Simply denoise: wavefield reconstruction via jittered undersampling

Recovery by sparsity-promoting inversion

Consider the following linear forward model for the recovery problem

$$\ensuremath{\mathbf{y}} = \tensor{R}\ensuremath{\mathbf{f}}_0,$$ (1)

where $\ensuremath{\mathbf{y}}\in\mathbb{R}^n$ represents the acquired data, $\ensuremath{\mathbf{f}}_0\in\mathbb{R}^N$ with $N \gg n$ the unaliased signal to be recovered, i.e., the model, and $\tensor{R}\in\mathbb{R}^{n\times N}$ the restriction operator that collects the acquired samples from the model. Assume that $\ensuremath{\mathbf{f}}_0$ has a sparse representation $\ensuremath{\mathbf{x}}_0\in\mathbb{C}^N$ in some known transform domain $\tensor{S}$ , equation 1 can now be reformulated as

$$\ensuremath{\mathbf{y}} = \tensor{A}\ensuremath{\mathbf{x}}_0 \quad\tensor{A}\; {\buildrel\rm def\over=}\;\tensor{RS}^H,$$ (2)

where the symbol   $^H$ represents the conjugate transpose. As a result, the sparsity of $\ensuremath{\mathbf{x}}_0$ can be used to overcome the singular nature of $\tensor{A}$ when estimating $\ensuremath{\mathbf{f}}_0$ from $\ensuremath{\mathbf{y}}$ . After sparsity-promoting inversion, the recovered signal is given by $\tilde{\ensuremath{\mathbf{f}}} = \tensor{S}^H\tilde{\ensuremath{\mathbf{x}}}$ with

$$\tilde{\ensuremath{\mathbf{x}}} = \arg\min_{\ensuremath{\mathbf{x}}}\vert\vert\ensuremath{\mathbf{x}}\vert\vert _1 \quad\ensuremath{\mathbf{y}}=\tensor{A}\ensuremath{\mathbf{x}}.$$ (3)

In these expressions, the symbol $\,\,\tilde{\mbox{}}\,\,$ represents estimated quantities and the $\ell_1$ norm is defined as $\Vert\ensuremath{\mathbf{x}}\Vert _1\; {\buildrel\rm def\over=}\; \sum_{i=1}^{N}\left\vert\ensuremath{\mathbf{x}}[i]\right\vert$ , where $\ensuremath{\mathbf{x}}[i]$ is the $i^{th}$ entry of the vector $\ensuremath{\mathbf{x}}$ .

Minimizing the $\ell_1$ norm in equation 3 promotes sparsity in $\ensuremath{\mathbf{x}}$ and the equality constraint ensures that the solution honors the acquired data. Among all possible solutions of the (severely) underdetermined system of linear equations ($n\ll N$ ) in equation 2, the optimization problem in equation 3 finds a sparse or, under certain conditions, the sparsest (Donoho and Huo, 2001) possible solution that explains the data.

Simply denoise: wavefield reconstruction via jittered undersampling