IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 5, OCTOBER 2001 1123

Cauchy-Filter-Based Algorithms for Reconstructionof Absorption Spectra

Piotr Sprze¸czak and Roman Z. Morawski, Senior Member, IEEE

Abstract—The computer-based interpretation of spectrometricdata ~ is aimed at identification of the main components ofan analyte. The first step of interpretation consists in estimationof its spectrum using an operator of (generalized) deconvolu-tion ^ = ~ ; p , where p is a vector of pa-rameters to be estimated during calibration of the spectrometer.A new structure of this operator, based on a nonlinear transfor-mation of the Cauchy filter, is proposed and studied in this paperusing both synthetic and real-world spectrophotometric data. Itssuperiority over existing algorithms for spectrum reconstructionis demonstrated both with respect to accuracy of spectrum recon-struction and numerical complexity.

Index Terms—Data processing, optical spectroscopy, parameterestimation, spectrochemical analysis.

I. INTRODUCTION

T HE COMPUTER-BASED interpretation of spectrometricdata is aimed at identification of the chemical com-

position of an analyte. The concentrations of its components areestimated on the basis of the data using some informationon the spectrometer, acquired during its calibration. The inter-pretation of spectrometric data is of great importance for appli-cations of various kinds of spectroscopy in science, biomedicalengineering, environmental engineering, and industry [1]–[11].The first step of interpretation usually consists in estimation ofthe spectrum of the analyte . Many methods for solvingthat problem have been developed [12]–[15]. The vast majorityof them are based on the assumption that the continuous enve-lope of the data may be adequately modeled witha convolution equation

(1)

where is wavelength and is theso-called apparatus function modeling the imperfections of thereal spectrometer. Under this assumption, a discrete estimateof can be found using an operator of (generalized)deconvolution [14]

(2)

Manuscript received May 4, 2000; revised February 24, 2001. This workwas supported by the State Committee of Scientific Research, Poland (Grants 8T10C 025 11 and 8 T10C 040 17).

The authors are with the Faculty of Electronics and Information Technology,Institute of Radioelectronics, Warsaw University of Technology, Warsaw,Poland (e-mail: r.morawski@elka.pw.edu.pl; p.sprzeczak@elka.pw.edu.pl).

Publisher Item Identifier S 0018-9456(01)08752-6.

where is a vector of parameters to be estimated during cal-ibration of the spectrometer. A new structure of this operator isproposed and studied in this paper. It is based on a nonlineartransformation of the following Cauchy filter we introduced[16]:

(3)

where .

Throughout this paper, the following notations are used:isthe wavelength, is the transmittance spectrum;

is the absorbance spec-trum; is the transmittance-domain data; and

is the transmittance-domain data.

II. PROPOSEDFAMILY OF THE ALGORITHMS FORSPECTRUM

RECONSTRUCTION

The operators of spectrum reconstruction, transforming thespectrometric data into an estimate of the discreterepresentation of the spectrum ,are assumed to be nonlinear in order to make possible compen-sation for nonlinear effects in the spectrometric measurementchannel. They are defined by the following formulas:

for (4)

where , , and for. Thus, the vector of parameters to be estimated

during calibration has the form

(5)

A genetic algorithm has turned out to be an appropriate toolfor estimation of this vector on the basis of the reference data

and via minimization of the criterion

(6)

0018–9456/01$10.00 © 2001 IEEE

1124 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 5, OCTOBER 2001

where

and is a regularization parameter optimized empiri-cally.The main advantage of employing a genetic algorithm forcalibration is the high probability of finding an estimate of theparameters corresponding to the global minimum of .

III. M ETHODOLOGY OFSTUDY

The Cauchy-filter-based algorithms of spectrum reconstruc-tion, introduced in the previous section and denoted hereinafterwith the symbol , were studied using both syntheticand real-world spectrometric data. The synthetic data were gen-erated according to the formulas

(7)

for (8)

where are zero-mean, uniformly distributed, pseudorandomnumbers with a known variance . The spectrum chosen forexperimentation had the form

for (9)

(10)

where for . The apparatusfunction chosen for experimentation had the form

for

for

(11)

where .The synthetic data for calibration, and

, were generated according to (7)–(11) usingand . An-

other set of data, and , designed for the as-sessment of the accuracy of spectrum reconstruction was gener-ated according to (7)–(11) usingand (see Fig. 1).

The real-world spectrometric data for the assessment of thestudied algorithms of spectrum reconstruction were acquiredby means of the high-resolution (0.3 nm) spectrometerU-3300(made byHitachi)—the spectra and , andthe low-resolution (5 nm) mini-spectrometerSpectraMatch GT(made byCVI Laser Corporation)—the spectra and

. The spectra and , representativeof a sample of Nb, were used for calibration, and the spectra

and , representative of a sample of HoO ,were used for validation (see Fig. 2).

Fig. 1. Synthetic spectrometric data used for validationfx g.

Fig. 2. Real-world spectrometric data used for validationfx g.

The accuracy of spectrum reconstruction was characterizedby the normalized mean-square error of spectrum reconstruction

(12)

defined using an estimate of the vector , determinedduring calibration. The values of this criterion were usedfor comparing the algorithms of spectrum reconstructioncorresponding to various combinations of the parameters,

, and . An indicator of numerical complexity of thosealgorithms—viz., the number of floating-point operation asmeasured by means of the MATLAB procedureflops—wasalso used for their comparison.

IV. RESULTS OFSTUDY

In Figs. 3–6, selected results of a systematic study of the pro-posed family of Cauchy-filter-based algorithms are presented.In Figs. 3–5, the dependence of the error of spectrum reconstruc-tion on the parameters , , and , respectively, is demon-strated. In Fig. 3, for each value of, the values of and arerepresented which minimize . Similarly, in Fig. 4, for eachvalue of , the values of and are represented which mini-mize , and in Fig. 5, for each value of, the values of and

minimizing are used. It may be seen from Figs. 3–5 thatthe best results may be obtained for the values of, , andclose to 3. In Fig. 6, the dependence of the error of spectrum re-construction on the level of disturbances in the spectrometricdata is illustrated. This figure clearly shows the robustnessof the Cauchy-filter-based algorithms to those disturbances: the

SPRZECZAK AND MORAWSKI: CAUCHY-FILTER-BASED ALGORITHMS 1125

Fig. 3. Dependence of the error of spectrum reconstruction@ on the filterparameterK.

Fig. 4. Dependence of the error of spectrum reconstruction@ on the filterparameterL.

Fig. 5. Dependence of the error of spectrum reconstruction@ on the filterparameterI .

Fig. 6. Dependence of the error of spectrum reconstruction@ on the level ofdisturbances in the spectrometric data� .

error of reconstruction is growing only times when thelevel of disturbances is increased times.

The proposed Cauchy-filter-based algorithms of spectrumreconstruction have been compared with the correspondingalgorithms based on the rational filters RF ,

TABLE ICOMPARISON OF THERESULTS OFSPECTRUMRECONSTRUCTIONOBTAINED

FOR THEREAL-WORLD DATA BY MEANS OF THECAUCHY FILTERS CRAND BY MEANS OF THERATIONAL FILTERS RF FORK = 1; . . . ; 5

described in [13], and currently considered to be the best refer-ence method for spectrum reconstruction. In Table I, the resultsof spectrum reconstruction obtained for the real-world data bymeans of the Cauchy filters CF and rational filters RF

are compared with respect to the error of spec-trum reconstruction and the number of floating-point opera-tions (flops) necessary for reconstruction. It may be noticed thatthe minimum error of reconstruction is attained for CF , andthat this filter requires only about 50% of the operations requiredby the corresponding rational filter RF. It may be shown that,in general, the proposed Cauchy filter CF requires the fol-lowing number of floating-point operations:

(13)

for reconstruction of the real-world spectrum shown in Fig. 2.

V. CONCLUSION

New algorithms for spectrum reconstruction,viz. the al-gorithms based on the Cauchy filters, have been proposedand studied using both synthetic and real-world spectrometricdata. Their superiority over existing algorithms has beendemonstrated both with respect to accuracy of spectrumreconstruction and numerical complexity. The proposed familyof algorithms seems to be an efficient tool for solving suchproblems of spectrometric data processing as those describedin [1]–[11]. However, it seems to be of particular usefulnessfor spectrophotometric data correction, and consequently,for designing procedures of low-cost compensation of lowresolution in integrated mini-spectrophotometers, such as thosedescribed in [17] and [18].

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1126 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 5, OCTOBER 2001

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Piotr Sprzeczak, photograph and biography not available at the time of publi-cation.

Roman Z. Morawski, (M’93–SM’99) photograph and biography not availableat the time of publication.