Gaussian process regression‐based detection and correction of disturbances in surface topography measurements

Abstract

Modern smart and intelligent manufacturing is characterised by an increasing use of highly engineered surfaces and quasi‐free form geometries, for example, by additive manufacturing, and the requirement for fast and informative measurement tools for quality controls in production. These have lately pulled towards the adoption of optical surface topography measuring instruments to qualify technological surfaces, which are core to being assessed to characterise the product and optimise the manufacturing process. Surface topography measurements performed by optical instruments are known to suffer from disturbances, such as non‐measured points and spikes. These are due to complex interactions between the measurand and the instrument and may result in poor measurement quality and biased characterisation results. Currently, the correction of measurement disturbances is carried out by empirical approaches which mostly involve thresholding and interpolations, whose sensitivity is often devolved to the operator. In this work, a Gaussian process regression‐based approach is outlined to identify and correct measurement disturbances exploiting spatial correlation properties of the measurements to provide a formal, robust, and univocally defined approach to manage measurement disturbances. The proposed approach, differently from currently available alternatives, is capable of managing at once the different type of disturbances by means of a supervised machine learning technique. The formal and practical advantages of the proposed method are discussed exploiting case studies of industrial interest applied on quasi‐flat surfaces to stress the disturbances effect.

Modeling the systematic behavior at the micro and nano length scales

“…A discussion on the effects of the measurement noise is beyond the scope of this work. While an innovative approach can be found elsewhere [60], the aim is to advise on the application of methods that are consolidated in many fields of metrology, although not verified and validated with micrographs acquisition due to the complexity of the operation of the instruments involved.…”

Optimisation of laser welding of deep drawing steel for automotive applications by Machine Learning: A comparison of different techniques

“…The solution, which minimises the mean squared prediction error, assumes the variable $$Y(\bm{x})={\bm{\beta}}^{T}\bm{f}(\bm{x})$$+Ψ( x ) consists of a regression, in particular a linear combination of m function f ( x ) with linear combination parameters β , and the spatially correlated regression error $$\mathrm{\Psi}(\bm{x})\sim N(0,{\sigma}_{Y}^{2}\bm{R}(\bm{h};\bm{\theta}))$$, having $$\bm{R}(\bm{h};\bm{\theta})$$ the correlation matrix dependent on the distance h and a set of parameters θ . The correction term depends on the residuals, with F is the matrix with entries $${F_{ik}} = {f_k}\;( {{x_i}} ),\;i \in \{ {1,2, \ldots ,n} \},\;k \in \{ {1,2, \ldots ,m} \}$$ and Y n the training set with n data, weighted by the correlation, let $${\bm{r}}_{0}=(\bm{R}({\bm{x}}_{0}-{\bm{x}}_{1}),\text{\ensuremath{\cdots}},\bm{R}({\bm{x}}_{0}-{\bm{x}}_{n}))^{T}$$ 39,45,46 . When training a GP model, it is crucial to determine the regression model F and the correlation parameters.…”

“…The correction term depends on the residuals, with F is the matrix with entries 𝐹 𝑖𝑘 = 𝑓 𝑘 (𝑥 𝑖 ), 𝑖 ∈ {1, 2, … , 𝑛}, 𝑘 ∈ {1, 2, … , 𝑚} and 𝒀 𝑛 the training set with n data, weighted by the correlation, let 𝒓 0 = (𝑹(𝒙 0 − 𝒙 1 ), ⋯, 𝑹(𝒙 0 − 𝒙 𝑛 )) 𝑇 . 39,45,46 When training a GP model, it is crucial to determine the regression model F and the correlation parameters. Thanks to the presence of the corrective term in the prediction, a possible solution is the ordinary kriging, which includes only a constant term rather than a linear combination of trend functions.…”

“…Gaussian Process Regression (GPR) is a stochastic regression method for interpolating and inferring models in sparse datasets, that is, investigating large portion of the domain in a non-necessarily structured nor densely filled way. 39,45,46 It relies on the assumption that the model response, that is, the prediction, depends on the correlation between the model response at two different evaluation points, and that the correlation is a function of the distance h between these evaluation points. In particular, under this assumption, the GP prediction can be written as:…”