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Connections between seislet frame and seislet-MCA algorithm
The complete data
is regarded to be superposition of several different geometrical components, and each component can be sparely represented using a seislet dictionary
, i.e.,
![\begin{displaymath}\begin{array}{ll} d&=\displaystyle \sum_{i=1}^N d_i=\sum_{i=1...
... \alpha_2\\ \vdots\\ \alpha_N \end{array} \right]\\ \end{array}\end{displaymath}](img100.png) |
(29) |
where
is a combined seislet dictionary (i.e. seislet frame), and the backward operator is chosen to be
![$\displaystyle B=\frac{1}{N}\,\left[\begin{array}{l} \Phi_1^{*}\\ \Phi_2^{*}\\ \vdots\\ \Phi_N^{*} \end{array} \right]$](img102.png) |
(30) |
in the sense that
![$\displaystyle FB=\displaystyle \frac{1}{N}\,\sum_{i=1}^N\Phi_i\Phi_i^{*}=\mathrm{Id}.$](img103.png) |
(31) |
The difference between seislet-MCA algorithm and seislet frame minimization is the use of BCR technique (Bruce et al., 1998): We sparsify one component while keeping all others fixed. At the
-th iteration applying the backward operator on the
-th component leads to
![$\displaystyle \tilde{\alpha}_i^{k+1}=\alpha_i^k+\sum_{i=1}^N\Phi_i^{*} r^k=\alpha_i^k+r_i^j+\sum_{j\neq i}\Phi_i^{*} r_j^k$](img106.png) |
(32) |
where the terms
are the crosstalk between the
-th component and the others. An intuitive approach to filter out the undesired crosstalk is shrinkage/thresholding. The proposed exponential shrinkage provides us a flexible control on the performance of the shrinkage/thresholding operator.
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2021-08-31