A new arbitrary region coding algorithm using a b-splinesurface is presented. Pixels in the coded region are regarded as valid image data, and the parameters of the b-splinesurface are calculated under certain optim...
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A new arbitrary region coding algorithm using a b-splinesurface is presented. Pixels in the coded region are regarded as valid image data, and the parameters of the b-splinesurface are calculated under certain optimisation criteria. Instead of the traditional least squares error (LSE) criterion, a least squares weighted error (LSWE) criterion is used as the approximation criterion. Simulation results show that the LSWE approach is much better for region coding than the LSE approach.
In the computer-controlled optical surfacing (CCOS) process, the accurate calculation of the dwell time map is essential for achieving the desired amount of material removal distribution. However, it remains challengi...
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In the computer-controlled optical surfacing (CCOS) process, the accurate calculation of the dwell time map is essential for achieving the desired amount of material removal distribution. However, it remains challenging for current methods to generate a smooth dwell time map that reduces the dynamic requirements of machine tools while maintaining the high finishing capability on freeform optics. This paper proposes a b-spline surface approximation method for obtaining the optimum dwell time in deterministic polishing. The b-splinesurface representation allows the dwell time map to be expressed by a small number of control points instead of a large number of discrete dwell points. The dwell time deconvolution then becomes a linear least-squares problem with only a few control points as variables, which significantly reduces the scale of the problem. To achieve a more accurate and smoother dwell time map, a knot position optimization algorithm is proposed to adapt the knots to the geometric features of the target removal surface. Simulation results demonstrate that the proposed b-spline-based dwell time algorithm significantly outperforms common dwell time algorithms in terms of the computa-tional accuracy, efficiency, as well as smoothness of the dwell time map. To validate the effectiveness and practicability of the proposed method, experimental polishing of a bi-sinusoidal optical surface has been carried out. The results show that the developed algorithm has greatly reduced the tracking error of the machine tool caused by dwell time fluctuations, and consequently, the corresponding residual error of material removal in deterministic polishing.
b-splinesurfaces possess attractive properties such as a high degree of continuity or the local support of their basis functions. One of the major applications of b-splinesurfaces in engineering geodesy is the least...
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b-splinesurfaces possess attractive properties such as a high degree of continuity or the local support of their basis functions. One of the major applications of b-splinesurfaces in engineering geodesy is the least-square (LS) fitting of surfaces from, e.g., 3D point clouds obtained from terrestrial laser scanners (TLS). Such mathematical approximations allow one to test rigorously with a given significance level the deformation magnitude between point clouds taken at different epochs. Indeed, statistical tests cannot be applied when point clouds are processed in commonly used software such as CloudCompare, which restrict the analysis of deformation to simple deformation maps based on distance computation. For a trustworthy test decision and a resulting risk management, the stochastic model of the underlying observations needs, however, to be optimally specified. Since b-spline surface approximations necessitate Cartesian coordinates of the TLS observations, the diagonal variance covariance matrix (VCM) of the raw TLS measurements has to be transformed by means of the error propagation law. Unfortunately, this procedure induces mathematical correlations, which can strongly affect the chosen test statistics to analyse deformation, if neglected. This may lead potentially to rejecting wrongly the null hypothesis of no-deformation, with risky and expensive consequences. In this contribution, we propose to investigate the impact of mathematical correlations on test statistics, using real TLS observations from a bridge under load. As besides TLS, a highly precise laser tracker (LT) was used, the significance of the difference of the test statistics when the stochastic model is misspecified can be assessed. However, the underlying test distribution is hardly tractable so that only an adapted bootstrapping allows the computation of trustworthy p-values. Consecutively, the extent to which heteroscedasticity and mathematical correlations can be neglected or simplified with
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