Sascha Eichstädt

Deconvolution filters for the analysis of dynamic measurement processes: a tutorial

Analysis of dynamic measurements is of growing importance in metrology as an increasing number of applications requires the determination of measurands showing a time-dependence. Often linear time-invariant (LTI) systems are appropriate for modelling the relation between the available measurement data and the required time-dependent values of the measurand. Estimation of the measurand is then carried out by deconvolution.This paper is a tutorial about the application of digital deconvolution filters to reconstruct a time-variable measurand from the measurement signal of a LTI measurement apparatus. The goal of the paper is to make metrologists aware of the potentialities of digital signal processing in such cases. A range of techniques is available for the construction of a digital deconvolution filter. Here we compare various approaches for a form of dynamic model that is relevant to many metrological applications and we discuss the consequences for these approaches of the different ways in which information about the LTI system may be expressed. We consider specifically the methods of minimum-phase all pass decomposition, asynchronous time reversal using the exact inverse filter and the construction of stable infinite impulse response and finite impulse response approximate inverse filters by a least squares approach in the frequency domain. The methods are compared qualitatively by assessing their numerical complexity and quantitatively in terms of their performance for a simulated measurement task.Taking into account numerical complexity and underlying assumptions of the methods, we conclude that when a continuous model of the LTI system is available, or when the starting point is a set of measurements of the frequency response of a system, application of least squares in the frequency domain for the construction of an approximate inverse filter is to be preferred. On the other hand, asynchronous time reversal filtering using the exact inverse filter appears superior when a discrete model of the LTI system is available and when causality of the deconvolution filter is not an issue.

Efficient implementation of a Monte Carlo method for uncertainty evaluation in dynamic measurements

Measurement of quantities having time-dependent values such as force, acceleration or pressure is a topic of growing importance in metrology. The application of the Guide to the Expression of Uncertainty in Measurement (GUM) and its Supplements to the evaluation of uncertainty for such quantities is challenging. We address the efficient implementation of the Monte Carlo method described in GUM Supplements 1 and 2 for this task. The starting point is a time-domain observation equation. The steps of deriving a corresponding measurement model, the assignment of probability distributions to the input quantities in the model, and the propagation of the distributions through the model are all considered. A direct implementation of a Monte Carlo method can be intractable on many computers since the storage requirement of the method can be large compared with the available computer memory. Two memory-efficient alternatives to the direct implementation are proposed. One approach is based on applying updating formulae for calculating means, variances and point-wise histograms. The second approach is based on evaluating the measurement model sequentially in time. A simulated example is used to compare the performance of the direct and alternative procedures.

Comparison of the Richardson–Lucy method and a classical approach for spectrometer bandpass correction

Bandpass correction in spectrometer measurements using monochromators is often necessary in order to obtain accurate measurement results. The classical approach of spectrometer bandpass correction is based on local polynomial approximations and the use of finite differences. Here we compare this approach with an extension of the Richardson–Lucy method, which is well known in image processing, but has not been applied to spectrum bandpass correction yet. Using an extensive simulation study and a practical example, we demonstrate the potential of the Richardson–Lucy method. In contrast to the classical approach, it is robust with respect to wavelength step size and measurement noise. In almost all cases the Richardson–Lucy method turns out to be superior to the classical approach both in terms of spectrum estimate and its associated uncertainties.

Mathematical modelling to support traceable dynamic calibration of pressure sensors

This paper focuses on the mathematical modelling required to support the development of new primary standard systems for traceable calibration of dynamic pressure sensors. We address two fundamentally different approaches to realizing primary standards, specifically the shock tube method and the drop-weight method. Focusing on the shock tube method, the paper presents first results of system identification and discusses future experimental work that is required to improve the mathematical and statistical models. We use simulations to identify differences between the shock tube and drop-weight methods, to investigate sources of uncertainty in the system identification process and to assist experimentalists in designing the required measuring systems. We demonstrate the identification method on experimental results and draw conclusions.

Dynamic Uncertainty for Compensated Second-Order Systems

The compensation of LTI systems and the evaluation of the according uncertainty is of growing interest in metrology. Uncertainty evaluation in metrology ought to follow specific guidelines, and recently two corresponding uncertainty evaluation schemes have been proposed for FIR and IIR filtering. We employ these schemes to compare an FIR and an IIR approach for compensating a second-order LTI system which has relevance in metrology. Our results suggest that the FIR approach is superior in the sense that it yields significantly smaller uncertainties when real-time evaluation of uncertainties is desired.

On challenges in the uncertainty evaluation for time-dependent measurements

The measurement of quantities with time-dependent values is a common task in many areas of metrology. Although well established techniques are available for the analysis of such measurements, serious scientific challenges remain to be solved to enable their routine use in metrology. In this paper we focus on the challenge of estimating a time-dependent measurand when the relationship between the value of the measurand and the indication is modeled by a convolution. Mathematically, deconvolution is an ill-posed inverse problem, requiring regularization to stabilize the inversion in the presence of noise. We present and discuss deconvolution in three practical applications: thrust-balance, ultra-fast sampling oscilloscopes and hydrophones. Each case study takes a different approach to modeling the convolution process and regularizing its inversion. Critically, all three examples lack the assignment of an uncertainty to the influence of the regularization on the estimation accuracy. This is a grand challenge for dynamic metrology, for which to date no generic solution exists. The case studies presented here cover a wide range of time scales and prior knowledge about the measurand, and they can thus serve as starting points for future developments in metrology. The aim of this work is to present the case studies and demonstrate the challenges they pose for metrology.

Improved estimation of reflectance spectra by utilizing prior knowledge.

Estimating spectral reflectance has attracted extensive research efforts in color science and machine learning, motivated through a wide range of applications. In many practical situations, prior knowledge is available that ought to be used. Here, we have developed a general Bayesian method that allows the incorporation of prior knowledge from previous monochromator and spectrophotometer measurements. The approach yields analytical expressions for fast and efficient estimation of spectral reflectance. In addition to point estimates, probability distributions are also obtained, which completely characterize the uncertainty associated with the reconstructed spectrum. We demonstrate that, through the incorporation of prior knowledge, our approach yields improved reconstruction results compared with methods that resort to training data only. Our method is particularly useful when the spectral reflectance to be recovered resides beyond the scope of the training data.

Linear Mixed Models: Gum and Beyond

Abstract In Annex H.5, the Guide to the Evaluation of Uncertainty in Measurement (GUM) [1] recognizes the necessity to analyze certain types of experiments by applying random effects ANOVA models. These belong to the more general family of linear mixed models that we focus on in the current paper. Extending the short introduction provided by the GUM, our aim is to show that the more general, linear mixed models cover a wider range of situations occurring in practice and can be beneficial when employed in data analysis of long-term repeated experiments. Namely, we point out their potential as an aid in establishing an uncertainty budget and as means for gaining more insight into the measurement process. We also comment on computational issues and to make the explanations less abstract, we illustrate all the concepts with the help of a measurement campaign conducted in order to challenge the uncertainty budget in calibration of accelerometers.