### Abstract

Original language | English |
---|---|

Pages (from-to) | 682-720 |

Number of pages | 39 |

Journal | SIAM Journal on Imaging Sciences |

Volume | 8 |

Issue number | 1 |

Early online date | 26 Mar 2015 |

DOIs | |

Publication status | Published - 2015 |

### Cite this

**On the role of total variation in compressed sensing.** / Poon, Clarice.

Research output: Contribution to journal › Article

*SIAM Journal on Imaging Sciences*, vol. 8, no. 1, pp. 682-720. https://doi.org/10.1137/140978569

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TY - JOUR

T1 - On the role of total variation in compressed sensing

AU - Poon, Clarice

PY - 2015

Y1 - 2015

N2 - This paper considers the problem of recovering a one- or two-dimensional discrete signal which is approximately sparse in its gradient from an incomplete subset of its Fourier coefficients which have been corrupted with noise. We prove that in order to obtain a reconstruction which is robust to noise and stable to inexact gradient sparsity of order $s$ with high probability, it suffices to draw ${\cal O}$ $\!{(s {\rm log} N)}$ of the available Fourier coefficients uniformly at random. However, we also show that if one draws ${\cal O}$ ${\!(s {\rm log} N)}$ samples in accordance with a particular distribution which concentrates on the low Fourier frequencies, then the stability bounds which can be guaranteed are optimal up to log factors. Finally, we prove that in the one-dimensional case where the underlying signal is gradient sparse and its sparsity pattern satisfies a minimum separation condition, to guarantee exact recovery with high probability, for some $M<N$, it suffices to draw ${\cal O}$ ${\!(s{\rm log} M {\rm log} s)}$ samples uniformly at random from the Fourier coefficients whose frequencies are no greater than $M$.

AB - This paper considers the problem of recovering a one- or two-dimensional discrete signal which is approximately sparse in its gradient from an incomplete subset of its Fourier coefficients which have been corrupted with noise. We prove that in order to obtain a reconstruction which is robust to noise and stable to inexact gradient sparsity of order $s$ with high probability, it suffices to draw ${\cal O}$ $\!{(s {\rm log} N)}$ of the available Fourier coefficients uniformly at random. However, we also show that if one draws ${\cal O}$ ${\!(s {\rm log} N)}$ samples in accordance with a particular distribution which concentrates on the low Fourier frequencies, then the stability bounds which can be guaranteed are optimal up to log factors. Finally, we prove that in the one-dimensional case where the underlying signal is gradient sparse and its sparsity pattern satisfies a minimum separation condition, to guarantee exact recovery with high probability, for some $M<N$, it suffices to draw ${\cal O}$ ${\!(s{\rm log} M {\rm log} s)}$ samples uniformly at random from the Fourier coefficients whose frequencies are no greater than $M$.

U2 - 10.1137/140978569

DO - 10.1137/140978569

M3 - Article

VL - 8

SP - 682

EP - 720

JO - SIAM Journal on Imaging Sciences

JF - SIAM Journal on Imaging Sciences

SN - 1936-4954

IS - 1

ER -