Double sparsity dictionary for seismic noise attenuation |

The minimization can be solved by updating coefficients of vector
and the learning-based dictionary
alternately. We adopt the following algorithm for solving problem 5:

Input: Base dictionary
, initial learning-based dictionary
.

- Transform data from data domain to model domain according to

**for**:- Fix the learning-based dictionary
, estimate the double-sparsity coefficients vector
by

- Given the double-sparsity coefficients vector
, update the learning-based dictionary
:

**end for**- Fix the learning-based dictionary
, estimate the double-sparsity coefficients vector
by

Assuming the size of the coefficients domain is , and , let a filter mapping be the block-wise Toeplitz matrix representing the convolution operator with a finitely supported 2D filter under the Newmann boundary condition. The learning based dictionary can be defined as

Each is a 2D filter associated with a tight frame and the columns of form a tight frame for . denotes the number of filters. The patch size discussed in the following examples corresponds to the size of each .

Liang et al. (2014) give an example of using spline wavelets for the initial and the finally learned dictionary . Following Liang et al. (2014), we also choose spline wavelets for the initial .

Double sparsity dictionary for seismic noise attenuation |

2016-02-27