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Connection with iterative shrinkage thresholding

The well-known iterative shrinkage-thresholding (IST) algorithm is used for solving equation 3 with a sparsity-promoting constraint:

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

where $ \mathbf{x}$ is the transformed domain data such that $ \mathbf{d}=\mathbf{A}\mathbf{x}$ , $ \mathbf{A}$ is a tight frame such that $ \mathbf{x}=\mathbf{A}^H\mathbf{d}$ and $ \mathbf{A}^{-1}=\mathbf{A}^{H}$ (e.g. Fourier transform), $ \mathbf{T}$ is a nonlinear thresholding operator, $ \mathbf{K}=\mathbf{MA}$ and $ [\cdot]^H$ denotes adjoint. Considering that $ \mathbf{d_{n+1}}=\mathbf{A}\mathbf{x}_{n+1}$ , $ \mathbf{M}^H=\mathbf{M}$ and $ \mathbf{MM}=\mathbf{M}$ , combined with equation 7, we get:

\begin{displaymath}\begin{split}\mathbf{d}_{n+1} &= \mathbf{A} \mathbf{x}_{n+1} ...
...thbf{d}_n+\mathbf{d}_{obs}-\mathbf{M}\mathbf{d}_n], \end{split}\end{displaymath} (8)

which is equal to equation 6 with $ \mathbf{S}$ chosen as $ \mathbf{ATA}^H$ and $ \mathbf{B}$ taken as an identity operator. Thus, we prove that the IST and shaping regularization are actually mathematically equivalent.


next up previous [pdf]

Next: Connection with projection onto Up: Theory Previous: Interpolation using shaping regularization

2015-11-24