Topological Data Analysis And Its Application In Image Classification

In recent years,with the development of new-generation information technologies such as the Internet and the Internet of Things,mankind has entered the "big data era".Digital information has penetrated into every aspect of human life,gradually affecting the way people communicate,learn and even think.The increasing demand for data analysis and processing has made the research on big data and its related technologies become a hot topic in various fields.Topological Data Analysis,a data analysis method,emerged as the times require.Topological Data Analysis is a technique that combines the theory of algebraic topology with the methods of statistical analysis.It is a powerful new tool for understanding the "shape"of data,based on continuous homology,by focusing on the relationships between data and the overall structure of data.Compared with traditional data analysis methods represented by principal component analysis,Topological Data Analysis has the following advantages:it does not need to reduce the dimension of high-dimensional data,and the analysis process does not produce information loss;its analysis objects are Metric-independent topological features and can integrate and analyze datasets under different metrics.The work of this paper is mainly divided into two parts:First,it introduces the visualization method of high dimensional data set structure by using topological data analysis method.Firstly,a series of simplicial complexes are constructed for the high-dimensional data set with continuous parameters by using the method of continuous homology,which can represent the topology of the data set.Secondly,calculating the homology information of these simplicial complexes.Thirdly,representing the homology information with "Barcode".Second,applying topological data analysis methods to the classification of simple natural images.Image similarity analysis is the key step to achieve image classification,we use topological data analysis method to extract topological features of images,and uses these features as the basis to measure their similarity.Firstly,converting the image data into point cloud data in Euclidean space.Secondly,on the basis of Javaplex program package,using MATLAB programming to construct a Witness complex on the point cloud data.Thirdly,using the principle of continuous homology to get its "Barcode",which can reflect its topological structure.In the process of analyzing the image data,in order to make the "Barcode" reflect its topological structure better,we make the following two efforts:(1)limiting the total number of colors in the image.(2)proposing an improved "max-min-pro" point selection method.On this basis,in order to quantitatively measure the similarity between images,we introduce the barcode distance formula,and through simple natural image experiments,it is found that the existing barcode distance formula has limitations,so that we propose a new barcode distance formula.By the contrastive experiment of simple natural images,we verify that the new barcode distance formula can reflect the difference in the topology of images better.