Sparse signal recovery: analysis and synthesis formulations with prior support information

The synthesis model for signal recovery has been the model of choice for many years in compressive sensing. Various weighting schemes using prior support information to adjust the objective function associated with the synthesis model have been shown to improve the recovery of the signal in terms of accuracy. Generally, even with no prior knowledge of the support, iterative methods can build support estimates and incorporate that into the recovery which has also been shown to increase the speed and accuracy of the recovery. However when the original signal is sparse with respect to a redundant dictionary (rather than an orthonormal basis) there is a counterpart model to synthesis, namely the analysis model, which has been less popular but has recently attracted more attention. The analysis model is much less understood and thus there are fewer theorems available in both the context of non-weighted and weighted signal recovery. In this thesis, we investigate weighting in both the analysis model and synthesis model in weighted $\ell_1$-minimization. Theoretical guarantees on reconstruction and various weighting strategies for each model are discussed. We give conditions for weighted synthesis recovery with frames which do not require strict incoherency conditions, this is based on recent results of regular synthesis with frames using optimal dual $\ell_1$ analysis. A novel weighting technique is introduced in the analysis case which outperforms its traditional counterparts in the case of seismic wavefield reconstruction. We also introduce a weighted split Bregman algorithm for analysis and optimal dual analysis. We then investigate these techniques on seismic data and synthetically created test data using a variety of frames.