- Introduction to the blog
- Basic topology with examples
- Applications to computer vision
- A plan of potential future posts
- References to books and surveys
Introduction to the blog
- Motivation: a few reasons to start the blog
- Comments and suggestions are welcome!
- What is topological data analysis about?
Basic topology with examples
Applications to computer vision
A plan of potential future posts
- Many types of graphs: combinatorial, topological, metric
- Homeomorphism is a topological equivalence of shapes
- Homotopy equivalence is like a continuous deformation
- Betti numbers are robust numerical descriptors of shapes
- Delaunay triangulation is a mesh on a finite sample of data
- Alpha-complexes represent offsets of a sample at all scales
- Persistent homology is a summary of data filtered at all scales
- Persistent homology is a stable descriptor of noisy raw data
- Union-find structure for 0-dimensional persistent homology
- Computing persistent 1-dimensional homology for 2D clouds
- Fast and robust algorithm to count holes in noisy 2D clouds
Motivation: a few reasons to start the blog
I was inspired by the excellent posts of Prof Gunnar Carlsson at the Ayasdi blog. My blog will contain a bit more technical details, especially on specific applications. My past research was in pure mathematics: geometric topology, non-commutative algebra, graphs, singularities.
However, in the academic year 2014-2015 I am on research leave in the Computer Vision group at Microsoft Research Cambridge, UK. Here is a short list of reasons to start this blog:
- promote topological data analysis in academia and industry
- present key concepts and results for new students and experts
in related areas: computer vision, machine learning, statistics - collect helpful references to surveys, tutorials and software
- initiate an introductory book in collaboration with colleagues.
Comments and suggestions are welcome!
Topological Data Analysis is a hot research area. New ideas will certainly emerge in the future. Current frameworks may also change since the theory is still being developed.
Hence I would appreciate suggestions from experts. If you don’t have your own blog and would like to explain your work in simple terms, then you are welcome as a guest blogger! You may also wish to add your book, survey or software to the list of references below.
Those who are new to this area, for example PhD students, could also help by letting me know which concepts deserve a separate post with more clarifications and examples. You may solve exercises at the end of every post and submit brief solutions or new questions in your comments. Your contributions will be highly appreciated.
For posting comments, an e-mail (not to be published) and a correct captcha are required only to filter our spammers. Anonymous e-mails are ok. Mine is vitaliy.kurlin(at)gmail.com.
References to books and surveys
- Book Computational Topology, An Introduction by H.Edelsbrunner, J.Harer.
- Survey Topology and Data by G.Carlsson.
- Survey Barcodes: the persistent topology of data by R.Ghrist.