• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.1065-1087
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• 1997

We prove that there exists a canonical level-preserving isomorphism between the stable Morse homology (or the Morse homology of generating functions) and the Floer homology on the cotangent bundle $T^*M$ for any closed submanifold $N \subset M$ for any compact manifold M.

• 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.

• Journal of the Korean Mathematical Society
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• v.48 no.4
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• pp.749-774
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• 2011

Given a Morse function on a manifold whose moduli spaces of gradient flow lines for each action window are compact up to breaking one gets a bidirect system of chain complexes. There are different possibilities to take limits of such a bidirect system. We discuss in this note the relation between these different limits.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.1-8
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• 2018

In this paper, we study some topological structure of a compact domain in ${\mathbb{R}}^n$ in terms of the curvature conditions and develop a geometric inequality involving the volume and the integral of mean curvatures over the boundary of the compact domain.

• 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.