The iterative shrinkage thresholding (IST) is one of the most effective methods to solve problem 2:
![$\displaystyle \mathbf{m}_{n+1} = \mathbf{T}_{\tau}\left[\mathbf{m}_{n} + (\math...
...bf{A}^{-1})^H(\mathbf{d}_{obs}-(\mathbf{S}\mathbf{A}^{-1})\mathbf{m}_n)\right],$](img22.png) |
(4) |
where
denotes the coefficients model after
th iteration,
denotes a thresholding operator (which is a nonlinear operator) with an input parameter
, and
denotes the adjoint operator. It's worth noting that
has different connections with
according to the thresholding type (Chen et al., 2014a).
Due to the
convergence of IST, implementing IST is usually very time-consuming in practice. Beck and Teboulle (2009) proposed the fast iterative shrinkage thresholding algorithm (FISTA) to improve the convergence rate:
![\begin{displaymath}\begin{split}
\mathbf{m}_{n}' &= \mathbf{m}_n + \frac{v_n-1}{...
...thbf{S}\mathbf{A}^{-1})\mathbf{m}_n'\right)\right],
\end{split}\end{displaymath}](img30.png) |
(5) |
where
is a controlling parameter with the initial value
and
.
The improved convergence rate is
, and thus becomes widely used in the image-processing field since its invention.
2020-04-11