Sparsity promoting regularization has been widely adopted in image processing and medical imaging. Successful examples include total variation, wavelet frames, nonlocal means, BM3Ds, data-driven transformations, etc. I will start with an overview of these methods and discuss the relations and differences among these methods. In particular, I will discuss discoveries on the deep connection between wavelet frame transforms and differential operators under variational and PDE framework. Then, I will present two of our recent work on CT image reconstruction using tight (wavelet) frame transforms with rather convincing numerical results. One is on joint image and Radon domain reconstruction using tight wavelet frames (TWF). Its main idea is to improve imaging quality by digitally increase the number of CT measurements using sparse approximation by WTF in both spatial and Radon domain. The other work is to further improve CT imaging quality by introducing a spatial-Radon domain data-driven tight frame (SRD-DDTF) regularization model. The main motivation is that a pre-constructed wavelet frame system may not provide ideal sparse approximation for a specific data. Therefore, in our SRD-DDTF model, the sparse promoting transformations are learned from the given data and updated iteratively. In recent years, we have witnessed rapid advances in information and computer technology, which contribute greatly to the exponential growth of data.  It has been noted that many of the current applications in data analysis are concerned with data that can be naturally organized on non-flat domains such as surfaces, manifolds, point clouds and graphs. In the final part of my talk, I will present our recent work on developing theory and applications of sparse approximation on these more general data sets by constructing tight wavelet frames on manifolds/graphs. Applications to high-dimensional classification will also be discussed. 

Prof. Dong received his B.S. from Peking University (PKU) in 2003, M.S degree from National University of Singapore (NUS) in 2005, and Ph.D degree from University of California Los Angeles (UCLA) in 2009. Then he spent 2 years at University of California San Diego (UCSD) as a visiting assistant professor. He was an assistant professor at University of Arizona since 2011 and joined Peking University as an associate professor in 2014. His research interest is in mathematical modeling and computations in imaging science and high dimensional data analysis, which includes (but not limited to) biological and medical imaging and image analysis , image guided diagnosis and treatment of disease, (semi-)supervised learning. A special feature of his research is a unique blending of wavelet frame theory, variational techniques and nonlinear PDEs. They are working on projects aiming at addressing new and fascinating connections among these subjects, which not only leads to new understandings of the subjects themselves, but also gives rise to new and effective mathematical tools for imaging/data science.