Simply denoise: wavefield reconstruction via jittered undersampling

Regular (under)sampling

When $\tensor{R}$ keeps all the data points of $\ensuremath{\mathbf{f}}_0$ , i.e., $\tensor{R}=\tensor{I}$ , the matrix $\tensor{A}^H\tensor{A}$ is the identity matrix, as depicted in Figure 2(a), $\tensor{L}=0$ , as plotted in Figure 2(d), and there is no spectral leakage. This property holds for any orthonormal sparsifying transform.

When $\tensor{R}$ corresponds to a regular undersampling scheme, the matrix $\tensor{A}^H\tensor{A}$ is no longer diagonal. It now also has a number of nonzero off-diagonals as depicted in Figure 2(b). These off-diagonals create aliases, i.e., undersampling artifacts that are the superposition of circular-shifted versions of the original spectrum. Since $\ensuremath{\mathbf{x}}_0$ is assumed to be sparse, these aliases are sparse as well. Therefore, they are also likely to enter in the solution $\tilde{\ensuremath{\mathbf{x}}}$ during sparsity-promoting inversion. Because the $\ell_1$ norm can not efficiently discriminate the original spectrum from its aliases, regular undersampling is the most challenging case for recovery.

In the seismic community, difficulties with regularly undersampled data are acknowledged when reconstructing by promoting sparsity in the Fourier domain. For example, Xu et al. (2005) write that the anti-leakage Fourier transform for seismic data regularization ``may fail to work when the input data has severe aliasing''.

Simply denoise: wavefield reconstruction via jittered undersampling