• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.25 no.4
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• pp.296-311
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• 2021

Topological Data Analysis (TDA), a relatively new field of data analysis, has proved very useful in a variety of applications. The main persistence tool from TDA is persistent homology in which data structure is examined at many scales. Representations of persistent homology include persistence barcodes and persistence diagrams, both of which are not straightforward to reconcile with traditional machine learning algorithms as they are sets of intervals or multisets. The problem of faithfully representing barcodes and persistent diagrams has been pursued along two main avenues: kernel methods and vectorizations. One vectorization is the Betti sequence, or Betti curve, derived from the persistence barcode. While the Betti sequence has been used in classification problems in various applications, to our knowledge, the stability of the sequence has never before been discussed. In this paper we show that the Betti sequence is unstable under the 1-Wasserstein metric with regards to small perturbations in the barcode from which it is calculated. In addition, we propose a novel stabilized version of the Betti sequence based on the Gaussian smoothing seen in the Stable Persistence Bag of Words for persistent homology. We then introduce the normalized cumulative Betti sequence and provide numerical examples that support the main statement of the paper.

• Journal of Korea Multimedia Society
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• v.21 no.1
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• pp.10-17
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• 2018

Persistence Betty numbers, which are the rank of the persistent homology, are a generalized version of the size theory widely known as a descriptor for shape analysis. They show robustness to both perturbations of the topological space that represents the object, and perturbations of the function that measures the shape properties of the object. In this paper, we present a shape matching algorithm which is based on the use of persistence Betty numbers. Experimental tests are performed with Kimia dataset to show the effectiveness of the proposed method.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.18 no.1
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• pp.223-229
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• 2018

This paper proposes a robust technique of image segmentation, which can be obtained if the topological persistence of each connected component is used as the feature vector for the graph-based image segmentation. The topological persistence of the components, which are obtained from the super-level set of the image, is computed from the morse function which is associated with the gray-level or color value of each pixel of the image. The procedure for the components to be born and be merged with the other components is presented in terms of zero-dimensional homology group. Extensive experiments are conducted with a variety of images to show the more correct image segmentation can be obtained by merging the components of small persistence into the adjacent components of large persistence.

• The Korean Journal of Applied Statistics
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• v.25 no.6
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• pp.1009-1018
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• 2012

Persistence homology (a type of methodology in computational algebraic topology) can be used to capture the topological characteristics of functional data. To visualize the characteristics, a persistence diagram is adopted by plotting baseline and the pairs that consist of local minimum and local maximum. We use the bottleneck distance to measure the topological distance between two different functions; in addition, this distance can be applied to multidimensional scaling(MDS) that visualizes the imaginary position based on the distance between functions. In this study, we use handwriting data (which has functional forms) to get persistence diagram and check differences between the observations by using bottleneck distance and the MDS.

• Journal of Korea Multimedia Society
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• v.22 no.1
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• pp.18-26
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• 2019

In this paper we propose how to detect circular objects in the gray scale image and segment them using the discrete Morse theory, which makes it possible to analyze the topology of a digital image, when it is transformed into the data structure of some combinatorial complex. It is possible to get meaningful information about how many connected components and topologically circular shapes are in the image by computing the persistent homology of the filtration using the Morse complex. We obtain a Morse complex by modeling an image as a cubical cellular complex. Each cell in the Morse complex is the critical point at which the topological structure changes in the filtration consisting of the level sets of the image. In this paper, we implement the proposed algorithm of segmenting the circularly shaped objects with a long persistence of homology as well as computing persistent homology along the filtration of the input image and displaying in the form of a persistence diagram.