Compressed sensing is applied to multiview image sets and the high degree of correlation between views is exploited to enhance recovery performance over straightforward independent view recovery. This gain in performance is obtained by recovering the difference between a set of acquired measurements and the projection of a prediction of the signal they represent. The recovered difference is then added back to the prediction, and the prediction and recovery procedure is repeated in an iterated fashion for each of the views in the multiview image set. The recovered multiview image set is then used as an initialization to repeat the entire process again to form a multistage refinement. Experimental results reveal substantial performance gains from the multistage reconstruction.