Iterative deblending with multiple constraints based on shaping regularization

Iterative seislet thresholding

When $\mathbf{S}=\mathbf{A}^{-1}\mathbf{T}_{\tau}\mathbf{A}$ , the iterative framework refers to the iterative seislet thresholding:

$$\mathbf{m}_{n+1} = \mathbf{A}^{-1}\mathbf{T}_{\tau_n}\mathbf{A}[\mathbf{m}_n+\lambda\Gamma^*[\mathbf{d}-\Gamma\mathbf{m}_n]],$$ (3)

where $\mathbf{T}_{\tau_n}$ denotes a thresholding operator with a threshold $\tau_n$ , $\mathbf{A}$ and $\mathbf{A}^{-1}$ are a pair of forward and inverse seislet transforms.

$\mathbf{T}_{\tau}$ can be either a soft-thresholding operator:

$$\mathbf{T}^s_{\tau}(x) = \left\{ \begin{array}{ll} (\vert x\vert-... ...vert x\vert\ge\tau 0 &\text{for} \quad \vert x\vert<\tau \end{array}\right.,$$ (4)

or a hard thresholding operator:

$$\mathbf{T}^h_{\tau}(x) = \left\{ \begin{array}{ll} x &\text{for} ... ...ert x\vert\ge\tau 0 &\text{for} \quad \vert x\vert<\tau \end{array}\right. .$$ (5)

In this paper, all the examples are based on soft thresholding. I iteratively decrease the threshold $\tau$ in the seislet domain during the iterations.