Paul J. Scott

Paradigm shifts in surface metrology. Part II. The current shift

This is the second part of the paper ‘Paradigm shifts in surface metrology’. In part I, the three historical paradigm shifts in surface metrology were brought together, and the subsequent evolution resulting from the shifts discussed. The historical philosophy highlighted the fact that the paradigm shifts must be robust and flexible, meaning that surface metrology must allow for full control of surface manufacture and provide an understanding of the surface functional performance. Part II presents the current paradigm shift as a ‘stepping stone’, building on the above historical context. Aspects of surface geometry will also have to cater for surfaces derived from disruptive application, i.e. structured and freeform surfaces are identified candidates. The current shift is presented in three aspects: from profile to areal characterization; from stochastic to structured surfaces; and from simple geometries to complex freeform geometries, all spanning the millimetre to sub-nanometre scales. In this paradigm shift, the scale of surface texture is beginning to approach some of the geometrical features in micro/nano electro-mechanical systems devices and is becoming one of the most important functionality indicators. Part II will contextualize the current shifts in the discipline of surface metrology, and cement surface metrology in place in the ultra precision and nanotechnology age.

Paradigm shifts in surface metrology. Part I. Historical philosophy

Surface texture and its measurement are becoming the most critical factors and important functionality indicators in the performance of high precision and nanoscale devices and components. Surface metrology as a discipline is currently undergoing a huge paradigm shift: from profile to areal characterization, from stochastic to structured surfaces, and from simple geometries to complex free-form geometries, all spanning the millimetre to sub-nanometre scales. This paper builds a complete philosophical framework for surface metrology through a review of the paradigm shifts that have occurred in the discipline of surface metrology, tracing the development of fundamental philosophies and techniques. The paper starts with a brief overview of the historical paradigm shifts and builds an up-to-date foundational philosophy, capable of rapid and effective development. The growth in interest in surface metrology stems mainly from the need to control the manufacture of armaments during the Second World War and the production of domestic goods and appliances since that time. The surfaces produced by manufacture seemed to offer the possibility of being useful for process control. Unfortunately, only a few tentative investigations had been carried out to establish usable relationships between the processes, the machine tools and the available surface parameters (with their limitations). Even fewer investigations had been carried out to relate surface geometry to the performance of manufactured products. The result was that the metrology was unprepared and, consequently, the progress was sporadic. This overall review is given in two parts. Part I focuses on the historical philosophy of surface metrology and Part II discusses the progress within the current paradigm shift.

Pattern analysis and metrology: the extraction of stable features from observable measurements

To extract patterns from observable measurements we need to be able to define and identify stable features in observable measurements that persist in the presence of small artificial features such as noise, measurement errors, etc. The representational theory of measurement is used to define the stability of a measurement procedure. A technique, ‘motif analysis’, is defined to identify and remove ‘insignificant’ features while leaving ‘significant’ features. This technique is formalized and three properties identified that ensure stability. The connection of motif analysis with morphological closing filters is established and used to prove the stability of motif analysis. Finally, a practical metrology example is given of motif analysis in surface texture. Here motif analysis is used to segment a surface into its significant features.

Template matching of freeform surfaces based on orthogonal distance fitting for precision metrology

Freeform surfaces are widely used in advanced optical and mechanical devices. In order to assess the form quality of a freeform surface, it is required to match the measurement surface with the design template. To improve the matching efficiency and accuracy, the whole procedure is divided into two stages: rough matching and final fitting. A new rough matching method, called the structured region signature, is proposed. The structured region signature is a generalized global feature which represents the surface shape by a one-dimensional function. The template location occupying the best matching signature is considered to be the correct rough position of the measurement surface. After that the motion parameters are updated iteratively based on the orthogonal distance fitting. The dependence between the foot-point parameters of the projection points and the motion parameters is derived from the closest-distance relationship between correspondence point pairs. Numerical experiments are given to demonstrate the validity of this approach.

An algorithm to extract critical points from lattice height data

The data measured on areal surface texture instruments often takes the form of height values on a discrete rectangular lattice. The definitions of critical points (peak, pit and saddle points) for this type of data is a very important issue for definitions and stable algorithms of areal texture parameters. Many of the published algorithms for critical points fail to meet some very simple topological properties which are always true for continuous data. Thus the collection of critical points produced from these published algorithms could not have come from a genuine continuous surface giving difficulties with interpretation of the results. An algorithm for critical points from lattice height data is presented which satisfies these topological properties and represents genuine features from the measured surface.

Freeform surface filtering using the diffusion equation

The measurement of texture for geometric surfaces is well established for surfaces that are of a planar (Euclidean) nature. Gaussian filtering is the fundamental base for scale-limited surfaces used in surface texture, but cannot be applied to non-Euclidean surfaces without distortion of the results. A link exists between Gaussian filtering and solutions of the PDE that models linear isotropic diffusion. In particular, an analytical solution of this diffusion equation over a planar region at a time t is given by the continuous convolution of the initial distribution of the diffused quantity with a Gaussian function of standard deviation . A practical implementation of the standard Gaussian filter on sampled data can be viewed as a discretization of this process. On a non-Euclidean surface, the diffusion equation is formulated by using the Laplace–Beltrami operator. Using this generalization, a method of Gaussian filtering for freeform surface data is proposed by solving the diffusion equation for approximation residuals defined on a freeform least-squares approximation of the measurement surface data. Results of the application of these methods to simulated and experimental data are presented.

Fast algorithm of the robust Gaussian regression filter for areal surface analysis

In this paper, the general model of the Gaussian regression filter for areal surface analysis is explored. The intrinsic relationships between the linear Gaussian filter and the robust filter are addressed. A general mathematical solution for this model is presented. Based on this technique, a fast algorithm is created. Both simulated and practical engineering data (stochastic and structured) have been used in the testing of the fast algorithm. Results show that with the same accuracy, the processing time of the second-order nonlinear regression filters for a dataset of 1024*1024 points has been reduced to several seconds from the several hours of traditional algorithms.

Intelligent sampling for the measurement of structured surfaces

Uniform sampling in metrology has known drawbacks such as coherent spectral aliasing and a lack of efficiency in terms of measuring time and data storage. The requirement for intelligent sampling strategies has been outlined over recent years, particularly where the measurement of structured surfaces is concerned. Most of the present research on intelligent sampling has focused on dimensional metrology using coordinate-measuring machines with little reported on the area of surface metrology. In the research reported here, potential intelligent sampling strategies for surface topography measurement of structured surfaces are investigated by using numerical simulation and experimental verification. The methods include the jittered uniform method, low-discrepancy pattern sampling and several adaptive methods which originate from computer graphics, coordinate metrology and previous research by the authors. By combining the use of advanced reconstruction methods and feature-based characterization techniques, the measurement performance of the sampling methods is studied using case studies. The advantages, stability and feasibility of these techniques for practical measurements are discussed.

The mathematics of motif combination and their use for functional simulation

Abstract The concept of Motif Combination was introduced in the French Automotive industrys R&W standard. There has however been no formal mathematical theory for the technique of motif combination. This paper sets out to develop this theory. The main result of the paper is a set of four properties that the motif combination rules must satisfy in order to have certain desirable metrological properties. Motif methods may well provide useful techniques which complement rather than replace the traditional approach based on a mean line. It was found that many functional problems could be simulated very efficiently using motif methods. Examples of functional simulation via motif combination are illustrated in the paper.

Morphological method for surface metrology and dimensional metrology based on the alpha shape

Morphological filters are useful tools as they are commonly employed in surface metrology and dimensional metrology, serving for surface texture analysis and data smoothing, respectively. Compared to the mean-line filtering techniques, such as the Gaussian filter, morphological filters have the merits of compact support, no need to remove form and being relevant to geometrical properties of surfaces. This paper proposes a novel morphological method based on the alpha shape. The proposed method has the advantages over the traditional methods that it runs relatively fast, enables arbitrary large ball radii and applies to freeform surfaces and nonuniformly sampled surfaces. The theory of basic morphological operations and the alpha shape is introduced and the theoretical link between the alpha hull and the morphological closing and opening operation is presented. A practical algorithm is developed that corrects possible singularities caused by data spikes and reduces the amount of calculation for open profiles/surfaces. Computer simulation is used to compare the results from the traditional algorithm and the proposed one. Experimental studies are conducted to demonstrate the feasibility and applicability of using the proposed method.