Imaging With Highly Incomplete And Corrupted Data

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 rho = b, where A is an N x K measurement matrix. We assume that the dimension of the unknown sparse vector rho is an element of C-K is much larger than the dimension of the data vector b is an element of C-N, i.e. K >> N. We provide a theoretical framework that allows us to examine under what conditions the-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 l(1)-minimization we propose to solve instead the augmented linear system vertical bar A vertical bar vertical bar C vertical bar rho = b where the N x Sigma 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 Sigma 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 Sigma less than or similar to 10K.