We treat the linear estimation problem with two simultaneous, competing objectives: minimum mean-squared error and minimum error-signal correlation. The latter objective minimizes the signal component in the error and maximizes the correlation of the estimator with the signal. The problem is solved, both for the scalar and stationary random process cases, as an optimal trade-off which produces a slightly higher mean-squared error and a much larger reduction in error-signal correlation over that of the minimum mean-squared error single objective solution. The optimal trade-off solution, which we call the mini-mum-error, minimum correlation (MEMC) filter is then applied to the problem of recovering space-invariant, blurred images with additive noise. As the theory predicts, the images restored through the MEMC filters are sharper and clearer than their minimum mean-squared error (Wiener) filter counterparts, but slightly noisier in appearance. Most viewers prefer the MEMC restorations to the Wiener ones, despite the noisier appearance.