The investigator and his colleagues work on designing Hilbert space frames for applications. This involves working directly with the groups needing such frames such as Siemens Corporate Research, the Bio-Medical Engineering Center at Carnegie Mellon University, and the Digital Signal Processing Center in the Electrical Engineering department at Rice University. One project involves designing frames and algorithms for doing signal reconstruction without phase. This project is designed to remove certain background noises from a signal so that the voice signals are clearer. Another project involves answering the question: What is the role and what can imaging do for systems biology? The goal is to have distributed yet integrated large bioimage databases that would allow researchers to upload their data, have it processed, share the data, download data as well as having platform-optimized code, all in a common format. A third project involves doing signal reconstruction with erasures. The investigator has designed a system that accounts for erasures without any extra coefficient calculations, which is both fast and accurate. This method needs to be custom designed for the myriad of applications where there are transmission losses or erasures. Another project involves designing "fusion frames" for problems in distributed processing. This requires designing localized frames that globally fuse data both quickly and accurately -- especially in the face of losses. These are used for a variety of problems including sensor networks. Another project is geared to finding equal-norm equi-angular Parseval frames and mutually unbiased bases. These frames are used in quantum state tomography, quantum cryptography and foundational issues in quantum mechanics.
The signal reconstruction project is designed to clean up certain types of audio signals that have noise embedded in their phase to present a clearer signal. Another project involves sensors that are used to observe the world around us. The amount of sensed data that is currently being collected is more than can be collected in one place for analysis. The investigator and his colleagues are designing intelligent ways to reduce the amount of raw data that is analyzed, and this necessitates understanding the nature and role of the redundancy in the measurements. Making mathematical contributions to our fundamental understanding of redundant information could advance the engineering solutions to problems in environmental monitoring (including agricultural technology), military surveillance, understanding the operation of sensory neural systems, and improving general data collection methods that efficiently turn the analog events in the natural world into digital signals ready for analysis. Bioimaging has a serious need for accurate classification, some of which involves life and death decisions and are currently very sensitive to misclassification (such as false negative decisions in detecting cancer). The goal here is to design systems that are able to classify with close-to-perfect accuracy.
