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

MCA using sparsity-promoting shaping

MCA considers the complete data $d$ to be the superposition of several morphologically distinct components: $d=\sum_{i=1}^Nd_i$ . For each component $d_i$ , MCA assumes there exists a transform $\Phi_i$ which can sparsely represent component $d_i$ by its coefficients $\alpha_i$ ( $\alpha_i=\Phi_i^{*}d_i$ should be sparse), and can not do so for the others. Mathematically,

$$\min_{\{d_i\}}\sum_{i=1}^{N}R(\Phi_i^{*}d_i), \mathrm{subject\; to}\; d_{obs}=M\sum_{i=1}^{N}d_i.$$ (11)

The above problem can be rewritten as

$$\min_{\{d_i\}}\frac{1}{2}\left\Vert d_{obs}-M\sum_{i=1}^Nd_i\right\Vert _2^2+\lambda \sum_{i=1}^NR(\Phi_i^{*}d_i).$$ (12)

We prefer to rewrite Eq. (12) as

$$\min_{\{d_i\}}\frac{1}{2}\left\Vert\left(d_{obs}-M\sum_{j\neq i}d_j\right)-Md_i\right\Vert _2^2$$

$$+\lambda R(\Phi_i^{*}d_i)+\lambda \sum_{j\neq i}R(\Phi_j^{*}d_j).$$ (13)

Thus, optimizing with respect to $d_i$ leads to the analysis IST shaping as Eq. (9). At the kth iteration, optimization is performed alternatively for many components using the block coordinate relaxation (BCR) technique (Bruce et al., 1998): for the ith component $d_i^{k}$ , $i=1,\ldots,N$ : $\Phi\leftarrow \Phi_i$ , $d^{k}\leftarrow d_i^{k}$ , $d^{k+1}\leftarrow d_i^{k+1}$ , $d_{obs}\leftarrow d_{obs}-M\sum_{j\neq i}d_j^{k}$ , yields the residual term $r^{k}=d_{obs}-M\sum_{i=1}^N d_i^{k}$ and the updating rule

$$\left\{ \begin{array}{l} r^{k}\leftarrow d_{obs}-M\sum_{i=1}^N d_... ...\\ d_i^{k+1}\leftarrow \Phi_i S(\Phi_i^{*}(d_i^{k}+r^{k})). \end{array} \right.$$ (14)

The final output of the above algorithm are the morphological components $\hat{d}_i,i=1,\ldots,N$ . The complete data can then be reconstructed via $\hat{d}=\sum_{i=1}^N \hat{d}_i$ . This is the main principle of the so-called MCA-based inpainting algorithm (Elad et al., 2005).

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