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A general inverse problem combined with a priori constraint
can be written as an optimization problem
![$\displaystyle \min_x \frac{1}{2}\Vert d_{obs}-F x\Vert _2^2+\lambda R(x),$](img7.png) |
(1) |
where
is the model to be inverted, and
is the observations. To solve the problem with sparsity constraint
, the iterative shrinkage-thresholding (IST) algorithm has been proposed (Daubechies et al., 2004), which can be generally formulated as
![$\displaystyle x^{k+1}=T_{\lambda}(x^{k}+F^*(d_{obs}-Fx^{k})),$](img11.png) |
(2) |
where
denotes the iteration number; and
indicates the adjoint of
.
is an element-wise shrinkage operator with threshold
:
![$\displaystyle T_{\lambda}(x)=(t_{\lambda}(x_1),t_{\lambda}(x_2),\ldots,t_{\lambda}(x_m))^T,$](img17.png) |
(3) |
in which the soft thresholding function (Donoho, 1995) is
![$\displaystyle t_{\lambda}(u)=\mathrm{Soft}_{\lambda}(u)=\left\{\begin{array}{ll...
...rt\leq \lambda. \end{array} \right. =u.*\max(1-\frac{\lambda}{\vert u\vert},0).$](img18.png) |
(4) |
Allowing for the missing elements in the data, the observations are connected to the complete data via the relation
![$\displaystyle d_{obs}=Md=M\Phi x=Fx, F=M\Phi.$](img19.png) |
(5) |
where
is an acquisition mask indicating the observed and missing values. Assume
is a tight frame such that
,
. It leads to
![\begin{displaymath}\begin{array}{ll} d^{k+1} & =\Phi x^{k+1} \\ & =\Phi T_{\lamb...
... =\Phi T_{\lambda}(\Phi^*(d^{k}+(d_{obs}-Md^{k}))), \end{array}\end{displaymath}](img24.png) |
(6) |
in which we use
and
. Now we define a residual term as
, thus Eq. (6) results in
![$\displaystyle \left\{ \begin{array}{l} r^{k}\leftarrow d_{obs}-Md^{k} \\ d^{k+1}\leftarrow \Phi T_{\lambda}(\Phi^*(d^{k}+r^{k})), \end{array} \right.$](img28.png) |
(7) |
which is equivalent to solving
![$\displaystyle \min\limits_{d}\frac{1}{2}\Vert d_{obs}-Md\Vert _2^2+\lambda R(\Phi^{*}d).$](img29.png) |
(8) |
Note that Eq. (8) analyzes the target unknown
directly, without resort to
and
. Eq. (6) is referred to as the analysis formula (Elad et al., 2007). In this paper, we used the analysis formula because it directly addresses the problem in the data domain for the convenience of interpolation and signal separation.
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2021-08-31