We use an extension of normalized convolution to smooth height maps from interferometry using confidence values. The latter are often used for dichotomous good/bad decisions only, with all bad data being discarded. To minimize loss of information, we weight each pixel individually by the inverse of its expected variance. The relation between supplied confidence values and empirical variances is found by regression. The width of the smoothing kernel—as small as possible to prevent loss of spatial resolution, as large as necessary to average out noise—is adjusted locally so as to yield a smoothed image with a prespecified uncertainty that is homogeneous throughout. In our experimental investigations using metrological data from a white light interferometric sensor, the variable-width mask leads to images with somewhat lower absolute deviation from an average image than the fixed-width masks we use for comparison.