We derive an optimal linear filter to suppress the noise from the COBE DMR sky maps for a given power spectrum. We then apply the filter to the first-year DMR data, after removing pixels within $20^circ$ of the Galactic plane from the data. The filtered data have uncertainties 12 times smaller than the noise level of the raw data. We use the formalism of constrained realizations of Gaussian random fields to assess the uncertainty in the filtered sky maps. In addition to improving the signal-to-noise ratio of the map as a whole, these techniques allow us to recover some information about the CMB anisotropy in the missing Galactic plane region. From these maps we are able to determine which hot and cold spots in the data are statistically significant, and which may have been produced by noise. In addition, the filtered maps can be used for comparison with other experiments on similar angular scales.
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