Moscoso, MiguelNovikov, AlexeiPapanicolau, GeorgeTsogka, Chrysoula2021-04-272021-04-272020-03Moscoso, M., Novikov, A., Papanicolaou, G. & Tsogka, C. (2020). Imaging with highly incomplete and corrupted data. Inverse Problems, 36(3), 035010.0266-5611https://hdl.handle.net/10016/32486We consider the problem of imaging sparse scenes from a few noisy data using an L1-minimization approach. This problem can be cast as a linear system of the form Ap = b, where A is an N x K measurement matrix. We assume that the dimension of the unknown sparse vector p E Ck is much larger than the dimension of the data vector b E Cn, i.e. K >>N. We provide a theoretical framework that allows us to examine under what conditions the L1-minimization problem admits a solution that is close to the exact one in the presence of noise. Our analysis shows that L1-minimization is not robust for imaging with noisy data when high resolution is required. To improve the performance of L1-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 eN 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.21eng© 2020 IOP Publishing Ltd.Array imagingL1-norm minimizationHighly corrupted dataresearch articleMaterialesQuímicahttps://doi.org/10.1088/1361-6420/ab5a21open access13(035010)21Inverse Problems36AR/0000026263