Imaging with highly incomplete and corrupted data
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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 A ρ =b, where A is an N× K measurement matrix. We assume that the dimension of the unknown sparse vector ρ∈C^K is much larger than the dimension of the data vector b ∈C^N, i.e, K ≫ 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] ρ =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≈ 10 K.
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