Implementation

The condition of using an overcomplete dictionary will hopefully be satisfied by choosing the number of model space variables to be roughly two times the number of data points. While Chen et al. (1999) uses a minimum of four times oversampling for impressive super resolution results with the Fourier transform, their examples are normally of the order n=1*10³. Larger dictionaries result in too slow processing for testing purposes.

Due to the fact that the LP infrastructure imposes a positivity constraint on the solution, we are forced to solve for a model space twice as large as we would choose with both negativity and positivity constraints, and then combine the two. We will use the hyperbolic radon transform (HRT) as the analysis operator, $\Phi$ in equation (2) or A in equation (3). This operator is normally approximately $1\%$ full and, therefore, a tractable operator to use for this method. As such, the programming is implemented with a sparse matrix approach rather than operators as it is traditional at SEP.