Xps Wavelet

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SURFACE AND INTERFACE ANALYSISSurf. Interface Anal. 2004; 36: 7180Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/sia.1650

Noise filtering and deconvolution of XPS data

by wavelets and Fourier transform

Catherine Charles,1

Gervais Leclerc,2

Pierre Louette,2

Jean-Paul Rasson1

and Jean-Jacques Pireaux2

1 Department of Mathematics, Facult es Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium2 Department of Physics, LISE Laboratory, Facult es Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium

Received 10 September 2002; Revised 15 October 2003; Accepted 15 October 2003

In experimental sciences, the recorded data are often modelled as the noisy convolution product of an

instrumental response with the true signal to find. Different models have been used for interpreting

x-ray photoelectron spectroscopy (XPS) spectra. This article suggests a method of estimate the true XPS

signal that relies upon the use of wavelets, which, because they exhibit simultaneous time and frequency

localization, are well suited to signal analysis.

First, a wavelet shrinkage algorithm is used to filter the noise. This is achieved by decomposing the

noisy signal into an appropriate wavelet basis and then thresholding the wavelet coefficients that contain

noise. This algorithm has a particular threshold related to frequency and time.

Secondly, the broadening due to the instrumental response is eliminated through a deconvolution

process similar to that developed in the previous paper in this series for the analysis of HREELS data. This

step mainly rests on least-squares and on the existing relation between the Fourier transform, the wavelet

transform and the convolution product. Copyright 2004 John Wiley & Sons, Ltd.

KEYWORDS: XPS; wavelets; deconvolution; Poisson noise

INTRODUCTIONWith reference to the two other papers in this series (this

issue), wavelet analysis of XPS data must concentrate on

different areas than applications to High-resolution elec-

tron energy-loss spectroscopy (HREELS) data, for differ-

ent reasons.

First, a simple consideration of the fundamental differ-

ences in the excitation processes reveals that in HREELS the

elemental vibrational excitations have in general an intrinsic

linewidth close to zero, at least at the scale of probe elec-

trons that have an energy (110 eV for example) defined

to 1 meV 8 cm1: Fourier transform infrared spectrashowing vibrational bands with a resolution of 0.1 cm

1

arenot uncommon, justifying the fact that a deconvolution pro-

cedure of HREELS data should consider the true signal as a

Dirac delta function. In XPS, in contrast, intrinsic core-level

linewidths are in the range of 0.1 (metals) to 0.4 eV (carbon),

while stand-alone spectrometers have intrinsic resolution

(monochromatized x-ray source and analyser contributions)

in the same energy range: peaks are naturally broad and

should be approximated by finite Gaussian or Lorentzian

shape (or a combination thereof).

Correspondence to: Jean-Jacques Pireaux, Department of Physics,

LISE Laboratory, Facultes Universitaires Notre-Dame de la Paix,B-5000 Namur, Belgium. E-mail: [email protected]/grant sponsor: Belgian FNRS.Contract/grant sponsor: Belgian Prime Ministers Office;Contract/grant number: PAI/UIAP(4/10).

Secondly, the literature is rich in contributions onspectra acquisition and data handling in XPS. All the

basic principles are illustrated in Sherwoods reference

review,1 while some state-of-the-art concerns have been

listed and commented more recently; as a result of an

international workshop entitled X-Ray photoelectron Spec-

troscopy: from Physics to Data, a compilation of recent

developments in available handbooks and databases, in

data processing software and standard test data is pre-

sented and well documented.2 But, when processing XPS

core-level spectra one should stay aware of the genuine

algorithms hypotheses, and include easy-to-interpret sta-

tistical diagnostics to judge objectively the quality of the

regression.3

The purpose of this contribution is therefore to explore

what specific contribution(s) wavelet analysis could bring

to XPS data: noise filtering and deconvolution (recovery

of the true signal) will be tested on synthetic (theoret-

ical) spectra and then applied to real spectra. As in the

previous paper in this series (this issue) for the work

on HREELS data processing, care will be taken to try

to recover real peak intensities, a prerequisite to keep

quantification through data analysis. Note that this con-

tribution will not discuss issues related to the choice and

use of routines (linear, polynomial, Shirley, Tougaard, etc.)

to estimate background, or the algorithms to allow com-positional depth profile data to be deduced from angle-

resolved XPS measurements, which are beyond the scope of

this work.

Copyright 2004 John Wiley & Sons, Ltd.


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72 C. Charles et al.

WAVELET ANALYSIS MODEL AND METHOD

As a mathematical model of the XPS spectrum, it is chosen

to use

tri Poi C aC bi with i D s li, i D 1 . . . n 1

where X Po

indicates that X is a Poisson randomvariable of parameter , s D sjlggjD1 is a Gaussian (orLorentzian or combination thereof) instrumental function

(transmission function of the spectrometer), lgg 2 IN, l Dlj

nlgg1jD1 refers to the true signal to be estimated and is

modelled as a sum of different Lorentzians, a is a constant

instrumental noise and b D biniD1 is a variable physicaland statistical noise (linear, Shirley, etc.); i is the channel

number. Note that this model considers the true signal

as a sum of Lorentzians but the method presented in this

paper to estimate this signal is easy to generalize to a sum

of Gaussians.

The non-ideality of the spectra arises through several

sources: the convolution of the electronic scattering functionwith the instrumental response, the constant noise, the vari-

able background and the occurrence of Poisson-distributed

random noise.

In this paper a new method is proposed that filters the

Poisson noise and reduces the broadening caused by the

instrumental response. The suggested methods rely mostly

upon wavelet signal processing. The choice of wavelets was

justified in the first paper of this series (this issue) and

also applied to HREELS spectra in the second paper of

this series (this issue) the reader who wishes to gain a

basic understanding of wavelet theory and wavelet signal

processing is referred to these papers.In the proposed method, the noise is filtered using a

wavelet shrinkage technique. Most publications concerned

with the filtering of XPS spectra have focused so far on esti-

mation of a signal to which an additive zero-mean normal

noise is superposed. However, XPS is a technique in which

the detection is performed by an electron counting device.

Particle counting usually follows a Poisson probability dis-

tribution. Consequently, the signal in each channel can be

modelled by this distribution. The proposed approach is

based on an idea by Donoho and Johnstone:4 6 decomposing

the noisy signal in an appropriate wavelet basis and then

thresholding the noisy coefficients. The DonohoJohnstone

technique was developed for Gaussian noise. Recently,Kolaczyk7 has suggested a similar wavelet shrinkage tech-

nique but for a specific case of Poisson-distributed noise. Our

second paper in this series has introduced a wavelet shrink-

age method related to the unconstrained Poisson process

generalizing Kolaczyks algorithm. The same methodology

will be applied to XPS spectra. The constant and variable

noise are removed by subtraction.

Once the filtering step is completed, the broadening due

to Gaussian instrumental function is eliminated via a similar

deconvolution process to that for HREELS. This will be

explained later. This article concludes with an analysis of the

results on the basis of the quadratic error, graphics and thetheory of XPS.

As in the second paper concerning HREELS, most of the

examples presented here use synthetic data.

NOISE FILTERING

Consider the model for an XPS spectrum described by

Eqn. (1). The first step consists of removing the random

(statistical) noise. The problem is then

given the observations

x D xin1iD0 where each xi Poi, estimate in1iD0 2

This problem is solved by using the wavelet shrinkage

method explained in the previous paper (this issue).

To summarize briefly the calculation, the noise removal

algorithm (LTFTIPSH) proceeds in three steps:

(1) Decomposition of the observations in the Haar wavelet

basis.

(2) Thresholding at each scale and at each time of the

coefficients relative to the details.

(3) Application of the inverse Haar transform to the

threshold coefficients.

We prefer using the translation-invariant Haar wavelet

transform to avoid a step estimator.

Below is a presentation of some results (real experimental

data) obtained so far with this LTFTIPSH algorithm. Figure 1

is a survey spectrum of a clean copper foil, with quite

good signal-to-noise ratio. Figure 2 reports a widescan of an

organic layer with more noise and Fig. 3 illustrates a narrow-

scan spectrum with (probably; see later) six core-level peaks.

The advantages and efficiency of this algorithm are

explained in detail in the second paper of this series. It is

clear from the displayed data that the LTFTIPSH algorithm

does not introduce any spurious information and operatesequally well in spectral areas with low and high intensity

counts. Some comparisons with other filtering methods are

made in Ref. 8.

The second step consists of removing the constant and

variable noise, as in conventional XPS data-handling codes.

Given the mathematical model of Eqn. (1), a is estimated as

the minimum oftr and b as linear of Shirley background, for

example. Figure 4 presents such a result for the real C1s C K2pexperimental spectrum shown previously.

Figure 5 illustrates the result of two filterings. The first

filters the random noise using the LTFTIPSH algorithm and

the second subtracts the constant and variable noise.

LOCALIZED LEAST-SQUARES

The previous section has explained how our XPS spectra

are filtered. After filtering, the problem stated in Eqn. (1) is

reduced to the following deconvolution problem

given tri D s li 1 i n, estimate l D linlggC1iD1 3

The reader is reminded of the following hypotheses:

(1) tri corresponds to the experimental spectrum, i.e. the

number of detected electrons versus binding energy.(2) s D silggiD1 is a Gaussian (or Lorentzian or combinationthereof) function featuring the spectrometer transmis-

sion function (a Gaussian function is 1/p

2ex2/22 ,

Copyright 2004 John Wiley & Sons, Ltd. Surf. Interface Anal. 2004; 36: 7180


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Deconvolution of XPS data by wavelets 73

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Figure 1. (a) Experimental XPS spectrum of copper foil after gentle ArC sputtering. (b) Noise filtering of (a) using theLTFTIPSH algorithm.

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Figure 2. (a) Experimental XPS spectrum of a thin organic layer on silicon wafer. (b) Noise filtering of (a) using the

LTFTIPSH algorithm.

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Figure 3. (a) Experimental XPS spectrum of carbon and potassium contamination on a dental titanium implant. (b) Noise filtering of

(a) using the LTFTIPSH algorithm.

where and are, respectively, the mean and the stan-

dard deviation; a Lorentzian function is 1 2 C x 2 ,where and are, respectively, the mean and the stan-

dard deviation).

(3) l is a sum of Lorentzian functions, representing the

true spectrum.

This problem is similar to the HREELS problem, con-

sequently it is proposed to resolve it by using the same



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Figure 4. The XPS spectrum of Fig. 3(a) and an estimation of

its constant and variable noise.

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Figure 5. The XPS spectrum of Fig. 3(a) after noise filtering

and background subtraction.

strategy: localization of peaks by wavelets and estimation of

their intensities by localized least-squares (LLS). This section

focuses on the LLS procedure.

Assume that the instrumental response s is known

and that the true spectrum is one Lorentzian function

or one ensemble of Lorentzian functions. Under this setof assumptions, it is not proposed to use the algorithm

suggested in the HREELS case

minl

i

[tri sli]2

because it does not impose the Lorentzian character of

the result. However, it would be interesting to take

into account this particular hypothesis to obtain a better

estimation. Moreover, this HREELS algorithm was based on

the resolution of a minimization problem that owns a small

number of variables (only some components ofl are differentto zero). However, there are a larger number of variables in

the XPS case, because a Lorentzian is never zero. Thus, the

previous algorithm risks being heavy on calculation time.

Consequently, it is proposed to minimize

min1,1,2,2,...

j

[trj s lj]2 where lt D

i

1

i

2i C t i24

In this case, theparameters i, i of the different Lorentzians

are the only variables and l is imposed to be one ensemble ofLorentzian functions. Therefore, if an initial point 0i ,

0i is

chosen for each Lorentzian sufficiently close to the solution,

this method is expected to converge towards the global

minimum that is the best solution to the problem.

It is therefore necessary to choose a good initialization.

The first idea has been to choose the inverse Fourier

Transform of TR/S, where TR (S) is the Fourier transform

of tr (s). If the deconvolution is realized by the Fourier

transform, its main disadvantage is a possible division by

zero. However, in the XPS case, the instrumental function s

consists of a Gaussian (or Lorentzian or combination thereof)

and it is known that its Fourier transform is never null.

For example

g1t D1p2

expt2/2 ! G1 D exp2/2

gt D1p2

expt2/22 ! G D exp22/25

However, even if this Fourier transform is theoretically never

zero, in practice, if or is sufficiently large the Fourier

transform of the Gaussian (Lorentzian) function becomes

very small for large (see Eqn. (5)).In this case, thecomputer

classifies it as zero. It is suggested bypassing this difficultyas follows. As in Eqn. (38) of the first paper in this series

dtrj,k D

l dsj,.

k; ctrj,k D

l csj,.

k 6

where dfj,k and c

fj,k are the wavelet coefficients related to

details and to a rough approximation, respectively, of the

functionf at scalej and time k. These formulae mean that the

convolution product is preserved by the wavelet transform.

Therefore, iftr and s are decomposed in a wavelet basis and

if the Fourier transform of dtrj,. is divided by the Fourier

transform of dsj,., the Fourier transform of an estimator of

l is obtained. Often, a quite good estimation is obtainedwith a good choice of wavelets. Between all the estimations

obtained with all the wavelets, the one that has the smallest

error is chosen and termed el. The error is defined as the

quadratic error between the trace and the convolution of the

estimated el and s. The parameters 0i and 0i relative to el are

estimated by

min1,1,2,2,...

j

i

1

i

2i C xj i2 elj

27

The initial point of the LLS method for XPS is 0i , 0i iD1,... .

Figure 6 illustrates the entire method. Figure 6(a) repre-sents a synthetic XPS spectrum generated by convoluting

l, the true signal (sum of two Lorentzians of parameters

1 D 3.54, 1 D 100, 2 D 4.55, 2 D 120), with s, the Gaussian



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(a) (b)

(d)(c)

Figure 6. (a) Synthetic XPS spectrum (total envelope) and the true signal l (two discrete peaks) to estimate. (b) Estimation el

(deconvolution by Fourier and wavelets) and the true signall. (c) The Lorentzians obtained with

0

i and 0

i (solution of theoptimization problem (Eqn. (7)) and the true signal l. (d) Final result (solution of the optimization problem (Eqn. (4)) and the

calculated true signal l.

instrumental function D 10. The convolution productis termed tr. First tr and s are decomposed in a wavelet

basis. Secondly, for each scale, the Fourier transform of the

wavelet coefficients of tr are divided by the Fourier trans-

form of the wavelet coefficients ofs; then the inverse Fourier

transform of this division is calculated. The results are some

estimations of l. Estimation el is that particular one giving

the smallest error between the trace and the convolution of

this estimation with s. In Fig. 6(b) it can be seen that thisestimation el recovers the two peak positions but also many

ghost peaks. Thirdly, in order to impose the Lorentzian char-

acter of the estimation of the true signal l, the parameters

01 , 01,

02 ,

02 of the significant peaks of el are estimated

by resolving the optimization problem (Eqn. (7)) Figure 6(c)

illustrates the sum of the two Lorentzians obtained with 01 ,

01, 02 ,

02 and l. Finally, the optimization problem (Eqn. (4))

is resolved with 01 , 01 ,

02 ,

02 as the initial point. The solution

of this final problem or deconvolution result for this local-

ized peak structure and the true signal l are represented in

Fig. 6(d).

The final result in Fig. 6(d) is satisfactory. The differencebetween l, thetrue signalto be calculated, andthe estimation

is small. This result is better evaluated by comparing

the positions and intensities of the peaks in Table 1. The

Table 1. Parameters and estimated parameters of the

Lorentzians of Fig. 6(d)

Estimation of Estimation of

100 3.54 100 3.57

120 4.55 120 4.03

positions are exactly recovered, the intensity of the first peakis well estimated and the intensity of the second peak is

underestimated by some 10%.

In order to analyse the efficiency of the method when

two Lorentzians form only one peak in tr, the results when

the distance between the two Lorentzians decreases while

the intensity is kept equal and constant (Table 2) have to be

analysed. As in HREELS, eval is defined as the FWHM of

the instrumental function divided by the distance between

the positions of the two Lorentzians. In the same way, inra

is defined as the ratio of the smallest Lorentzian parameter

to the largest Lorentzian parameter. The positions of the two

Lorentzians are exactly recovered and the intensities are wellestimated even when the peaks are very close to each other.

Note that the two Lorentzians visually form only one peak

in tr in all these cases.



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Table 2. Parameters and estimated parameters of two

Lorentzians with the same intensity but with a different

interdistance

eval

Estimation

of

Estimation

of

2.35 100 3 100 3.13110 3 110 2.83

1.567 100 3 100 3

115 3 115 2.93

1.175 100 3 100 2.98

120 3 120 2.98

Table 3. Parameters and estimated parameters of

Lorentzians with the same position

inra

Estimation

of

Estimation

of

1/2 100 1 100 0.94

120 2 120 1.28

2/3 100 2 100 1.97

120 3 120 2.53

3/5 100 3 100 2.95

120 5 120 3.97

Table 3 analyses the results when the distance between

the two Lorentzians is constant and the intensity of the two

Lorentzians differs. The positions of the two Lorentzians are

exactly recovered, the smallest intensity is well estimated

but the largest intensity is not so well estimated. Note thatthe two Lorentzians visually form only one peak in tr for all

these cases.

DECONVOLUTION WITH WAVELETS

The procedure for solving the deconvolution problem of

Eqn. (1)consistsof applying theLLS approach to thedifferent

features of the filtered spectrum. Detection of the position of

the peaks and estimation of the instrumental function g are

well explained and discussed in the two other papers of this

series. Below is a presentation of some results of this new

method applied to synthetic filtered spectra.

In Fig. 7 is presented an illustration of the methodonce the position of the peaks in tr is detected and the

instrumental function is estimated. In Fig. 7(a), a synthetic

XPS spectrum has been generated by convoluting the true

signal l (sum of five Lorentzians of different intensities) with

Gaussian instrumental function s. The convolution product

is termed tr. First tr and s are decomposed in a wavelet basis.

Secondly, for each scale, the Fourier transform of the wavelet

coefficients of tr is divided by the Fourier transform of the

wavelet coefficients ofs and the inverse Fourier transform of

this division is taken. These results are some estimations ofl.

Estimation el is the one that gives the smallest error between

the trace tr and the convolution of this estimation with s.Figure 7(b) presents the estimated el and the true signal l.

Thirdly, in order to impose the Lorentzian character of the

estimated el, the parameters 0i , 0i i of the significant peaks

of el are calculated by resolving the optimization problem

(Eqn. (7)) for each peak of tr. Figure 7(c) illustrates the sum

of the Lorentzians obtained with 0i , 0i and l. Finally, for

each peakthe optimization problem (Eqn. (4)) is resolved with

0i , 0i as initial point. The solution of this final problem and

the true signal l are represented in Fig. 7(d). This result is

excellent as allthe intensities andpositions of theLorentziansare recovered.

Figure 8 presents another synthetic spectrum obtained

with five Lorentzians of different intensities convoluted with

a Gaussian; the same strategy as for Fig. 7 has been applied

to this spectrum. The true signal (the sum of the five

Lorentzians) and their estimation are shown in Fig. 8(b).

For better evaluation of the result, parameters and of

the true Lorentzians and their estimation via the wavelet

algorithm are presented in Table 4.

With the graphics in Fig. 8 and the values in Table 4 it

can be seen that the positions of the Lorentzians are well

recovered and estimation of the true signal is satisfactory.

Only Lorentzian 4 (that, with another Lorentzian, leads to

one single peak in tr) has its intensity not so well estimated

(eval D 1.175 and inra D 1/2).

OTHER RESULTS

Some results corresponding to experimental XPS spectra are

reported in Figs 9 13. The noise filtering algorithm and the

deconvolution method presented earlier have been applied.

In Fig. 9, the same experimental spectrum as that shown

and filtered in Figs 35 is presented; the estimated original

XPS function (represented multiplied by a factor of 0.153)

has been added. In order to evaluate the result obtained withthe new method, the result has been compared with data

from a standard curve-fitting procedure. The curve-fitting

method estimates the Lorentzians once the number of peaks,

their positions, their widths and the type of noise are chosen.

This comparison is reported in Table 5. The positions of the

peaks are very similar and the intensity of the peaks differs

slightly. Note that one additional peak is discovered with the

wavelet-based code.

The spectral analysis is completed in 12 min on a

Pentium III biprocessor (900 MHz). In fact, thenoise filtering,

the estimation of s and the estimation by Fourier and

wavelets take

1 min. Only theresolution of theoptimization

problems is time consuming.Below are two other results of the new procedure applied

to very noisy experimental XPS spectra: a single Ca 2p

doublet (Figs 10 and 11) and a structured C 1s spectrum

(Figs 12 and 13).

Table 4. Parameters and estimated parameters of

the Lorentzians of Fig. 8

Estimation of Estimation of

1 3 1 2.95

50 2 50 1.74

100 1 100 0.83120 2 119 1.14 x

220 3 220 2.97



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Figure 7. (a) Synthetic XPS spectrum and the true signal l to estimate. (b) Estimation el (deconvolution by Fourier and wavelets)

and the true signal l. (c) The Lorentzians obtained with 0i

and 0i

(solution of the optimization problem (Eqn. (7))) and the true

signal l. (d) Final result (solution of the optimization problem (Eqn. (4)) for each peak) and the true signal l.

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Figure 8. (a) Synthetic XPS spectrum trand the true signal l to estimate. (b) The true signal land its estimation.

Note that in Fig. 10 the estimated original XPS function

is represented multiplied by a factor of 0.338. In order

to appreciate the result, the estimated Lorentzians are

reconvoluted with the calculated instrumental function, the

constant andShirley noise added;the final resultis compared

with the experimental spectrum without any further curve-

fitting. It can be seen in Fig. 11 that the reconvolution isindeed very satisfactory.

A similar type of very good result is obtained in Figs 12

and 13. Note that in Fig. 12 the estimated original XPS

function is represented multiplied by a factor of 0.272. In

order to appreciate the result, the estimated Lorentzians

are reconvoluted with the estimated instrumental function,

the constant and Shirley noise added; the final result is

compared with the experimental spectrum without any further

curve-fitting (Fig. 13).

The National Institute of Standards and Technology(NIST) provide well-characterized spectral data for assessing

quality in computer-based data analysis procedures.911 The

NIST data analysed here are simulations of XPS spectra.



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Figure 9. The XPS spectrum and its estimated original

XPS function.

Table 5. Parameters of the Lorentzians of Fig. 9

estimated via the wavelet method (W) and

curve-fitting (CF)

W IntensityW CF IntensityCF

286.25 2107.84 286.25 2107.84

287.37 667.20 287.49 492.22

289.29 352.45 289.28 305.68

291.05 51.74

293.93 1542.34 293.91 1531.50

296.57 783.83 296.41 781.25

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Figure 10. The XPS Ca 2p spectrum and its estimated original

XPS function.

Because the XPS spectra are simulations, peak parameters

are known from the models and are not estimates. The

test data are designed to help the analyst to assess data

analysis procedures from the errors in the analysts peak

parameter determinations. Performance is measured bydeviations of the analysts parameter estimates from the true

values for these parameters. The noise filtering algorithm

and the deconvolution method are applied to 22 spectra

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Figure 11. The XPS Ca 2p spectrum and its estimation with

reconvolution.

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Figure 12. The XPS C 1s spectrum and its estimated original

XPS function.

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Figure 13. The XPS C 1s spectrum and its estimation with

reconvolution.

(18 doublets and 4 singlets). Following a factorial design,

the doublet spectra present varying degrees of overlap



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between peaks (valley, shoulder, no shoulder), varying

levels of relative intensity between peaks (Ii II iI) and

varying levels of Poisson noise (low noise, high noise).

The results are encouraging. Figure 14 presents the result

of deconvolution and the reconvolution without any other

curve-fitting for these 22 spectra. As the reader can see,

the program recovers the 18 doublets and the 4 singlets. Theintensities are well estimatedexceptfor thespectra generated

by two peaks of the same intensity (II). In these cases, the

algorithm tends to the solution of one large peak and a very

small peak.

CONCLUSION

This article has tried to show how wavelets can be used

to estimate the original XPS function or the true signal.

The first part of the method consists of filtering the noisy

spectrum to eliminate the Poisson noise. It has be seen

in the second paper of this series that the LTFTIPSH

algorithm is a wavelet shrinkage technique specific to this

kind of signal. Subtraction of the constant and variable

noise is done traditionally. The second step consists of

deconvoluting the filtered spectrum. It is proposed touse a method similar to the HREELS method presented

in the second paper of this series. The positions of the

Lorentzians are estimated via the wavelets. In order to

deconvolute each peak, the localized least-squares method

is used with parameters and of the Lorentzians as

variables. Initialization of this optimization problem is done

by means of combined wavelets and Fourier transform.

The results for XPS spectra are satisfactory. The positions

and the intensity of the peaks are well recovered. This

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Figure 14. Twenty-two XPS spectra: their estimated original XPS function and their reconvolution.



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Figure 14. (Continued).

method possesses the advantage of having a fast algorithm

(implemented in MatLab from WaveLab Toolbox available

at http://playfair.stanford.edu/wavelab).In this series of three papers the potential, if not the

real efficiency, of the wavelet shrinkage technique in the

field of noise filtering of surface science experimental data

has been demonstrated. It has also been illustrated that the

wavelets are well adapted to the detection of singularities.

However, because the convolution product is preserved

by the wavelet transform, the use of wavelets is not yet

the general solution to the deconvolution problem but

the wavelet transform can be combined with the Fourier

transform for some improvements, as was seen with the

XPS data.

AcknowledgementsC.C. is indebted to the Belgian FNRS for a Research Fellow grant.Part of the research at LISE is financed through the PAI/UIAP(4/10)

programme of the Belgian Prime Ministers Office (Federal Servicesfor Scientific, Technical and Cultural Affairs).

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2. Kover L. Surf. Interface Anal. 2000; 29: 671.3. Leclerc G, Pireaux JJ. J. Electron Spectrosc. 1995; 71: 141.4. Donoho DL, Johnstone IM. Biometrika 1994; 81: 425.5. Donoho DL, Johnstone IM, Kerkyacharian G, Picard D.J. R. Stat.

Soc. Ser. B 1995; 57: 301.6. Donoho DL, Johnstone IM, Kerkyacharian G, Picard D. C. R.

Acad. Sci. Paris (A) 1992; 315: 211.7. Kolaczyk ED. Biometrika 1996; 46: 352.8. Charles C, Rasson J-P. Computational Statistics and Data Analysis

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