Simply denoise: wavefield reconstruction via jittered undersampling

Sparsity-promoting solvers and jittered undersampling

The applicability of CS to the large-scale problems of exploration geophysics heavily relies on the implementation of an efficient $\ell_1$ solver. Despite several recent attempts to overcome this bottleneck (van den Berg and Friedlander, 2007; Tibshirani, 1996; Figueiredo et al., 2007), a wide range of large-scale applications still uses approximate $\ell_1$ solvers such as iterated re-weighted least-squares (IRLS - Gersztenkorn et al., 1986), stage-wise orthogonal matching pursuit (StOMP - Donoho et al., 2006), and iterative soft-thresholding with cooling (Hennenfent and Herrmann, 2005; Herrmann and Hennenfent, 2007) derived from Daubechies et al. (2004). The success and/or efficiency of these approximate solvers depends upon the implicit-or-explicit assumption that the MAI is incoherent. Because optimally-jittered undersampling creates such a MAI, these solvers can be used for the sparsity-promoting reconstruction with curvelets or other localized Fourier-based transforms. More importantly, jittered undersampling can be useful to evaluate the efficiency/robustness of (approximate) $\ell_1$ solvers since the jitter parameter controls the amount of coherent energy that enters the MAI.