Roman Z. Morawski

Incorporation of a positivity constraint into a Kalman-filter-based algorithm for correction of spectrophotometric data

The results of spectrophotometric measurements are subject to systematic errors of instrumental type which may be partially corrected provided a mathematical model of the instrumental imperfections is identified. It is assumed that this model has the form of an integral, convolution-type equation of the first kind. The correction of the results of the measurements subject to random measurement errors consists in the numerical solution of this equation on the basis of these results. A new method for solving the problem of correction is proposed; it is based on application of the Kalman filter modified in such a way that the negative values of the solution are prohibited. The efficiency of this regularization method is demonstrated. It is studied using both synthetic and real data. >

Unified approach to measurand reconstruction

The basic notions of measurement science are overviewed. A general scheme of measurement is proposed that emphasizes the key role of measurand reconstruction. The problems of measurand reconstruction are classified. Two classes of them, being of particular importance in practical applications, are identified and discussed in more detail. These are the nonlinear reconstruction of a scalar static measurand in the presence of a scalar influence quantity, and the linear reconstruction of a scalar dynamic measurand. Considerations of a tutorial and theoretical nature are illustrated with practical examples. >

Spectrophotometric applications of digital signal processing

Spectrophotometry is more and more often the method of choice not only in analysis of (bio)chemical substances, but also in the identification of physical properties of various objects and their classification. The applications of spectrophotometry include such diversified tasks as monitoring of optical telecommunications links, assessment of eating quality of food, forensic classification of papers, biometric identification of individuals, detection of insect infestation of seeds and classification of textiles. In all those applications, large numbers of data, generated by spectrophotometers, are processed by various digital means in order to extract measurement information. The main objective of this paper is to review the state-of-the-art methodology for digital signal processing (DSP) when applied to data provided by spectrophotometric transducers and spectrophotometers. First, a general methodology of DSP applications in spectrophotometry, based on DSP-oriented models of spectrophotometric data, is outlined. Then, the most important classes of DSP methods for processing spectrophotometric data—the methods for DSP-aided calibration of spectrophotometric instrumentation, the methods for the estimation of spectra on the basis of spectrophotometric data, the methods for the estimation of spectrum-related measurands on the basis of spectrophotometric data—are presented. Finally, the methods for preprocessing and postprocessing of spectrophotometric data are overviewed. Throughout the review, the applications of DSP are illustrated with numerous examples related to broadly understood spectrophotometry.

The use of deconvolution and iterative optimization for spectrogram interpretation

The problem of spectrogram interpretation is considered under the assumption that the parameters of spectral peaks-their positions and magnitudes-contain the information essential for spectrometric analysis. The subsequent use of Tikhonov deconvolution and iterative correction of the estimates of those parameters is proposed. Deconvolution is used for transforming the processed spectrogram in such a way as to facilitate finding initial estimates of its parameters. The advantages of the proposed approach, i.e., gains in accuracy of estimating the parameters of peaks, are demonstrated using both synthetic and real-world spectrophotometric data.

Unified approach to measurement signal reconstruction

Abstract Any measurement process may be logically decomposed into two parts: conversion and reconstruction. The first one comprises transformation of a signal to be measured, x , into a raw result of measurement, y , which represents x in the domain of ‘easily interpretable’ phenomena, such as electrical signals and digital codes. Having an adequate mathematical model of the relationship between x and y , one can attempt to reconstruct x . Since such a numerical task is ill-conditioned, it requires regularisation. In the paper, a systematisation of the methods of reconstruction is proposed, based on the mechanism of regularisation as the criterion. It is shown that such an approach is productive both in the domain of theory and application of measurement signal reconstruction.

Combined use of Tikhonov deconvolution and curve fitting for spectrogram interpretation

Abstract The problem of numerical correction of spectrograms is addressed. A new method of correction is developed which consists of sequential use of the Tikhonov deconvolution algorithm, for estimating the positions of spectral peaks, and a curve-fitting algorithm, for estimating their magnitudes. The metrological and numerical properties of the proposed method for spectrogram interpretation are assessed by means of spectrometry-based criteria, using synthetic and real-world spectrograms. Conclusions are drawn concerning computational complexity and accuracy of the proposed method and its metrological applicability.

Kalman-filter-based algorithms of spectrometric data correction-Part I: an iterative algorithm of deconvolution

This series of two papers aims to present the different solutions of the problem of improving the resolution of spectrometric measurements via numerical processing of spectrometric data subject both to systematic instrumental errors and to random measurement errors. It is assumed that the model of the spectrometric data has the form of a convolution-type equation of the first kind. The method for improving the resolution consists in numerically solving this equation using the acquired data. In this first paper of the series, an algorithm of correction is proposed which is based on the iterative use of the Kalman filter incorporating a non-negativity constraint. Its applicability to the problem of correction is assessed not only from a purely metrological point of view (accuracy, resolution) but also with respect to its suitability for implementation as a VLSI processor dedicated to measuring systems. For this latter reason a time-invariant model of the data and a steady-state version of the Kalman filter is used. The efficiency of this approach to correction is demonstrated using both synthetic and real-world data.

Dynamic calibration of measurement channels using algorithms of nondifferentiable optimization

The problem of dynamic calibration of measurement channels is considered under an assumption that the relationship between the measurand and the raw result of measurement is adequately modeled by a linear, fixed-parameter difference equation. The parameters of this model are estimated on the basis of the reference data using an algorithm of nondifferentiable optimization and criteria of the quality of calibration defined in the domain of the measurand. The results of calibration are evaluated using the errors of the results of reconstruction performed by means of the model resulting from calibration. Some conclusions on the metrological applicability of the proposed approach are formulated. >

Calibration of a spectrometer using a genetic algorithm

This paper is on digital processing of data acquired by means of low-resolution spectrometers. A procedure for calibration of such spectrometers is proposed which provides parameters of operators for correcting effects of imperfect spectrometric hardware. The procedure, based on the use of a genetic algorithm of global optimization, is studied using both synthetic and real-world data, viz, it is used for estimating parameters of three nonlinear operators of data correction: a rational filter (published in a previous paper), a Cauchy filter (CF) (proposed in this paper), and a superposition of two Cauchy filters. The results of study are presented that confirm the practical usefulness of the proposed procedure for calibration of spectrometers and of the proposed Cauchy filter.

Numerical correction of spectrometric data using a rational filter

Raw spectrometric data are subject to systematic errors of an instrumental type that may be reduced provided a mathematical model of the spectrometer used, or its pseudoinverse, i.e. an operator of reconstruction, is identified. The idea to identify this operator directly during calibration of the spectrometer is developed in this paper. The applicability of an operator of reconstruction having the form of a rational filter is studied when it is used for correction of the instrumental errors introduced by an absorption spectrometer. An algorithm for identification of this operator is proposed and studied using spectrometric‐type synthetic data and real‐world spectrometric data. The proposed operator is compared with some existing algorithms of spectrometric data correction, using accuracy‐related criteria, both non‐specific and specific to spectrometry. Copyright