Persistence diagrams with linear machine learning models

Prediction of nanocomposite properties and process optimization using persistent homology and machine learning

Physical property prediction and synthesis process optimization are key targets in material informatics. In this study, we propose a machine learning approach that utilizes ridge regression to predict the oxygen permeability at fuel cell electrode surfaces and determine the optimal process temperature. These predictions are based on a persistence diagram derived from tomographic images captured using transmission electron microscopy (TEM). Through machine learning analysis of the complex structures present in the Pt/CeO2 nanocomposites, we discovered that l2 regularization considering diverse structural elements is more appropriate than l1 regularization (sparse modeling). Notably, our model successfully captured the activation energy of oxygen permeability, a phenomenon that could not be solely explained by the geometric feature of the Betti numbers, as demonstrated in a previous study. The correspondence between the ridge regression coefficient and persistence diagram revealed the formation process of the local and three-dimensional structures of CeO2 and their contributions to pre-exponential factor and activation energies. This analysis facilitated the determination of the annealing temperature required to achieve the optimal structure and accurately predict the physical properties.

Descriptor generation from Morgan fingerprint using persistent homology

In cheminformatics, molecular fingerprints (FPs) are used in various tasks such as regression and classification. However, predictive models often underutilize Morgan FP for regression and related tasks in machine learning. This study introduced descriptors derived from reshaped Morgan FPs using persistent homology for the predictive accuracy improvement. In the solvation free energy (FreeSolv) and water solubility (ESOL) datasets, persistent homology was found to enhance predictive accuracy compared to the use of only Morgan FPs. Notably, using the first-order persistence diagram (PD1) for descriptor generation resulted in more significant improvements than using the zeroth-order persistence diagram (PD0). Combining 4096 bits Morgan FPs with PD1-generated descriptors increased the average coefficient of determination in the Gaussian process regression from 0.597 to 0.667 for FreeSolv and from 0.629 to 0.654 for ESOL. Adjusting the grid size parameter during PD-based descriptor generation is crucial, as finer grids, especially with PD0, generate more descriptors but reduce predictive accuracy. Coarsening the grid or applying principal component analysis (PCA) mitigates overfitting and enhances accuracy. When descriptors were generated from Morgan FPs with randomly shuffled bit positions, coarsening the grid and/or applying PCA achieved similar accuracy improvements as when the persistent homology of the original Morgan FPs was used.

Causal analysis and visualization of magnetization reversal using feature extended landau free energy

The magnetization reversal in nanomagnets is causally analyzed using an extended Landau free-energy model. This model draws an energy landscape in the information space using physics-based features. Thus, the origin of the magnetic effect in macroscopic pinning phenomena can be identified. The microscopic magnetic domain beyond the hierarchy can be explained using energy gradient analysis and its decomposition. Structural features from the magnetic domains are extracted using persistent homology. Extended energy is visualized using ridge regression, principal component analysis, and Hadamard products. We found that the demagnetization energy concentration near a defect causes the demagnetization effect, which quantitatively dominates the pinning phenomenon. The exchange energy inhibits pinning, promotes saturation, and shows slight interactions with the defect. Furthermore, the energy distributions are visualized in real space. Left-position defects reduce the energy barrier and are useful for the topological inverse design of recording devices.

Pore-network extraction using discrete Morse theory: Preserving the topology of the pore space

Pore-scale modeling based on the 3D structural information of porous materials has enormous potential in assessing physical properties beyond the capabilities of laboratory methods. Such capabilities are pricey in terms of computational expenses, and this limits the applicability of the direct simulations to a small volume and requires high-performance computational resources, especially for multiphase flow simulations. The only pore-scale technique capable of dealing with large representative volumes of porous samples is pore-network (PNM) based modeling. The problem of the PNM approach is that 3D pore geometry first needs to be simplified into a graph of pores and throats that conserve topological and geometrical properties of the original 3D image. While significant progress has been achieved in terms of geometry representation, no methodology provides full conservation of the topological features of the pore structure. In this paper we present a pore-network extraction algorithm for binary 3D images based on discrete Morse theory and persistent homology that by design targets topology preservation. In addition to methodological developments, we also clarify the relationship between topological characteristics of constructed Morse chain complex and pore-network elements. We show that the Euler numbers calculated for PNMs based on our methodology coincide with those obtained using the direct topological analysis. The characteristics of the extracted pore network are calculated for several 3D porous binary images and compared with the results of maximum inscribed balls-based and watershed-based approaches as well as a hybrid approach to support our methodology.

Can Persistent Homology Features Capture More Intrinsic Information about Tumors from 18F-Fluorodeoxyglucose Positron Emission Tomography/Computed Tomography Images of Head and Neck Cancer Patients?

This study hypothesized that persistent homology (PH) features could capture more intrinsic information about the metabolism and morphology of tumors from ¹⁸F-fluorodeoxyglucose positron emission tomography (PET)/computed tomography (CT) images of patients with head and neck (HN) cancer than other conventional features. PET/CT images and clinical variables of 207 patients were selected from the publicly available dataset of the Cancer Imaging Archive. PH images were generated from persistent diagrams obtained from PET/CT images. The PH features were derived from the PH PET/CT images. The signatures were constructed in a training cohort from features from CT, PET, PH-CT, and PH-PET images; clinical variables; and the combination of features and clinical variables. Signatures were evaluated using statistically significant differences (p-value, log-rank test) between survival curves for low- and high-risk groups and the C-index. In an independent test cohort, the signature consisting of PH-PET features and clinical variables exhibited the lowest log-rank p-value of 3.30 × 10⁻⁵ and C-index of 0.80, compared with log-rank p-values from 3.52 × 10⁻² to 1.15 × 10⁻⁴ and C-indices from 0.34 to 0.79 for other signatures. This result suggests that PH features can capture the intrinsic information of tumors and predict prognosis in patients with HN cancer.

Topological descriptor of thermal conductivity in amorphous Si

Quantifying the correlation between the complex structures of amorphous materials and their physical properties has been a longstanding problem in materials science. In amorphous Si, a representative covalent amorphous solid, the presence of a medium-range order (MRO) has been intensively discussed. However, the specific atomic arrangement corresponding to the MRO and its relationship with physical properties, such as thermal conductivity, remains elusive. We solved this problem by combining topological data analysis, machine learning, and molecular dynamics simulations. Using persistent homology, we constructed a topological descriptor that can predict thermal conductivity. Moreover, from the inverse analysis of the descriptor, we determined the typical ring features correlated with both the thermal conductivity and MRO. The results could provide an avenue for controlling material characteristics through the topology of the nanostructures.

ASPECTS OF TOPOLOGICAL APPROACHES FOR DATA SCIENCE

We establish a new theory which unifies various aspects of topological approaches for data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.

A multi-parameter persistence framework for mathematical morphology

The field of mathematical morphology offers well-studied techniques for image processing and is applicable for studies ranging from materials science to ecological pattern formation. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis. We demonstrate that morphological operations naturally form a multiparameter filtration and that persistent homology can then be used to extract information about both topology and geometry in the images as well as to automate methods for optimizing the study and rendering of structure in images. For illustration, we develop an automated approach that utilizes this framework to denoise binary, grayscale, and color images with salt and pepper and larger spatial scale noise. We measure our example unsupervised denoising approach to state-of-the-art supervised, deep learning methods to show that our results are comparable.

Classification for Transmission Electron Microscope Images from Different Amorphous States Using Persistent Homology

It is difficult to discriminate the amorphous state using a transmission electron microscope (TEM). We discriminated different amorphous states on 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 molecular dynamic simulation. TEM images in several defocus conditions were simulated by the multi-slice method using the created amorphous and liquid states, and their persistent diagrams were calculated. Finally, logistic regression and support vector classification machine learning algorithms were applied for discrimination. Consequently, we found that the amorphous and liquid phases can be discriminated by more than 85%. Because the 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.

Analysis on Microstructure-Property Linkages of Filled Rubber Using Machine Learning and Molecular Dynamics Simulations

A better understanding of the microstructure–property relationship can be achieved by sampling and analyzing a microstructure leading to a desired material property. During the simulation of filled rubber, this approach includes extracting common aggregates from a complex filler morphology consisting of hundreds of filler particles. However, a method for extracting a core structure that determines the rubber mechanical properties has not been established yet. In this study, we analyzed complex filler morphologies that generated extremely high stress using two machine learning techniques. First, filler morphology was quantified by persistent homology and then vectorized using persistence image as the input data. After that, a binary classification model involving logistic regression analysis was developed by training a dataset consisting of the vectorized morphology and stress-based class. The filler aggregates contributing to the desired mechanical properties were extracted based on the trained regression coefficients. Second, a convolutional neural network was employed to establish a classification model by training a dataset containing the imaged filler morphology and class. The aggregates strongly contributing to stress generation were extracted by a kernel. The aggregates extracted by both models were compared, and their shapes and distributions producing high stress levels were discussed. Finally, we confirmed the effects of the extracted aggregates on the mechanical property, namely the validity of the proposed method for extracting stress-contributing fillers, by performing coarse-grained molecular dynamics simulations.

Applications of Topological Data Analysis in Oncology

The emergence of the information age in the last few decades brought with it an explosion of biomedical data. But with great power comes great responsibility: there is now a pressing need for new data analysis algorithms to be developed to make sense of the data and transform this information into knowledge which can be directly translated into the clinic. Topological data analysis (TDA) provides a promising path forward: using tools from the mathematical field of algebraic topology, TDA provides a framework to extract insights into the often high-dimensional, incomplete, and noisy nature of biomedical data. Nowhere is this more evident than in the field of oncology, where patient-specific data is routinely presented to clinicians in a variety of forms, from imaging to single cell genomic sequencing. In this review, we focus on applications involving persistent homology, one of the main tools of TDA. We describe some recent successes of TDA in oncology, specifically in predicting treatment responses and prognosis, tumor segmentation and computer-aided diagnosis, disease classification, and cellular architecture determination. We also provide suggestions on avenues for future research including utilizing TDA to analyze cancer time-series data such as gene expression changes during pathogenesis, investigation of the relation between angiogenic vessel structure and treatment efficacy from imaging data, and experimental confirmation that geometric and topological connectivity implies functional connectivity in the context of cancer.

Higher-order structure of polymer melt described by persistent homology

The optimal method of the polymer Materials Informatics (MI) has not been developed because the amorphous nature of the higher-order structure affects these properties. We have now tried to develop the polymer MI’s descriptor of the higher-order structure using persistent homology as the topological method. We have experimentally studied the influence of the MD simulation cell size as the higher-order structure of the polymer on its electrical properties important for a soft material sensor or actuator device. The all-atom MD simulation of the polymer has been calculated and the obtained atomic coordinate has been analyzed by the persistent homology. The change in the higher-order structure by different cell size simulations affects the dielectric constant, although these changes are not described by a radial distribution function (RDF). On the other hand, using the 2nd order persistent diagram (PD), it was found that when the cell size is small, the island-shaped distribution become smoother as the cell size increased. There is the same tendency for the condition of change in the monomer ratio, the polymer chain length or temperature. As a result, the persistent homology may express the higher-order structure generated by the MD simulation as a descriptor of the polymer MI.

Assessment of skin barrier function using skin images with topological data analysis

Recent developments of molecular biology have revealed diverse mechanisms of skin diseases, and precision medicine considering these mechanisms requires the frequent objective evaluation of skin phenotypes. Transepidermal water loss (TEWL) is commonly used for evaluating skin barrier function; however, direct measurement of TEWL is time-consuming and is not convenient for daily clinical practice. Here, we propose a new skin barrier assessment method using skin images with topological data analysis (TDA). TDA enabled efficient identification of structural features from a skin image taken by a microscope. These features reflected the regularity of the skin texture. We found a significant correlation between the topological features and TEWL. Moreover, using the features as input, we trained machine-learning models to predict TEWL and obtained good accuracy (R2 = 0.524). Our results suggest that assessment of skin barrier function by topological image analysis is promising.

Pattern detection in colloidal assembly: A mosaic of analysis techniques

Characterization of the morphology, identification of patterns and quantification of order encountered in colloidal assemblies is essential for several reasons. First of all, it is useful to compare different self-assembly methods and assess the influence of different process parameters on the final colloidal pattern. In addition, casting light on the structures formed by colloidal particles can help to get better insight into colloidal interactions and understand phase transitions. Finally, the growing interest in colloidal assemblies in materials science for practical applications going from optoelectronics to biosensing imposes a thorough characterization of the morphology of colloidal assemblies because of the intimate relationship between morphology and physical properties (e.g. optical and mechanical) of a material. Several image analysis techniques developed to investigate images (acquired via scanning electron microscopy, digital video microscopy and other imaging methods) provide variegated and complementary information on the colloidal structures under scrutiny. However, understanding how to use such image analysis tools to get information on the characteristics of the colloidal assemblies may represent a non-trivial task, because it requires the combination of approaches drawn from diverse disciplines such as image processing, computational geometry and computational topology and their application to a primarily physico-chemical process. Moreover, the lack of a systematic description of such analysis tools makes it difficult to select the ones more suitable for the features of the colloidal assembly under examination. In this review we provide a methodical and extensive description of real-space image analysis tools by explaining their principles and their application to the investigation of two-dimensional colloidal assemblies with different morphological characteristics.

Computer-aided classification of hepatocellular ballooning in liver biopsies from patients with NASH using persistent homology

BACKGROUND AND OBJECTIVE

Hepatocellular ballooning is an important histological parameter in the diagnosis of nonalcoholic steatohepatitis (NASH), and it is considered to be a morphological pattern that indicates the severity and the progression to cirrhosis and liver-related deaths. There remains uncertainty about the pathological criteria for evaluating the spectrum of non-alcoholic fatty liver disease (NAFLD) in liver biopsies. We introduce persistence images as novel mathematical descriptors for the classification of ballooning degeneration in the pathological diagnosis.

METHODS

We implemented and tested a topological data analysis methodology combined with linear machine learning techniques and applied this to the classification of tissue images into NAFLD subtypes using Matteoni classification in liver biopsies.

RESULTS

Digital images of hematoxylin- and eosin-stained specimens with a pathologist's visual assessment were obtained from 79 patients who were clinically diagnosed with NAFLD. We obtained accuracy rates of more than 90% for the classification between NASH and non-NASH NAFLD groups. The highest area under the curve from the receiver operating characteristic analysis was 0.946 for the classification of NASH and NAFL2 (type 2 of Matteoni classification), when both 0- and 1-dimensional persistence images were used.

CONCLUSIONS

Our methodology using persistent homology provides quantitative measurements of the topological features in liver biopsies of NAFLD groups with considerable accuracy.

Hepatic tumor classification using texture and topology analysis of non-contrast-enhanced three-dimensional T1-weighted MR images with a radiomics approach

The purpose of this study is to evaluate the accuracy for classification of hepatic tumors by characterization of T1-weighted magnetic resonance (MR) images using two radiomics approaches with machine learning models: texture analysis and topological data analysis using persistent homology. This study assessed non-contrast-enhanced fat-suppressed three-dimensional (3D) T1-weighted images of 150 hepatic tumors. The lesions included 50 hepatocellular carcinomas (HCCs), 50 metastatic tumors (MTs), and 50 hepatic hemangiomas (HHs) found respectively in 37, 23, and 33 patients. For classification, texture features were calculated, and also persistence images of three types (degree 0, degree 1 and degree 2) were obtained for each lesion from the 3D MR imaging data. We used three classification models. In the classification of HCC and MT (resp. HCC and HH, HH and MT), we obtained accuracy of 92% (resp. 90%, 73%) by texture analysis, and the highest accuracy of 85% (resp. 84%, 74%) when degree 1 (resp. degree 1, degree 2) persistence images were used. Our methods using texture analysis or topological data analysis allow for classification of the three hepatic tumors with considerable accuracy, and thus might be useful when applied for computer-aided diagnosis with MR images.

Finding Degradation Trigger Sites of Structural Materials for Airplanes Using X-Ray Microscopy

Macroscopic properties of carbon fiber-reinforced plastic (CFRP) and environmental barrier coating (EBC), widely used for airplanes, can be deteriorated by local cracks or degradation ("trigger sites"). We have tried to find these trigger sites using x-ray microscopy (XM), which can provide the 2D or 3D images of the chemical states and microstructures. Crack initiation in CFRP was observed in a non-destructive manner in multi-scales (nm-mm). 3D chemical-state mapping of Yb in EBC was achieved with high resolution (<50 nm). In addition to XM, in-situ observations at high temperatures were conducted for obtaining complementary information. X-ray absorption spectroscopy (XAS) and x-ray diffraction (XRD) analysis were performed simultaneously up to 1773 K. Dynamic XAS with short time-resolution (<10 ns) was conducted to investigate changes in the local structure of metal. These approaches can help us identify degradation trigger sites in the materials.