In conventional surface estimation using white light interferometry, data acquisition is followed by a preprocessing step in which the 3-D (space-time) data are reduced to a 2-D surface or height map. Finally, a postprocessing step eliminates the height outliers that arise from the preprocessing under ordinary experimental conditions. We introduce a Bayesian approach that unifies pre- and postprocessing by considering simultaneously both the full 3-D data set and knowledge concerning the surface smoothness. This knowledge is coded into the prior probability of local height configurations. The surface is estimated as the mode of the marginal posterior at each pixel. An adept formulation of the prior allows for the exact computation of the estimate, obviating the need to sample from the posterior using Markov chain Monte Carlo methods. A complete surface can thus be obtained in 3 to 30 s. A quantitative comparison with an (adaptive) median filter shows that all three approaches decimate outliers, but that the Bayesian estimation leads to smaller average absolute errors. For low scanning speeds and good raw data, this is due to a reduced tendency to oversmooth; for poor input data, the enhancement is explained by the more complete exploitation of the observations.
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.