Iterative shrinkage thresholding

In order to reconstruct the missing traces in the seismic data, one can use a sparsity-promoting transform to precondition $\mathbf{d}$ in equation 1, that is:

$\displaystyle \mathbf{d}=\mathbf{Ax},$ (4)

where $\mathbf{A}$ is a tight frame such that $\mathbf{x}=\mathbf{A}^{H}\mathbf{d}$ and $\mathbf{A}^{-1}=\mathbf{A}^{H}$, and $[\cdot]^H$ denotes adjoint. A common selection for $\mathbf{A}$ is the Fourier transform. Inserting equation 4 into equation 1, and let $\mathbf{K}=\mathbf{MA}$, we obtain

$\displaystyle \mathbf{d}_{obs} = \mathbf{Kx}.$ (5)

Correspondingly, equation 3 turns to:

$\displaystyle \hat{\mathbf{x}} = \arg\min_{\mathbf{x}} \Arrowvert \mathbf{d}_{obs}-\mathbf{Kx} \Arrowvert_2 + \mathbf{R}'(\mathbf{x})$ (6)

The well-known IST algorithm is used for solving equation 6 with a sparsity constraint:

$\displaystyle \mathbf{x}_{n+1} = \mathbf{T}_{\gamma(\tau,p)}[\mathbf{x}_n+\mathbf{K}^{H}(\mathbf{d}_{obs}-\mathbf{Kx}_n)].$ (7)

Here $\mathbf{T}_{\gamma(\tau,p)}$ corresponds to a thresholding operator performed element-wise with threshold $\gamma(\tau,p)$ (Yang et al., 2013a). When $p=1$, $\gamma(\tau,1)=\tau$, $\mathbf{R}'(\cdot)=\tau \Arrowvert \cdot \Arrowvert_1$, where $\tau$ is a regularization parameter which controls the weight between misfit and constraint in the minimization problem, $\mathbf{T}_{\gamma(\tau,1)}$ corresponds to a soft-thresholding operator:

$\displaystyle \mathbf{T}_{\gamma(\tau,1)}[v(\mathbf{x})] = \left\{ \begin{array...
...\text{for}\quad \vert v(\mathbf{x})\vert \le \gamma(\tau,1)
\end{array}\right.,$ (8)

where $v(\mathbf{x})$ denotes the amplitude of each position-coordinate vector $\mathbf{x}$. When $p=0$, $\gamma(\tau,0)=\sqrt{2\tau}$, $\mathbf{R}'(\cdot)=\tau \Arrowvert \cdot \Arrowvert_0$, $\mathbf{T}_{\gamma(\tau,0)}$ corresponds to a hard-thresholding operator:

$\displaystyle \mathbf{T}_{\gamma(\tau,0)}[v(\mathbf{x})] = \left\{ \begin{array...
...\text{for}\quad \vert v(\mathbf{x})\vert \le \gamma(\tau,0)
\end{array}\right.,$ (9)