We will be considering analysis sparsity based regularization for Magnetic Resonance Imaging reconstruction. The analysis sparsity regularization is based on the recently introduced Transform Learning framework, which has reduced complexity regarding other sparse regularization methods. We will formulate a variational reconstruction problem which utilizes the analysis sparsity regularization together with an l(1) norm based data fidelity term. The use of the non-smooth data fidelity term results in robustness against outliers and impulsive noise in the observed data. The resulting algorithm with the l(1) observation fidelity showcases enhanced performance under impulsive observation noise when compared to a similar algorithm utilizing the conventional quadratic error term.