Imaging with incomplete and corrupted data
We 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 Ax=b. The dimension of the unknown sparse vector x is much larger than the dimension of the data vector b. The l1-minimization alone, however, is not robust for imaging with noisy data. To improve its performance we propose to solve instead the augmented linear system [A|C]x=b, where the matrix C is a noise collector. It is constructed so as its column vectors provide a frame on which the noise of the data can be well approximated with high probability. This approach gives rise to a new parameter-free imaging method that has a zero false discovery rate for any level of noise. We also obtain exact support recovery if the noise is not too large.