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Uncertainty of NURBS surface fit by Monte Carlo simulations

Abstract A free-form surface expressed by NURBS (nonuniform rational B-splines) is fitted to the measured coordinates of points by the lofting method. The unknown control points of the free-form surface are therefore not simultaneously estimated but determined by cross-sectional curve fits. This uses much less computer time than the simultaneous estimation and gives identical results. The free-form surface should be determined with an uncertainty which does not considerably surpass the uncertainty of the measurements. This is investigated here for the example of a free-form surface for a pothole in a road determined by the measurements of a laserscanner. The uncertainties are expressed by standard deviations and confidence intervals. They are computed using Monte Carlo simulations for the positioning of a point by the measured coordinates and by fitting a free-form surface. The resulting uncertainties agree. In addition, the uncertainties of quantities characterizing the shape and the slope of the surface are determined by Monte Carlo simulations. It turns out that the uncertainties resulting from the measurements and from the free-form surface fit are approximately identical.