Optimally Sparse Frames

Frames have established themselves as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. However, when the signal dimension is large, the computation of the frame measurements of a signal typically requires a large number of additions and multiplications, and this makes a frame decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we introduce sparsity of a frame as a new paradigm. In our terminology, a sparse frame is a frame whose elements have a sparse representation in an orthonormal basis, thereby enabling low-complexity frame decompositions. To introduce a precise meaning of optimality, we take the sum of the numbers of vectors needed of this orthonormal basis when expanding each frame vector as sparsity measure. We then analyze the recently introduced algorithm Spectral Tetris for construction of unit norm tight frames and prove that t...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Peter G. Casazza, Andreas Heinecke, Felix Krahmer, Gitta Kutyniok
• Keywords: CORR 2010, Education, Frame Decompositions, Sparse, Sparse Frames