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• W2038441709 abstract "This paper discusses reconstruction of a signal from undersampled data in the situation that the signal is sparse or approximately sparse in terms of a (possibly) highly overcomplete and coherent tight frame <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>D</i> via the <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>l</i> <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> -analysis optimization problem. Some new sufficient conditions on the <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>D</i> -restricted isometry property are given to guarantee stable recovery of signals which are nearly sparse in terms of <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>D</i> , from undersampled data with minimal <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>l</i> <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> -norm of transform coefficients. One of the main results of this paper shows that if the <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>D</i> -restricted isometry constant δ <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>s</i> of the measurement matrix <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>A</i> satisfies δ <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>s</i> <; 0.307 , then signals which are approximately <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>s</i> -sparse in terms of <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>D</i> are guaranteed to be stably recovered via the <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>l</i> <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> -analysis optimization problem. We point out that with the lemmas and the proof techniques developed in this paper, most of the sufficient conditions on the standard restricted isometry property for stable recovery of nearly sparse signals via standard <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>l</i> <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> -minimization, can be similarly extended to the general case of the <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>D</i> -restricted isometry property for stable recovery of signals that are nearly sparse in terms of <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>D</i> via the <i xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>l</i> <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> -analysis optimization problem, yielding weaker conditions than previously available in the literature." @default.
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• W2038441709 date "2013-02-01" @default.
• W2038441709 modified "2023-09-25" @default.
• W2038441709 title "New Bounds for Restricted Isometry Constants With Coherent Tight Frames" @default.
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• W2038441709 doi "https://doi.org/10.1109/tsp.2012.2226171" @default.
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