Topological Data Analysis Methods Research Articles

Enhanced MRI brain tumor detection and classification via topological data analysis and low-rank tensor decomposition

The advent of artificial intelligence in medical imaging has paved the way for significant advancements in the diagnosis of brain tumors. This study presents a novel ensemble approach that uses magnetic resonance imaging (MRI) to identify and categorize common brain cancers, such as pituitary, meningioma, and glioma. The proposed workflow is composed of a two-fold approach: firstly, it employs non-trivial image enhancement techniques in data preprocessing, low-rank Tucker decomposition for dimensionality reduction, and machine learning (ML) classifiers to detect and predict the type of brain tumor. Secondly, persistent homology (PH), a topological data analysis (TDA) technique, is exploited to extract potential critical areas in MRI scans. When paired with the ML classifier output, this additional information can help domain experts to identify areas of interest that might contain tumor signatures, improving the interpretability of ML predictions. When compared to automated diagnoses, this transparency adds another level of confidence and is essential for clinical acceptance. The performance of the system was quantitatively evaluated on a well-known MRI dataset, with an overall classification accuracy of 97.28% using an extremely randomized trees model. The promising results show that the integration of TDA, ML, and low-rank approximation methods is a successful approach for brain tumor identification and categorization, providing a solid foundation for further study and clinical application.

Towards Analysis of Multivariate Time Series Using Topological Data Analysis

Topological data analysis (TDA) has proven to be a potent approach for extracting intricate topological structures from complex and high-dimensional data. In this paper, we propose a TDA-based processing pipeline for analyzing multi-channel scalp EEG data. The pipeline starts with extracting both frequency and temporal information from the signals via the Hilbert–Huang Transform. The sequences of instantaneous frequency and instantaneous amplitude across all electrode channels are treated as approximations of curves in the high-dimensional space. TDA features, which represent the local topological structure of the curves, are further extracted and used in the classification models. Three sets of scalp EEG data, including one collected in a lab and two Brain–computer Interface (BCI) competition data, were used to validate the proposed methods, and compare with other state-of-art TDA methods. The proposed TDA-based approach shows superior performance and outperform the winner of the BCI competition. Besides BCI, the proposed method can also be applied to spatial and temporal data in other domains such as computer vision, remote sensing, and medical imaging.

Topological Data Analysis Combined with High-Throughput Computational Screening of Hydrophobic Metal–Organic Frameworks: Application to the Adsorptive Separation of C3 Components

The shape and topology of pores have significant impacts on the gas storage properties of nanoporous materials. Metal–organic frameworks (MOFs) are ideal materials with which to tailor to the needs of specific applications, due to properties such as their tunable structure and high specific surface area. It is, therefore, particularly important to develop descriptors that accurately identify the topological features of MOF pores. In this work, a topological data analysis method was used to develop a topological descriptor, based on the pore topology, which was combined with the Extreme Gradient Boosting (XGBoost) algorithm to predict the adsorption performance of MOFs for methane/ethane/propane. The final results show that this descriptor can accurately predict the performance of MOFs, and the introduction of the topological descriptor also significantly improves the accuracy of the model, resulting in an increase of up to 17.55% in the R2 value of the model and a decrease of up to 46.1% in the RMSE, compared to commonly used models that are based on the structural descriptor. The results of this study contribute to a deeper understanding of the relationship between the performance and structure of MOFs and provide useful guidelines and strategies for the design of high-performance separation materials.

Topological data analysis expands the genotype to phenotype map for 3D maize root system architecture.

A central goal of biology is to understand how genetic variation produces phenotypic variation, which has been described as a genotype to phenotype (G to P) map. The plant form is continuously shaped by intrinsic developmental and extrinsic environmental inputs, and therefore plant phenomes are highly multivariate and require comprehensive approaches to fully quantify. Yet a common assumption in plant phenotyping efforts is that a few pre-selected measurements can adequately describe the relevant phenome space. Our poor understanding of the genetic basis of root system architecture is at least partially a result of this incongruence. Root systems are complex 3D structures that are most often studied as 2D representations measured with relatively simple univariate traits. In prior work, we showed that persistent homology, a topological data analysis method that does not pre-suppose the salient features of the data, could expand the phenotypic trait space and identify new G to P relations from a commonly used 2D root phenotyping platform. Here we extend the work to entire 3D root system architectures of maize seedlings from a mapping population that was designed to understand the genetic basis of maize-nitrogen relations. Using a panel of 84 univariate traits, persistent homology methods developed for 3D branching, and multivariate vectors of the collective trait space, we found that each method captures distinct information about root system variation as evidenced by the majority of non-overlapping QTL, and hence that root phenotypic trait space is not easily exhausted. The work offers a data-driven method for assessing 3D root structure and highlights the importance of non-canonical phenotypes for more accurate representations of the G to P map.

Exploring Contemporary Calligraphic Creation and Aesthetic Characteristics Using Topological Data Analysis Methods

Abstract In the process of digitization in today's society, the organic combination of technological means and traditional arts, such as calligraphy culture, is a topic that China is facing. The topological data analysis method is used in this paper to conduct an in-depth study on the creation and aesthetic characteristics of contemporary calligraphy, taking into account the situation above. In this paper, the topological data analysis method is first utilized to calculate the position of the critical points in the vector field and draw the topological structure of the vector field. The Jacobi matrix is computed, and the numerical solutions of the eigenvalues and eigenvectors in the matrix are used to pinpoint critical points. The TDA algorithm allows for the visualization and modeling of aesthetic features in contemporary calligraphy creation using the data required for the acquired bulk body model. The aesthetic features of calligraphy creation are analyzed through simulation experiments in the end. The results of simulation experiments show that the TDA algorithm completes the convergence before 500 iterations under a 4-fold filter, the nodes are mainly concentrated in the range of [500, 1000], and the convergence speed is the fastest among the five algorithms. Testing the topological data analysis algorithm for the recognition performance of aesthetic features of calligraphy creation, the recognition rate of TDA is as high as 98.30%, which has a good recognition effect.

Research on digital rural landscape garden landscape planning under rural revitalization strategy based on topological data analysis method

Abstract The advancement of rural revitalization strategies underscores the increasing significance of landscape garden construction. This study optimizes scenic road routing within rural garden planning to support these strategies. By refining the topological data analysis method with Particle Swarm Optimization (PSO), we develop a model for rural scenic road routing. The model utilizes selected suitability indexes, organized through factor analysis into four categories: scenic, land, industrial resources, and infrastructure. Applied to a case study (Route Z), the model identifies Route 2 as the most suitable option, scoring 2.933. Our findings offer innovative insights and technical approaches for effectively planning rural scenic roads, contributing to countryside revitalization efforts.

Topological persistence guided knowledge distillation for wearable sensor data

Deep learning methods have achieved a lot of success in various applications involving converting wearable sensor data to actionable health insights. A common application areas is activity recognition, where deep-learning methods still suffer from limitations such as sensitivity to signal quality, sensor characteristic variations, and variability between subjects. To mitigate these issues, robust features obtained by topological data analysis (TDA) have been suggested as a potential solution. However, there are two significant obstacles to using topological features in deep learning: (1) large computational load to extract topological features using TDA, and (2) different signal representations obtained from deep learning and TDA which makes fusion difficult. In this paper, to enable integration of the strengths of topological methods in deep-learning for time-series data, we propose to use two teacher networks — one trained on the raw time-series data, and another trained on persistence images generated by TDA methods. These two teachers are jointly used to distill a single student model, which utilizes only the raw time-series data at test-time. This approach addresses both issues. The use of KD with multiple teachers utilizes complementary information, and results in a compact model with strong supervisory features and an integrated richer representation. To assimilate desirable information from different modalities, we design new constraints, including orthogonality imposed on feature correlation maps for improving feature expressiveness and allowing the student to easily learn from the teacher. Also, we apply an annealing strategy in KD for fast saturation and better accommodation from different features, while the knowledge gap between the teachers and student is reduced. Finally, a robust student model is distilled, which can at test-time uses only the time-series data as an input, while implicitly preserving topological features. The experimental results demonstrate the effectiveness of the proposed method on wearable sensor data. The proposed method shows 71.74% in classification accuracy on GENEActiv with WRN16-1 (1D CNNs) student, which outperforms baselines and takes much less processing time (less than 17 sec) than teachers on 6k testing samples.

Statistical inference for dependence networks in topological data analysis.

Topological data analysis (TDA) provide tools that are becoming increasingly popular for analyzing multivariate time series data. One key aspect in analyzing multivariate time series is dependence between components. One application is on brain signal analysis. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes. These dependence patterns may be altered by various neurological and cognitive impairments such as Alzheimer's and Parkinson's diseases, as well as attention deficit hyperactivity disorder (ADHD). Because there is no ground-truth with known dependence patterns in real brain signals, testing new TDA methods on multivariate time series is still a challenge. Our goal here is to develop novel statistical inference procedures via simulations. Simulations are useful for generating some null distributions of a test statistic (for hypothesis testing), forming confidence regions, and for evaluating the performance of proposed TDA methods. To the best of our knowledge, there are no methods that simulate multivariate time series data with potentially complex user-specified connectivity patterns. In this paper we present a novel approach to simulate multivariate time series with specific number of cycles/holes in its dependence network. Furthermore, we also provide a procedure for generating higher dimensional topological features.

Asymptotics of lower dimensional zero-density regions

Topological data analysis (TDA) allows us to explore the topological features of a dataset. Among topological features, lower dimensional ones have recently drawn the attention of practitioners in mathematics and statistics due to their potential to aid the discovery of low dimensional structure in a data set. However, lower dimensional features are usually challenging to detect based on finite samples and using TDA methods that ignore the probabilistic mechanism that generates the data. In this paper, lower dimensional topological features occurring as zero-density regions of density functions are introduced and thoroughly investigated. Specifically, we consider sequences of coverings for the support of a density function in which the coverings are comprised of balls with shrinking radii. We show that, when these coverings satisfy certain sufficient conditions as the sample size goes to infinity, we can detect lower dimensional, zero-density regions with increasingly higher probability while guarding against false detection. We supplement the theoretical developments with the discussion of simulated experiments that elucidate the behaviour of the methodology for different choices of the tuning parameters that govern the construction of the covering sequences and characterize the asymptotic results.

Application of topological data analysis method based on calligraphy creation schema and its limitation analysis

Abstract This paper first describes the principle of topological data and the basic steps of topological data analysis. It then outputs images using parametric filters and establishes the expressions of differential evolution. The system is designed using a Gaussian mixture model, the stroke matching module is categorized, and the composition execution diagram of the system is designed. Finally, the calligraphy creation system based on topological data analysis is completed, and the three evaluation results are 62.6%, 58.9%, and 59.2%, all of which are more than 50%, which is better than the traditional creation system. The topological data-based oriented calligraphy creation system designed in this paper is proven effective in achieving calligraphy creation and has good application.

Topological data analysis for revealing dynamic brain reconfiguration in MEG data.

In recent years, the focus of the functional connectivity community has shifted from stationary approaches to the ones that include temporal dynamics. Especially, non-invasive electrophysiological data (magnetoencephalography/electroencephalography (MEG/EEG)) with high temporal resolution and good spatial coverage have made it possible to measure the fast alterations in the neural activity in the brain during ongoing cognition. In this article, we analyze dynamic brain reconfiguration using MEG images collected from subjects during the rest and the cognitive tasks. Our proposed topological data analysis method, called Mapper, produces biomarkers that differentiate cognitive tasks without prior spatial and temporal collapse of the data. The suggested method provides an interactive visualization of the rapid fluctuations in electrophysiological data during motor and cognitive tasks; hence, it has the potential to extract clinically relevant information at an individual level without temporal and spatial collapse.

Dynamic ECG signal quality evaluation based on persistent homology and GoogLeNet method.

Cardiovascular disease is a serious health problem. Continuous Electrocardiograph (ECG) monitoring plays a vital role in the early detection of cardiovascular disease. As the Internet of Things technology continues to mature, wearable ECG signal monitors have been widely used. However, dynamic ECG signals are extremely susceptible to contamination. Therefore, it is necessary to evaluate the quality of wearable dynamic ECG signals. The topological data analysis method (TDA) with persistent homology, which can effectively capture the topological information of high-dimensional data space, has been widely studied. In this study, a brand-new quality assessment method of wearable dynamic ECG signals was proposed based on the TDA with persistent homology method. The point cloud of an ECG signal was constructed, and then the complex sequence was generated and displayed as a persistent barcode. Finally, GoogLeNet based on the transfer learning model with a 10-fold cross-validation method was used to train the classification model. A total of 12-leads ECGs Dataset and single-lead ECGs Dataset, established based on the 2011 PhysioNet/CinC challenge dataset, were both used to verify the performance of this method. In the study, 773 "acceptable" and 225 "unacceptable" signals were used as 12-leads ECGs Dataset. We relabeled 12,000 ECG signals in the challenge dataset, and treated them as single-lead ECGs Dataset after empty lead detection and balance datasets. Compared with the traditional ECG signal quality assessment method mainly based on waveform characteristics and time-frequency characteristics, the performance of the quality assessment method proposed. In this study, the classification performance of the proposed method are fairly great, mAcc = 98.04%, F1 = 98.40%, Se = 97.15%, Sp = 98.93% for 12-leads ECGs Dataset and mAcc = 98.55%, F1 = 98.62%, Se = 98.37%, Sp = 98.85% for single-lead ECGs Dataset.

Topological data analysis for geographical information science using persistent homology

Topological data analysis (TDA) is an emerging field of research, which considers the application of topology to data analysis. Recently, these methods have been successfully applied to research problems in the field of geographical information science (GIS) and there is much potential for future applications. In this article, we provide an introduction to the fundamentals of TDA for GIS researchers and practitioners and highlight specific benefits that TDA methods provide relative to some conventional methods. We focus on the method of persistent homology, which is the most commonly used TDA method. We describe how persistent homology can be applied to data types commonly encountered in the GIScience domain, namely sets of points, networks and sequences of images. We also describe the application of persistent homology to two specific GIS problems, which are the point pattern analysis of UK city pubs and the analysis of UK rainfall radar imagery. In each case we stress the specific benefits of TDA methods that include, for example, generating an output signature in a form that can be subject to subsequent analyses; identification of void regions in point patterns; and providing a relatively simple method to track objects in spatio-temporal images.

V-Mapper: topological data analysis for high-dimensional data with velocity

Mapper, a topological data analysis method for high-dimensional data, represents a topological structure as a simplicial complex or graph based on the nerve of clusters. We propose V-Mapper (velocity Mapper), an extension of Mapper, for high-dimensional data with velocity. V-Mapper simultaneously describes a topological structure and flow as a weighted directed graph (V-Mapper graph) by embedding velocity in the edges of the Mapper graph. We apply V-Mapper to single-cell gene expression data using a method for inferring the velocity of gene expression. Moreover, the application of the Hodge decomposition on graph enhances the interpretation of the flow within V-Mapper graph.

University Academic Performance Development Prediction Based on TDA.

With the rapid development of higher education, the evaluation of the academic growth potential of universities has received extensive attention from scholars and educational administrators. Although the number of papers on university academic evaluation is increasing, few scholars have conducted research on the changing trend of university academic performance. Because traditional statistical methods and deep learning techniques have proven to be incapable of handling short time series data well, this paper proposes to adopt topological data analysis (TDA) to extract specified features from short time series data and then construct the model for the prediction of trend of university academic performance. The performance of the proposed method is evaluated by experiments on a real-world university academic performance dataset. By comparing the prediction results given by the Markov chain as well as SVM on the original data and TDA statistics, respectively, we demonstrate that the data generated by TDA methods can help construct very discriminative models and have a great advantage over the traditional models. In addition, this paper gives the prediction results as a reference, which provides a new perspective for the development evaluation of the academic performance of colleges and universities.

Topological Knowledge Distillation for Wearable Sensor Data.

Converting wearable sensor data to actionable health insights has witnessed large interest in recent years. Deep learning methods have been utilized in and have achieved a lot of successes in various applications involving wearables fields. However, wearable sensor data has unique issues related to sensitivity and variability between subjects, and dependency on sampling-rate for analysis. To mitigate these issues, a different type of analysis using topological data analysis has shown promise as well. Topological data analysis (TDA) captures robust features, such as persistence images (PI), in complex data through the persistent homology algorithm, which holds the promise of boosting machine learning performance. However, because of the computational load required by TDA methods for large-scale data, integration and implementation has lagged behind. Further, many applications involving wearables require models to be compact enough to allow deployment on edge-devices. In this context, knowledge distillation (KD) has been widely applied to generate a small model (student model), using a pre-trained high-capacity network (teacher model). In this paper, we propose a new KD strategy using two teacher models - one that uses the raw time-series and another that uses persistence images from the time-series. These two teachers then train a student using KD. In essence, the student learns from heterogeneous teachers providing different knowledge. To consider different properties in features from teachers, we apply an annealing strategy and adaptive temperature in KD. Finally, a robust student model is distilled, which utilizes the time series data only. We find that incorporation of persistence features via second teacher leads to significantly improved performance. This approach provides a unique way of fusing deep-learning with topological features to develop effective models.

Measuring population-level plant gene flow with topological data analysis

Hybridization is an important factor influencing the evolution of many plant species. Measurement of interspecific gene flow and quantification of recombination provide insight into the processes that shape population genetic diversity. Recently, a new approach called topological data analysis (TDA) has begun to be implemented to track genomic recombination between species and populations. However, existing TDA methods tend to underestimate gene flow in cross-population models. Here, we present a novel graph-based approach to measuring gene flow, which occurs during ongoing speciation and hybridization in plants in various mating systems. We combine minimal spanning networks with topological filtering and then test the resulting parameter measurements on simulated primary divergence and secondary contact processes under different mating conditions. The resulting parameters, based on the first Betti number (total number of one-dimensional cycles), showed a high positive correlation with the number of migrants (Nm) in almost all cases of secondary contact. Considering the modularity of MSN networks, we have devised a parameter that works well in the event of primary divergence, where gene flow is at its highest level. This approach is suitable for various types of data, including SNPs, microsatellites, and binary restriction fragment data.

Detecting Carbon Nanotube Orientation with Topological Analysis of Scanning Electron Micrographs.

As the aerospace industry is increasingly demanding stronger, lightweight materials, ultra-strong carbon nanotube (CNT) composites with highly aligned CNT network structures could be the answer. In this work, a novel methodology applying topological data analysis (TDA) to scanning electron microscope (SEM) images was developed to detect CNT orientation. The CNT bundle extensions in certain directions were summarized algebraically and expressed as visible barcodes. The barcodes were then calculated and converted into the total spread function, V(X, θ), from which the alignment fraction and the preferred direction could be determined. For validation purposes, the random CNT sheets were mechanically stretched at various strain ratios ranging from 0 to 40%, and quantitative TDA was conducted based on the SEM images taken at random positions. The results showed high consistency (R2 = 0.972) compared to Herman’s orientation factors derived from polarized Raman spectroscopy and wide-angle X-ray scattering analysis. Additionally, the TDA method presented great robustness with varying SEM acceleration voltages and magnifications, which might alter the scope of alignment detection. With potential applications in nanofiber systems, this study offers a rapid and simple way to quantify CNT alignment, which plays a crucial role in transferring the CNT properties into engineering products.

Optimal Base Station Network Based on Topological Data Analysis

The decision of which base stations need to be removed due to the cost is always a difficult problem, because the influence on the cover rate of the network caused by the removal should be kept to a minimum. However, the common methods to solve this problem such as K-means Clustering show a low accuracy. Barcode, which belongs to TDA, has the possibility to show the result by identifying the Persistent Homology of base station network. This essay mainly illustrates the specific problem of optimal base station network, which applies the TDA(Topological Data Analysis) methods to find which base stations need removing due to the cost K-means Clustering and Topological Data Analysis methods were mainly used. With the simulated distribution of telecommunication users, K-means Clustering algorithm was used to locate 30 best base stations. By comparing the minimum distance between the results (K=25 and K=30), K-means Clustering was used again to decide base station points to be removed. Then TDA was used to select which 5 base stations should be removed through observing barcode. By repeating above steps five times, Finally the average and variance of cover area in original network, K-means Clustering and TDA were compared. The experiment showed that the average cover rate of original network was 81.20% while the result of TDA and K-means Clustering were 92.13% and 89.87%. It was proved by simulation that it is more efficient to use TDA methods to construct the optimal base station network.

Persistence Curves: A canonical framework for summarizing persistence diagrams

Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space lacks an inner product. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important topic currently researched in TDA. In this paper, our main contribution consists of three components. First, we develop a general and unifying framework of vectorizing diagrams that we call the Persistence Curves (PCs), and show that several well-known summaries, such as Persistence Landscapes, fall under the PC framework. Second, we propose several new summaries based on PC framework and provide a theoretical foundation for their stability analysis. Finally, we apply proposed PCs to two applications—texture classification and determining the parameters of a discrete dynamical system; their performances are competitive with other TDA methods.