{"id":22,"date":"2012-09-05T11:22:33","date_gmt":"2012-09-05T15:22:33","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/template\/?page_id=22"},"modified":"2026-08-01T23:34:22","modified_gmt":"2026-08-02T03:34:22","slug":"research","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/jiuncho\/research\/","title":{"rendered":"Research"},"content":{"rendered":"\n
\n

Research<\/h1>\n\n\n\n

My research is in Topological Data Analysis (TDA). I mainly work on persistent homology for image and voxel data, cubical filtrations, and faster algorithms for computing persistence in these special cases. I am also interested in applications to scientific and medical data.<\/p>\n\n\n\n

Persistent Homology for Images<\/strong>: <\/p>\n\n\n\n

For two-dimensional image data, standard persistent homology computations often require a sorting step over all pixels or edges before tracking how connected components appear and merge. This leads to an n<\/em> log n<\/em> running time, even though most pixels and edges sorted in this step do not contribute to non-ephemeral features in the final persistence diagram.<\/p>\n\n\n\n

My current work avoids sorting the entire image by first reducing it to a much smaller filtered graph. The resulting graph has a number of vertices and edges proportional to the number of non-ephemeral features in the output, so the final sorting step is performed only on this reduced graph. This gives an output-sensitive running time of k<\/em> log k<\/em>, where k<\/em> is the much more modest size of the output, plus an almost-linear factor in terms of the size of the input. This algorithm is currently being implemented as a software package for Python, written in C++.<\/p>\n\n\n\n

Cubical Filtrations<\/strong>: <\/p>\n\n\n\n

We study and characterize local configurations within cubical filtrations, which are used to model voxel data in the context of persistent homology. These local configurations arise from adding a single voxel (and its faces) to a cubical complex, and determine the possible topology changes from this addition.
We show the number of such configurations in dimensions one
to four. These results are efficiently computed using a simple but subtle algorithm exploiting an interesting duality. Our results explain why pre-computing these local configurations is feasible in dimensions up to three, while indicating that new techniques are needed in dimension four and above. These insights already play an important role in state of the art software in dimension two and three, and will inform design of software in dimension four and above.<\/p>\n\n\n\n

*to be presented at presented at CCCG 2026<\/p>\n\n\n\n

Applications<\/strong><\/p>\n\n\n\n

Applications of topology to science are always exciting! Here I will describe two that I’m currently involved in. In project one, we study high-dimensional probability landscapes of stochastic gene regulatory networks. Some methods rely on low dimensional projections that may obscure or distort topological structure. With the help of Cubicle, a software my advisor Hubert Wagner developed that can handle large voxel data, we use its persistent homology to study these landscapes directly in their original high-dimensional state spaces.<\/p>\n\n\n\n

My involvement on this project mainly concerns the mathematical interpretation of these topological features and their relationship to the underlying biological dynamics. The project is a continuation of earlier work by Farid Manuchehrfar, Huiyu Li, Wei Tian, Ao Ma, and Jie Liang on Exact Topology of the Dynamic Probability Surface of an Activated Process by Persistent Homology<\/em><\/a>.<\/p>\n\n\n\n

Another project I’m currently involved in concerns data analysis and classification of three-dimensional medical images using persistence homology summaries. <\/p>\n<\/div><\/div><\/div><\/section>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":11,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-22","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/jiuncho\/wp-json\/wp\/v2\/pages\/22","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/jiuncho\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/jiuncho\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/jiuncho\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/jiuncho\/wp-json\/wp\/v2\/comments?post=22"}],"version-history":[{"count":10,"href":"https:\/\/people.clas.ufl.edu\/jiuncho\/wp-json\/wp\/v2\/pages\/22\/revisions"}],"predecessor-version":[{"id":259,"href":"https:\/\/people.clas.ufl.edu\/jiuncho\/wp-json\/wp\/v2\/pages\/22\/revisions\/259"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/jiuncho\/wp-json\/wp\/v2\/media?parent=22"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}