Decomposition and analysis of massive point-sources in motion

Although the term ``Big Data" is often used as hype for advertisement, the rapidly increasing amount and complexity of data to be analyzed and understood in any scientific endeavor is a big challenge to the communities of mathematicians and other scientists.

This proposal is motivated by the need of innovative mathematical theory along with effective computational schemes, to extract data information for super- resolved fluorescence microscopy in cell biology, and to facilitate the understanding of the birth of the first stars in cosmology, with focus on decomposition and analysis of living cells and stars while in motion. Indeed, motion of cell migration is key to the identification of tumor cells, particularly cancer cells that move by using different internal mechanism, movement of RNA molecules in cell nuclei enables eukaryotic gene readouts, and irregular movement of young galaxies facilitates identification of the first-generation stars for the discovery of chemical elements of life. This proposed research is concerned with decomposition of unknown point-sources in motion. The main objective is to develop a mathematical theory along with effective computational schemes and visualization tools, for predicting, analyzing and displaying motion activities. To accomplish this goal, a one-pass solution of the super-resolution inverse problem, based on our recent work [11,12] and the PDE model of mean- curvature flow, will be developed. Visualization algorithms will be extension of our previous papers [16,17] in computer graphics.

The long-term impact of the proposed research is not limited to analysis of point- sources in motion, but also to providing a powerful mathematical tool for big data decomposition and understanding. Furthermore, in view of the current exciting development of super-resolved light microscopy with sub-cellular nano-scale capability for viewing molecular activities within a living cell, and the forthcoming launch of the James Webb space telescope in 2018, with the capability of seeing the earliest stars and galaxies and to looking deep into nearby dust clouds to study the formation of stars and planets, the timing and research findings of this proposed project should have significant great impact to the research advancement of cell biology and cosmology, as well as the advancement of science and technology in general.