Tracing Hα Fibrils through Bayesian Deep Learning

Figure 2 explains how FibrilNet works. Training Hα images are preprocessed in steps 1 and 2, and then used to train the Bayesian deep-learning model (step 3). The trained model takes as input a test Hα image (step 4) and produces as output a predicted mask accompanied with results for quantifying aleatoric uncertainty and epistemic uncertainty (step 5). In the post-processing phase (step 6), based on the predicted mask, fibrils on the test Hα image are detected and highlighted by thin red curves. Furthermore, the orientations of the detected fibrils are determined based on a fibril-fitting algorithm, where the orientations are shown by different colors.

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Specifically, in step 1, we apply the threshold-based tool developed by Jing et al. (2011) to each training Hα image described in Section 2 to obtain a corresponding fibril mask. Fibril patterns on this mask are very thick, containing a lot of noise. In step 2, we refine the fibril mask via a skeletonization procedure to obtain a fibril skeleton in which fibrils are marked by black and regions without fibrils are marked by white. The skeletonization procedure works by extracting a region-based shape feature representing the general form of the fibrils. This skeletonization procedure results in better and cleaner images suitable for model training (Umbaugh 2010).

The training Hα images and fibril skeletons are then used to train the Bayesian deep-learning model for probabilistic image segmentation and uncertainty quantification (step 3). During training, in order to obtain a robust model, we use the data augmentation technique described in Jiang et al. (2020) to expand the training set by shifting, rotating, flipping, and scaling the training images. In step 4, a test Hα image is fed to the trained Bayesian deep-learning model. During testing, we use the Monte Carlo (MC) dropout sampling technique described in Section 3.2 to produce a predicted mask of the test Hα image accompanied with aleatoric uncertainty and epistemic uncertainty results (step 5). In step 6, by using the fibril-fitting algorithm based on the polynomial regression model described in Section 3.3, our FibrilNet tool outputs detected fibrils marked by red color on the test Hα image and their orientations represented by different colors.

The Bayesian deep-learning model used by FibrilNet is similar to the model used in SolarUnet (Jiang et al. 2020) for tracking magnetic flux elements. Both models have four encoder blocks (E1, E2, E3, and E4) and four decoder blocks (D1, D2, D3, and D4), mediated by a bottleneck (Bot). See Figure 3 and Jiang et al. (2020) for the configuration and parameter settings of the models. While both models are based on an encoder–decoder convolutional neural network, they differ in three ways. First, in performing 2 × 2 max pooling, represented by a red arrow in Figure 3, the corresponding max pooling indices are stored. During decoding, the max pooling indices at the corresponding encoder layer are recalled, represented by an orange arrow, to upsample, represented by a green arrow, as done in Badrinarayanan et al. (2017). This upsampling technique used by FibrilNet, designed to reduce the number of trainable parameters in the model (network) and hence save memory, is different from the upconvolution layers used in SolarUnet. Second, since fibril patterns are relatively vague and harder to identify than magnetic flux elements, FibrilNet uses twice as many kernels as SolarUnet in all of the blocks in the encoder and decoder, as well as in the bottleneck. Finally, during testing, instead of using the trained model (network) directly to produce segmentation results as done in SolarUnet, FibrilNet employs an MC dropout sampling technique, detailed below, to produce, for a test Hα image, a predicted mask accompanied with aleatoric uncertainty and epistemic uncertainty results (see Figure 2). This MC dropout sampling technique allows FibrilNet to perform probabilistic image segmentation with uncertainty quantification, which is lacking in SolarUnet.

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Specifically, to quantify uncertainty with the convolutional neural network, we use a prior probability, P( W ), over the network's weights, W . During training, pairs of Hα images and their corresponding fibril skeletons, collectively referred to as D , are used to train the network. According to Bayes' theorem,

$$\begin{eqnarray}&&P({\boldsymbol{W}}| {\boldsymbol{D}})=\frac{P({\boldsymbol{D}}| {\boldsymbol{W}})P({\boldsymbol{W}})}{P({\boldsymbol{D}})}.\end{eqnarray} \tag{ 1 }$$

Computation of the exact posterior probability, P( W ∣ D ), is intractable (Denker & LeCun 1990). Nevertheless, we can use variational inference (Graves 2011) to learn the variational distribution over the network's weights parameterized by θ, q_θ ( W ), by minimizing the Kullback–Leibler (KL) divergence of q_θ ( W ) and P( W ∣ D ) (Blei et al. 2017). It is known that training a network with dropout is equivalent to a variational approximation on the network (Gal & Ghahramani 2016). Furthermore, minimizing the cross-entropy loss of the network is equivalent to the minimization of the KL divergence (Goodfellow et al. 2016). Therefore, we use a binary cross-entropy loss function and the adaptive moment estimation (Adam) optimizer (Goodfellow et al. 2016) with a learning rate of 0.0001 to train our model (network). Let $\hat{\theta }$ denote the optimized variational parameter obtained by training the model (network); we use ${q}_{\hat{\theta }}({\boldsymbol{W}})$ to represent the optimized weight distribution.

In deep learning, dropout is mainly used to prevent overfitting, where a trained model overfits training data and hence cannot be generalized to make predictions on unseen test data. During training, dropout refers to ignoring or dropping out units (i.e., neurons) of a certain set of neurons that is chosen randomly. During testing, dropout can be used to retrieve T MC samples by processing the input test Hα image T times (Gal & Ghahramani 2016). (In the study presented here, T is set to 50.) Each time, a set of weights is randomly drawn from ${q}_{\hat{\theta }}({\boldsymbol{W}})$. Each pixel in the predicted mask, shown in step 5 of Figure 2, gets a mean and variance over the T samples. If the mean is greater than or equal to a threshold, the pixel is marked by black indicating that the pixel is part of a fibril; otherwise the pixel is marked by white indicating that the pixel is not part of a fibril. (In the study presented here, the threshold is set to 0.5). Following Kwon et al. (2020), we decompose the variance into aleatoric uncertainty and epistemic uncertainty at the pixel. The aleatoric uncertainty captures the inherent randomness of the predicted result, which comes from the input test Hα image, while the epistemic uncertainty comes from the variability of W , which accounts for the uncertainty in the model parameters (weights).

In the post-processing phase, we use a connected-component labeling algorithm (He et al. 2009) to group all adjacent black segments if their pixels in the edges or corners touch each other. For each resulting group, which represents a fibril, we locate its pixels in the predicted mask and highlight their corresponding pixels in the test Hα image by red. (Resulting groups containing fewer than 10 pixels are considered noise and filtered out.) We then output the detected fibrils highlighted by red color in the test Hα image, as shown in step 6 of Figure 2.

Most of the detected fibrils are lines or curves. In contrast to Jing et al. (2011), who used a quadratic function to fit the detected fibrils, we adopt a polynomial regression model here. Specifically, our regression model is a polynomial function with varying degrees capable of fitting the detected fibrils with different curvatures. In general, regression analysis investigates the relationship between a dependent variable and an independent variable (Bishop 2006). We model a detected fibril as an nth-degree polynomial function as follows:

$$\begin{eqnarray}&&y={\gamma }_{0}+{\gamma }_{1}x+{\gamma }_{2}{x}^{2}+\ldots +{\gamma }_{n}{x}^{n}+\epsilon ,\end{eqnarray} \tag{ 2 }$$

where γ_i denotes the coefficients and is a random error term. In Equation (2), the independent variable x represents the x coordinate of a pixel in the detected fibril and the dependent variable y represents the y coordinate of the same pixel, where the x-axis represents the E–W direction and the y-axis represents the S–N direction (see Figure 1). When the degree n equals 1, Equation (2) represents a linear regression model, meaning that the detected fibril is represented by a straight line. In our work, n ranges from 1 to 10.

We then use the least squares method (Ostertagová 2012) to find the optimal γ_i values. There are 10 candidate polynomial functions for representing the detected fibril. We use the R-squared score (Ostertagová 2012) to assess the feasibility of these 10 candidate polynomial functions. Specifically, we choose the candidate polynomial function yielding the largest R-squared score, and use this polynomial function to represent the detected fibril. To determine the orientation of the detected fibril, we calculate the derivative of the chosen polynomial function. For each pixel on the detected fibril, we thus obtain the slope of the tangent at the pixel, leading to the orientation angle of the pixel, denoted by θ_f, with respect to the x-axis. Notice that the orientation angle θ_f is in the 0°−180° range, as two directions differing by 180° are indistinguishable here because the detected fibril in Hα does not carry information on the vertical dimension. Thus, θ_f represents the direction of the detected fibril with a 180° ambiguity (Jing et al. 2011).

In this series of experiments, we used the 241 Hα line center images from 20:16:32 UT to 22:41:30 UT on 2017 July 13 mentioned in Section 2 along with their corresponding fibril skeletons to train the FibrilNet tool as described in Section 3. We then used the trained tool to predict and trace fibrils on the five test images at 0.0 Å, +0.4 Å, +0.6 Å, −0.4 Å, and −0.6 Å from the Hα line center 6563 Å with a 70'' circular FOV collected in AR 12665 on 2017 July 13 20:15:58 UT (see Figure 1). Figure 4 presents the tracing results on the test image at 0.0 Å; the tracing results on the other four test images can be found in the Appendix. In all of the tracing results, fibrils containing 10 or fewer pixels were treated as noise and excluded.

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Figure 4(A) shows the original test Hα image. Figure 4(B) shows the enlarged FOV of the region highlighted by white box 1 in Figure 4(A). It can be seen from Figure 4(B) that there are salt-and-pepper noise pixels in the region highlighted by white box 2, which are caused by image reconstruction limitations. Figure 4(C) shows the fibrils (red curves) on the test Hα image detected by the tool (after skeletonization) presented in Jing et al. (2011). Figure 4(D) shows the fibrils (red curves) predicted by FibrilNet. FibrilNet uses the images processed by the tool in Jing et al. (2011) as training data. The results in Figures 4(C) and (D) are quite similar, demonstrating the good learning capability of FibrilNet.

Figures 4(E) and (F) show the aleatoric uncertainty (data uncertainty) and epistemic uncertainty (model uncertainty) maps, respectively, produced by FibrilNet. Regions predicted with less uncertainty and higher confidence are colored blue. Regions predicted with more uncertainty and lower confidence are colored red. We can see that the main source of uncertainty is the data rather than the model. Specifically, the values in the data uncertainty map in Figure 4(E) range from 0 to 0.246 while the values in the model uncertainty map in Figure 4(F) range from 0 to 0.086. Furthermore, we observe that the ends of a detected fibril are often associated with higher uncertainty. This happens because there is ambiguity surrounding the transition from the fibril body to the nonfibril background area, a finding consistent with that in object detection with uncertainty quantification reported in the literature (Kendall & Gal 2017; Kwon et al. 2020).

Notice also that the map in Figure 4(E) shows higher uncertainty in the noisy region inside white box 2 compared to that in the region outside white box 2. Specifically, in the noisy region inside white box 2, 90% of the values are contained in the range [0.00014 (5%), 0.20528 (95%)]. By contrast, in the region outside white box 2, 90% of the values are contained in the range [0 (5%), 0.18969 (95%)]. Furthermore, it is observed in Figure 4(C) that the tool in Jing et al. (2011) misses some fibril structures with at least 15 pixels inside white box 3. These fibril structures are not present in the mask predicted by FibrilNet either, as shown inside white box 3 in Figure 4(D). Nevertheless, the uncertainty maps of FibrilNet are able to catch and display these missed fibril structures with higher uncertainty by Bayesian inference, as shown inside white box 3 in Figures 4(E) and (F). This finding demonstrates the usefulness of the uncertainty maps, as they not only provide a quantitative way to measure the confidence on each predicted fibril, but also help identify fibril structures that are not detected by the tool in Jing et al. (2011) but are inferred through machine learning. It should be pointed out that the previous fibril-tracing tool in Jing et al. (2011) does not have the capability of producing these uncertainty maps as described here.

Figure 5 compares the orientation angles of the fibrils found by the tool in Jing et al. (2011) and by FibrilNet. The colors of angles between 0° and 90° range from dark blue to green. The colors of angles between 90° and 180° range from green to dark red. It can be seen from Figure 5 that the orientation angles found by the two tools mostly agree with each other, though the angles detected by FibrilNet tend to be smoother. This happens because FibrilNet uses polynomial functions of varying degrees, as opposed to the quadratic function employed by the tool in Jing et al. (2011), to better fit the detected fibrils with different curvatures. Notice also that the quadratic function used by the tool in Jing et al. (2011) may produce wrong angles, which are calculated correctly by the polynomial regression model of FibrilNet. For example, the orientation angle of the fibril at E–W = 425'' and S–N = −190'', which is highlighted by a small red circle in Figure 5, is roughly 90°. It is calculated incorrectly by the tool in Jing et al. (2011), as shown in Figure 5(A). On the other hand, FibrilNet calculates the orientation angle of this fibril correctly, as shown in Figure 5(B). It should be pointed out that the smoother and more accurate orientation angles detected by FibrilNet are due to the better fibril-fitting algorithm used by the tool, as explained above. They are not caused by FibrilNet's Bayesian deep-learning model, whose purpose is mainly image segmentation (i.e., marking each pixel by black indicating the pixel is part of a fibril or by white indicating the pixel is not part of a fibril as shown in the predicted mask in Figure 2) with uncertainty quantification (i.e., producing the uncertainty maps as shown in Figure 4).

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As mentioned above, FibrilNet has two parts: (i) a Bayesian deep-learning model for predicting fibrils with probabilistic image segmentation, and (ii) a fibril-fitting algorithm for determining the orientations of predicted fibrils based on the polynomial regression function in Equation (2). Here, we adopt four measures, defined below, to quantitatively assess the first part, comparing the image segmentation algorithms employed by FibrilNet and the tool (after skeletonization) in Jing et al. (2011), based on the same data from AR 12665 used in Section 4.1. Unlike FibrilNet, which employs deep learning for image segmentation, the tool in Jing et al. (2011) used a threshold-based algorithm rather than machine learning for image segmentation.

Let A (B) denote the set of 720 × 720 = 518,400 pixels in the mask (skeleton) predicted by FibrilNet (calculated by the tool in Jing et al. 2011) for a test image. Let p ∈ A be a pixel in A and let q ∈ B be p's corresponding pixel in B, i.e., q is at the same position as p. We use A ∩_A B to represent a subset of pixels in A such that for each pixel p in A ∩_A B and p's corresponding pixel q in B, p and q are marked by the same color. That is, p and q are both marked by black indicating p and q are part of a fibril in A and B, respectively, or p and q are both marked by white indicating p and q are not part of a fibril in A and B, respectively. Similarly, we use A ∩_B B to represent a subset of pixels in B such that for each pixel q in A ∩_B B and q's corresponding pixel p in A, q and p are marked by the same color. The first quantitative measure is the pixel similarity (PS), also known as the global accuracy (Badrinarayanan et al. 2017), which is defined as

$$\begin{eqnarray}&&\mathrm{PS}=\displaystyle \frac{| A{\cap }_{A}B| +| A{\cap }_{B}B| }{| A| +| B| },\end{eqnarray} \tag{ 3 }$$

where ∣ · ∣ is the cardinality of the indicated set. PS is used to assess the pixel-level similarity between the mask A predicted by FibrilNet and the skeleton B calculated by the tool in Jing et al. (2011) for the test image. The value of PS ranges from 0 to 1. The larger (i.e., the closer to 1) the PS value, the greater the pixel-level similarity between the mask A and the skeleton B.

Let A_F (B_F) denote the set of pixels on the fibrils in A (B). Thus, in A, the pixels in A_F are marked by black while the pixels not in A_F are marked by white. Similarly, in B, the pixels in B_F are marked by black while the pixels not in B_F are marked by white. We use ${A}_{{\rm{F}}}{\cap }_{{A}_{{\rm{F}}}}{B}_{{\rm{F}}}$ to represent a subset of pixels in A_F such that for each black pixel p in ${A}_{{\rm{F}}}{\cap }_{{A}_{{\rm{F}}}}{B}_{{\rm{F}}}$, p's corresponding pixel q is also black, i.e., q is in B_F. Similarly, we use ${A}_{{\rm{F}}}{\cap }_{{B}_{{\rm{F}}}}{B}_{{\rm{F}}}$ to represent a subset of pixels in B_F such that for each black pixel q in ${A}_{{\rm{F}}}{\cap }_{{B}_{{\rm{F}}}}{B}_{{\rm{F}}}$, q's corresponding pixel p is also black, i.e., p is in A_F. The second quantitative measure is the fraction of common fibril pixels (FCFP), defined as

$$\begin{eqnarray}&&\mathrm{FCFP}=\frac{| {A}_{{\rm{F}}}{\cap }_{{A}_{{\rm{F}}}}{B}_{{\rm{F}}}| +| {A}_{{\rm{F}}}{\cap }_{{B}_{{\rm{F}}}}{B}_{{\rm{F}}}| }{| {A}_{{\rm{F}}}| +| {B}_{{\rm{F}}}| }.\end{eqnarray} \tag{ 4 }$$

FCFP is used to measure the pixel-level similarity between the fibrils predicted by FibrilNet and those found by the tool in Jing et al. (2011). The value of FCFP ranges from 0 to 1. The larger (i.e., the closer to 1) the FCFP value, the greater the pixel-level similarity between the fibrils predicted by FibrilNet and those found by the tool in Jing et al. (2011).

The third quantitative measure is the fraction of disjunct fibril pixels (FDFP), defined as

$$\begin{eqnarray}&&\mathrm{FDFP}=1-\mathrm{FCFP}.\end{eqnarray} \tag{ 5 }$$

FDFP is used to measure the pixel-level dissimilarity (distance) between the fibrils predicted by FibrilNet and those found by the tool in Jing et al. (2011). The value of FDFP ranges from 0 to 1. The smaller (i.e., the closer to 0) the FDFP value, the greater the pixel-level similarity between the fibrils predicted by FibrilNet and those found by the tool in Jing et al. (2011).

The fourth quantitative measure is the Rand index (RI; Rand 1971; Unnikrishnan et al. 2005), which calculates the ratio of pairs of pixels whose colors (black or white) are consistent between the mask A predicted by FibrilNet and the skeleton B calculated by the tool in Jing et al. (2011) for the test image. RI accommodates the inherent ambiguity in image segmentation, and provides region sensitivity and compensation for coloring errors near the ends of detected fibrils. For example, consider a wider fibril. FibrilNet may detect the portion to the left of the center of the fibril and highlight this portion by red. The tool in Jing et al. (2011) may detect the portion to the right of the center of the fibril and highlight that portion by red. Under this circumstance, FCFP does not consider there are common pixels between the two red curves, though RI treats the two red curves as consistent curves. Visually the fibril is indeed found by both tools. As a consequence, RI is often used in comparing image segmentation algorithms. The value of RI also ranges from 0 to 1. The larger (i.e., the closer to 1) the RI value, the greater the visual similarity between the fibrils predicted by FibrilNet and those found by the tool in Jing et al. (2011).

Table 1 presents the quantitative measure values of FibrilNet based on the five test images in Figure 1. It can be seen from the table that the mask predicted by FibrilNet and the skeleton calculated by the tool in Jing et al. (2011) are very similar at pixel level, with PS ≥ 95% on the test images. FCFP is about 80%. However, visually, the similarity/consistency between the fibrils predicted by FibrilNet and those found by the tool in Jing et al. (2011) is much greater, where it is quantitatively assessed with RI ≥ 91% on the test images. This finding is consistent with the results presented in Figures 4(C) and (D).

Next, we quantitatively assess the second part of FibrilNet, comparing the fibril-fitting algorithms employed by FibrilNet and the tool (after skeletonization) in Jing et al. (2011), based on the same data from AR 12665 described in Section 4.1. The fibril-fitting algorithms are used to determine the orientations of detected fibrils. Let θ_f represent the fibril orientation angle of a pixel calculated by the polynomial regression function in FibrilNet, and let θ_j represent the fibril orientation angle of the same pixel calculated by the quadratic function in the tool of Jing et al. (2011). The acute angle difference between θ_f and θ_j, denoted as δ(θ_f, θ_j), is defined as

$$\begin{eqnarray}\delta ({\theta }_{{\rm{f}}},{\theta }_{{\rm{j}}})=\left\{\begin{array}{ll}| {\theta }_{{\rm{f}}}-{\theta }_{{\rm{j}}}| & \mathrm{if}\,| {\theta }_{{\rm{f}}}-{\theta }_{{\rm{j}}}| \leqslant {90}^{\circ }\\ 180^\circ -| {\theta }_{{\rm{f}}}-{\theta }_{{\rm{j}}}| & \mathrm{otherwise}\end{array}\right..\end{eqnarray} \tag{ 6 }$$

The angle difference is decided in favor of an acute or right angle, i.e., 0° ≤ δ(θ_f, θ_j) ≤ 90°.

Figure 6 quantitatively compares the orientation angles of common fibril pixels calculated by the fibril-fitting algorithms used in FibrilNet and the tool of Jing et al. (2011) based on the test image at 0.0 Å from the Hα line center 6563 Å with a 70'' circular FOV collected in AR 12665 on 2017 July 13 20:15:58 UT. Figure 6(A) shows a 2D histogram of the orientation angles of common fibril pixels produced by the two tools, where the x-axis (y-axis) represents the orientation angles calculated by FibrilNet (the tool of Jing et al. 2011). The 2D histogram is computed by grouping common fibril pixels whose orientation angles are specified by their x and y coordinates into bins, and counting the common fibril pixels in a bin to compute the color of the tile representing the bin. The width of each bin equals 2°. It can be seen from Figure 6(A) that the orientation angles of common fibril pixels calculated by the two tools mostly agree with each other, which is consistent with the findings shown in Figure 5. Figure 6(B) shows the differences of the orientation angles of common fibril pixels produced by the two tools. It can be seen from Figure 6(B) that most of the common fibril pixels have very small orientation angle differences, displayed by a purple color. For the common fibril pixels with large orientation angle differences, the orientation angles calculated by the quadratic function used in the tool of Jing et al. (2011) are often incorrect (see, for example, the fibrils highlighted by small red circles in Figure 6(B) and Figure 5).

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In this series of experiments, we applied FibrilNet to other types of test images, including (i) a full-disk image from the Global Oscillation Network Group (GONG; Harvey et al. 1996; Plowman & Berger 2020) at the National Solar Observatory, (ii) a full-disk image from the Kanzelhöhe Solar Observatory (KSO; Otruba 1999; Otruba et al. 2008), (iii) high-resolution superpenumbral fibrils from BBSO (Jing et al. 2019), and (iv) two high-resolution quiet-Sun regions from BBSO. The GONG full-disk Learmonth reduced Hα data in (i) was collected on 2015 September 28 00:01:34 UT. The KSO full-disk raw-image Hα data in (ii) was collected on 2015 September 14 09:14:20 UT. The GONG and KSO full-disk images have relatively low resolution. The BBSO superpenumbra of sunspots in (iii) was collected at Hα −0.6 Å from AR 12661 (501E, 95N) on 2017 June 4 19:08:44 UT. The two BBSO quiet-Sun regions in (iv) were collected on 2018 July 29 16:33:12 UT and 2020 June 10 16:10:25 UT at Hα −0.6 Å from (604E, 125S) and Hα 0.0 Å from (283E, 789N), respectively. The FibrilNet tool was trained using the same 241 Hα line center images described in Section 2. Here we present results without uncertainty maps. Results with uncertainty maps can be generated as done in Section 4.1.

Figure 7 shows the fibrils (red curves) predicted by FibrilNet on the GONG and KSO test images. Figure 7(A) presents the GONG full-disk Hα image. Figure 7(B) shows an enlarged view of the region highlighted by the white box in Figure 7(A). In Figure 7(C), we see that FibrilNet detects many fibrils on the GONG image. Figure 7(D) presents the KSO full-disk Hα image. Figure 7(E) shows an enlarged view of the region highlighted by the white box in Figure 7(D). Figure 7(F) clearly demonstrates that FibrilNet detects threads of filaments and fibrils on the KSO image.

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Figure 8 presents the fibril prediction results on the BBSO high-resolution test Hα images. Figure 8(A) shows the BBSO superpenumbra of sunspots image used in the study. It can be seen that there are superpenumbral fibrils around the sunspot in the center of the image. Figure 8(D) shows the predicted superpenumbral fibrils (red curves) produced by FibrilNet on the image in Figure 8(A). We see in Figure 8(D) that FibrilNet can distinguish the superpenumbral fibrils from the clusters of spicules nearby. Figures 8(B) and (C) present the two BBSO quiet-Sun regions. Figures 8(E) and (F) show the predicted mottles in the quiet-Sun rosette structures in Figures 8(B) and (C), respectively. These high-resolution Hα images clearly demonstrate the good fibril prediction capability of our tool.

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