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

Understanding iterative thresholding as shaping regularization

Note that at each iteration soft thresholding is the only nonlinear operation corresponding to the $\ell_1$ constraint for the model $x$ , i.e., $R(x)=\Vert x\Vert _1$ . Shaping regularization (Fomel, 2007,2008) provides a general and flexible framework for inversion without the need for a specific penalty function $R(x)$ when a particular kind of shaping operator is used. The iterative shaping process can be expressed as

$$x^{k+1}=S(x^{k}+B(d_{obs}-Fx^{k})),$$ (9)

where the shaping operator $S$ can be a smoothing operator (Fomel, 2007), or a more general operator even a nonlinear sparsity-promoting shrinkage/thresholding operator (Fomel, 2008). It can be thought of a type of Landweber iteration followed by projection, which is conducted via the shaping operator $S$ . Instead of finding the formula of gradient with a known regularization penalty, we have to focus on the design of shaping operator in shaping regularization. In gradient-based Landweber iteration the backward operator $B$ is required to be the adjoint of the forward mapping $F$ , i.e., $B=F^*$ ; in shaping regularization however, it is not necessarily required. Shaping regularization gives us more freedom to choose a form of $B$ to approximate the inverse of $F$ so that shaping regularization enjoys faster convergence rate in practice. In the language of shaping regularization, the updating rule in Eq. (7) becomes

$$\left\{ \begin{array}{l} r^{k}\leftarrow d_{obs}-Md^{k}, \\ d^{k+1}\leftarrow \Phi S(\Phi^{*}(d^{k}+r^{k})), \end{array} \right.$$ (10)

where the backward operator is chosen to be the inverse of the forward mapping.

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