Painless approximation of dual frames, with applications to shift-invariant systems

Thomas Strohmer

doi:10.1117/12.366819

26 October 1999 Painless approximation of dual frames, with applications to shift-invariant systems

Thomas Strohmer

Proceedings Volume 3813, Wavelet Applications in Signal and Image Processing VII; (1999) https://doi.org/10.1117/12.366819
Event: SPIE's International Symposium on Optical Science, Engineering, and Instrumentation, 1999, Denver, CO, United States

Abstract

We analyze the relation between infinite-dimensional frame theory and finite-dimensional models for frames as they are used for numerical algorithms. Special emphasis in this paper is on perfect reconstruction oversampled filter banks, also known as shift-invariant frames. For certain finite- dimensional models it is shown that the corresponding finite dual frame provides indeed an approximation of the dual frame of the original infinite-dimensional dual frame. For filter banks on l² (Z) we derive error estimates for the approximation of the synthesis filter bank when the analysis filter bank satisfies certain decay conditions. We show how one has to design the finite-dimensional model to preserve important structural properties of filter banks, such as polyphase representation. Finally an efficient regularization method is presented to solve the ill-posed problem arising when approximating the dual frame on L²(R) via truncated Gram matrix.

Citation Download Citation

Thomas Strohmer "Painless approximation of dual frames, with applications to shift-invariant systems", Proc. SPIE 3813, Wavelet Applications in Signal and Image Processing VII, (26 October 1999); https://doi.org/10.1117/12.366819

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