Purpose Compressed sensing (CS) reduces the measurement time of magnetic resonance (MR) imaging, where the use of regularizers or image priors are key techniques to boost reconstruction precision. The optimal prior generally depends on the subject and the hand-building of priors is hard. A methodology of combining priors to create a better one would be useful for various forms of image processing that use image priors. Methods We propose a theory, called prior ensemble learning (PEL), which combines many weak priors (not limited to images) efficiently and approximates the posterior mean (PM) estimate, which is Bayes optimal for minimizing the mean squared error (MSE). The way of combining priors is changed from that of an exponential family to a mixture family. We applied PEL to an undersampled (10%) multicoil MR image reconstruction task. Results We demonstrated that PEL could combine 136 image priors (norm-based priors such as total variation (TV) and wavelets with various regularization coefficient (RC) values) from only two training samples and that it was superior to the CS-SENSE-based method in terms of the MSE of the reconstructed image. The resulting combining weights were sparse (18% of the weak priors remained), as expected. Conclusion By the theory, the PM estimator was decomposed into the sparse weighted sum of each weak prior's PM estimator, and the exponential computational complexity for RCs was reduced to polynomial order w.r.t. the number of weak priors. PEL is feasible and effective for a practical MR image reconstruction task.
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