🍩 Database of Original & Non-Theoretical Uses of Topology
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Topology of Frame Field Meshing (2020)
Piotr BebenAbstract
In the past decade frame fields have emerged as a promising approach for generating hexahedral meshes for CFD and CAE applications. One important problem asks for construction of a boundary aligned frame field with prescribed singularity constraints that correspond to a valid hexahedral mesh. We give a necessary and sufficient condition in terms of solutions to a system of monomial equations whose variables are in the binary octahedral group. Along the way we look at frame field design from an algebraic topological perspective, proving various results, some known, some new. -
Topological Differential Testing (2020)
Kristopher Ambrose, Steve Huntsman, Michael Robinson, Matvey YutinAbstract
We introduce topological differential testing (TDT), an approach to extracting the consensus behavior of a set of programs on a corpus of inputs. TDT uses the topological notion of a simplicial complex (and implicitly draws on richer topological notions such as sheaves and persistence) to determine inputs that cause inconsistent behavior and in turn reveal \emph\de facto\ input specifications. We gently introduce TDT with a toy example before detailing its application to understanding the PDF file format from the behavior of various parsers. Finally, we discuss theoretical details and other possible applications. -
Topological Data Analysis: Concepts, Computation, and Applications in Chemical Engineering (2021)
Alexander D. Smith, Paweł Dłotko, Victor M. ZavalaAbstract
A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g., convolutional neural nets, Fourier series). Topological Data Analysis (TDA) is a field of mathematics that analyzes data from a fundamentally different perspective. TDA represents datasets as geometric objects and provides dimensionality reduction techniques that project such objects onto low-dimensional descriptors. The key properties of these descriptors (also known as topological features) are that they provide multiscale information and that they are stable under perturbations (e.g., noise, translation, and rotation). In this work, we review the key mathematical concepts and methods of TDA and present different applications in chemical engineering. -
The Euler Characteristic: A General Topological Descriptor for Complex Data (2021)
Alexander Smith, Victor ZavalaAbstract
Datasets are mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape encodes important knowledge about the system under study. Topology is an area of mathematics that provides diverse tools to characterize the shape of data objects. In this work, we study a specific tool known as the Euler characteristic (EC). The EC is a general, low-dimensional, and interpretable descriptor of topological spaces defined by data objects. We revise the mathematical foundations of the EC and highlight its connections with statistics, linear algebra, field theory, and graph theory. We discuss advantages offered by the use of the EC in the characterization of complex datasets; to do so, we illustrate its use in different applications of interest in chemical engineering such as process monitoring, flow cytometry, and microscopy. We show that the EC provides a descriptor that effectively reduces complex datasets and that this reduction facilitates tasks such as visualization, regression, classification, and clustering. -
Severe Slugging Flow Identification From Topological Indicators (2022)
Simone CasoloAbstract
In this work, topological data analysis is used to identify the onset of severe slug flow in offshore petroleum production systems. Severe slugging is a multiphase flow regime known to be very inefficient and potentially harmful to process equipment and it is characterized by large oscillations in the production fluid pressure. Time series from pressure sensors in subsea oil wells are processed by means of Takens embedding to produce point clouds of data. Embedded sensor data is then analyzed using persistent homology to obtain topological indicators capable of revealing the occurrence of severe slugging in a condition-based monitoring approach. A large dataset of well events consisting of both real and simulated data is used to demonstrate the possibilty of authomatizing severe slugging detection from live data via topological data analysis. Methods based on persistence diagrams are shown to accurately identify severe slugging and to classify different flow regimes from pressure signals of producing wells with supervised machine learning. -
Raman Spectroscopy and Topological Machine Learning for Cancer Grading (2023)
Francesco Conti, Mario D’Acunto, Claudia Caudai, Sara Colantonio, Raffaele Gaeta, Davide Moroni, Maria Antonietta PascaliAbstract
In the last decade, Raman Spectroscopy is establishing itself as a highly promising technique for the classification of tumour tissues as it allows to obtain the biochemical maps of the tissues under investigation, making it possible to observe changes among different tissues in terms of biochemical constituents (proteins, lipid structures, DNA, vitamins, and so on). In this paper, we aim to show that techniques emerging from the cross-fertilization of persistent homology and machine learning can support the classification of Raman spectra extracted from cancerous tissues for tumour grading. In more detail, topological features of Raman spectra and machine learning classifiers are trained in combination as an automatic classification pipeline in order to select the best-performing pair. The case study is the grading of chondrosarcoma in four classes: cross and leave-one-patient-out validations have been used to assess the classification accuracy of the method. The binary classification achieves a validation accuracy of 81% and a test accuracy of 90%. Moreover, the test dataset has been collected at a different time and with different equipment. Such results are achieved by a support vector classifier trained with the Betti Curve representation of the topological features extracted from the Raman spectra, and are excellent compared with the existing literature. The added value of such results is that the model for the prediction of the chondrosarcoma grading could easily be implemented in clinical practice, possibly integrated into the acquisition system. -
Testing Topological Data Analysis for Condition Monitoring of Wind Turbines (2024)
Simone Casolo, Alexander Stasik, Zhenyou Zhang, Signe Riemer-SørensenAbstract
We present an investigation of how topological data analysis (TDA) can be applied to condition-based monitoring (CBM) of wind turbines for energy generation.TDA is a branch of data analysis focusing on extracting mean- ingful information from complex datasets by analyzing their structure in state space and computing their underlying topo- logical features. By representing data in a high-dimensional state space, TDA enables the identification of patterns, anoma- lies, and trends in the data that may not be apparent through traditional signal processing methods. For this study, wind turbine data was acquired from a wind park in Norway via standard vibration sensors at different lo- cations of the turbine’s gearbox. Both the vibration acceler- ation data and its frequency spectra were recorded at infre- quent intervals for a few seconds at high frequency and fail- ure events were labelled as either gear-tooth or ball-bearing failures. The data processing and analysis are based on a pipeline where the time series data is first split into intervals and then transformed into multi-dimensional point clouds via a time-delay embedding. The shape of the point cloud is an- alyzed with topological methods such as persistent homol- ogy to generate topology-based key health indicators based on Betti numbers, information entropy and signal persistence. Such indicators are tested for CBM and diagnosis (fault de- tection) to identify faults in wind turbines and classify them accordingly. Topological indicators are shown to be an in- teresting alternative for failure identification and diagnosis of operational failures in wind turbines.