Problem formulation

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

$\displaystyle \forall_i \mathbf{m}_i ^n$ $\displaystyle = \arg \min_{\mathbf{m}_i} \parallel \mathbf{D} - \mathbf{F}^n\mathbf{M} \parallel_F^2, s.t. \forall_i \parallel \mathbf{m}_i \parallel_0 \le T,$ (3)
$\displaystyle \mathbf{F}^{n+1}$ $\displaystyle = \arg\min_{\mathbf{F}} \parallel \mathbf{D} - \mathbf{FM}^n \parallel_F^2,$ (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.