We address the problem of estimating a random vector X from two sets of measurements Y and Z, such that the estimator is linear in Y . We show that the partially linear minimum mean squared error (PLMMSE) estimator requires knowing only the second-order moments of X and Y , making it of potential interest in various applications. We demonstrate the utility of PLMMSE estimation in recovering a signal, which is sparse in a unitary dictionary, from noisy observations of it and of a filtered

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... of it. We apply the method to the problem of image enhancement from blurred/noisy image pairs. In this setting the PLMMSE estimator performs better than denoising or deblurring alone, compared to state-of-the-art algorithms. Its performance is slightly worse than joint denoising/deblurring methods, but it runs an order of magnitude faster.