Double sparsity dictionary for seismic noise attenuation

Learning DSD in data and model domains - synthesis and analysis models

Provided that the learning-based dictionary is a tight frame such that $(\mathbf{W}^T)^T\mathbf{W}^T=\mathbf{W}\mathbf{W}^T=\mathbf{I}$ , DSD can be learned by considering the following problem:

$$\begin{split}&\min_{\mathbf{W,m}} \frac{1}{2}\parallel \mathb... ...quad and \quad \mathbf{W}\mathbf{W}^T = \mathbf{I}, \end{split}$$ (3)

where $t$ denotes the sparsity constrained coefficient.

Equation 3 can be solved alternatively by minimizing

$$\begin{split}\min_{\mathbf{W},\mathbf{m}} \frac{1}{2}&\parall... ...el_0, \ &s.t. \mathbf{W}\mathbf{W}^T = \mathbf{I}, \end{split}$$ (4)

where $\lambda$ denotes the damping factor, which is connected with the sparsity constrained coefficient $t$ . Equation 4 is called synthesis model for DSD.

Assuming the base dictionary $\mathbf{B}$ is invertible, the synthesis-based DSD model as shown in equation 3 is equivalent to:

$$\begin{split}\min_{\mathbf{W,m}} \frac{1}{2}&\parallel \mathb... ...lel_0 \ &s.t. \mathbf{W}\mathbf{W}^T = \mathbf{I}, \end{split}$$ (5)

where $\mathbf{B}^{-1}$ denotes the forward base transform. Equation 5 suggests that DSD can also be learned in the model domain of the multi-scale decomposition operator $\mathbf{B}$ instead of in the data domain. Equation 5 is called analysis model for DSD.

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