Abstract
Frame theory is closely intertwined with signal processing through a canon of methodologies for the analysis of signals using (redundant) linear measurements. The canonical dual frame associated with a frame provides a means for reconstruction by a least squares approach, but other dual frames yield alternative reconstruction procedures. The novel paradigm of sparsity has recently entered the area of frame theory in various ways. Of those different sparsity perspectives, we will focus on the situations where frames and (not necessarily canonical) dual frames can be written as sparse matrices. The objective for this approach is to ensure not only low-complexity computations, but also high compressibility. We will discuss both existence results and explicit constructions.
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Notes
For the definition of the Zariski topology, we refer to the book by Hartshorne (1977).
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Acknowledgments
The authors would like to thank the referees whose detailed reports have significantly improved the presentation of the paper. The second author acknowledges support by the Einstein Foundation Berlin, by Deutsche Forschungsgemeinschaft (DFG) Grant SPP-1324 KU 1446/13 and DFG Grant KU 1446/14, by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”, and by the DFG Research Center Matheon “Mathematics for Key Technologies” in Berlin.
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Krahmer, F., Kutyniok, G. & Lemvig, J. Sparse matrices in frame theory. Comput Stat 29, 547–568 (2014). https://doi.org/10.1007/s00180-013-0446-1
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DOI: https://doi.org/10.1007/s00180-013-0446-1