Comparison of 3D Surface Reconstruction Data from Certified Depth Standards Obtained by SEM and an Infinite Focus Measurement Machine (IFM)

The Scanning Electron Microscope (SEM) is an excellent tool for the characterization of surfaces. In order to evaluate the accuracy of the reconstruction of a 3D dataset, a known depth standard for tactile depth measurement devices with a step height of 92 μm was investigated by SEM. The reconstruction of surfaces using two tilted images (StereoCreator) or three different tilted images (TriCreator) is affected by the tilt angle itself, the image resolution and the working distance. We compare the results of the obtained data from different combinations of tilt angles and working distances in relation with the type of reconstruction…. To compare the accuracy and reliability of the 3D datasets from the new Infinite Focus Measurement Machine (IFM) with the datasets reconstructed from the well known SEM technique, we utilized a groove standard with six trapezoidal grooves with known width and depth certified by the PTB – Braunschweig – Germany (Physikalisch – Technische Bundesanstalt Braunschweig – calibration mark 035PTB04 – measured with a traceable stylus instrument according to DIN EN ISO 3274). The great advantage of the SEM/ESEM (Environmental Scanning Electron Microscope) and IFM technique compared to tactile-profile-measurement instruments is that it is a non-tactile and a non-destructive method. The potential and accuracy of these methods in comparison to tactile measurement systems is shown and discussed in this paper.     

A dual open coloring axiom

We discuss a dual of the Open Coloring Axiom introduced by Abraham et al. [U. Abraham, M. Rubin, S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ₁-dense real order types, Ann. Pure Appl. Logic 29 (2) (1985) 123–206] and show that it follows from a statement about continuous colorings on Polish spaces that is known to be consistent. We mention some consequences of the new axiom and show that implies that all cardinal invariants in Cichoń’s diagram are at least ₂.     

The Construction of cubature rules for multivariate highly oscillatory integrals

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule.

The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.