This project has three main goals: (i) Develop the theory and algorithms for optimal dual (and fusion) frames for erasures and for sparse signal reconstruction. (ii) Develop the theory of the frame spectrum for fractal measures that can provide much faster reconstruction (for example, compared to the classical Shannon sampling method) for functions that are concentrated on fractal-like regions. (iii) Investigate the connections between the Feichtinger frame conjecture for group representation frames and related problems such as the Kadison-Singer problem.
In today's digital world efficient data processing has become more demanding in all kinds of applications. The principal investigator studies fundamental questions in harmonic and functional analysis that are applicable in areas such as wireless communications, radar, seismic sensing, image processing, reliable classification of features in biomedical images, and algorithms for the compensation of data loss in digital transmission of signals. The optimal dual frame design problems for erasures and sparsity are directly related to basic issues in redundant representation and transformation of signals. The investigation leads to applicable tools for problems in engineering and information applications, and provides theoretical analysis and algorithmic construction for computationally efficient frames.