### Abstract

We consider the problem of imaging sparse scenes from a few noisy data using an ℓ_{1}-minimization approach. This problem can be cast as a linear system of the form Ap = b, where A is an N × K measurement matrix. We assume that the dimension of the unknown sparse vector p ϵ ℂ^{k} is much larger than the dimension of the data vector b ϵ ℂ^{N}, i.e. K Gt; N. We provide a theoretical framework that allows us to examine under what conditions the ℓ_{1}-minimization problem admits a solution that is close to the exact one in the presence of noise. Our analysis shows that ℓ_{1}-minimization is not robust for imaging with noisy data when high resolution is required. To improve the performance of ℓ_{1}-minimization we propose to solve instead the augmented linear system [A|C]p = b, where the N × σ matrix C is a noise collector. It is constructed so as its column vectors provide a frame on which the noise of the data, a vector of dimension N, can be well approximated. Theoretically, the dimension σ of the noise collector should be e^{N} which would make its use not practical. However, our numerical results illustrate that robust results in the presence of noise can be obtained with a large enough number of columns σ ≺∼ 10K.

Original language | English (US) |
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Article number | 035010 |

Journal | Inverse Problems |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2020 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Signal Processing
- Mathematical Physics
- Computer Science Applications
- Applied Mathematics

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## Cite this

*Inverse Problems*,

*36*(3), [035010]. https://doi.org/10.1088/1361-6420/ab5a21