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

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

where the shaping operator 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 . 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 is required to be the adjoint of the forward mapping , i.e., ; in shaping regularization however, it is not necessarily required. Shaping regularization gives us more freedom to choose a form of to approximate the inverse of so that shaping regularization enjoys faster convergence rate in practice. In the language of shaping regularization, the updating rule in Eq. (7) becomes

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

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

2021-08-31