Seislet-based morphological component analysis using scale-dependent exponential shrinkage

Connections between seislet frame and seislet-MCA algorithm

The complete data $d$ is regarded to be superposition of several different geometrical components, and each component can be sparely represented using a seislet dictionary $\Phi_i$ , i.e.,

$$\begin{array}{ll} d&=\displaystyle \sum_{i=1}^N d_i=\sum_{i=1... ... \alpha_2\\ \vdots\\ \alpha_N \end{array} \right]\\ \end{array}$$ (29)

where $F=[\Phi_1,\Phi_2,\cdots,\Phi_N]$ is a combined seislet dictionary (i.e. seislet frame), and the backward operator is chosen to be

$$B=\frac{1}{N}\,\left[\begin{array}{l} \Phi_1^{*}\\ \Phi_2^{*}\\ \vdots\\ \Phi_N^{*} \end{array} \right]$$ (30)

in the sense that

$$FB=\displaystyle \frac{1}{N}\,\sum_{i=1}^N\Phi_i\Phi_i^{*}=\mathrm{Id}.$$ (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 $k+1$ -th iteration applying the backward operator on the $i$ -th component leads to

$$\tilde{\alpha}_i^{k+1}=\alpha_i^k+\sum_{i=1}^N\Phi_i^{*} r^k=\alpha_i^k+r_i^j+\sum_{j\neq i}\Phi_i^{*} r_j^k$$ (32)

where the terms $\sum_{j\neq i}\Phi_i^{*}r_j^k$ are the crosstalk between the $i$ -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.

Seislet-based morphological component analysis using scale-dependent exponential shrinkage