@mastersthesis {hargreaves2014THssr,
	title = {Sparse signal recovery: analysis and synthesis formulations with prior support information},
	year = {2014},
	note = {(MSc)},
	month = {04},
	school = {The University of British Columbia},
	type = {masters},
	address = {Vancouver},
	abstract = {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.},
	keywords = {analysis, MSc, sparse, synthesis, thesis, weighted $\ell_1$},
	url = {https://slim.gatech.edu/Publications/Public/Thesis/2014/hargreaves2014THssr/hargreaves2014THssr.pdf},
	author = {Brock Hargreaves}
}
@presentation {hargreaves2013SINBADbfo,
	title = {The bridge from orthogonal to redundant transforms and weighted $\ell_1$ optimization},
	year = {2013},
	publisher = {SINBAD},
	abstract = {Traditional arguments in synthesis $\ell_1$ -optimization require our forward operator to be orthogonal, though we use redundant transforms in practice. These traditional arguments do not translate to redundant transforms, and other arguments require impractical conditions on our effective measurement matrix. Recent theory in one-norm analysis, namely the optimal dual $\ell_1$ analysis of Shidong et al, have provided point-wise reconstruction error estimates for synthesis using an equivalence relationship where we can use weaker assumptions. This exposes an important model assumption indicating why analysis might outperform synthesis, for which careful consideration in seismic is necessary, and the need for models such as the cosparse model. In this talk we will discuss these ideas, provide evidence which indicates this theory should generalize to uniform error estimates(and thus not signal dependent), and how redundancy, support information, and weighting play important roles.},
	keywords = {Presentation, SINBAD, SINBADFALL2013, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Fall/hargreaves2013SINBADbfo/hargreaves2013SINBADbfo.pdf},
	author = {Brock Hargreaves and Ozgur Yilmaz and Felix J. Herrmann}
}
@presentation {hargreaves2012SINBADavs,
	title = {Analysis versus synthesis in weighted sparse recovery},
	year = {2012},
	publisher = {SINBAD},
	abstract = {The synthesis model for compressive sensing has been the model of choice for many years and various weighting schemes have been shown to improve it{\textquoteright}s performance (see Yilmaz, Mansour, and Ghadermarzy talks). However, there is a counterpart model to synthesis, namely the analysis model, which has been less popular but recently attracted more attention (see Lin{\textquoteright}s talk). In this talk, weighting in the analysis model is discussed and applied to the seismic trace interpolation problem.},
	keywords = {Presentation, SINBAD, SINBADFALL2012, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2012/Fall/hargreaves2012SINBADavs/hargreaves2012SINBADavs_pres.pdf},
	author = {Brock Hargreaves}
}
