Compressed-sensing reconstruction of still images and video sequences driven by multihypothesis predictions is considered. Specifically, for still images, multiple predictions drawn for an image block are made from spatially surrounding blocks within an initial non-predicted reconstruction. For video, multihypothesis predictions of the current frame are generated from one or more previously reconstructed reference frames. In each case, the predictions are used to generate a residual in the domain of the compressed-sensing random projections. This residual being typically more compressible than the original signal leads to improved reconstruction quality. To appropriately weight the hypothesis predictions, a Tikhonov regularization to an ill-posed least-squares optimization is proposed. Experimental results demonstrate that the proposed reconstructions outperform alternative strategies not employing multihypothesis predictions.