In this talk, we look at the results of various studies in which computational mathematics is used in medical imaging. Through the various scope of research from mathematical modeling to data-based methodology, we can think about the future direction by examining what we can do in data science can contribute and what contribution we can make to medical imaging.

Symmetric spaces from Lie theory and differential geometry are often represented by special set of structured matrices. The Cartan decomposition and its generalization of symmetric spaces and classical Lie groups recover many of the known matrix factorizations in numerical linear algebra, such as the singular value decomposition, CS decomposition, generalized SVD and many more. We discuss a blueprint for generating fifty-three matrix factorizations from the generalized Cartan decomposition, most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978). This is joint work with Alan Edelman.