Experimental Verification of Formulas for Variances of Plane Parameters Fitted to Three-Dimensional Imaging Data

Nonlinear least squares method is applied to fit a plane to 3-D imaging data. Two different error functions used in fitting are tested: orthogonal and directional. Variances of fitted parameters are calculated either from a single data set using derived earlier analytical formulas or by repeating the scans at the same experimental settings. In the latter case, the variance is equal to the square of the standard deviation from the repeated scans. The results show that the orthogonal error function severely underestimates the variances calculated by using the mathematical formulas when compared to the variances calculated from multiple scans. The variances calculated from formulas based on the directional error function are in good agreement with the estimates calculated from multiple scans. Thus, the variances of the fitted plane parameter can be determined correctly from a single scan when proper formulas are used. This is important because multiple scans from the same location are typically not performed in the field.