Estimating Volumes of Near-Spherical Molded Artifacts.

The Food and Drug Administration (FDA) is conducting research on developing reference lung cancer lesions, called phantoms, to test computed tomography (CT) scanners and their software. FDA loaned two semi-spherical phantoms to the National Institute of Standards and Technology (NIST), called Green and Pink, and asked to have the phantoms' volumes estimated. This report describes in detail both the metrology and computational methods used to estimate the phantoms' volumes. Three sets of coordinate measuring machine (CMM) measured data were produced. One set of data involved reference surface data measurements of a known calibrated metal sphere. The other two sets were measurements of the two FDA phantoms at two densities, called the coarse set and the dense set. Two computational approaches were applied to the data. In the first approach spherical models were fit to the calibrated sphere data and to the phantom data. The second approach was to model the data points on the boundaries of the spheres with surface B-splines and then use the Divergence Theorem to estimate the volumes. Fitting a B-spline model to the calibrated sphere data was done as a reference check on the algorithm performance. It gave assurance that the volumes estimated for the phantoms would be meaningful. The results for the coarse and dense data sets tended to predict the volumes as expected and the results did show that the Green phantom was very near spherical. This was confirmed by both computational methods. The spherical model did not fit the Pink phantom as well and the B-spline approach provided a better estimate of the volume in that case.

An Examination of New Paradigms for Spline Approximations.

Lavery splines are examined in the univariate and bivariate cases. In both instances relaxation based algorithms for approximate calculation of Lavery splines are proposed. Following previous work Gilsinn, et al. [7] addressing the bivariate case, a rotationally invariant functional is assumed. The version of bivariate splines proposed in this paper also aims at irregularly spaced data and uses Hseih-Clough-Tocher elements based on the triangulated irregular network (TIN) concept. In this paper, the univariate case, however, is investigated in greater detail so as to further the understanding of the bivariate case.