scholarly journals Topological data analysis of vascular disease: A theoretical framework

2019 ◽  
Author(s):  
John Nicponski ◽  
Jae-Hun Jung
Keyword(s):  
Data Analysis ◽  
Vascular Disease ◽  
Projection Method ◽  
Persistent Homology ◽  
Theoretical Development ◽  
Topological Data Analysis ◽  
Real Patient ◽  
Different Types ◽  
Spherical Projection ◽  
Topological Data

AbstractVascular disease is a leading cause of death world wide and therefore the treatment thereof is critical. Understanding and classifying the types and levels of stenosis can lead to more accurate and better treatment of vascular disease. Some clinical techniques to measure stenosis from real patient data are invasive or of low accuracy.In this paper, we propose a new methodology, which can serve as a supplementary way of diagnosis to existing methods, to measure the degree of vascular disease using topological data analysis. We first proposed the critical failure value, which is an application of the 1-dimensional homology group to stenotic vessels as a generalization of the percent stenosis. We demonstrated that one can take important geometric data including size information from the persistent homology of a topological space. We conjecture that we may use persistent homology as a general tool to measure stenosis levels for many different types of stenotic vessels.We also proposed the spherical projection method, which is meant to allow for future classification of different types and levels of stenosis. We showed empirically using the spectral approximation of different vasculatures that this projection could provide a new medical index that measures the degree of vascular disease. Such a new index is obtained by calculating the persistence of the 2-dimensional homology of flows. We showed that the spherical projection method can differentiate between different cases of flows and reveal hidden patterns about the underlying blood flow characteristics, that is not apparent in the raw data. We showed that persistent homology can be used in conjunction with this technique to classify levels of stenosis.The main interest of this paper is to focus on the theoretical development of the framework for the proposed method using a simple set of vascular data.

scholarly journals Preprocessing of general stenotic vascular flow data for topological data analysis

2021 ◽  
Author(s):  
Christopher Bresten ◽  
Jihoon Kweon ◽  
Xinjuan Chen ◽  
Young-Hak Kim ◽  
Jae-Hun Jung
Keyword(s):  
Data Analysis ◽  
Velocity Field ◽  
Vascular Disease ◽  
Fluid Dynamic ◽  
Persistent Homology ◽  
Original Method ◽  
Topological Data Analysis ◽  
Hemodynamic Variables ◽  
Vascular Flow ◽  
Topological Data

A new analysis and classification method of vascular disease based on topological data analysis (TDA) has been proposed in [1]. The proposed method utilizes the application of persistent homology to hemodynamic variables. Particularly, 2D homology is obtained from the velocity field of the flow projected onto the unit sphere, known as so-called the S2 projection. It was shown that such homology is closely related to the degree of vascular disease. The original method was developed based on the  computational fluid dynamic (CFD) solutions of the straight stenotic vessels. In this paper, we develop a preprocessing method that enables the proposed TDA method to be applied to general stenotic vessels of irregular geometry. The velocity field is subject to a coordinate transformation correcting for orientation and curved geometry. The preprocessed data is projected onto S2 and the corresponding homology is calculated. We show that this preprocessing is necessary for the proposed TDA method to be successfully applied to general types of stenotic vessels. Validation was performed on a set of clinical data including reconstructed vascular geometry with corresponding diagnostic indices.

scholarly journals Classification of apatite structures via topological data analysis: a framework for a ‘Materials Barcode’ representation of structure maps

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Scott Broderick ◽  
Ruhil Dongol ◽  
Tianmu Zhang ◽  
Krishna Rajan
Keyword(s):  
Machine Learning ◽  
Data Analysis ◽  
Crystal Chemistry ◽  
Persistent Homology ◽  
Hierarchical Classification ◽  
Topological Data Analysis ◽  
Learning Tool ◽  
Coordination Polyhedra ◽  
Machine Learning Tool ◽  
Topological Data

AbstractThis paper introduces the use of topological data analysis (TDA) as an unsupervised machine learning tool to uncover classification criteria in complex inorganic crystal chemistries. Using the apatite chemistry as a template, we track through the use of persistent homology the topological connectivity of input crystal chemistry descriptors on defining similarity between different stoichiometries of apatites. It is shown that TDA automatically identifies a hierarchical classification scheme within apatites based on the commonality of the number of discrete coordination polyhedra that constitute the structural building units common among the compounds. This information is presented in the form of a visualization scheme of a barcode of homology classifications, where the persistence of similarity between compounds is tracked. Unlike traditional perspectives of structure maps, this new “Materials Barcode” schema serves as an automated exploratory machine learning tool that can uncover structural associations from crystal chemistry databases, as well as to achieve a more nuanced insight into what defines similarity among homologous compounds.

scholarly journals Topological data analysis for true step detection in periodic piecewise constant signals

2018 ◽  
Vol 474 (2218) ◽  
pp. 20180027 ◽  
Author(s):  
Firas A. Khasawneh ◽  
Elizabeth Munch
Keyword(s):  
Data Analysis ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Future Research ◽  
Accurate Identification ◽  
Piecewise Constant ◽  
Higher Dimensional ◽  
Powerful Approach ◽  
Hadamard Transforms ◽  
Topological Data

This paper introduces a simple yet powerful approach based on topological data analysis for detecting true steps in a periodic, piecewise constant (PWC) signal. The signal is a two-state square wave with randomly varying in-between-pulse spacing, subject to spurious steps at the rising or falling edges which we call digital ringing. We use persistent homology to derive mathematical guarantees for the resulting change detection which enables accurate identification and counting of the true pulses. The approach is tested using both synthetic and experimental data obtained using an engine lathe instrumented with a laser tachometer. The described algorithm enables accurate and automatic calculations of the spindle speed without any choice of parameters. The results are compared with the frequency and sequency methods of the Fourier and Walsh–Hadamard transforms, respectively. Both our approach and the Fourier analysis yield comparable results for pulses with regular spacing and digital ringing while the latter causes large errors using the Walsh–Hadamard method. Further, the described approach significantly outperforms the frequency/sequency analyses when the spacing between the peaks is varied. We discuss generalizing the approach to higher dimensional PWC signals, although using this extension remains an interesting question for future research.

scholarly journals Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace

2018 ◽  
Vol 9 ◽  
Author(s):  
Mao Li ◽  
Hong An ◽  
Ruthie Angelovici ◽  
Clement Bagaza ◽  
Albert Batushansky ◽  
...  
Keyword(s):  
Data Analysis ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Morphometric Method ◽  
Topological Data

scholarly journals Flow Estimation Only from Image Data, based on Persistent Homology

2021 ◽  
Author(s):  
Anna Suzuki ◽  
Miyuki Miyazawa ◽  
James Minto ◽  
Takeshi Tsuji ◽  
Ippei Obayashi ◽  
...  
Keyword(s):  
Data Analysis ◽  
Persistent Homology ◽  
Image Data ◽  
Fracture Network ◽  
Topological Data Analysis ◽  
Geometric Information ◽  
Flow Estimation ◽  
Flow Phenomena ◽  
Network Patterns ◽  
Topological Data

Abstract Topological data analysis is an emerging concept of data analysis for characterizing shapes. A state-of-the-art tool in topological data analysis is persistent homology, which is expected to summarize quantified topological and geometric features. Although persistent homology is useful for revealing the topological and geometric information, it is difficult to interpret the parameters of persistent homology themselves and difficult to directly relate the parameters to physical properties. In this study, we focus on connectivity and apertures of flow channels detected from persistent homology analysis. We propose a method to estimate permeability in fracture networks from parameters of persistent homology. Synthetic 3D fracture network patterns and their direct flow simulations are used for the validation. The results suggest that the persistent homology can estimate fluid flow in fracture network based on the image data. This method can easily derive the flow phenomena based on the information of the structure.

scholarly journals Using Topological Data Analysis (TDA) and Persistent Homology to Analyze the Stock Markets in Singapore and Taiwan

2021 ◽  
Vol 9 ◽  
Author(s):  
Peter Tsung-Wen Yen ◽  
Siew Ann Cheong
Keyword(s):  
Data Analysis ◽  
Stock Markets ◽  
Persistent Homology ◽  
Betti Numbers ◽  
Stock Index ◽  
Topological Data Analysis ◽  
Series Data ◽  
Topological Features ◽  
Market Crashes ◽  
Topological Data

In recent years, persistent homology (PH) and topological data analysis (TDA) have gained increasing attention in the fields of shape recognition, image analysis, data analysis, machine learning, computer vision, computational biology, brain functional networks, financial networks, haze detection, etc. In this article, we will focus on stock markets and demonstrate how TDA can be useful in this regard. We first explain signatures that can be detected using TDA, for three toy models of topological changes. We then showed how to go beyond network concepts like nodes (0-simplex) and links (1-simplex), and the standard minimal spanning tree or planar maximally filtered graph picture of the cross correlations in stock markets, to work with faces (2-simplex) or any k-dim simplex in TDA. By scanning through a full range of correlation thresholds in a procedure called filtration, we were able to examine robust topological features (i.e. less susceptible to random noise) in higher dimensions. To demonstrate the advantages of TDA, we collected time-series data from the Straits Times Index and Taiwan Capitalization Weighted Stock Index (TAIEX), and then computed barcodes, persistence diagrams, persistent entropy, the bottleneck distance, Betti numbers, and Euler characteristic. We found that during the periods of market crashes, the homology groups become less persistent as we vary the characteristic correlation. For both markets, we found consistent signatures associated with market crashes in the Betti numbers, Euler characteristics, and persistent entropy, in agreement with our theoretical expectations.

scholarly journals Persistent homology approach distinguishes potential pattern between “Early” and “Not Early” hepatic decompensation groups using MRI modalities

2021 ◽  
Vol 7 (2) ◽  
pp. 488-491
Author(s):  
Yashbir Singh ◽  
William Jons ◽  
Gian Marco Conte ◽  
Jaidip Jagtap ◽  
Kuan Zhang ◽  
...  
Keyword(s):  
Data Analysis ◽  
Liver Failure ◽  
Algebraic Topology ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Topological Representation ◽  
Hepatic Decompensation ◽  
Topological Data

Abstract Primary sclerosis cholangitis (PSC) predisposes individuals to liver failure, but it is challenging for radiologists examining radiologic images to predict which patients with PSC will ultimately develop liver failure. Motivated by algebraic topology, a topological data analysis - inspired framework was adopted in the study of the imaging pattern between the “Early Decompensation” and “Not Early” groups. The results demonstrate that the proposed methodology discriminates “Early Decompensation” and “Not Early” groups. Our study is the first attempt to provide a topological representation-based method into early hepatic decompensation and not early groups.

scholarly journals Revealing hidden medium-range order in amorphous materials using topological data analysis

2020 ◽  
Vol 6 (37) ◽  
pp. eabc2320
Author(s):  
Søren S. Sørensen ◽  
Christophe A. N. Biscio ◽  
Mathieu Bauchy ◽  
Lisbeth Fajstrup ◽  
Morten M. Smedskjaer
Keyword(s):  
Data Analysis ◽  
Amorphous Materials ◽  
Persistent Homology ◽  
Amorphous Solids ◽  
Range Order ◽  
Silicate Glasses ◽  
Topological Data Analysis ◽  
Medium Range Order ◽  
Medium Range ◽  
Topological Data

Despite the numerous technological applications of amorphous materials, such as glasses, the understanding of their medium-range order (MRO) structure—and particularly the origin of the first sharp diffraction peak (FSDP) in the structure factor—remains elusive. Here, we use persistent homology, an emergent type of topological data analysis, to understand MRO structure in sodium silicate glasses. To enable this analysis, we introduce a self-consistent categorization of rings with rigorous geometrical definitions of the structural entities. Furthermore, we enable quantitative comparison of the persistence diagrams by computing the cumulative sum of all points weighted by their lifetime. On the basis of these analysis methods, we show that the approach can be used to deconvolute the contributions of various MRO features to the FSDP. More generally, the developed methodology can be applied to analyze and categorize molecular dynamics data and understand MRO structure in any class of amorphous solids.

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