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[Fumihiko Uesugi](https://orcid.org/0000-0003-3346-4218), [Masashi Ishii](https://orcid.org/0000-0003-0357-2832)

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[Classification for transmission electron microscope images from different amorphous states using persistent homology](https://mdr.nims.go.jp/datasets/c906f5c4-7e26-475c-8477-f6e6840503ea)

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Microscopy, 2022, 71(3), 161–168DOI: https://doi.org/10.1093/jmicro/dfac008Advance Access Publication Date: 16 February 2022ArticleClassification for transmission electron microscope imagesfrom different amorphous states using persistenthomologyFumihiko Uesugi1,* and Masashi Ishii21Electron Microscopy Analysis Station, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan2Materials Data Platform Center, 1-1 Namiki Tsukuba Ibaraki 305-0044, Japan*To whom correspondence should be addressed. E-mail: UESUGI.Fumihiko@nims.go.jpAbstractIt is difficult to discriminate the amorphous state using a transmission electron microscope (TEM). We discriminated different amorphous stateson TEM images using persistent homology, which is a mathematical analysis technique that employs the homology concept and focuses on‘holes’. The structural models of the different amorphous states, that is, amorphous and liquid states, were created using classical moleculardynamic simulation. TEM images in several defocus conditions were simulated by themulti-slice method using the created amorphous and liquidstates, and their persistent diagrams were calculated. Finally, logistic regression and support vector classification machine learning algorithmswere applied for discrimination. Consequently, we found that the amorphous and liquid phases can be discriminated bymore than 85%. Becausethe contrast of TEM images depends on sample thickness, focus, lens aberration, etc., radial distribution function cannot be classified; however,the persistent homology can discriminate different amorphous states in a wide focus range.Key words: amorphous structure, TEM image simulation, GaN, persistent homologyIntroductionInterpreting amorphous images using transmission electronmicroscopy (TEM) is difficult owing to the lack of periodicityor symmetry. When analyzing a crystal structure with peri-odicity or symmetry using TEM, features can be extractedrelatively easily using the Fourier transform. The contrastobtained in the TEM image significantly changes depend-ing on the defocus, aberrations and sample thickness. Inthe analysis using the diffractogram obtained by a Fouriertransform of the TEM images, the effects of these defocusand aberrations cannot be well distinguished. In the anal-ysis of amorphous structures, the structure may be deter-mined by performing a radial distribution function (RDF)analysis on the halo pattern obtained using selected areadiffraction or nanobeam diffraction. To evaluate a smallarea, a nanobeam or angstrom beam that converges the elec-tron beam is used; thus, the sample may be damaged, andit is unlikely that data showing the original structure canbe obtained. Therefore, for material analysis, it is impor-tant to have a method that can identify regions with dif-ferent sample states from TEM images with lower electronbeam density and less damage to the sample than that ofnanobeams.Persistent homology (PH) is a concept of mathematicalhomology and is a data analysis method focusing on ‘holes’[1,2]. Using PH, extracting information quantitatively in theform of data becomes possible. In PH, the circles are contin-uously enlarged from particular points scattered in the space.When the circles come into contact with each other and forma ‘hole’, the time of occurrence (birth time) and the time whenthe circles are further enlarged to form the inner ring increases.The time when it disappears is recorded as the death time.The graph exhibiting this is called a persistent diagram (PD).The PD represents the birth–death time as points scatteredfrom the diagonal of the graph. Differences in the degree ofdispersion of the starting points appeared in the PD. Its appli-cations are being promoted in fields such as material science[3], molecular genetics and biochemistry. Algorithms suitablefor machine learning (ML) using PD have also been devel-oped, and Obayashi et al. developed and published them asHomCloud [4].In the present work, we examinedwhether the TEM imagesof the amorphous state can be distinguished from imagesof liquids using PH. Because this is the first attempt, weconsidered the ideal state of a binary problem. Specifically,amorphous and liquid structures were created using classi-cal molecular dynamics (MD) calculations. TEM simulationimages were created using this output, andMLwas performedusing the PD obtained from the images. The samples wererun on a binary GaN compound. Consequently, the accu-racy was greater than 80%. It was found that PH is effectivefor identifying amorphous TEM images, such as amorphousand liquid phases. PH is often used for the analysis of three-dimensional data, such as the amorphous state, but in thisstudy, we showed that it is also effective for two-dimensionaldata, such as TEM images.Received 8 November 2021; Revised 9 February 2022; Editorial Decision 13 February 2022; Accepted 15 February 2022© The Author(s) 2022. Published by Oxford University Press on behalf of The Japanese Society of Microscopy.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/),which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.Downloaded from https://academic.oup.com/jmicro/article/71/3/161/6529715 by NATIONAL INSTITUTE FOR MATERIALS SCIENCE user on 02 July 2024https://orcid.org/0000-0003-3346-4218https://orcid.org/0000-0003-0357-2832mailto:UESUGI.Fumihiko@nims.go.jphttps://creativecommons.org/licenses/by/4.0/162 Microscopy, 2022, Vol. 71, No. 3Fig. 1. Results of molecular dynamics of 1000K, 4000K and 6000Ktreatment. Perspective views (left column), TEM images (center) and PD(right) are shown. In the same temperature, the upper row is without RTtreatment while the lower one is with RT treatment.Data preparationGeneration of amorphous and liquid structureusing molecular dynamics simulationAmorphous and liquid structure data for TEM simula-tion were generated using a classical MD simulation code,LAMMPS [5,6], which is distributed by Sandia National Lab-oratories. LAMMPS was operated via Pyiron [7,8], which isa Python library created by Janssen of Max Planck Institutionand executed on a PC.The present MD simulations adapted the whole direc-tion periodic boundary condition and the isothermal–isobaricensemble, which is suitable for structural transition, glasstransition, crystallization, melting simulation, etc. The pres-sure and temperature of the model were kept constant underthe isothermal–isobaric ensemble condition. Wurtzite GaN[9] crystals (29 × 24 × 10) were prepared as input data becausean area of 50Å×50Å× 50Å was used for the TEM simu-lation. The Tersoff style [10] potential presented by Nordet al. [11] was adopted. The MD simulation for creatingmelting conditions was conducted under several temperaturesbetween 1000K and 6000K. The calculation time was 10 000steps (1 step=1[fs]). Additionally, 5000 steps at 300K toFig. 2. Atomic arrangements of liquid (a) and amorphous (b). Optimalvolumes corresponding to the peak position indicated by red arrow in theright bottom PD of Fig. 1 are indicated with green and red lines. Forvisibility, atomic radii are made smaller than the calculated value.Fig. 3. AADI of simulated TEM images in various defocus. Solid lines areliquid (heat treatment without RT), and dashed lines are amorphous(with RT).produce an amorphous structure (for quench treatment) wereadded to the above liquid conditions. The MD temperatureis unrealistic, and the melting temperature is higher than theactual one [12,13].Perspective views of the 20Å×20Å×20Å center regionof the calculated structure in the [001] direction are shownin the left column of Fig. 1. It was found that the crystalstructure and symmetry were maintained except for 6000K,but as the temperature increased from 1000 to 4000K,the structural disorder also increased. In structures treatedwith additional heat treatment at room temperature (RT),there was less disturbance and the crystal structure wasalmost normal. However, the symmetry of the structuretreated at 6000K was completely disturbed, which cannotbe assumed to have a crystal structure. Furthermore, struc-tures treated with additional heat treatment at RT couldnot recover the crystal structure. In the case of 6000K,the structure in which additional RT treatment was added(or not) could not be distinguished from the formerstructure. Alternatively, the amorphous or liquid struc-tures cannot be distinguished only from these perspectiveviews.Generation of TEM simulation imagesA commercial soft electron beam and image simulator (ELBIS)[14], which adopts the multi-slice method [15] and transmis-sion cross coefficient (TCC) [16,17], was used to generateDownloaded from https://academic.oup.com/jmicro/article/71/3/161/6529715 by NATIONAL INSTITUTE FOR MATERIALS SCIENCE user on 02 July 2024F. Uesugi and M. Ishii Classification for transmission electron microscope images 163TEM simulation images. The input sample size for MLwas 50Å×50Å×50Å, and the simulated image size was1024 × 1024 pixels (0.0488Å/pixel) in nine defocus condi-tions as follows: −100, −50, −25, −13, 0, +13, +25,+50 and +100Å. The defocus range of the present studyis thought to be valid because the focus is adjustable inthe defocus range in the case of Cs-corrected TEM; more-over, the TEM image contrast changes largely and inversesat defocus 0Å. Small images for ML were cut into 256 × 256pixels from the original simulated image. The slice thick-ness used to calculate the potential map is 1Å. The Cs andCc aberrations were −0.00050mm and 1.37mm, respec-tively. The other higher aberration coefficients were zero.The calculated images were saved in a 16-bit Tiff for-mat. Because the Homcloud library uses 8 bit in grayscale,these images were transformed to an 8-bit format using thecontrast limited adaptive histogram equalization (OpenCV)library.One of the simulated images using former MD resultsis shown in the center column of Fig. 1. The image sizewas 20Å×20Å in the [001] direction same as the perspec-tive view. The trend of the simulated TEM image changewith temperature was similar to that of perspective views.Regardless of RT treatment, TEM images from MD resultstreated at 1000K and 4000K had a small disturbance butpreserved the symmetry of the pristine crystal structure. How-ever, at 6000K treatment, it was found that the symmetrydisappeared regardless of RT treatment. They had a crys-tal structure and were believed to be amorphous and liquidstates.Fig. 4. Filtration of cubical set using image intensity. (a) Example using a small set. Numbers in cells are assumed to be image intensity. Changing theintensity (i) from high to low, expand the areas. Zeroth simplices (i.e. islands= connected components) and first ones (i.e. rings) are born or diedepending on the intensity. Blue and red indexes represent zeroth- and first-order simplices, respectively. The zeroth- and first-order PD are shown inthe bottom right of (a). When the islands come into contact with each other, they die, and another island is born. (At i=7, islands D01 and D02 died oncontact with each other and D03 was born.) (b) and (c) show amorphous and liquid filtration. White areas expand depending on the intensity.(The image size is 12.5Å×12.5Å. Numbers indicate intensity lower limit for filtration.).Downloaded from https://academic.oup.com/jmicro/article/71/3/161/6529715 by NATIONAL INSTITUTE FOR MATERIALS SCIENCE user on 02 July 2024164 Microscopy, 2022, Vol. 71, No. 3Persistent diagram (PD) from MD 3D dataPDs were generated using the HomCloud [4] library ofPython. The input data were former MD results of Ga andN radii calculated using values obtained by Ishimaru et al.[18] and were 1.5625Å and 0.49Å, respectively. PDs from the3D structure data are shown in the right column of Fig. 1. InPDs from structure data without RT treatment, it was foundthat localized birth–death pairs were dispersed as the temper-ature increased and each birth–death pair peak disappearedat 6000K. In the data for the 1000K+RT and 4000K+RTdata, peaks appear clearly, but in the case of 6000K+RT,peaks do not appear. The situation shown by the PD at eachtemperature is consistent with the perspective view and TEMimage. The peaks near (birth, death)= (1,1.7) and (2,2.4)were due to the existence of a surface in the structural modelused in this study and did not appear in an infinitely largemodel.Figure 2 represents the optimal volumes of the peak posi-tion indicated by the red arrow in the bottom-right PD ofFig. 1. The optimal volumes are represented by the greenand red lines, respectively. It can be observed that there aremore atomic arrangements satisfied with the optimal volumein Fig. 2b than in Fig. 2a.The state of 6000K+RT was relaxed by changing theliquid state to RT and had a state different from the liquidstate. This state was defined as an amorphous state becauseit was clearly different from the liquid state when comparedby PD. Hereafter, data in 6000K treatment with/withoutRT was used as an amorphous state and liquid in common.Here, 6000K+RTwas not intended to reproduce quenching.The combination can be regarded as an amorphous struc-ture resulting from pure structural stabilization by relaxationfrom a random atomic arrangement. In other words, it isconsidered to be the most stable structure determined by theatomic potential, independent of the quenching rate. Simi-larly, the simulation of 6000K without RT did not assumedifficult ultra-high temperature TEM observations but ratheran ideal random atomic arrangement, perhaps obtained byion implantation.Examination by each identification methodAnnular averaged diffraction intensity (AADI)We attempted to classify the states using the AADIs. AADI isobtained by averaging Fourier-transformed TEM image in theradial direction and is approximately equivalent to the radialdistribution in reciprocal lattice space. AADIs from simulatedimages of various foci are shown in Fig. 3. The abscissa inFig. 3 is labeled as scattering angle, which can be considered asdistance in the RDF, although it is the reciprocal. The profileschanged in a complicated manner depending on their defocus,regardless of whether they were amorphous or liquid. There-fore, it was difficult to determine whether the image was takenfrom an RT-treated sample or not.ML using persistent homologyAs a present ML method for PH, we applied Obayashi’smethod [19], which uses persistent images (PI) [20] for learn-ing objects. The transformation from PD to PI was performedusing the method described by Adams et al. [20]. In the PI,each birth–death point is assigned a weight depending on thedistance from the diagonal of the PD. The larger the distance,the larger the weight.Because the learning method is detailed in [19], only asimple procedure is shown below.1. The zeroth- and first-order PDs were calculated usingsimulated TEM images. Since the filtrations were donefrom high to low intensity, the points that representbirth–death pairs are displayed under the diagonal inthe PD. 0th order simplices are connected components,and the first ones are rings. An example is shown inFig. 4a.2. Both PIs are transformed from PDs.3. Logistic regression and support vector classification fortwo-class classification with cross validation were con-ducted using zeroth- and first-order PIs separately or incombination. The PI data were divided into 75% fortraining and 25% for testing for the cross validation.The results of filtration on amorphous and liquid TEMimages are shown in Fig. 4b and c, respectively. In the caseof the amorphous results in Fig. 4b, it seems that the chang-ing manner of the white area is rather gradual and individualcomponents grow larger than the liquid ones. However, inFig. 5. TEM images and PDs of zeroth- and first-order.Downloaded from https://academic.oup.com/jmicro/article/71/3/161/6529715 by NATIONAL INSTITUTE FOR MATERIALS SCIENCE user on 02 July 2024F. Uesugi and M. Ishii Classification for transmission electron microscope images 165Fig. 6. Visualized learning results using PD. (a)–(d) are results of logistic regression and (e)–(h) support vector classification. The upper rows show resultsusing zeroth- and first-order PI data separately. The lower rows combined zeroth- and first-order PI data.the case of liquid, smaller islands are born and die more fre-quently, and they have finer structures than amorphous ones.According to examples (b) and (c), it seems that the arrange-ment of atoms in amorphous has a certain structure, while itis more random in liquids than in amorphous.Figure 5 shows a part of the TEM image and the PD usedfor ML. According to the results of the previous filtration,there seems to be a difference in the way islands and rings areformed and disappear between amorphous and liquid; how-ever, because of the focus dependence of the image, it is notclear at a glance where the characteristics of PD exist. Becausethe actual data is slightly out of focus, it is necessary to obtainthe features using the data of many focus sets. ML is effectivefor this purpose, and it was executed by combining the scikit-learn and Homcloud [4] mentioned above. The Homcloud[4] library was used to create PD and PI from TEM images,while the scikit-learn library [21–22] was used for ML usingPIs. The logistic regression (LR) and support vector classifica-tion (SV) results were compared. MLs were conducted usingeach order PI separately or using a combined zeroth-order andfirst-order PI. This learning method is a supervised learning,in which all amorphous and liquid images were labeled as 0 or1. In both methods, the regularization parameters were 0.01,and the hinge function was used in SV. Accuracy was definedas the ratio between the correct number and the total imagenumber.For supervised learning insensitive to focusing, the amor-phous and liquid phases can be discriminated by 0.773 withLR and 0.817 with SV. In the case of learning the zeroth-orderPIs and first-order ones separately, the zeroth and first accu-racies were 0.649 and 0.756 for LR and 0.672 and 0.790 forSV, respectively.The training results for GaN are presented in Fig. 6. Theblue and red areas represent the features inherent to amor-phous and liquid phases, respectively. Both LR and SV classifyPI by substituting the value of the function h(x) to Sigmoidfunction, where the vector x is the vectorized PI. Specifically,h(x) is obtained by operating the input vector x with theweight vector w that is obtained using ML.h(x) =w ·x (omitted constant for simplicity)Substituting h(x) to Sigmoid function, makes it correspondto the label 0 for amorphous and 1 for liquid.Fig. 7. Inverse analysis result using just-focused amorphous and liquidTEM image. The areas of the inverse analysis from the selectedbirth–death pairs of amorphous are shown in the first and third rows;those of liquid are shown in the second and fourth rows. The zeroth andfirst PDs are shown in the left column. The blue (red) areas on the TEMimage in the second (third) column are from the birth–death pairs in theamorphous (liquid) superior area in Fig. 6g and h. The green points arethe birth points.If h(x)< 0, then Sigmoid function >=0.5, and ML predictsthe label 0. If not, Sigmoid function <=0.5, and it predicts thelabel 1. Each point of PI is a one-dimensional vector with aDownloaded from https://academic.oup.com/jmicro/article/71/3/161/6529715 by NATIONAL INSTITUTE FOR MATERIALS SCIENCE user on 02 July 2024166 Microscopy, 2022, Vol. 71, No. 3nonnegative value. Each element of w represents the contri-bution of each element of x. Since x is a nonnegative vector,each element of weight w must be minus (plus) for amor-phous (liquid). Figure 6 is drawn with the weight w . Theseareas allow the machine to determine the difference betweenboth phases. The upper row shows the classification resultsusing the zeroth- and first-order PI data separately ((a) and(e) for zeroth and (b) and (f) for first). The lower row (c),(d), (g) and (h) show the results using the combined zeroth-and first-order PI data. Note that even when the zeroth- andfirst-order data are combined, the features of zeroth and firstare obtained separately, as shown in the lower row. Fromthese figures, it can be observed that the blue areas are far-ther away from the diagonal than the red areas in both thezeroth-order and first-order PDs. Since points in a PD rep-resent lifetimes (= death time−birth time) of simplices, a longdistance from the diagonal means a long lifetime i.e. the longerthe distance from the diagonal, the larger is the structure. Sucha clear difference favors the use of PDs to detect the differencebetween amorphous and liquid images. As described above,when the zeroth- and first-order data were trained separately,the ability to discriminate between amorphous and liquid wasapproximately 65% and 75%, respectively, indicating thatthe contribution to discrimination ability was higher for thefirst-order training results. This suggests that the structure ofthe ‘ring’ in amorphous materials, as discussed in Fig. 4, iseffective for discrimination. Although it is difficult to discrimi-nate amorphous and liquid phases using only the zeroth-orderPD, which does not reflect the short-range structural order,the ability to discriminate each phase is enhanced when com-bined with the first-order PD. This finding indicates that theability to discriminate can be further improved by incorpo-rating the rings as well as the formation and disappearanceof their connecting components. Therefore, in the case ofamorphous materials, it can be considered that the con-necting components are formed at the initial state of ringgeneration.Figure 7 illustrates the inverse analysis results from Fig. 6.,in which amorphous and liquid TEM images of just focus areused as examples. Inverse analysis represents the pattern in aTEM image that a particular region of PD corresponds to. Theupper two rows show the zeroth-order results, and the lowertwo rows show the first-order ones. In each result, upper(lower) shows amorphous (liquid) results. The left columnshows the PD superimposed on the contour map convertedfrom the results in Fig. 6. The second and third columnsshow inverse analysis results and blue (red) patterns corre-sponding to purple (red) circled region in Fig. 6g and h. Thepurple (red) circled regions are amorphous (liquid) superiorones in the PD. The green points are their birth points. Inthe zeroth-order results, there was only a slight differencebetween amorphous and liquid in terms of the number ofregions corresponding to the regions surrounded by the purpleand red lines. In the first-order results, the pattern corre-sponding to the area surrounded by the purple (red) line wasmore in Amorphous (Liquid) than in Liquid (Amorphous).Although the correct answer ratio learned using only thezeroth-order PD is lower than using only the first-order PD asmentioned above, the inverse analysis results are in agreementwith it.Figure 8 shows the dependence of the accuracy on theamount of defocus for LR and SV. Figure 8 shows the pre-dicted accuracy using (a) only the zeroth-, (b) first- and(c) zeroth- and first-order PDs. The average accuracy ofeach of these (a)–(c) is summarized by the dashed linein Fig. 8d. In these figures, the closed and open circlesFig. 8. Accuracy of amorphous and liquid TEM image using logistic regression or support vector classification for (a) zeroth-order, (b) first-order, and(c) a combination of zeroth- and first-orders.Downloaded from https://academic.oup.com/jmicro/article/71/3/161/6529715 by NATIONAL INSTITUTE FOR MATERIALS SCIENCE user on 02 July 2024F. Uesugi and M. Ishii Classification for transmission electron microscope images 167represent predictions using LR and SV, respectively. FromFig. 8d, it can be observed that SV is superior to LR.Furthermore, the improvement in accuracy discussed above,zeroth-order < first-order < zeroth- and first-order (denoted by‘0th + 1st’ in this figure), is graphically summarized.Importantly, we found that the prediction accuracy is dif-ferent when the defocus region in Fig. 8a–c is limited to±25Å(in-focus) from the results when the defocus region is consid-ered to be up to ±100Å (defocus). The results of the in-focusare summarized as solid lines in Fig. 8d. From a comparisonbetween the solid and dashed lines in this figure, the accuracyof the in-focus is higher than that of the defocus. Consideringthis result from a crystallographic point of view, it may corre-spond to the fact that the contrast transfer function differencesare not large around the focus, to ensure that the structuralinformation is easily reflected in the image, whereas in thecase of defocus, the structural information is lost due to theoverlapping interference fringes. Considering it from the per-spective of ML, it indicates that the structural information isconsequently reflected in the features of Fig. 6, even thoughthe model does not contain any prior information about thestructure.Finally, we reconsidered the accuracy from a practicalpoint of view. In this case, we discuss a relatively largedefocus amount (±100Å). However, in recent TEMs, it ispossible to adjust the focus within ±25Å using a Fouriertransform during operation. The solid line presented in Fig. 8dshows that when the amount of defocus is within ±25Å,the accuracy improves to 0.803 for LR and 0.851 for SV.Therefore, we may identify the amorphous state with a highdiscrimination accuracy of more than 85%, even in normaloperation.Concluding remarksWe studied the effectiveness of the PH to classify the TEMimages from two different amorphous states: liquid and amor-phous. After transforming PI from PD using simulated TEMimages of each state with different defocus, two ML methods,LR and SV, were applied to discriminate them. Consequently,in the case of GaN, the accuracy was over 85% in the SVfor defocus between −25Å and 25Å. PH is known to beeffective for three-dimensional data, but it has also beenfound to be effective even in less dimensional data, such astwo-dimensional image data. However, the three-dimensionalstructure/information, which changes in a complicated man-ner owing to the difference depending on the defocus inTEM, still exists. Two-dimensional PD is created using thepixel intensity of the image. It is not possible to simply asso-ciate a PD created from three-dimensional structural datawith one created from two-dimensional image data on a one-to-one basis. Therefore, PH also has a difficult phase ininterpretation. Although not limited to TEM images, actualmeasurement data contain noise. As PH is believed to be vul-nerable to noise, its application to TEM images might notbe suitable. However, a low-pass filter is often applied toTEM images to avoid dropping information, thus noise isnot considered a big problem for PH applications. In thepresent work, we considered only the change in defocus,but PH can be a new tool for analyzing amorphous TEMimages.FundingThis research was carried out within the framework offundamental research enhancement programs of NIMS onadvanced materials analysis and data-driven materials devel-opment.Conflict of InterestThe authors declare that they have no conflict of interest.References1. Edelsbrunner H, Letscher D, and Zomorodian A (2002) Topolog-ical persistence and simplification. Discret. Comput. Geom. 28:511–533.2. Zomorodian A and Carlsson G (2004) Computing persistenthomology. Proc. Annu. Symp. Comput. Geom. 274: 347–356.3. Hiraoka Y, Nakamura T, Hirata A, Escolar E G, Matsue K, andNishiura Y (2016) Hierarchical structures of amorphous solidscharacterized by persistent homology. Proc. Natl. Acad. Sci. U.S. A. 113: 7035–7040.4. Obayashi I, Wada T, Tunhua T, Jun Miyanaga Y, and Hiraoka HHomCloud. https://homcloud.dev/index.html.5. 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