A Novel Technique for the Extraction of Dynamic Events in Extreme Ultraviolet Solar Images

A Novel Technique for the Extraction of Dynamic Events in Extreme Ultraviolet Solar Images Petra Kosová Kalenská 1 , Pavel Rajmic 2 , Karolína Gebrtová 1 , and Miloslav Druckmüller 1 1 Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, 616 69 Brno, Czech Republic; [email protected] 2 Dept. of Telecommunications, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic Received 2024 August 28; revised 2024 October 8; accepted 2024 October 9; published 2024 November 7 Abstract High-spatial-resolution images of the solar corona acquired in the extreme ultraviolet (EUV), most notably with the Atmospheric Imaging Assembly (AIA)instrument on the Solar Dynamics Observatory (SDO)reveal the abundance of dynamic events which range from flaring bright points and jets to erupting prominences and coronal mass ejections (CMEs). In this work we present novel techniques to extract such dynamic events from the more steady background corona using 17.1 nm SDO-AIA images. The techniques presented here treat any time series of coronal images as a matrix that can be decomposed into two matrices representing the background and the dynamic component, respectively. The latter has the properties of a so-called sparse matrix, and the proposed methods are classified as methods based on sparse representations. The proposed methods are the median-filter method, the principal component pursuit, and the dynamic-mode decomposition, all of which include data pre-processing using the noise-adaptive fuzzy equalization method. The study reveals that the median-filter method and the dynamicmode decomposition enhance all motions in the time series and produce similar results. On the other hand, the principal component pursuit enables the clear differentiation of CMEs from the background corona, thus providing a valuable tool for the characterization of their acceleration profiles in the low corona as seen in the EUV. Unified Astronomy Thesaurus concepts: Solar coronal mass ejections (310);Solar corona (1483);Analytical mathematics (38);Astronomy software (1855);Computational methods (1965) 1. Introduction Coronal mass ejections (CMEs)are the largest eruptions of complex plasma and magnetic field structures, originating in the corona and expanding into interplanetary space. They were first discovered in time-series coronagraphic white-light observations (G. F. Moreton 1961; H. E. Ramsey & S. F. Smith 1966). Interest in their dynamical properties stems from their potentially hazardous impact on Earthʼs magnetic environment (D. F. Webb 2000). High-resolution EUV observations such as from Solar Dynamics Observatory’s Atmospheric Imaging Assembly (SDO-AIA)reveal their origin at the Sun and their frequent association with filament eruptions (see the review by D. F. Webb & T. A. Howard 2012). Since CMEs expand throughout the environment of filamentary coronal structures, they cannot be observed in isolation (H. Morgan et al. 2012). Hence to analyze the dynamic behavior and underlying changes in their structures, one has to extract them from the background corona. One software package that has been developed to detect and extract large dynamic events in EUV image time series is described in T. Williams & H. Morgan (2022). There are currently a number of decomposition techniques for time-series observations, referred to hereafter as videos. These include phase correlation, running difference, base difference, running center median, wavelet decomposition etc., which are described in more detail in the next section. However, the background corona is not properly extracted in these techniques. The novel approach proposed here relies on a decomposition technique to achieve this goal. This technique uses methods based on sparse representations for differentiation between CMEs and other phenomena observed in solar videos. Included in this approach is the preprocessing of the data by noise-adaptive fuzzy equalization algorithm (NAFE; M. Druckmüller 2013). In this work, the technique is applied to the 17.1 nm SDO-AIA images where the emergence of CMEs at their sources at the Sun can be explored. These sources are excluded from white-light coronagraphic images, which reveal CMEs once they have already undergone significant evolution. The choice of the 17.1 nm filterband images is dictated by their much higher quality compared to the other AIA bands whose temperature-response curves are not as discriminate as the 17.1 nm one, characterized by a single temperature peak, rather than multiple ones. This paper is organized as follows: Section 2presents an overview of existing decomposition techniques, and their limitations. The concept behind the proposed image decomposition techniques, namely the median-filter method (MFM),principal component pursuit (PCP), and dynamic-mode decomposition (DMD), are given in Section 3.(All abbreviations used throughout this paper are summarized in Table 1.)The preprocessing (NAFE) and postprocessing enhancing techniques (adaptive histogram equalization)are described in Section 4.Thisisfollowedby comparison of all three methods in Section 5. A discussion of the results and their application for CME acceleration measurement are in Section 6. The summary of the main results and potential future work are given in Section 7. 2. Overview of Existing Decomposition Techniques Several techniques for the extraction of CMEs from the background corona have been proposed. Here we give a brief overview of the most commonly used ones. The Astrophysical Journal Supplement Series, 275:15 (10pp), 2024 November https://doi.org/10.3847/1538-4365/ad8633 © 2024. The Author(s). Published by the American Astronomical Society. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s)and the title of the work, journal citation and DOI. 1 The phase correlation method is one of the simpler methods. It is based on finding the shift between two images using the Fourier transform (M. Peeters 2003). Unfortunately, phase correlation is limited only to videos with entirely static backgrounds. Modification of this method was proposed by V. Argyriou & T. Vlachos (2005). Individual frames of the video are decomposed into squares of different sizes according to the estimated movement. The phase correlation is then applied only to these segments. Another approach, called phaseamplified correlation, can be found in D. Konstantinidis et al. (2019). This method integrates motion magnification, or more precisely the Fourier transform of an image of spatial translation magnified by a specific factor, which causes this method to be able to identify very subtle movements in the video. The weakness of methods based on phase correlation is the inability to work with videos that contain different dynamic motions in the foreground and the background since the dominant peaks in the phase spectrum do not have to be the relevant ones. Consequently, the results of phase correlation are not applicable without a perfect knowledge of the phenomenon in the video and their effect on the shape of the phase correlation. The concept of optical flow is another approach to handle object movement in a time series of images. This approach is best described by B. K. P. Horn & B. G. Schunck (1981)as the distribution of the velocity of specific elements in the series of images is based on their brightness. The methods for estimation of the optical flow use partial derivatives of the image brightness in time. The Lucas–Kanade method is based on the affine model of the flow (G. Zhang & H. Chanson 2018). The Horn–Schunck method is based on preserving the brightness and smoothness of the flow (G. Zhang & H. Chanson 2018), and the Buxton–Buxton method is based on edge detection and motion tracking of these edges (V. Bruce & G. W. Humphreys 1989). One shortcoming of these methods, however, is the assumption of brightness consistency through the series of images, which is not fulfilled in a time series of solar observations. The running-difference method is a well-established technique for separating components of solar observation videos (B. J. Thompson et al. 1999; Sheeley et al. 2014; J. Yin et al. 2017). It is based on the subtraction of the previous image in time from the current image. This method has the advantage of being very easy to implement with a clear mathematical definition. However, it has two disadvantages. First, this method is sensitive to a wide variety of signals, making it difficult to interpret the results (J. Plowman 2016). Second, it shows a time derivative rather than the true CME structure, making it difficult to use for structural interpretation of CMEs (H. Morgan et al. 2012). An example of this technique is shown in Figure 1. It should be noted that the decomposed image shows motions in the inner corona other than the desired CMEs. The base-difference images method addresses the issue of running differences to a certain extent, as evidenced by the findings of D. M. Long et al. (2014). The method involves selecting a frame from the time series and using it as a reference image. The differences between this reference image and each of the other frames are then computed. This process allows for the identification of whether the image returns to the background level or not. However, this method is limited by its sensitivity to any drift of the image with respect to the reference. Consequently, after a certain period of time, the ability to identify small and rapid variations is reduced. Figure 1. Example of the running-difference image applied to the image sequence (left and middle)taken by SDO-AIA 17.1 nm on 2012 April 16, solar north is to the right. The right-hand image shows the result of the running-difference method, the gray color represents the background, and the black pixels represent the different brightness levels between the two images. It can be seen that the black pixels occur in the whole image, which means that this method separates all motions in the images, not only the CME. Table 1 List of Abbreviations Abbreviation Meaning CME Coronal mass ejection SDO Solar Dynamics Observatory AIA Atmospheric Imaging Assembly NAFE Noise-adaptive fuzzy equalization MFM Median-filter method PCP Principal component pursuit DMD Dynamical mode decomposition SVD Singular value decomposition ADMM Alternating direction method of multipliers 2 The Astrophysical Journal Supplement Series, 275:15 (10pp), 2024 November Kalenská et al. One method based on the previous two is called the running center median filter (J. Plowman 2016). It calculates the difference between the current image and the per-pixel median of all images falling within a given time interval of the current image. The elements of the series falling in this interval are referred to as a window. Using the median avoids the limitations of the running-difference and base-difference methods described above. The principal limitation of this method is its relative insensitivity in comparison to the running difference. Indeed, this method is sensitive to transient variation from timescales of approximately half the window width down to the frame-to-frame scale, whereas the running difference is most effective for detecting variations at the halfwindow-width scale. The multiscale wavelet decomposition, which is an alternative approach, is also effective in revealing CME structure (G. Stenborg & P. J. Cobelli 2003). It is a more sophisticated method than running differences, but unfortunately it also enhances other high-frequency components, such as the edges of streamers. The method presented by H. Morgan et al. (2012)aims to improve the automated detection and cataloging of CMEs from coronagraph images. The described process is based on an iterative deconvolution in time and space that aims to isolate the CME from the background in a white-light image, resulting in separate images of the dynamic and quiescent components of the corona. According to H. Morgan et al. (2012), one of the limitations of this technique is that any feature in the coronagraphic data that is not smooth in the radial and temporal dimensions, e.g., noise, will be seen in the final separated CME component. Even very faint dynamical events are therefore present in the CME component images. 3. Overview of the Proposed Decomposition Technique The concept and methodology underlying the proposed decomposition technique explored in this work are described in what follows. 3.1. Preprocessing of the Data The data utilized are first processed. Assume that the input is a composite of Îmconsecutive images. Each image can be interpreted as a matrix of pixel values, M i , for Îá ñim1, . The matrix M, which will be used in the decomposition technique is obtained as follows: []()=MMMMvec , vec , ,vec , 1 m12 where Îá ñMimvec , 1, istands for the vectorization of the matrix, i.e., making a single column vector out of the matrix by concatenation. Now each row of the matrix Mrepresents values of one pixel in the course of time. 3.2. Basic Idea Assuming that there are no rapid motions in the video, the columns of matrix Mare alike and the matrix is recognized as a so-called low-rank matrix (G. Allaire & S. M. Kaber 2008).On the other hand, if there is a movement, a few pixels per row will differ from the surroundings. If only those pixels are kept and the rest are changed to zero, then the generated matrix is called a sparse matrix (G. Allaire & S. M. Kaber 2008). Based on these ideas the matrix Mcan be decomposed to a low-rank matrix L(static component)and a sparse matrix S(dynamic component)such that M=L+S. Thus the suitable methods for our purpose could be the ones based on sparse representations. There are several approaches to obtain the matrices Land S, e.g., MFM, DMD, and PCP. Only the last one is effective for the separation of CMEs from solar observation videos, as described in the following sections. The other two are briefly discussed below, with the detailed descriptions in P. Kalenska (2024). In the following, the matrix Lshall be referred to as the “background matrix.” 3.3. MFM: Median-filter Method The MFM is the easiest way to decompose matrix Minto the low-rank and sparse matrices. The classical model works with the assumption that the columns of the matrix Mare alike and the only differences between them are elements of matrix S (dynamic component). It also assumes that individual pixels have the same value in more than half of the frames. Then, since the median is the value that halves a series of ascending data, the background matrix Lis obtained as the median over rows of M, and the matrix Sis a difference of Mand L, i.e., S=M−L. The primary benefit of applying a median filter is its ability to withstand extreme values, which is a result of the median’s inherent properties. 3.4. DMD: Dynamical Mode Decomposition The DMD method (J. Grosek & J. N. Kutz 2014; S. L. S. L. Brunton 2019)is a recent, data-driven method, first used in hydromechanics. The main concept of DMD is to simplify the intricate workings of a system using a linear model. This allows us to extract the key modes that are responsible for its behavior. DMD derives these modes and their respective frequencies by taking measurements that depict the system’s behavior across time. By analyzing these dominant modes, we gain insight into the dynamics of the system and can even predict its future behavior. 3.5. PCP: Principal Component Pursuit 3.5.1. Basics of the PCP Method This simultaneously minimizes the number of different columns in the matrix L, and the number of nonzero elements in the matrix S. For our data set, this method calculates the matrices L(background)and S(dynamic component)in parallel. In other words, it tries to minimize both the almost static motion associated with the background and the fast motion associated with the dynamic component. Mathematically speaking, the problem of decomposing M into the low-rank matrix Land sparse matrix Scan be written as the following optimization problem: () ()++=LS LSMmin rank subject to , 2 LS,0 since the ℓ 0 norm (∥·∥ 0 )implies the number of nonzero elements of the matrix and rank corresponds to the maximal number of linearly independent columns (G. Allaire & S. M. Kaber 2008). Unfortunately, both rank and ℓ 0 norm make this problem nonconvex. Therefore, it cannot be solved in a feasible time. According to E. J. Candès et al. (2011)the low-rank plus sparse decomposition (Equation (2)) can be approximated by 3 The Astrophysical Journal Supplement Series, 275:15 (10pp), 2024 November Kalenská et al. the convex optimization problem ()  *l++=LS LSMmin subject to , 3 LS,1 where ∥L∥ * denotes the nuclear norm of the matrix L (L. N. L. N. Trefethen 1997), which is defined as the sum of the singular values of the matrix L, and ∥S∥ 1 denotes the ℓ 1 norm of the matrix (L. N. L. N. Trefethen 1997), which is defined as the sum of the absolute value of each element from matrix S. Finally, λis a regularization parameter. BasedonE.J.Candèsetal.(2011), Equation (3)has an exact solution under the assumption that Mis not low rank and sparse simultaneously. Further, it is found that there exists a universal λ, precisely l =n 1,where ()= n mnmax , for matrix L m,n .The presence of this universal tuning parameter λrepresents a significant advantage of the PCP algorithm. This is because the parameter guarantees the convergence of the problem, regardless of the matrices Land S. One disadvantage of the PCP algorithm is the high time consumption caused by the computation of the singular value decomposition (SVD)at each iteration. 3.5.2. The PCP Algorithm The PCP algorithm is derived from the alternating direction method of multipliers (ADMM)applied to Equation (3). The principle of ADMM is described, for example, in P. Záviška et al. (2018). The method exploits the augmented Lagrangian, which for problem (3)takes the form () ()    *l m =+ +á--ñ +-- mLSY L S YM L S MLS L,, , 2,4 1 F 2 where Yis a matrix of the Lagrange multipliers, ·· á ñ,is the scalar product, μis the step size, and ∥·∥ F denotes the Frobenius norm, which is a matrix analogy of the Euclidean norm (G. Allaire & S. M. Kaber 2008). Equation (4)can be reformulated to represent the augmented Lagrangian in its scaled form, which is more appropriate for the following calculations: () ()      *lmm mm =+ + --+ - mLSY L S M L S Y Y L,, 2 2. 5 12 2 2 2 Since Equation (5)is a convex problem, the ADMM is eligible to find the global minimum. The ADMM design for solving the problem (5)is ⎛ ⎝ ) () ( ) () ()  * mm=--++ + 6 LMLSLYarg min 2 L kkk1 2 2 ⎛ ⎝ ) () () () ()  mml=--++ ++ 7 SMLSSYarg min 2 S kkk11 2 21 ()() () () () () m=+ - - +++ YY MLS.8 kk kk111 The (k) in the ADMM design represents the iteration counter. Proximal operators can be utilized for numerically solving convex optimization problems (P. L. Combettes & J.-C. Pesquet 2011). In the case of problem (3),soft thresholding and singular value thresholding are utilized, which correspond to ∥·∥ 1 and ∥·∥ * , respectively. Their definitions are as follows: () ∣∣ (∣ ∣ ) ( )tt=-> txx xxsoft max , 0 ; 0 9 ii i i () () ( )* ås= tt = Xuvsvt soft , 10 j n jjj 1 where τis a regularization parameter. In Equation (10),u j and v j * are the vectors of matrices U,V * stemming from the SVD of X, i.e., X=UΣV * , and σ j is the singular value of X, and therefore the element on the diagonal of the matrix Σ (L. N. L. N. Trefethen 1997). Now by using the proximal operators (9)and (10)the PCP algorithm in Algorithm 1is obtained. (Based on E. J. Candès et al. 2011.) Algorithm 1. Principal Component Pursuit Result: L, S initialization: m== >SY0, 0 00 ; while  d-- > ++ ML S M kk11F F do ()m=-+ +m LMSYsvt kkk11; ( ) m=-+ ++ l m SMLYsoft kkk11 ; ( ) m=+ - - +++ YY ML S kk kk111 ; end The PCP algorithm contains two optional parameters, namely the accuracy parameter δand step size μ. According to E. J. Candès et al. (2011), the suitable choices are ()d=- 10 , 11 5 ()  m=M M mn 4,for . 12 mn 1 , 4. Preprocessing and Postprocessing of EUV Images 4.1. Data The MFM, DMD, and PCP methods were implemented in MATLAB R2019b. Input data were series of level 1 images from SDO-AIA. The DN units were not changed, since they do not have an impact on the results. We use a time series of 500 consecutive 4096 ×4096 pixel images; thus one video is approximately 67 GB. Because of the limited size of RAM on our computer (see Table 4in the Appendix), only one-quarter of each image is used, i.e., the methods are computed on sections of size 2048 ×2048 pixels. As noted by M. Druckmüller (2013), the most suitable images for our approach are from the 17.1 nm AIA wavelength channel. All three methods were tested on three dates containing CMEs, namely from 2011 June 7, 2012 April 16, and 2014 October 2. Information about the data set is presented in Table 2. 4.2. NAFE Enhancement SDO-AIA images are typically characterized by the extreme contrast between bright features in active regions and dark backgrounds containing low-contrast structures (M. Druckmüller 2013). In order to fully exploit the information contained in 4 The Astrophysical Journal Supplement Series, 275:15 (10pp), 2024 November Kalenská et al. these images, we preprocessed them using the NAFE. NAFE is particularly suitable for SDO-AIA images with an extremely high dynamic range. This method visualizes fine structures in EUV images of the corona and produces artifact-free images. The output of this method can be influenced by three parameters, γ,w, and σ. The γparameter affects the brightness of the image. Lower values for γresult in darker images and higher values result in brighter images. The constant wis called the NAFE weight and its specification allows you to control the enhancement of the structures. The σis the standard deviation of the Gaussian distribution of the added noise. The higher the value of σ, the lower the amount of noise in the low-contrast parts of the image. It is not correct to suppress the noise too much because the low-contrast details will be lost. A more detailed description is given by M. Druckmüller (2013). The values we have chosen for these parameters are written in Table 3. The preprocessing of the data was done as a linear combination of images from our data set and images enhanced by NAFE. Thus, in the following way ·()· ()=+-MM Mww1, 13 new NAFE where the value of wwas chosen as 1, 0.8, 0.6, 0.4, 0.2, and 0. The images M NAFE and Mare shown in Figure 2. 4.3. Postprocessing of the Final Images The images that have undergone preprocessing in MATLAB are stored in format uint8. Unfortunately during the computation, the data have to be converted into format double, since the algorithm contains subtraction and division. Therefore, the output data needs to be transformed back into the interval [] 0, 255 . The transformation is done as follows: ⎧ ⎨ ⎪ ⎩ ⎪·()= < - -<< > F BFB FB WB BF W WFW if , 255 if , if , 14 ij ij ij ij ij , , , , , where Fstands for matrices M,L, and S.Bis the 0.001 quantile from Fand Wis the 0.999 quantile from F. The resulting videos lack depth, thus the accuracy of the methods cannot be sufficiently evaluated. Hence, it is necessary to apply a contrast enhancement method. First we applied the method described in H. Morgan & M. Druckmüller (2014), which is based on enhancing the images by using the arctan function. This method was found to be unsuitable as it resulted in the suppression of those areas of the image where significant changes were occurring. It can be reasonably assumed, therefore, that the optimal method is the one that makes optimal use of all available brightness levels. Therefore, we have chosen to implement the adaptive histogram equalization by W. K. Pratt (2007). The histogram equalization process alters the histogram of the image to make it as uniform as possible. The adaptivity of this process is based on the progressive modification of the histogram on segments of a given size, which is a square of size 50 ×50 pixels in our case. A comparison between the resulting image with and without the use of an adaptive histogram is shown in Figure 3. 5. Comparison between the DMD, MFM, and PCP A quantitative comparison between the three methods is illustrated in the histograms presented in Figure 4. These have been computed from the images that have undergone the postprocessing transformation described in Equation (14). Consequently, the “background”after the decomposition is not equal to zero, but is shifted to higher values. (The word Table 2 Data Set Specifications Parameter Value Observation time 1 hr and 40 minutes Cadence 12 s Number of frames 500 Size of real frame 4096 ×4096 Size of used frame 2048 ×2048 2011 Jun 7 Acquisition dates 2012 Apr 16 2014 Oct 2 Figure 2. Comparison between an unprocessed image (left)and an NAFEprocessed image (right). The images were taken by SDO-AIA 17.1 nm on 2012 April 16 at 17:00:00 UTC. Figure 3. Comparison between resulting images with (left)and without (right) the use of adaptive histogram equalization. The images are from SDO-AIA 17.1 nm on 2012 April 16 at 17:33:50 UTC. Table 3 NAFE Parameter Selection Parameter Value γ2.6 w0.2 σ15 5 The Astrophysical Journal Supplement Series, 275:15 (10pp), 2024 November Kalenská et al. background used in this paragraph is the image-processing terminology, it has nothing to do with our matrix L). The PCP histogram is dominated by the “background”values given by the very sharp peak between pixel values 150 and 200, with only a few corresponding to the main dynamic event (at pixel values 50 and 250). In contrast, the histograms of MFM and DMD demonstrate a greater dispersion in pixel values, indicating a significant contribution from a larger number of small-scale dynamic events other than the primary CMEs. In other words, the PCP histogram allows one to readily identify the threshold between the residual and the event. This is not possible with MFM and DMD. A qualitative assessment of the accuracy and quality of each method is conducted exclusively through visual analysis and is based on the observation of whether large dynamic events appear as isolated structures. The use of objective methods is precluded due to the inability to precisely define the correctly extracted object. The comparison of all methods is shown in the videos given in Figure 5. The black and white pixels observed in the dynamic component images cannot be associated with any tangible physical properties. They merely demonstrate the most significant changes in the dynamic events. The MFM output in the video is a single image of the background (i.e., a constant video), and a video of the dynamic component. It can be seen in the videos in Figure 5that this method also detects additional motions in the inner corona besides the desired CMEs, so it does not bring any improvement compared to the running-difference method frequently used, even with or without the NAFE enhancement, precisely for w=0.2 and w=0. The other weights cause deterioration of the results since the noise is now more in contrast and the method evaluates it as a movement. The computational time of this method is 18–20 s. The DMD appears to produce comparable results to those obtained with MFM. Figure 5shows that DMD also identifies other motions in the inner corona. Furthermore, just like for the MFM, NAFE enhancement does not improve the results. The best outcome is obtained from unprocessed images. NAFE processing causes deterioration of the results because of enhanced noise. The computational time of this method is 200–300 s (approximately 3–5 minutes). However, one of the the key advantages of DMD over PCP is its relatively low computational complexity. Its disadvantage is its high noise sensitivity, as discussed by Z. Wu et al. (2021). With the PCP method large dynamic events appear as isolated structures, as can be seen in Figure 5.Beforethe occurrence of the CME, the algorithm searches for the fastest movement, so it highlights the noise in the image. During the eruption, the noise almost disappears and the resulting images contain only the CME and few fast movements in the inner corona. It is worth noting that the results do not depend on the position of the CME in the video. The same outcome is found whether the CME is observed entirely in projection on the solar disk or extends off the limb. Careful inspection of the PCP video shows the coronal activity preceding, during, and following the CME that is not evident in the original videos. For the PCP method, NAFE enhancement improves the results, unlike for the DMD and MFM techniques. As can be seen in Figure 6the best outcome is for w=0.2. The movements in the inner corona are better disregarded for w=0.4, but the CME in this case is not correctly separated (the lower part is missing). The computational time of PCP is heavily dependent on the number of iterations. One iteration takes 80–100 s. In most cases, the PCP algorithm terminated after four iterations, corresponding to a computational time of around 350 s (approximately 6 minutes). The deviation from this time occurs for the video from 2012 April 16 with Figure 4. Histograms of the number of occurrences of a given pixel value vs. pixel value computed from the 250th video frame from 2012 April 16, taken from the resulting video of dynamic component (matrix S). The scale on the left correlates with the blue histogram, while the right logarithmic scale correlates with the gray histogram. 6 The Astrophysical Journal Supplement Series, 275:15 (10pp), 2024 November Kalenská et al. Figure 5. Comparison of MFM, PCP, and DMD. Upper images in each year correspond to the background matrix (L), the lower images correspond to matrix of dynamic component (S). The 1.7 minute duration videos can be obtained from Zenodo at doi:10.5281/zenodo.13919522. They cover 1.7 hr of observations starting at 06:00:00 UTC on 2011 June 7, 16:50:00 UTC on 2012 April 16, and 18:30:11 UTC on 2014 October 2. 7 The Astrophysical Journal Supplement Series, 275:15 (10pp), 2024 November Kalenská et al. w=0.2, its computational time is 10,542 s (approximately 3hr), and the algorithm finished after 104 iterations. 6. Discussion All the examples presented so far for the three different methods pertained to 17.1 nm AIA images. However, all three methods are applicable to the other AIA filter bands. In principle, the He II 30.4 nm should be particularly attractive given its sensitivity to emission between 10 4 and 4 ×10 5 K, where prominences, which are often associated with CMEs, are most visible. (The results of applying these methods to the He II 30.4 nm band are given in P. Kalenska 2024.)Unfortunately, the He II 30.4 nm filter band is unable to properly distinguish between prominence and coronal material because of its very broad temperature-response curve extending to >10 7 K(see P. Antolin et al. 2024). This is illustrated in the comparison of applying PCP to 17.1 and 30.4 nm, shown in Figure 7for the same dates studied earlier, with the exception of 2011 when the scanning cadence for 30.4 nm in 2011 was one frame per minute, while the scanning cadence for 17.1 nm remained consistent at 12 s per frame per year. (Note that the white vertical line observed intermittently in the video depicted in Figure 7is attributed to the absence of an antiblooming system on the chip.) These two examples show clear differences between the dynamic behavior in 30.4 and 17.1 nm. In 2012 (left), the spatial extent of the 30.4 nm emission is much smaller than 17.1 nm, which could be due to hotter, low-lying loops with plasma temperatures >5×10 6 K, whose emission dominates the 30.4 nm band. They would not be prominent in 17.1 nm. On the other hand the 2014 (right)event is different. Here the rise of the strong cool prominence emission in 30.4 nm is absent in 17.1 nm, which is instead tracing the dynamic evolution of the edge of the CME loop. One straightforward application of the PCP algorithm, other than the extraction of CMEs, is the measurement of their acceleration profiles, as illustrated in Figure 8. This application is of particular importance for studies of the dynamics of CMEs at their sources in the low corona below heliocentric distances of 1.5 R s . The distances were measured manually, without the use of automated tools, and are only intended for illustrative purposes. The interpolation line corresponds to a second-order polynomial. It is clear that each CME has a different acceleration profile. Of particular interest is the 2011 CME with the fastest acceleration, which, however, ended up being erroneously labeled as a failed event. 3 It can be seen in video at http:// www.zam.fme.vutbr.cz/~druck/SDO/Pm-nafe/2011_06_07/ 0-info.htm, and also in the videos presented here that part of the CME returned back to the Sun, while other parts continued along their original trajectory. 7. Summary and Conclusions This paper employs methods based on sparse representation to separate CMEs from the background corona in time-series images taken by SDO-AIA. While these methods are widely known, their application to this specific type of video has not been explored before. The problem with the currently dominant method, known as the running-difference method, is that it detects also additional activity within the inner corona. The MFM is the fastest proposed method. Regrettably, it is not effective for videos exhibiting disparate movement dynamics. It separates all motions in the video, not merely the desired CMEs. Nevertheless, if the objective of a given study is to ascertain all behavior associated with the corona during eruptions, this method is a viable option. The DMD algorithm produces comparable outcomes to the MFM. While the CME is decomposed, the slower movements within the inner corona remain in the decomposition. This method yields slightly better results than MFM, as the matrix representing the nearly static motions, referred to as the background, is not a constant but rather exhibits slight temporal variation. As with the MFM, this approach can be employed to ascertain all behaviors associated with the corona during eruptions. The PCP algorithm decomposes the video effectively as it separates the fastest dynamic image component. Unfortunately, the resulting video includes motion in the inner corona, which has the same velocity as the CME, because the velocity of the CME is not always unique at the time the image is taken. This technique resolves the issue of the running-difference method primarily during the fastest CME movements. Most importantly, the method yields superior outcomes compared to the state-of-the-art method. The PCP method reveals only those changes that are significant, whereas the running-difference method reveals all motions. The only drawback of this method is a high computational time. One limitation of this work is the inability to employ objective methods for assessing accuracy, as the results cannot be compared with any external standard. This issue will be addressed in future work employing machine learning techniques, with the results of the PCP algorithm serving as a reference. The next approach for future work is to find the correlation between the CME and other small dynamic events on the Sun using the results of all three proposed methods. Figure 6. Comparison of outcomes of the PCP method with different NAFE enhancement weights. It shows the 250th frame from the video of the dynamic component. The input images were taken by SDO-AIA 17.1 nm on 2012 April 16. See text for more details. 3 https://www.nasa.gov/image-article/spectacular-solar-cme-06-07-2011/ 8 The Astrophysical Journal Supplement Series, 275:15 (10pp), 2024 November Kalenská et al. Acknowledgments We thank the anonymous reviewer for their careful reading of our manuscript and their many insightful comments and suggestions. The images were used with courtesy of NASA/ SDO and the AIA, EVE, and HMI science teams. The work of P.K. and M.D. was supported by the Grant Agency of Brno University of Technology, project No. FSI-S-23-8161. Source Codes and Data The source codes for MFM, PCP, and DMD can be downloaded from P. Kalenska (2024). Testing images from SDO-AIA can be obtained from https://sdac.virtualsolar.org/cgi/search. NAFE-enhanced images can be obtained from http://www. zam.fme.vutbr.cz/~druck/Nafe/Index.htm. Appendix Computer Specifications Table 4shows the computer specification used for the calculations. The RAM specification is also a minimum requirement for reproducing results on the same data set. Figure 7. Comparison of PCP results applied to the 30.4 and 17.1 nm AIA bandpasses. The 1 minute and 50 s duration videos starting at 16:50:00 UTC on 2012 April 16 and 18:30:11 UTC on 2014 October 2, respectively, cover 1 hr of observations. They can be obtained from from Zenodo at doi: 10.5281/zenodo.13919522. Figure 8. Plot of heliocentric distance vs. time for the three CMEs presented in this work. The dotted lines represent the interpolation of a second-degree polynomial for each date. Each CME has a different acceleration profile, with the 2011 case eventually turning into a failed CME. 9 The Astrophysical Journal Supplement Series, 275:15 (10pp), 2024 November Kalenská et al.