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The moving least squares (MLS) method has been developed for the fitting of measured data contaminated with random error. The local approximants of MLS method only take the error of dependent variable into account, whereas the independent variable of measured data always contains random error. Considering the errors of all variables, this paper presents an improved moving least squares (IMLS) method to generate curve and surface for the measured data. In IMLS method, total least squares (TLS) with a parameterλbased on singular value decomposition is introduced to the local approximants. A procedure is developed to determine the parameterλ. Numerical examples for curve and surface fitting are given to prove the performance of IMLS method.

The Moving Least Squares (MLS) method has been developed for fitting of the measurement data contaminated with errors. The local approximants of the MLS method only take the random errors of the dependent variable into account, whereas the independent variables of measurement data always contain errors. To consider the influence of errors of dependent and independent variables, the Moving Total Least Squares (MTLS) offers a better choice. However, both MLS and MTLS method are sensitive to outliers, which greatly affects the fitting accuracy and robustness. This paper presents an improved method-Trimmed Moving Total Least Squares (TrMTLS) method, in which Total Least Squares (TLS) method with truncation procedure is adopted to determine the local coefficients in the influence domain. This method can deal with outliers and random errors of all variables without setting the threshold or adding small weights subjectively. The numerical simulation and measurement experiments results indicate that the proposed algorithm has better fitting accuracy and robustness compared with the MTLS and MLS method.

The total least square method based on singular value decomposition for fitting straight line and plane surface has been developed to deal with the straightness calibration problem. Different from the ordinary least square method only taking into account the error of the dependent variable, total least square method considers the errors of all the variables in a symmetrical way. However, in practice, it is difficult to choose an optimal method for the variable errors of measurement data in an asymmetric way. This article presents an improved calibration method for straightness error of a coordinate measuring machine. The proposed method, named as improved total least square, could fit straight line and plane surface when the variables are in an asymmetric way. In improved total least square method, weight matrices with parameter λ set between the independent and dependent variables are introduced to augmented matrix. A procedure is developed to determine the parameter λ. Numerical cases and measurement experiment are given to prove the performance of improved total least square method.

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