🍩 Database of Original & Non-Theoretical Uses of Topology

(found 3 matches in 0.000842s)

  1. Persistent Homology for Breast Tumor Classification Using Mammogram Scans (2022)

    Aras Asaad, Dashti Ali, Taban Majeed, Rasber Rashid
    Abstract An Important tool in the field topological data analysis is known as persistent Homology (PH) which is used to encode abstract representation of the homology of data at different resolutions in the form of persistence diagram (PD). In this work we build more than one PD representation of a single image based on a landmark selection method, known as local binary patterns, that encode different types of local textures from images. We employed different PD vectorizations using persistence landscapes, persistence images, persistence binning (Betti Curve) and statistics. We tested the effectiveness of proposed landmark based PH on two publicly available breast abnormality detection datasets using mammogram scans. Sensitivity of landmark based PH obtained is over 90% in both datasets for the detection of abnormal breast scans. Finally, experimental results give new insights on using different types of PD vectorizations which help in utilising PH in conjunction with machine learning classifiers.

  2. Extracting Insights From the Shape of Complex Data Using Topology (2013)

    P. Y. Lum, G. Singh, A. Lehman, T. Ishkanov, M. Vejdemo-Johansson, M. Alagappan, J. Carlsson, G. Carlsson
    Abstract This paper applies topological methods to study complex high dimensional data sets by extracting shapes (patterns) and obtaining insights about them. Our method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets. Through this hybrid method, we often find subgroups in data sets that traditional methodologies fail to find. Our method also permits the analysis of individual data sets as well as the analysis of relationships between related data sets. We illustrate the use of our method by applying it to three very different kinds of data, namely gene expression from breast tumors, voting data from the United States House of Representatives and player performance data from the NBA, in each case finding stratifications of the data which are more refined than those produced by standard methods.

  3. Topological Analysis of Gene Expression Arrays Identifies High Risk Molecular Subtypes in Breast Cancer (2012)

    Javier Arsuaga, Nils A. Baas, Daniel DeWoskin, Hideaki Mizuno, Aleksandr Pankov, Catherine Park
    Abstract Genomic technologies measure thousands of molecular signals with the goal of understanding complex biological processes. In cancer these molecular signals have been used to characterize disease subtypes, signaling pathways and to identify subsets of patients with specific prognosis. However molecular signals for any disease type are so vast and complex that novel mathematical approaches are required for further analyses. Persistent and computational homology provide a new method for these analyses. In our previous work we presented a new homology-based supervised classification method to identify copy number aberrations from comparative genomic hybridization arrays. In this work we first propose a theoretical framework for our classification method and second we extend our analysis to gene expression data. We analyze a published breast cancer data set and find that that our method can distinguish most, but not all, different breast cancer subtypes. This result suggests that specific relationships between genes, captured by our algorithm, help distinguish between breast cancer subtypes. We propose that topological methods can be used for the classification and clustering of gene expression profiles.