Simply denoise: wavefield reconstruction via jittered undersampling

Undersampled data contaminated by noise

Although we focused on a noise-free (severely) underdetermined system of linear equations, the CS theory, and hence our work, both extend to the recovery from undersampled data contaminated by noise (Candès et al., 2005b). In this case, the noise $\ensuremath{\mathbf{e}}$ that corrupts the data adds to the undersampling artifacts in the sparsifying domain. The quantity that relates to the recoverability is now given by $\tensor{A}^H\left(\tensor{A}\ensuremath{\mathbf{x}}_0+\ensuremath{\mathbf{e}}\right)-\alpha\ensuremath{\mathbf{x}}_0$ as opposed to $\tensor{A}^H\tensor{A}\ensuremath{\mathbf{x}}_0-\alpha\ensuremath{\mathbf{x}}_0$ in the noise-free case. Consequently, the undersampling artifacts $\ensuremath{\mathbf{z}}$ and the imprint of the contaminating noise in the sparsifying domain, i.e., $\tensor{A}^H\ensuremath{\mathbf{e}}$ , have to be studied jointly.