during the two stages.
The real-time duration of the animation is 14 seconds.  Figure 4: (a) The
temporal variation of the 85% contours of the sources at the three NRH
frequencies to give the slit (or the moving direction) for further analysis,
(b) the distance-time ($d-t$) maps with the 193 Å running-difference data and
(c) the direct 94 Å data. The triangle in panel (a) represents the source of
the flare. The diamonds with bars in panel (b) represent the $d-t$ data of the
source centroid locations at 445 MHz, the bars are given by the dimension of
the 85% contours of the source. In panels (b)-(c), the $d-t$ data are given
along the same slit presented in panel (a), and the dashed lines represent the
linear fittings to calculate the speeds. Figure 5: The AIA data of the event:
(a) The 171, 193, and 94 Å images observed at the start of stage I and (b)
around the end of stage II, (c) the running-difference images observed around
the end of stage II. The dashed line represents the slit taken from Figure
4(a) to give the $d-t$ maps of Figures 4(b)-(c), the triangle represents the
flare source. The arrows point to the hot structure (yellow), the EUV wave
(red), and the jets (orange). An animation from 12:05 to 12:35 UT is available
online, which shows the brightenings and jets, and the evolutions of the hot
structure and the EUV wave at 171, 193, and 94 Å during the two stages. The
real-time duration of the animation is 5 seconds.  Figure 6: The combined AIA
and NRH data at different times from 12:10 to 12: 34 UT. The running-
difference images are at 193 Å, and the NRH contours are at the 50% levels for
six NRH frequencies. The white circles in panel (g) delineate the front of the
steepening EUV wave. An animation from 12:05 to 12:35 UT of the running
difference images is available online. The animation shows the simultaneous
evolution of the BBP sources and the EUV wave at 193 Å during the two stages.
The bottom portion of the animation includes the corresponding dynamic spectra
showing the time line (the vertical line) of the radio images. This part of
the animation is not shown in the Figure. The real-time duration of the
animation is 5 seconds.  Figure 7: (a) The extrapolated PFSS field lines
superposed onto the HMI magnetogram data, the contours are given by the 50%
level of the corresponding maximum $T_{b}$ at the six NRH frequencies, (b) the
schematic to illustrate the physical origin of the BBPs. The white circles in
panel (a) are taken from Figure 6(g) to show the front of the EUV wave.