Generalized form characterization of ultra-precision freeform surfaces using an invariant feature-based pattern analysis

Abstract

Ultra-precision freeform surfaces are complex surfaces that possess non-rotational symmetry and are widely used in many industries, such as advanced optics and biomedical implants, due to their superior optical and mechanical properties. In view of the geometrical complexity of freeform surfaces, there is no international standard for the traceable measurement and characterization of machined ultra-precision freeform surfaces with sub-micrometre form accuracy and nanomatric surface finishing. Motivated by the need for such a standard, this thesis presents an Invariant Feature-based Pattern Analysis Method (IFPAM) for the generalized form characterization of ultra-precision freeform surfaces. The IFPAM makes use of intrinsic surface properties, such as Gaussian curvature, to map the surface into a special image to form an orientation invariant feature pattern (IFP) for the representation of the surface geometry. The digital image processing techniques are then employed to conduct the IFP registration and correspondence searching for the form characterization of the surface. Compared with traditional freeform form characterization methods, such as least squares or minimum zone methods, the IFPAM is not only independent of the type of the surface being characterized but also from the coordinate frame which brings many difficulties and uncertainties for the characterization of freeform surfaces. The calculation of the intrinsic surface features from a machined freeform surface is susceptible to the sampling strategy and the measurement noise and outliers associated in the measured data. To address these problems, a bidirectional curve network based sampling strategy (BCNSS) combined with a robust surface fitting and reconstruction algorithm (RSFRA) are developed for ensuring accurate extraction of the intrinsic surface features from a machined freeform surface. The BCNSS is based on scanning two sets of curves on the measured surface along two different directions to form a curve network which is used to construct a substitute surface to represent the measured surface. The RSFRA is developed to reconstruct a high fidelity surface from measured discrete points while the surface smoothness can be ensured as well. A fitting threshold, named the confidence interval of fitting error, is used to strike a balance between fitting accuracy and surface smoothness in the fitting process. Experimental study confirms that the BCNSS and RSFRA provide an effective means for the improvement of the efficiency in data sampling and in increasing the accuracy of the surface representation for the measurement of ultra-precision freeform surfaces. To access the reliability and accuracy of the IFPAM, a task specific uncertainty analysis model is built based on a Monte Carlo method to estimate the uncertainty associated in the results of the form characterization of ultra-precision freeform surfaces. Three influential factors are identified and considered in the model, including measurement error, surface form error, and sample size. Fractional Brownian motion is used to quantify the random surface form error while the measurement error is modeled by multivariable random noise. Rather than relying on intuition, the study is more focused on mathematical modeling of the relationship between the influential factors and the resulted uncertainty so that a prediction can be made to estimate the uncertainty in the form characterization of a specific freeform surface. The developed uncertainty analysis model is helpful for control and optimization of the IFPAM so as to provide more reliable form characterization results. The IFPAM substantially addresses the deficiencies and limitations of traditional freeform surface characterization methods, which are more susceptible to embedded coordinated systems and possess uncertainty due to the geometrical complicity and form variety of freeform surfaces. The outcome of this study not only significantly contributes to the state-of-the-art of measurement science and technology but also provides approaches that can be used in the standardization of measurement and characterization of freeform surfaces.