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[Ryuto Eguchi](https://orcid.org/0009-0003-2859-6934), [Yu Wen](https://orcid.org/0000-0001-7809-5941), [Hideki Abe](https://orcid.org/0000-0002-8392-7586), [Ayako Hashimoto](https://orcid.org/0000-0002-1985-7667)

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[Interpretable Structural Evaluation of Metal-Oxide Nanostructures in Scanning Transmission Electron Microscopy (STEM) Images via Persistent Homology](https://mdr.nims.go.jp/datasets/da718d39-f10f-4bc4-aaab-82b812b431eb)

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Interpretable Structural Evaluation of Metal-Oxide Nanostructures in Scanning Transmission Electron Microscopy (STEM) Images via Persistent HomologyCitation: Eguchi, R.; Wen, Y.; Abe, H.;Hashimoto, A. InterpretableStructural Evaluation of Metal-OxideNanostructures in ScanningTransmission Electron Microscopy(STEM) Images via PersistentHomology. Nanomaterials 2024, 14,1413. https://doi.org/10.3390/nano14171413Academic Editor: Jakob BirkedalWagnerReceived: 19 July 2024Revised: 16 August 2024Accepted: 25 August 2024Published: 29 August 2024Copyright: © 2024 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).nanomaterialsArticleInterpretable Structural Evaluation of Metal-OxideNanostructures in Scanning Transmission Electron Microscopy(STEM) Images via Persistent HomologyRyuto Eguchi 1,2,* , Yu Wen 1,2 , Hideki Abe 1,3 and Ayako Hashimoto 1,2,*1 National Institute for Materials Science, 1-2-1 Sengen, Tsukuba 305-0047, Ibaraki, Japan;yu.wen.lw@alumni.tsukuba.ac.jp (Y.W.); abe.hideki@nims.go.jp (H.A.)2 Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-2-1 Sengen,Tsukuba 305-0047, Ibaraki, Japan3 Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Saitama 338-8570, Japan* Correspondence: eguchi.ryuto@nims.go.jp (R.E.); hashimoto.ayako@nims.go.jp (A.H.)Abstract: Persistent homology is a powerful tool for quantifying various structures, but it is equallycrucial to maintain its interpretability. In this study, we extracted interpretable geometric featuresfrom the persistent diagrams (PDs) of scanning transmission electron microscopy (STEM) imagesof self-assembled Pt-CeO2 nanostructures synthesized under different annealing conditions. Wefocused on PD quadrants and extracted five interpretable features from the zeroth and first PDs ofnanostructures ranging from maze-like to striped patterns. A combination of hierarchical clusteringand inverse analysis of PDs reconstructed by principal component analysis through vectorizationof the PDs highlighted the importance of the number of arc-like structures of the CeO2 phase in thefirst PDs, particularly those that were smaller than a characteristic size. This descriptor enabled usto quantify the degree of disorder, namely the density of bends, in nanostructures formed underdifferent conditions. By using this descriptor along with the width of the CeO2 phase, we classified12 Pt-CeO2 nanostructures in an interpretable way.Keywords: persistent homology; metal-oxide nanostructures; scanning transmission electronmicroscopy (STEM)1. IntroductionMetal–oxide nanocomposites with strong interfacial interactions have been studiedextensively because their excellent chemical and physical properties make them attractiveas catalysts and electrode materials [1,2]. As the structure of metal–oxide nanocompositesaffects their performance, investigating the structure–property relationship of nanocompos-ites can provide information to improve materials design. Traditionally, geometric factorssuch as size [3–6], density [7,8], area, and shape are useful for describing the structureof materials quantitatively [9,10]. However, because most nanostructures do not have aregular structure, discussions on their relationships to physical properties based on suchgeometric factors are limited. Therefore, more suitable geometrical features are requiredfor property predictions.Topological data analysis (TDA) [11,12] is one of the powerful tools that have thepossibility to provide alternative structural features in materials science. With its versatilityand time efficiency, TDA has been used to characterize the structure of organic species,proteins [13], inorganic materials, and amorphous materials [14]. Unlike the traditionalgeometric measurements of the dimensions, TDA can quantify the multi-dimensionaland multi-scale topological intrinsic features. The features based on the homology cap-ture the data shape as N-dimensional holes, including connected components as zero-dimensional (0D) holes and loops or tunnels as one-dimensional (1D) holes. The numberNanomaterials 2024, 14, 1413. https://doi.org/10.3390/nano14171413 https://www.mdpi.com/journal/nanomaterialshttps://doi.org/10.3390/nano14171413https://doi.org/10.3390/nano14171413https://creativecommons.org/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://www.mdpi.com/journal/nanomaterialshttps://www.mdpi.comhttps://orcid.org/0009-0003-2859-6934https://orcid.org/0000-0001-7809-5941https://orcid.org/0000-0002-8392-7586https://orcid.org/0000-0002-1985-7667https://doi.org/10.3390/nano14171413https://www.mdpi.com/journal/nanomaterialshttps://www.mdpi.com/article/10.3390/nano14171413?type=check_update&version=1Nanomaterials 2024, 14, 1413 2 of 12of N-dimensional holes, called the Betti number (βN), is known for topological invariants.Using this basic descriptor, it becomes easy to tell the difference between two complicatedstructures and deduce the quantitative relationship between structure and function. In ourprevious work [15,16], we applied β0 and β1 to scanning transmission electron microscopy(STEM) and STEM tomography for analysis of both the two-dimensional (2D) and three-dimensional (3D) structures of various self-assembled Pt-CeO2 nanocomposites that weresynthesized under different annealing conditions. The quantitative relationship betweenthe CeO2 phase connectivity and oxygen-ion conductivity was successfully established byusing β0 of CeO2 phases as a structural descriptor.Persistent homology (PH) includes specific and local geometric features besides βNwith rough and global approximations. The main concept of PH lies in tracking the scalerequired for the appearance and disappearance of N-dimensional holes through continuousdeformation. Each homological feature that appears at birth time b and disappears atdeath time d is recorded. The birth and death positions of a structure can be identified byinverse analysis. Therefore, PH is a competent approach for structure description and hasbeen applied to many fields, such as amorphous structures [17,18], magnetic domain struc-tures [19,20], and spin systems [21]. In recent years, studies have also predicted materialproperties using PH-based machine learning. Minamitani and co-workers predicted thethermal conductivity of amorphous carbon [22,23]. Sato et al. [24] evaluated the activationenergy of ionic conductivity in silver iodide (AgI) by investigating its atomic configurationsconstructed with molecular dynamics simulations. Uesugi et al. [25] created the ridgeregression model that predicted the pre-exponential factor and activation energy of thePt-CeO2 nanocomposites. Thus, PH successfully generated quantitative geometric featuresand thereby revealed new relationships to physical properties. Then, the physical meaningof the features should be understood in order to examine the validity of the models for prop-erty predictions. However, useful structural descriptors from PH-based machine learningare not always interpretable. Furthermore, when analyzing experimental data for materialdesign, machine learning may be hardly able to be performed or hardly produce expectedresults due to the limited number of data. The interpretability of structural descriptorswould promote revealing the relationships between synthesis conditions, structures, andfunctions even from the limited data.In this study, we aimed to extract interpretable structural descriptors to facilitatedirect structural classification for the Pt-CeO2 nanocomposites from the STEM images byusing PH. We could extract the averaged structural features from the PDs of the STEMimages. Further step-by-step analysis found new important interpretable features with thecharacteristic size information. Finally, the two effective descriptors were determined by arandom forest classification, which quantified the scale and the degree of disorder, namelythe density of bends. Their scatter plots provided simple and interpretable classification.This approach would be able to contribute to identifying the more important relationshipbetween structure and transport properties, such as ionic conductivity.2. Materials and MethodsPreparation of Pt-CeO2 nanocomposites, acquisition of STEM images, and pre-processingof STEM images were conducted according to our previous reports [15,16]. Twelve phase-separated Pt-CeO2 nanocomposites were synthesized by annealing Pt5Ce alloy at differenttemperatures (500, 600, and 700 ◦C) and syngas ratios (CO:O2 = 0:1, 1:1, 2:1, and 3:1).The nanostructures of the Pt-CeO2 nanocomposites were characterized using high-angle annular dark-field STEM (HAADF-STEM; JEM-2100F, JEOL, Tokyo, Japan). Sevenimages of each nanocomposite were selected and trimmed to the same size and scale(1024 × 1024 pixels, 406 × 406 nm). The Pt and CeO2 phases were identified by their brightand dark contrast, respectively. The STEM images were processed by background removaland binarization (Figures S1 and S2).Figure 1 shows a conceptual diagram for the extraction of interpretable structuralfeatures from STEM images by using PH and the classification of nanostructures by usingNanomaterials 2024, 14, 1413 3 of 12the selected effective features. We extracted the averaged structural features from the PDsof the STEM images, while further step-by-step analysis by vectorization of all the PDs wasconducted to add alternative important features with the characteristic size information.As a final target, the effective descriptors were determined by a random forest classification,which was used for the interpretable classification.Nanomaterials 2024, 14, x FOR PEER REVIEW 3 of 13   dark contrast, respectively. The STEM images were processed by background removal and binarization (Figures S1 and S2). Figure 1 shows a conceptual diagram for the extraction of interpretable structural features from STEM images by using PH and the classification of nanostructures by using the selected effective features. We extracted the averaged structural features from the PDs of the STEM images, while further step-by-step analysis by vectorization of all the PDs was conducted to add alternative important features with the characteristic size infor-mation. As a final target, the effective descriptors were determined by a random forest classification, which was used for the interpretable classification.  Figure 1. Conceptual diagram for extraction of interpretable structural features from STEM images by using PH and classification of nanostructures by using the determined effective features. To obtain PDs, first, each pixel value in the binarized STEM images was assigned by a signed Euclidean distance (SED) from the boundaries between the black (CeO2) and white (Pt) phases. Each pixel was given a sign: negative for the black domains and positive for the white ones (Figure S3). Here, we focused on the homology of the CeO2 phase be-cause of its correlation with oxygen ionic conductivity as well as the previous work [15,16]. Second, a filtration was processed to calculate b and d, where the threshold for binarization increased, persisting from the minimum to the maximum value. When N-dimensional holes appeared and disappeared during the filtration, the SED thresholds were defined as b and d, respectively. Then, 2D histograms of birth and death pairs (b-d pairs) were obtained as PDs. The difference between birth and death time (d − b) is defined as lifetime. We calculated the zeroth PD for 0D holes and the first PD for 1D holes. Next, we conducted hierarchical clustering and PCA for each vectorized PD to clas-sify the different Pt-CeO2 nanostructures and identify important features. The persistence image (𝜌) was used for vectorization of PDs [26]. Because a PD consists of a set of b-d pairs {(𝑏𝑘, 𝑑𝑘): 𝑘 = 1,2, … , 𝑙}, where 𝑏𝑘 (𝑑𝑘) is from the 𝑘-th hole and 𝑙 is the total number of holes, 𝜌 is defined as 𝜌(𝑥, 𝑦) = Σ𝑘=1𝑙 𝑤(𝑏𝑘, 𝑑𝑘) exp (−(𝑏𝑘 − 𝑥)2 + (𝑑𝑘 − 𝑦)22𝜎2), (1) 𝑤(𝑏𝑘, 𝑑𝑘) = arctan(𝐶(𝑑𝑘 − 𝑏𝑘)𝑝)  (2) Figure 1. Conceptual diagram for extraction of interpretable structural features from STEM imagesby using PH and classification of nanostructures by using the determined effective features.To obtain PDs, first, each pixel value in the binarized STEM images was assigned by asigned Euclidean distance (SED) from the boundaries between the black (CeO2) and white(Pt) phases. Each pixel was given a sign: negative for the black domains and positive forthe white ones (Figure S3). Here, we focused on the homology of the CeO2 phase becauseof its correlation with oxygen ionic conductivity as well as the previous work [15,16].Second, a filtration was processed to calculate b and d, where the threshold for binarizationincreased, persisting from the minimum to the maximum value. When N-dimensionalholes appeared and disappeared during the filtration, the SED thresholds were defined as band d, respectively. Then, 2D histograms of birth and death pairs (b-d pairs) were obtainedas PDs. The difference between birth and death time (d − b) is defined as lifetime. Wecalculated the zeroth PD for 0D holes and the first PD for 1D holes.Next, we conducted hierarchical clustering and PCA for each vectorized PD to classifythe different Pt-CeO2 nanostructures and identify important features. The persistenceimage (ρ) was used for vectorization of PDs [26]. Because a PD consists of a set of b-d pairs{(b k, dk): k = 1, 2, . . . , l}, where bk (dk) is from the k-th hole and l is the total number ofholes, ρ is defined asρ(x, y) = ∑lk=1 w(bk, dk)exp(− (bk − x)2 + (dk − y)22σ2), (1)w(bk, dk) = arctan(C(dk − bk)p) (2)Here, the variables C, p, and σ were set as 2.0, 0.5, and 1.0, respectively. We discretizedthe region [−12, 6] × [−12, 6] in the zeroth PD and [−6, 12] × [−6, 12] in the first PD into90 × 90 grids and then evaluated ρ (x, y) of the respective grids. Vectorized PDs werecalculated by organizing each ρ (x, y ) in a prefixed order.Hierarchical clustering was performed using the cosine similarity and dissimilarity ofvectorized zeroth PDs to evaluate the closeness of each PD [27]. We calculated the aver-Nanomaterials 2024, 14, 1413 4 of 12aged vectors from the seven vectorized PDs with the same synthesis conditions, resultingin 12 vectors for each set of conditions. Subsequently, cosine dissimilarities for all thecombinations of 12 vectors were calculated and used as the matrix distance. A weightedmethod was used for calculating the distance between the clusters as the averaged cosinedissimilarities for all possible pairs [28]. PCA decreases the dimensionality of datasets andmaximizes their variance, thereby allowing us to extract critical information and emphasizethe differences between input vectors [29]. A set of 84 vectors (seven images × 12 sets ofconditions) was processed for each of the zeroth and first PDs. Furthermore, the PDs werereconstructed with the first principal component (PC1) obtained through PCA by project-ing PC1 onto the grid used in the PD vectorization. The procedures for PD calculation,visualization, vectorization, and reconstruction by PCA were conducted using the dataanalysis software ‘HomCloud’ [30,31].Finally, the random forest method was used to quantify the importance of PH featuresin classification tasks [32]. The image sets were augmented to encompass a thirty-sixfoldincrease in the total image number by segmenting the images into 3 × 3 pieces and rotatingthese pieces every 90◦. We extracted the five interpretable features from the PD quadrantsof all augmented images and adapted them as explanatory variables for the classificationof four types of sample sets, which were chosen from the result of hierarchical clustering.Thus, we evaluated the importance of the extracted features based on their contribution tothe decrease in a Gini impurity, that is, a decrease in the number of misclassifications. Thenumber of trees was set to 100 and the maximum depth of trees was set to five to preventthe overfitting.3. Results and Discussion3.1. Extraction of Interpretable Features from Persistent Diagram QuadrantsFigure 2a–c show binarized HAADF-STEM images of the structural transformation ofPt-CeO2 nanostructures with rising formation temperature from 500 to 700 ◦C. The structurechanged from maze-like to striped as temperature increased. Figure 2d–f show the zerothand Figure 2g–i show the first PDs derived from the CeO2 phase in the correspondingnanostructures in Figure 2a–c, respectively. The distribution of individual b–d pairs changesin the zeroth and first PDs, reflecting the structural changes; however, the quadrants inwhich the b-d pairs are located remain the same in both PDs. Therefore, we first focused onthe PD quadrants, namely, the signs of b and d which allowed us to extract five interpretablefeatures. In the zeroth PDs (Figure 2d–f), because new domains can appear only throughshrinkage, b is inevitably negative and captures the scale required for the appearance ofa domain during shrinkage. Hence, for striped domains, the absolute value of b almostcorresponds to half of the stripe width (Figure S4a,c). In contrast, d exhibits both negativeand positive signs, which describe the splitting of the connected domains through shrinkageand the merging of isolated domains through expansion, respectively. In the region withpositive d, the number of b-d pairs represents the number of isolated domains minus one,which additionally corresponds to the frequency of merging (Figure S4b). Thus, we couldderive the trends in the width and number of CeO2 domains from the positive d region.The absolute b values, which were estimated by Gaussian fitting as shown in Figure 3a–c,increased with the annealing temperature, directly implying the average width of theCeO2 phase increased with temperature. In Figure 3d, the number of the CeO2 domainsdecreased with rising fabrication temperature. These results from the PD analysis coincidewith the visual analysis of the STEM images (Figure 2a–c). Additionally, the total lengths ofstripes from all domains in one image could be estimated by counting the number of b-dpoints with short lifetimes (Figure S5). In the first PDs (Figure 2g–i), because ring-shapedCeO2 domains can disappear only through expansion, d was exclusively positive andcaptured the scale required for the merge of ring-shaped domains, which, therefore, couldbe regarded as the diameter of ring-shaped structures (Figure S4b). In the first PDs, b waseither negative or positive, which described the splitting of the rings through shrinkageand the evolution from arc-like structures to rings through expansion, respectively. Thus,Nanomaterials 2024, 14, 1413 5 of 12the number of b-d pairs in the respective regions represents those of rings and arc-likestructures (Figure S6).Nanomaterials 2024, 14, x FOR PEER REVIEW 6 of 13    Figure 2. Binarized high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images of Pt-CeO2 nanostructures synthesized at (a) 500 °C, (b) 600 °C, and (c) 700 °C with a CO:O2 gas ratio of 2:1. Scale bars represent 100 nm. (d), (e), and (f) Zeroth and (g), (h), and (i) first persistent diagrams (PDs) of CeO2 phases in the nanostructures corresponding to (a), (b), and (c), respectively. The color bars indicate the number of b-d pairs. Figure 2. Binarized high-angle annular dark-field scanning transmission electron microscopy(HAADF-STEM) images of Pt-CeO2 nanostructures synthesized at (a) 500 ◦C, (b) 600 ◦C, and (c) 700 ◦Cwith a CO:O2 gas ratio of 2:1. Scale bars represent 100 nm. (d), (e), and (f) Zeroth and (g), (h), and(i) first persistent diagrams (PDs) of CeO2 phases in the nanostructures corresponding to (a), (b), and(c), respectively. The color bars indicate the number of b-d pairs.Table 1 summarizes the relationships between the five interpretable features and PDquadrants. The average and error values for three Pt-CeO2 nanostructures fabricatedat different temperatures calculated from seven images of structures formed under thesame synthesis conditions are also listed in Table 1. The interpretable features of the Ptphase were obtained from the PDs after reversing black and white regions in the binarizedimages. For width measurements, only b-d pairs near b = d were used because they mainlyoriginated from the striped structure rather than particle- and dumbbell-like structures(Figure S4d). The total length of the CeO2 and Pt stripes from all domains decreased asthe synthesis temperature increased. This trend agrees with the actual phase change frommaze-like to striped structure observed in the STEM images. The number of arcs in bothphases decreased with rising synthesis temperature, quantitatively indicating the decreaseof curved structure in the samples. The number of CeO2 domains decreased, whereasthat of Pt domains remained almost the same as the synthesis temperature increased.Conversely, the number of CeO2 rings remained almost the same, whereas the numberof Pt rings decreased as the fabrication temperature rose. These findings indicate that Ptinhibits the diffusion of Ce during the structural formation process [33], resulting in thedevelopment of many isolated CeO2 domains at low fabrication temperatures. At thispoint, the inhibitor Pt surrounds CeO2, thus forming many Pt rings. Conversely, at highsynthesis temperatures, where the diffusion rate of Ce is high, it becomes less inhibitedby Pt than is the case at lower temperatures, allowing isolated Ce domains to connect anddecreasing the number of domains. Under these conditions, Pt no longer easily surroundsCeO2, resulting in a decrease in the number of Pt rings.Nanomaterials 2024, 14, 1413 6 of 12Nanomaterials 2024, 14, x FOR PEER REVIEW 7 of 13    Figure 3. Sum of the frequency distributions of the number of b-d pairs from the seven images as a function of b determined from zeroth PDs of nanostructures synthesized at (a) 500 °C, (b) 600 °C, and (c) 700 °C with a CO:O2 gas ratio of 2:1. The blue solid lines indicate the Gaussian fitting at the center of each binary region. (d) Dependence of the number of b-d pairs with positive d values de-rived from the zeroth PDs of the nanostructures on synthesis temperature. Table 1. Summary of five interpretable features, corresponding to zeroth and first PD quadrants, and the estimated average and error (in parentheses) for the Pt-CeO2 nanostructures synthesized at different temperatures with a CO:O2 gas ratio of 2:1. Each average and error was calculated using standard deviation from the seven images of structures formed under the same synthesis condition. Geometric Information Representation in PD 500 °C CeO₂ 500 °C Pt 600 °C CeO₂ 600 °C Pt 700 °C CeO₂ 700 °C Pt Number of domains Number of b-d pairs in positive death region of zeroth PD 272 (31) 71 (11) 158 (18) 67 (7) 70 (5) 68 (9) Width of stripes [nm] Average of twice the value of absolute birth value in zeroth PD 4.83 (8) 5.04 (7) 5.61 (12) 6.21 (16) 6.21 (13) 6.91 (17) Total length of stripes [nm] Number of b-d pairs with short lifetime of zeroth PD 2805 (185) 3073 (172) 2458 (245) 2642 (231) 1466 (211) 1473 (217) Number of rings Number of b-d pairs in negative birth region of first PD 34 (8) 186 (29) 34 (3) 92 (14) 25 (7) 25 (3) Number of arcs Number of b-d pairs in positive birth region of first PD 550 (27) 446 (38) 390 (27) 352 (18) 203 (16) 193 (16) Figure 3. Sum of the frequency distributions of the number of b-d pairs from the seven images as afunction of b determined from zeroth PDs of nanostructures synthesized at (a) 500 ◦C, (b) 600 ◦C, and(c) 700 ◦C with a CO:O2 gas ratio of 2:1. The blue solid lines indicate the Gaussian fitting at the centerof each binary region. (d) Dependence of the number of b-d pairs with positive d values derived fromthe zeroth PDs of the nanostructures on synthesis temperature.Table 1. Summary of five interpretable features, corresponding to zeroth and first PD quadrants,and the estimated average and error (in parentheses) for the Pt-CeO2 nanostructures synthesized atdifferent temperatures with a CO:O2 gas ratio of 2:1. Each average and error was calculated usingstandard deviation from the seven images of structures formed under the same synthesis condition.GeometricInformation Representation in PD 500 ◦CCeO2500 ◦CPt600 ◦CCeO2600 ◦CPt700 ◦CCeO2700 ◦CPtNumber ofdomainsNumber of b-d pairs in positive deathregion of zeroth PD 272 (31) 71 (11) 158 (18) 67 (7) 70 (5) 68 (9)Width ofstripes [nm]Average of twice the value of absolutebirth value in zeroth PD 4.83 (8) 5.04 (7) 5.61 (12) 6.21 (16) 6.21 (13) 6.91 (17)Total length ofstripes [nm]Number of b-d pairs with short lifetimeof zeroth PD 2805 (185) 3073 (172) 2458 (245) 2642 (231) 1466 (211) 1473 (217)Number ofringsNumber of b-d pairs in negative birthregion of first PD 34 (8) 186 (29) 34 (3) 92 (14) 25 (7) 25 (3)Number of arcs Number of b-d pairs in positive birthregion of first PD 550 (27) 446 (38) 390 (27) 352 (18) 203 (16) 193 (16)Nanomaterials 2024, 14, 1413 7 of 123.2. Visualization of Interpretable Features by Inverse Analysis of Reconstructed PersistentDiagrams Based on Principal Component AnalysisAlthough five geometric features were extracted by interpreting the PD quadrants,they were averaged or summed values that did not keep the information of each PD.Therefore, all 84 PDs of the Pt-CeO2 nanostructures were examined by vectorization todeepen our knowledge of their features. We visualized the interpretable features by acombination of hierarchical clustering and inverse analysis of reconstructed PDs based onPCA. First, all the mean cosine similarities were calculated from the vectorized PDs forhierarchical clustering. The mean cosine similarities exceeded 0.99 within the same setof synthesis conditions, and their maximum deviation was 0.007148 (Table S1). The highsimilarities and small deviation implied that the Pt-CeO2 nanostructures formed underthe same conditions were almost identical. Figure 4a shows the hierarchical clusteringwith cosine dissimilarities for 12 different nanostructures. At the first branch point, all thenanostructures are divided into two clusters: one cluster (G1–G2–G3) for structures formedusing CO:O2 gas ratios of 0:1 and 1:1 and the other (G4–G5–G6) for structures formed usingCO:O2 gas ratios of 2:1 and 3:1, except for the sample fabricated at 500 ◦C with a CO:O2 gasratio of 2:1. The binarized STEM images in Figure 4b clearly show a structural differencebetween the two clusters in the stripe width. While the average annealing temperatures forthe two groups were calculated to be 586 and 620 ◦C, the average CO:O2 gas ratios of thetwo groups were 0.714:1 and 2.60:1. The larger relative difference between the gas ratiosof the two groups than between temperatures could indicate that gas ratio has a strongereffect on the structure of the materials than temperature. At the second and third branchpoints, the two clusters were further divided into two at each point. This clustering mightbe based on temperature differences rather than the gas ratios. However, specific geometricdifferences are challenging to discern directly from STEM images. After the fifth branchpoint, the cosine dissimilarities fell within the range of the maximum deviation, i.e., thethreshold for clustering, as indicated by the dashed line. Thus, all 12 nanostructures couldbe clustered into six groups (G1–G6).Nanomaterials 2024, 14, x FOR PEER REVIEW 8 of 13   3.2. Visualization of Interpretable Features by Inverse Analysis of Reconstructed Persistent Diagrams Based on Principal Component Analysis Although five geometric features were extracted by interpreting the PD quadrants, they were averaged or summed values that did not keep the information of each PD. Therefore, all 84 PDs of the Pt-CeO2 nanostructures were examined by vectorization to deepen our knowledge of their features. We visualized the interpretable features by a combination of hierarchical clustering and inverse analysis of reconstructed PDs based on PCA. First, all the mean cosine similarities were calculated from the vectorized PDs for hierarchical clustering. The mean cosine similarities exceeded 0.99 within the same set of synthesis conditions, and their maximum deviation was 0.007148 (Table S1). The high sim-ilarities and small deviation implied that the Pt-CeO2 nanostructures formed under the same conditions were almost identical. Figure 4a shows the hierarchical clustering with cosine dissimilarities for 12 different nanostructures. At the first branch point, all the nanostructures are divided into two clusters: one cluster (G1–G2–G3) for structures formed using CO:O2 gas ratios of 0:1 and 1:1 and the other (G4–G5–G6) for structures formed using CO:O2 gas ratios of 2:1 and 3:1, except for the sample fabricated at 500 °C with a CO:O2 gas ratio of 2:1. The binarized STEM images in Figure 4b clearly show a structural difference between the two clusters in the stripe width. While the average an-nealing temperatures for the two groups were calculated to be 586 and 620 °C, the average CO:O2 gas ratios of the two groups were 0.714:1 and 2.60:1. The larger relative difference between the gas ratios of the two groups than between temperatures could indicate that gas ratio has a stronger effect on the structure of the materials than temperature. At the second and third branch points, the two clusters were further divided into two at each point. This clustering might be based on temperature differences rather than the gas ratios. However, specific geometric differences are challenging to discern directly from STEM images. After the fifth branch point, the cosine dissimilarities fell within the range of the maximum deviation, i.e., the threshold for clustering, as indicated by the dashed line. Thus, all 12 nanostructures could be clustered into six groups (G1–G6).  Figure 4. (a) Hierarchical clustering of Pt-CeO2 nanostructures with cosine dissimilarity calculated from zeroth vectorized PDs. The branch points are numbered as 1, 2, … in descending order of cosine dissimilarity. The dashed line is the threshold for clustering and is the maximum deviation calculated from the mean of cosine dissimilarity for the same set of synthesis conditions. The 12 nanostructures were finally clustered into six groups (G1–G6). (b) Binarized STEM images of Pt-CeO2 nanostructures surrounded by the colors corresponding to those of the six groups (G1–G6) in (a). Second, an inverse analysis of the PDs reconstructed from PCA was conducted to visualize the details of the structural differences between the groups. In the PCA of the Figure 4. (a) Hierarchical clustering of Pt-CeO2 nanostructures with cosine dissimilarity calculatedfrom zeroth vectorized PDs. The branch points are numbered as 1, 2, . . . in descending order of cosinedissimilarity. The dashed line is the threshold for clustering and is the maximum deviation calculatedfrom the mean of cosine dissimilarity for the same set of synthesis conditions. The 12 nanostructureswere finally clustered into six groups (G1–G6). (b) Binarized STEM images of Pt-CeO2 nanostructuressurrounded by the colors corresponding to those of the six groups (G1–G6) in (a).Second, an inverse analysis of the PDs reconstructed from PCA was conducted tovisualize the details of the structural differences between the groups. In the PCA ofthe original PDs, the contribution ratios of principal component 1 (PC1) and principalcomponent 2 (PC2) were 0.862 and 0.073 for the zeroth PDs, respectively, and 0.864 andNanomaterials 2024, 14, 1413 8 of 120.105 for the first PDs, respectively. The large PC1 contribution ratio indicates that PC1was sufficient to represent the nano-structural differences between the groups. Therefore,we focused on only PC1 and obtained the zeroth and first reconstructed PDs, which areshown in Figure 5a,c, respectively. The b-d pairs in the red and blue regions contributepositively and negatively to PC1, respectively. Figure 5b,d plot the PC1 values from all84 vectorized PDs categorized by the six groups G1–G6 through hierarchical clustering(Figure 4) to show how PC1 contributes to the clustering. The PC1 values for both thezeroth and first reconstructed PDs have obvious dependence on each group, which furtherconfirms that PC1 can effectively describe the nanostructures. All 84 PDs of the Pt-CeO2nanostructures were examined by vectorization and we were able to visualize interpretablefeatures by a combination of hierarchical clustering and inverse analysis of reconstructedPDs based on PCA.Nanomaterials 2024, 14, x FOR PEER REVIEW 9 of 13   original PDs, the contribution ratios of principal component 1 (PC1) and principal com-ponent 2 (PC2) were 0.862 and 0.073 for the zeroth PDs, respectively, and 0.864 and 0.105 for the first PDs, respectively. The large PC1 contribution ratio indicates that PC1 was sufficient to represent the nano-structural differences between the groups. Therefore, we focused on only PC1 and obtained the zeroth and first reconstructed PDs, which are shown in Figure 5a,c, respectively. The b-d pairs in the red and blue regions contribute positively and negatively to PC1, respectively. Figure 5b,d plot the PC1 values from all 84 vectorized PDs categorized by the six groups G1–G6 through hierarchical clustering (Fig-ure 4) to show how PC1 contributes to the clustering. The PC1 values for both the zeroth and first reconstructed PDs have obvious dependence on each group, which further con-firms that PC1 can effectively describe the nanostructures. All 84 PDs of the Pt-CeO2 nanostructures were examined by vectorization and we were able to visualize interpreta-ble features by a combination of hierarchical clustering and inverse analysis of recon-structed PDs based on PCA.  Figure 5. Zeroth (a) and first (c) reconstructed PDs of Pt-CeO2 nanostructures obtained from PC1 of PCA. The color bars indicate the PC1 coefficients. PC1 values from all 84 vectorized (b) zeroth and (d) first PDs categorized into six groups through hierarchical clustering. Inset images show the in-verse analysis results obtained from the (a) zeroth and (c) first reconstructed PDs for the nanostruc-tures synthesized at 700 °C with a CO:O2 gas ratio of 1:1 (left) and at 700 °C with a CO:O2 gas ratio of 2:1 (right). The blue and red regions in the image in (b) correspond to the nanostructures from the b-d pairs with −4 nm ≤ b, d < 0 and b < −4 nm, d < 0 in the zeroth PDs, respectively, and those in (d) correspond to the structures from the b-d pairs with 0 < b,  d ≤ 4 nm and 0 < b, 4 nm ≤  d in the first PDs, respectively.  Then, the geometric feature related to PC1 was revealed through inverse analysis of the reconstructed PDs, which enabled us to identify the corresponding N-dimensional holes to the red (positive) and blue (negative) regions in the reconstructed PDs (Figure 5a,c). Here, to maintain simplicity and interpretability, we set a characteristic scale of 4 nm to define specific blue and non-blue regions by the characteristic scale of b in the zeroth reconstructed PDs and d in the first ones because the blue and non-blue regions can be separated by this characteristics value (Figure S7). Furthermore, we confined b-d pairs with −4 nm ≤ b, d < 0 in the zeroth PDs for the connected domains and those with 0 < b, d < 4 nm in the first PDs for arc-like structures, which correspond to the blue (negative) regions. The inset images in Figure 5b,d show the inverse analysis results obtained from Figure 5. Zeroth (a) and first (c) reconstructed PDs of Pt-CeO2 nanostructures obtained from PC1 ofPCA. The color bars indicate the PC1 coefficients. PC1 values from all 84 vectorized (b) zeroth and(d) first PDs categorized into six groups through hierarchical clustering. Inset images show the inverseanalysis results obtained from the (a) zeroth and (c) first reconstructed PDs for the nanostructuressynthesized at 700 ◦C with a CO:O2 gas ratio of 1:1 (left) and at 700 ◦C with a CO:O2 gas ratio of2:1 (right). The blue and red regions in the image in (b) correspond to the nanostructures from theb-d pairs with −4 nm ≤ b, d < 0 and b < −4 nm, d < 0 in the zeroth PDs, respectively, and those in(d) correspond to the structures from the b-d pairs with 0 < b, d ≤ 4 nm and 0 < b, 4 nm ≤ d in thefirst PDs, respectively.Then, the geometric feature related to PC1 was revealed through inverse analysis of thereconstructed PDs, which enabled us to identify the corresponding N-dimensional holes tothe red (positive) and blue (negative) regions in the reconstructed PDs (Figure 5a,c). Here,to maintain simplicity and interpretability, we set a characteristic scale of 4 nm to definespecific blue and non-blue regions by the characteristic scale of b in the zeroth reconstructedPDs and d in the first ones because the blue and non-blue regions can be separated bythis characteristics value (Figure S7). Furthermore, we confined b-d pairs with −4 nm ≤ b,d < 0 in the zeroth PDs for the connected domains and those with 0 < b, d < 4 nm in thefirst PDs for arc-like structures, which correspond to the blue (negative) regions. The insetimages in Figure 5b,d show the inverse analysis results obtained from the zeroth and firstreconstructed PDs for the nanostructure synthesized at 700 ◦C with a 1:1 syngas ratio andat 700 ◦C with a 2:1 syngas ratio, which had a large difference in PC1 values. Their blueNanomaterials 2024, 14, 1413 9 of 12and red phases correspond to the blue and non-blue regions in the reconstructed PDs,respectively. We can clearly observe that the area fraction of the blue phase decreasesas PC1 increases for both PDs. This implies that the difference in hierarchical clusteringbetween the six groups could mainly be explained by the number of narrower domainsand smaller arcs rather than the characteristic size. Another important finding here isthat the number of small arcs can represent the degree of disorder because it reflects thedensity of local bends in the nanostructures. The number of domains was interpreted as theconnectivity in a previous study [15]. Even though the stripes are maintained, they oftenbend to form maze-like structures. That is, the number of small arcs can act as an effectivefeature to describe the nanostructure rather than the number of domains. The visualizationof the extracted features by a combination of hierarchical clustering and inverse analysisrevealed that considering even the characteristic size of the features made the meaning ofthe features clearer and improved their interpretability.3.3. Determination and Use of Effective Interpretable DescriptorsA random forest classification was performed to evaluate the five features obtained anddetermine the most effective descriptors for structural classification as the final objective.Here, from the feature visualization, the number of arcs was replaced by that of small arcswith d < 4 nm. To improve the reliability for evaluation of the feature importance, wemade four sample sets consisting of the three nanostructures confirmed to be classifiable byhierarchical clustering (Table S2). In all the sample sets, the number of small arcs and thewidth of the CeO2 phase were the most and second-most important features, respectively,among the five features used for the random forest classification (Figure S8). Interestingly,the number of small arcs was the most important feature in sample set 1, which consistedof nanostructures fabricated at different temperatures with the same gas ratio (Figure 2).This result is consistent with that of inverse analysis using the PC1 contribution (Figure 5).It was confirmed that binarization procedures with different parameters had little effect onthe determination of effective descriptors through random forest classification (Figure S9).Figure 6 displays a scatter plot of the number of small arcs with the characteristicsize and width of the CeO2 phases identified as the effective descriptors for 12 Pt-CeO2nanostructures. The scatter plots clearly classify the various nanostructures synthesizedunder different conditions. The width (vertical axis) behaves like a scale factor, and thenumber of small arcs (horizontal axis) behaves as a descriptor of disorder. For comparison,the well-classified nanostructures synthesized at 500, 600, and 700 ◦C with a gas ratio of 2:1are shown in blue, green, and red, respectively. As observed in the corresponding binarizedSTEM images (inset), while the nanostructures at the top left (red) are striped, those atthe bottom right (blue) are maze-like, with more small arcs present in the samples formedat lower temperatures, as mentioned in Section 3.1. Notably, PCA could also classify the12 nanostructures but provided little direct interpretability (Figure S10).In addition, these two simple and interpretable descriptors could be utilized formaterial design to control the ionic conductivity of Pt/CeO2. The lower activation energyshould increase ionic conductivity. In our previous measurements, the nanostructuressynthesized at 500, 600, and 700 ◦C with a gas ratio of 2:1 had activation energies of 0.74 eV,1.2 eV, and 1.24 eV, respectively [16]. Comparing them with the results in Figure 6, thesmaller the width and the more the number of small arcs, the lower the activation energy.This trend could indicate that increasing the interfacial area between Pt and CeO2 phaseswith the smaller width and the number of small arcs led to an increase in ionic conductivity.This assumption agrees with the fact that oxygen ions diffuse rapidly at the interfacebetween metal and oxygen [34]. Thus, we could not only classify the nanocomposites butalso directly visualize the structural differences, contributing to establishing the guidelinesfor materials design by the two interpretable descriptors.Nanomaterials 2024, 14, 1413 10 of 12Nanomaterials 2024, 14, x FOR PEER REVIEW 11 of 13    Figure 6. Scatter plot of the number of small arcs with the characteristic size and the width of the CeO2 phase for 12 Pt-CeO2 nanostructures fabricated at different annealing temperatures and gas ratios. The colors show that there are three well-classified groups that were fabricated at different temperatures but with the same gas ratio. Inset images are corresponding binarized STEM images for each group. 4. Conclusions We quantitatively evaluated Pt-CeO2 nanostructures ranging from maze-like to striped structures by using the PDs of their STEM images. We aimed to effectively extract interpretable features to enable structural classification. Analysis of the PD quadrants pro-vided five features: average width and total length of the striped CeO2 phases, the number of CeO2 phases from the zeroth PDs, and the numbers of ring- and arc-like structures from first PDs. The combination of hierarchical clustering and inverse analysis of each vector-ized PD revealed that the addition of the characteristic size to the features effectively im-proved the interpretability; that is, the number of small arc-like structures represented the density of bends in the nanostructures. Finally, the two most important interpretable de-scriptors among the five features were extracted by a random forest classification: the number of small arcs and the width of the CeO2 phase. A simple 2D scatter plot based on these two descriptors could effectively classify the nanocomposites and visualize their structure differences via the two metrics of degree of disorder and scale factor. Thus, PH facilitates the assessment of the structural differences that arise from variations in synthe-sis conditions/methods. Furthermore, the extraction of the interpretable descriptors in this way was effective and convenient when quantitatively evaluating the relationship be-tween quantitative properties such as ionic conductivity. Using this approach, it is possi-ble to predict properties from structural descriptors. Supplementary Materials: The following supporting information can be downloaded at www.mdpi.com/xxx/s1, Figure S1. Binarization and post-processing of the background removal im-age for the STEM images before the PH analysis. Figure S2. Original HAADF-STEM images of Pt-CeO2 nanostructures synthesized at (a) 500 °C, (b) 600 °C, and (c) 700 °C with a CO:O2 gas ratio of Figure 6. Scatter plot of the number of small arcs with the characteristic size and the width of theCeO2 phase for 12 Pt-CeO2 nanostructures fabricated at different annealing temperatures and gasratios. The colors show that there are three well-classified groups that were fabricated at differenttemperatures but with the same gas ratio. Inset images are corresponding binarized STEM images foreach group.4. ConclusionsWe quantitatively evaluated Pt-CeO2 nanostructures ranging from maze-like to stripedstructures by using the PDs of their STEM images. We aimed to effectively extract inter-pretable features to enable structural classification. Analysis of the PD quadrants providedfive features: average width and total length of the striped CeO2 phases, the numberof CeO2 phases from the zeroth PDs, and the numbers of ring- and arc-like structuresfrom first PDs. The combination of hierarchical clustering and inverse analysis of eachvectorized PD revealed that the addition of the characteristic size to the features effectivelyimproved the interpretability; that is, the number of small arc-like structures representedthe density of bends in the nanostructures. Finally, the two most important interpretabledescriptors among the five features were extracted by a random forest classification: thenumber of small arcs and the width of the CeO2 phase. A simple 2D scatter plot basedon these two descriptors could effectively classify the nanocomposites and visualize theirstructure differences via the two metrics of degree of disorder and scale factor. Thus, PHfacilitates the assessment of the structural differences that arise from variations in synthesisconditions/methods. Furthermore, the extraction of the interpretable descriptors in thisway was effective and convenient when quantitatively evaluating the relationship betweenquantitative properties such as ionic conductivity. Using this approach, it is possible topredict properties from structural descriptors.Supplementary Materials: The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/nano14171413/s1, Figure S1. Binarization and post-processing ofthe background removal image for the STEM images before the PH analysis. Figure S2. OriginalHAADF-STEM images of Pt-CeO2 nanostructures synthesized at (a) 500 ◦C, (b) 600 ◦C, and (c) 700 ◦Cwith a CO:O2 gas ratio of 2:1. (d), (e), and (f) Binarized images corresponding to (a), (b), and (c),https://www.mdpi.com/article/10.3390/nano14171413/s1https://www.mdpi.com/article/10.3390/nano14171413/s1Nanomaterials 2024, 14, 1413 11 of 12respectively, which are identical to Figure 2a–c. Scale bars represent 100 nm. Figure S3. Assignmentof distance and filtration for zero-dimensional holes in binary image contrast. Figure S4. Filtrationprocesses of (a) connected and (b) isolated domains for the zeroth PDs. Figure S5. (a–c) BinarizedHAADF-STEM images of Pt-CeO2 nanostructures (the same as in Figure 1a–c) with birth points ofthe zeroth PDs (Figure 1d–f) with lifetimes shorter than 0.2 nm (indicated by red points). Figure S6.Filtration process of a (a) ring and (b) arc for the first PDs. Table S1. Mean cosine similarities andtheir deviations of vectorized PDs for Pt-CeO2 nanostructures synthesized under different annealingconditions. Figure S7. Frequency distributions of the number of grids with negative (blue) andpositive (red) PC1 values for (a) b in the reconstructed zeroth PD and (b) d in the first PD. Table S2.Four sample sets for random forest classification and their classification accuracy for each sampleset with different synthesis conditions. Figure S8. Feature importance for sample set (a) 1, (b) 2,(c) 3, and (d) 4 in Table S2 determined from random forest classification. Figure S9. Changes of(a) binarized STEM images of the Pt-CeO2 nanostructure formed at an annealing temperature of600 ◦C and syngas ratio of 2:1, (b) feature importance in random forest classification, and (c) scatterplots of the width and number of CeO2 phases caused by the two binarization parameters of blocksize in the adaptive-threshold method and kernel size in opening–closing processing. Figure S10.PCA for the (a) zeroth and (b) first PDs of the 12 Pt-CeO2 nanostructures produced under differentsynthesis conditions.Author Contributions: Conceptualization, R.E., Y.W. and A.H.; methodology, R.E.; software, R.E.;validation, R.E. and Y.W.; formal analysis, R.E. and Y.W.; investigation, R.E. and Y.W.; resources,A.H.; data curation, Y.W.; writing—original draft preparation, R.E. and Y.W.; writing—review andediting, A.H., Y.W. and H.A.; visualization, R.E. and Y.W.; supervision, A.H.; project administration,A.H.; funding acquisition, A.H. All authors have read and agreed to the published version ofthe manuscript.Funding: AH acknowledges JST FOREST Program (Grant Number JPMJFR213U, Japan), JST PRESTO(Grant Number JPMJPR17S7), and ARIM (Proposal numbers JPMXP1223NM5325, JPMXP1224NM5122)for financial support. HA acknowledges JSPS Grant-in-Aid for Scientific Research (B) (KAKENHI)(Grant Numbers 22H01799, 23K23067).Data Availability Statement: Data are contained within the article and Supplementary Materials.Acknowledgments: A part of this work was supported by “Advanced Research Infrastructure forMaterials and Nanotechnology in Japan (ARIM)” of the Ministry of Education, Culture, Sports,Science and Technology (MEXT). We thank Akihiko Hirata in Univ. of Waseda for useful discussionon homology analysis. We thank Natasha Lundin for editing a draft of this manuscript.Conflicts of Interest: The authors declare no conflicts of interest.References1. Liu, X.; Iocozzia, J.; Wang, Y.; Cui, X.; Chen, Y.; Zhao, S.; Li, Z.; Lin, Z. 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