Request for some references exploring the connections of Riemann surfaces with medical imaging

Question

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!

Answer 1

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

Answer 2

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.