# Sparse Representation based on Dictionary Learning

This work presents the use of the Dictionary Learning method for a sparse representation of 4D seismic data. We consider a trade-off between the number of nonzero coefficients retained in the sparse data representation, the computational cost, and how well we can capture the main features of the original 4D seismic signal. K-SVD is an iterative algorithm used in Dictionary Learning that alternates between the calculation of the sparse representation vector and dictionary update. The algorithm starts with the definition of an initial dictionary (Discrete Cosine Transform, for instance). To calculate the sparse representation vector, one can use the Orthogonal Matching Pursuit (OMP) algorithm and constrain the problem into two distinct approaches: (1) sparsity-constrained; and (2) error-constrained. Finally, it is possible to update the dictionary through SVD. We evaluated the influence of critical parameters of the algorithm (dictionary size, number of iterations, patch size, and training dataset size). Results showed that the dictionary learning method can capture the main features of the original 4D seismic signal with the sparse representation. However, the number of nonzero coefficients retained is highly dependent on the selected parameters. Therefore, they need to be carefully determined to obtain a reasonable amount of nonzero coefficients retained.