Data-driven Atomic Decomposition by Getting Around the Uncertainty Barriers

While “time–frequency analysis” is arguably the most important approach for innovative development of powerful mathematical tools for signal and image processing as well as for data analysis in general, there are certain “scientific barriers” to this endeavor, most notably: the "Heisenberg uncertainty principle” for time-frequency localization and the “Rayleigh criterion” that gives rise to an “uncertainty limit of imaging resolution”. On the other hand, clever ways have been introduced recently in experimental physical sciences to get around these two uncertainty limits, notably: a group of researchers at Rochester and Ottawa have a way to bypass the Heisenberg uncertainty principle; and the three 2014 Nobel laureates in Chemistry, Eric Betzig, Stefan Hell and William Moerner, have developed ways to capture images of living cells at nanoscale resolution.

In computational mathematics, a naiive way to bypass the Heisenberg uncertainty principle is to design parallel computational schemes. But getting around the uncertainty resolution limit would take much more sophisticated imagination. Indeed, in our recent paper [15], while carrying out localization in both time and frequency domains in our proofs, the idea of getting around the resolution limit was unfortunately buried among the highly technical details, in that by considering any time instance as the center of a frequency-window with adjustable width allows us to separate all distinct frequency contents at this time instance. In other words, using the time instant (or image pixel) as the center of an adaptive frequency-window is one way to bypass the resolution limit, allowing the design of the zoom-in/zoom-out feature to arbitrarily high or low resolutions, thereby the capability to capture signal or image contents in super- resolution.

In this proposal, we will proceed to pose several outstanding problem areas that can be solved by exploring around this innovation. One of the problems is blind-source decomposition of a given real-world signal, from arbitrary (non-uniform) digital samples, into its primary building blocks, called atoms. Another problem is to incorporate the techniques from curvelets and shearlets in the wavelet literature with our blind source decomposition paradigm for extracting atoms that contain arbitrarily fine image features. This investigation gives rise to many image-based applications, including image understanding, image classification, image enhancement, imagery recovery and inpainting. Our ultimate goal along this line is to identify tiny metastatic tumors from MRI images most accurately, in order to assist brain surgeons to detect and remove such tumors without introducing unnecessary damages.