Problem formulation

Sparse representation via learning based dictionary consists of two main steps.

• Sparse coding. Given the observed data $\mathbf{d}$, sparse coding aims at solving the optimization problem:

  $$\mathbf{m}^n = \arg \min_{\mathbf{m}} \parallel \mathbf{d} - \mathbf{F}^n\mathbf{m} \parallel_2^2, s.t. \parallel \mathbf{m} \parallel_0 \le T,$$ (1)

  
  
  where $\parallel\cdot\parallel_2$ and $\parallel\cdot\parallel_0$ denote the $L_2$ and $L_0$ norms of an input vector, respectively. $T$ is the number of non-zero coefficients. $\mathbf{F}$ is the learned dictionary and $\mathbf{m}$ is the sparse representation of $\mathbf{d}$.
• Dictionary updating. For the obtained $\mathbf{m}^n$, update $\mathbf{F}^n$ such that

  $$\mathbf{F}^{n+1} = \arg \min_{\mathbf{F}} \parallel \mathbf{d} - \mathbf{Fm}^n \parallel _2^2.$$ (2)

  
  

Equations 1 and 2 are iterated $Niter$ times to learn the optimal dictionary and the sparest representation.

The multidimensional seismic data is first reformulated into patch form $\mathbf{D}$. Each column vector in $\mathbf{D}$ is extracted from the multidimensional seismic data matrix. An example is given in Yu et al. (2015) and Chen et al. (2016a). Equations 1 and 2 then become

$$\forall_i \mathbf{m}_i ^n = \arg \min_{\mathbf{m}_i} \parallel \mathbf{D} - \mathbf{F}^n\mathbf{M} \parallel_F^2, s.t. \forall_i \parallel \mathbf{m}_i \parallel_0 \le T,$$ (3)

$$\mathbf{F}^{n+1} = \arg\min_{\mathbf{F}} \parallel \mathbf{D} - \mathbf{FM}^n \parallel_F^2,$$ (4)

where $\parallel\cdot\parallel_F$ denotes the Frobenius norm of an input matrix.

Problem 3 is a NP-hard problem, and directly finding the truly optimal $\mathbf{M}$ is impossible and is usually solved by an approximation pursuit method, such as the orthogonal matching pursuit (OMP) algorithm. To solve problem 4 for the adaptive dictionary $\mathbf{F}$, there are several different algorithms.