Double sparsity dictionary for seismic noise attenuation |

where denotes the sparsity constrained coefficient.

Equation 3 can be solved alternatively by minimizing

where denotes the damping factor, which is connected with the sparsity constrained coefficient . Equation 4 is called

Assuming the base dictionary is invertible, the synthesis-based DSD model as shown in equation 3 is equivalent to:

where denotes the forward base transform. Equation 5 suggests that DSD can also be learned in the model domain of the multi-scale decomposition operator instead of in the data domain. Equation 5 is called

The synthesis model (equation 4) offers more flexibility for learning DSD because there are many sparsity-promoting transforms that are not exactly invertible. The analysis model, on the other hand, offers more convenience for constructing DSD when an invertible sparsity-promoting base transform is available.

Double sparsity dictionary for seismic noise attenuation |

2016-02-27