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Digital Images with 3D Geometry from Data Compression by Multi-scale Representations of B-Spline SurfacesTo build up a 3D (three-dimensional) model of the surface of an object, the heights of points on the surface are measured, for instance, by a laser scanner. The intensities of the reflected laser beam of the points can be used to visualize the 3D model as range image. It is proposed here to fit a two-dimensional B-spline surface to the measured heights and intensities by the lofting method. To fully use the geometric information of the laser scanning, points on the fitted surface with their intensities are computed with a density higher than that of the measurements. This gives a 3D model of high resolution which is visualized by the intensities of the points on the B-spline surface. For a realistic view of the 3D model, the coordinates of a digital photo of the object are transformed to the coordinate system of the 3D model so that the points get the colors of the digital image. To efficiently compute and store the 3D model, data compression is applied. It is derived from the multi-scale representation of the dense grid of points on the B-spline surface. The proposed method is demonstrated for an example.

N-dimensional B-spline surface estimated by lofting for locally improving IRIN-dimensional surfaces are defined by the tensor product of B-spline basis functions. To estimate the unknown control points of these B-spline surfaces, the lofting method also called skinning method by cross-sectional curve fits is applied. It is shown by an analytical proof and numerically confirmed by the example of a four-dimensional surface that the results of the lofting method agree with the ones of the simultaneous estimation of the unknown control points. The numerical complexity for estimating vn control points by the lofting method is O(vn+1) while it results in O(v3n) for the simultaneous estimation. It is also shown that a B-spline surface estimated by a simultaneous estimation can be extended to higher dimensions by the lofting method, thus saving computer time.An application of this method is the local improvement of the International Reference Ionosphere (IRI), e.g. by the slant total electron content (STEC) obtained by dual-frequency observations of the Global Navigation Satellite System (GNSS). Three-dimensional B-spline surfaces at different time epochs have to be determined by the simultaneous estimation of the control points for this improvement. A four-dimensional representation in space and time of the electron density of the ionosphere is desirable. It can be obtained by the lofting method. This takes less computer time than determining the four-dimensional surface solely by a simultaneous estimation.

A new method for dealing with systematic effects in laser scanning and visualizing them by confidence regions is derived. The standard deviations of the systematic effects are obtained by repeatedly measuring three-dimensional coordinates by the laser scanner. In addition, autocovariance and cross-covariance functions are computed by the repeated measurements and give the correlations of the systematic effects. The normal distribution for the measurements and the multivariate uniform distribution for the systematic effects are applied to generate random variates for the measurements and random variates for the measurements plus systematic effects. Monte Carlo estimates of the expectations and the covariance matrix of the measurements with systematic effects are computed. The densities for the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects are obtained by relative frequencies. They only depend on the size of the rectangular volume elements for which the densities are determined. The problem of sorting the densities is solved by sorting distances together with the densities. This allows a visualization of the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects.

To visualize the surface of an object, laser scanners determine the rectangular coordinates of points of a grid on the surface of the object in a local coordinate system. Vertical angles, horizontal angles and distances of a polar coordinate system are measured with the scanning. Outliers generally occur as gross errors in the distances. It is therefore investigated here whether rectangular or polar coordinates are better suited for the detection of outliers. The parameters of a surface represented by a polynomial are estimated in the nonlinear Gauss Helmert (GH) model and in a linear model. Rectangular and polar coordinates are used, and it is shown that the results for both coordinate systems are identical. It turns out that the linear model is sufficient to estimate the parameters of the polynomial surface. Outliers are therefore identified in the linear model by the expectation maximization (EM) algorithm for the variance-inflation model and are confirmed by the EM algorithm for the mean-shift model. Again, rectangular and polar coordinates are used. The same outliers are identified in both coordinate systems.

Best invariant quadratic unbiased estimates (BIQUE) of the variance and covariance components for a nonlinear Gauss Helmert (GH) model are derived. To detect outliers, the expectation maximization (EM) algorithm based on the variance-inflation model and the mean-shift model is applied, which results in an iterative reweighting least squares. Each step of the iterations for the EM algorithm therefore includes first the iterations for linearizing the GH model and then the iterations for estimating the variance components. The method is applied to fit a surface in three-dimensional space to the three coordinates of points measured, for instance, by a laser scanner. The surface is represented by a polynomial of second degree and the variance components of the three coordinates are estimated. Outliers are detected by the EM algorithm based on the variance-inflation model and identified by the EM algorithm for the mean-shift model.