2
$\begingroup$

I'd like to know some references for a beginner who has basic background in Riemann surfaces and differential geometry, and would like to start learning/working on more applied areas, medical imaging/imaging problems in particular. I searched it online, but it was not so productive for me.

I was also wondering whether one must learn the theory of discrete differential geometry/discrete Riemann surfaces in order to work in these areas; I've sometimes seen faculty webpages mentioning their research in both areas.

Thanks in advance!

$\endgroup$
3
$\begingroup$

Here are some papers--accessible to beginners-- relating circle packings (which themselves are related to triangulations of Riemann surfaces) and image processing:

MR2492509 Williams, G. Brock: Circle packings, quasiconformal mappings and applications. Quasiconformal mappings and their applications, 327–346, Narosa, New Delhi, 2007.

MR2011604 Stephenson, Kenneth: Circle packing: a mathematical tale. Notices Amer. Math. Soc. 50 (2003), no. 11, 1376–1388. http://www.ams.org/notices/200311/fea-stephenson.pdf

$\endgroup$
3
$\begingroup$

Just a little update, during the years, even now, this topic is being updated. I would like to recommend two more papers. The first one, with the title "Brain Morphometry Analysis with Surface Foliation Theory", it is proposed a novel method for brain surface morphometry analysis based on surface foliation theory [1]. The second one is an elder but still updated, titled as "FLASH: Fast landmark aligned spherical harmonic parameterization for genus-0 closed brain surfaces", it is an example of a proposed algorithm with a quasi-conformal map defined between two Riemann surfaces [2]. Hope it helps you.

[1] Wen, Chengfeng, Na Lei, Ming Ma, Xin Qi, Wen Zhang, Yalin Wang, and David Xianfeng Gu. "Brain Morphometry Analysis with Surface Foliation Theory." arXiv preprint arXiv:1809.02870 (2018).

[2] Choi, Pui Tung, Ka Chun Lam, and Lok Ming Lui. "FLASH: Fast landmark aligned spherical harmonic parameterization for genus-0 closed brain surfaces." SIAM Journal on Imaging Sciences 8, no. 1 (2015): 67-94.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.