Root canal therapy is the most fundamental and effective approach for treating endodontics and periapicalitis. The length of the root canal must be accurately measured to clean the pathogenic substances in it. This study aims to present a multifrequency impedance method based on a neural network for root canal length measurement. A circuit system was designed which generates a current of frequencies from 100 Hz to 20 kHz in order to augment the data of impedance ratios with different combinations of frequencies. Several impedance ratios and other quantified characteristics, such as the type of tooth and file, were selected as features to train a neural network model that could predict the distance between the file and apical foramen. The model uses leave-one-out cross-validation, adopts the Adam optimizer and regularization, and has two hidden layers with nine and five nodes, respectively. The neural network-based multifrequency impedance method exhibits nearly 95% accuracy, compared with the dual-frequency impedance ratio method (which demonstrated no more than 85% accuracy in some situations). This method may eliminate the influence of human and environmental factors on measurement of the root canal length, thereby increasing measurement robustness.

A laser Fizeau interferometer measures surface form by comparing the wavefronts reflected from a reference surface and the surface under test. The measurement accuracy is limited by the unknown absolute form error of the reference flat. Absolute testing offers a strategy to break through the limitation. In this work, we propose a three-flat absolute testing method using minimum norm least squares solutions (MNLS). After acquiring three measurement sets from pairwise combinations among the flats, one flat is rotated by 90° or 180° for the fourth measurement. The simulation demonstrates the effectiveness of the proposed MNLS method and its robustness against rotational angle errors. Our absolute testing method is experimentally validated using a home-built and a commercial Fizeau interferometers. Correcting the absolute forms of the home-built Fizeau interferometer’s reference flat improves its measurement accuracy. The experimental outcomes of the MNLS absolute testing technique are compared with two alternative absolute testing methods. The MNLS method’s advantage is demonstrated.

As one of the core components of ultra-precision machining and inspection equipment, the 2D stage plays a major role in sophisticated technology. Self-calibration is considered an excellent method to separate stage system errors because of its low cost and high efficiency. In this paper, the self-calibration model is represented using an overdetermined equations system. A weight matrix was used to solve the system through least squares with unequal accuracy. The error propagation ratios increase with a decrease in the number of positions or increase in the number of points, however, all the ratios are less than one. Self-calibration based on weighted least squares can suppress the noise, which was proved through simulation. Experiments were conducted to confirm that self-calibration separated the stage system errors successfully. The effectiveness and repeatability of the method were confirmed by the fact that the standard deviation among the plates returning to the initial position was of tens of nanometers. Two methods of self-calibration, namely self-calibration based on least squares and optimization self-calibration, were verified against each other. The differences between the stage system errors separated in the two methods were only a few nanometers.

Two-dimensional (2D) self-calibration is more suitable for ultra-precision engineering than conventional calibration, as it does not require higher-precision device calibration. A grid plate is translated or rotated in various positions on a stage, and the translation and rotation can be combined into a hybrid position. Using 2D self-calibration, the stage and plate errors are separated from the overall measurement errors obtained by measuring the grid plate on the stage. During this process, random noise propagates to the separated stage errors through the self-calibration model, causing propagation of the uncertainty of the errors. Accordingly, in this study, least squares-based 2D self-calibration was investigated. An overdetermined linear equation system was established based on the relationships among the variables for self-calibration. The Guide to the Expression of Uncertainty in Measurement was used to calculate the uncertainty propagation ratio after obtaining the least squares solutions for the self-calibration model, and the uncertainty was further analyzed using the Monte Carlo method. The uncertainty propagation ratios were less than 1, indicating that the self-calibration model had a favourable noise-suppression ability. The robustness of this self-calibration approach was mathematically determined. The effects of various position schemes (with and without a hybrid position) on the uncertainty propagation ratio were examined to support the design optimisation of the self-calibration program. The experiments revealed that the use of hybrid position schemes was advantageous over not using such schemes, although the self-calibration performance was comparable under both types of schemes.

The lateral distortion of a surface measuring Fizeau interferometer may cause distorted image features in the lateral direction, as well as the surface form error in the axial direction (which is a source of the retrace error). Traditional method for lateral distortion measurement requires a high-accuracy calibration plate featuring a grid pattern. Such a calibration plate is not always available, especially when the required accuracy of the grid pattern comes to the order of sub-micrometer or even nanometer level. To remove the dependence on the plate accuracy, we propose a self-calibration method for the measurement and correction of lateral distortion in Fizeau interferometer. The self-calibration technique may separate the lateral distortion and the geometric error of the calibration plate. This method is verified using a 108-mm-aperture Fizeau interferometer. The experiments show that the form measurement error of a surface tilted at approximately 5° and 16° can be reduced from 92 nm to 43 nm and from 251 nm to 144 nm (peak-to-valley value), respectively, after the distortion correction.

Three-dimensional self-calibration for high-precision 3-D measurement instruments can separate the system errors using an auxiliary artifact, with accuracy comparable to that of the instrument to be calibrated, having significant advantages in terms of production costs and realization conditions. The auxiliary artifact is rotated and/or translated to different positions. From measurements of the artifact in various positions, the system errors of the measurement instrument are separated after the elimination of the artifact’s errors through the self-calibration model. In this study, an iterative optimization method in 2-D self-calibration was extended to 3-D and compared with an established 3-D self-calibration method, namely the equation method based on least squares. The optimization method involves an optimization of 3-D instrument to eliminate its system errors by a correction function during an iterative loop. The equation method is used to establish a 3-D self-calibration model and solve the overdetermined equation model by least squares. The uncertainties of these two methods were analyzed using different approaches. The two methods were compared by simulation with the separated system errors and the effect of calibration when the artifact position scheme changed. Experiments on a computed tomography (CT) instrument were conducted to confirm the effectiveness of the two methods of 3-D self-calibration, through the mutual verification of results.