Simply denoise: wavefield reconstruction via jittered undersampling

Favorable recovery conditions

Following Verdu (1998) and Donoho et al. (2006), we define the matrix $\tensor{L}\; {\buildrel\rm def\over=}\; \tensor{A}^H\tensor{A}-\alpha\tensor{I}$ to study the undersampling artifacts $\ensuremath{\mathbf{z}}\; {\buildrel\rm def\over=}\; \tensor{L}\ensuremath{\mathbf{x}}_0$ . The matrix $\tensor{I}$ is the identity matrix and the parameter $\alpha$ is a scaling factor such that diag$(\tensor{L}) = \ensuremath{\mathbf{0}}$ . For more general problems and in particular in the field of digital communications, these undersampling artifacts $\ensuremath{\mathbf{z}}$ are referred to as Multiple-Access Interference (MAI).

According to the CS theory (Donoho, 2006; Candès et al., 2006), the solution $\tilde{\ensuremath{\mathbf{x}}}$ in equation 3 and $\ensuremath{\mathbf{x}}_0$ coincide when two conditions are met, namely 1) $\ensuremath{\mathbf{x}}_0$ is sufficiently sparse, i.e., $\ensuremath{\mathbf{x}}_0$ has few nonzero entries, and 2) the undersampling artifacts are incoherent, i.e., $\ensuremath{\mathbf{z}}$ does not contain coherent energy. The first condition of sparsity requires that the energy of $\ensuremath{\mathbf{f}}_0$ is well concentrated in the sparsifying domain. The second condition of incoherent random undersampling artifacts involves the study of the sparsifying transform $\tensor{S}$ in conjunction with the restriction operator $\tensor{R}$ . Intuitively, it requires that the artifacts $\ensuremath{\mathbf{z}}$ introduced by undersampling the original signal $\ensuremath{\mathbf{f}}_0$ are not sparse in the $\tensor{S}$ domain. When this condition on $\ensuremath{\mathbf{z}}$ is not met, sparsity alone is no longer an effective prior to solve the recovery problem. Albeit qualitative, the second condition provides a fundamental insight in choosing undersampling schemes that favor recovery by sparsity-promoting inversion.

Simply denoise: wavefield reconstruction via jittered undersampling