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Note that at each iteration soft thresholding is the only nonlinear operation corresponding to the
constraint for the model
, i.e.,
.
Shaping regularization (Fomel, 2007,2008) provides a general and flexible framework for inversion without the need for a specific penalty function
when a particular kind of shaping operator is used. The iterative shaping process can be expressed as
![$\displaystyle x^{k+1}=S(x^{k}+B(d_{obs}-Fx^{k})),$](img33.png) |
(9) |
where the shaping operator
can be a smoothing operator (Fomel, 2007), or a more general operator even a nonlinear sparsity-promoting shrinkage/thresholding operator (Fomel, 2008). It can be thought of a type of Landweber iteration followed by projection, which is conducted via the shaping operator
. Instead of finding the formula of gradient with a known regularization penalty, we have to focus on the design of shaping operator in shaping regularization. In gradient-based Landweber iteration the backward operator
is required to be the adjoint of the forward mapping
, i.e.,
; in shaping regularization however, it is not necessarily required. Shaping regularization gives us more freedom to choose a form of
to approximate the inverse of
so that shaping regularization enjoys faster convergence rate in practice. In the language of shaping regularization, the updating rule in Eq. (7) becomes
![$\displaystyle \left\{ \begin{array}{l} r^{k}\leftarrow d_{obs}-Md^{k}, \\ d^{k+1}\leftarrow \Phi S(\Phi^{*}(d^{k}+r^{k})), \end{array} \right.$](img37.png) |
(10) |
where the backward operator is chosen to be the inverse of the forward mapping.
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Previous: Analysis-based iterative thresholding
2021-08-31