Bayesian machine learning for measurement noise estimation of focus variation microscopy instrument

ISO 25178-600 is an international standard explaining metrological characteristics that are applied to calibrate the optical areal surface topography measuring instruments. The determination of measurement noise is typically one of the metrological characteristics to be estimated for calibrating areal surface topography measuring instruments. However, the metrological characteristic estimation, that is, specific to the vertical heights of the optical instrument stage, requires calibration time. Commonly, the estimation of measurement noise is only applied for a specific range from the total vertical range of an instrument and measurement parameters. The noise can significantly affect and increase the measurement uncertainty of measurements. Reducing measurement noise is crucial to reduce the overall measurement uncertainty and then improve the measurement reliability. Calibrating the measuring instruments can help quantify this noise contribution to the uncertainty and hence can develop good measurement practices on how to minimize or reduce the measurement noise. Machine learning (ML) can estimate measurement noise for uncalibrated vertical axis heights by using known measurement noise from a specific vertical range and specific measurement parameters. This process of machine learning estimation can significantly reduce the overall time of performing the measurement-noise calibration for all vertical axis ranges of optical measuring instruments. This paper applies Bayesian machine learning to predict the measurement noise of a focus variation microscopy (FVM) on overall height locations based on measurement noise estimations from a few specific height locations of the instrument. This method can reduce the overall measurement operational cost. Different measurement parameters are used to quantify the measurement noise and the prediction test with the proposed machine learning method. The parameters include vertical and lateral resolutions, image contrast, exposure time, objective lens, illumination type and measurement position of the vertical motion stage axis. The measurement noise estimation is performed by measuring a standard flat-surface artefact and applying a subtraction method. The Bayesian machine learning estimation is optimized by finding the optimal hyper-parameter of the Bayesian model. Results show that Bayesian machine learning can predict unknown measurement noise at different vertical heights and with different measurement parameters at the accuracy up to 94%. In addition, the uncertainty of prediction is provided for reliable comparison with known data.