Measurement Error Correction for Distributed Properties – Filter Cake Thickness Measurements

Measurement Error Correction for Distributed Properties –Filter Cake Thickness Measurements

Michael Koch*, Mahmood Saleem**, Gernot Krammer*(Received: 7 August 2006; in revised form: 17 July 2007; accepted: 18 July 2007)

DOI: 10.1002/ppsc.200601068

1 Introduction

Distributed properties of systems are often encounteredin engineering applications. Information on such proper-ties is commonly handled by treating the results as dis-tributions. In the field of particle technology, e.g., parti-cle size distributions are used widely, reflecting thedistributed nature of the property size [1].Experimental data is often affected by a measurementerror. Such an error typically affects each single datapoint. The statistical model of an error affecting a mea-surement is used following ISO guideline 5725-1 [2].Each measurement, i, of the true value, m, is affected byan independent error, yi, and the observed value, zi, isgenerated by Eq. (1):

zi � m � yi (1)

This model is suitable for the handling of measurementsof a single value m and gives an extensive overview ofthe accuracy of such measurements, based mainly on re-peated measurements [2].When the measured property does not have a single truevalue, m, but instead, each measurement, i, has an un-derlying true value, xi, then the model must be modifiedto:

zi � xi � yi (2)

In this case, it is the distribution of the xi’s rather than asingle value xi that is of major interest. Taking xi, yi, andzi as realizations of the corresponding random variablesX, Y, and Z, the model can be written as:

Z � X � Y (3)

In practice, a problem arises that when realizations forZ are observed and some information is available on Y,a distribution of X, e.g., the density function, must beestimated.In Figure 1, the consequences of a measurement erroron a discrete distribution are illustrated schematically.The true distribution of X, shown in Figure 1a, spansover only four classes (III–VI). In Figure 1b, each value,

Part. Part. Syst. Charact. 24 (2007) 431–439 431

* Dipl.-Ing. M. Koch (corresponding author), Prof. Dr. G.Krammer, Department of Energy and Process Engineering,Norwegian University of Science and Technology, KolbjørnHejes vei 1, NO-7491 Trondheim (Norway).E-mail: [email protected].

** Dr. M. Saleem, Department of Chemical Apparatus Design,Particle Technology, and Combustion, Graz University ofTechnology, Inffeldgasse 25B, A-8010 Graz (Austria).

Abstract

The measurement of a distributed property is frequentlyencountered in engineering applications. The impact ofa possible measurement error on the measured distribu-tion is described and two methods to account for suchan error are detailed: A Fourier transform based decon-volution procedure and a moment based deconvolutionapproach via kernel estimators are presented. The de-

convolution methods can be used in many particle char-acterization and process applications where particle fea-tures or properties are commonly presented in adistributed form. In the current instance, these methodsare carefully explored by using artificially generateddata and applied to experimental data obtained from fil-ter cake thickness measurements.

Keywords: deconvolution, distributed characteristic, measurement error

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim http://www.ppsc-journal.com

432 Part. Part. Syst. Charact. 24 (2007) 431–439

xi, is affected by a measurement error according to Eq.(3). The assumed error in this example is zero biasedand normally distributed with a standard deviationequal to one class width. Thus, the columns in Figure 1b

exhibit a normal distribution around the original classcentroid, but also populate the neighboring classes. Fi-nally, the columns in each class are added up and givethe histogram in Figure 1c.From a comparison of Figures 1a and 1c, it can be seenthat the classes are blended to obtain the resulting distri-bution of Z. In statistical terms this means that the or-derings (histograms are a representation of ordereddata) of the xi’s and zi’s are different, i.e., their ranks inthe respective order statistics differ [3].Although almost 62% of the samples i change class, i.e.,they leave their original class when becoming zi’s, the ef-fect of these class changes is camouflaged by other samplesentering the original class. The net effect of these changesis observed when measuring the zi’s. It is mainly the classestowards the tails of the distribution that are affected bysuch a change, but the peak heights in the center of the dis-tribution are also reduced. The distribution changes shape,and the measured distribution of Z is not a good estima-tor for the underlying distribution of X by itself.In this work, independent xi’s and yi’s are considered,which is reflected by each measurement being affectedby an identically distributed error. If the error distribu-tion were dependent, then each column in the histogramof the zi’s would consist of parts being affected by differ-ent error distributions.In comparison to the case of the measurement of a sin-gle value m, where an estimator for m is usually readilyobtained by just taking the mean of several zi’s for azero-biased error, the problem of estimating the densityof a distribution is not as straightforward.In statistical literature the problem outlined is termeddensity estimation with a measurement error which isencompassed by the field of measurement error models.Textbooks by Fuller [4] and Carroll et al. [5], give a gen-eral overview on the available methods and literature inthe field. The latter explicitly tackles the problem ofdensity estimation. A more recent review article in thefield is given by Stefanski [6].In statistics, one finds a general discrimination betweenparametric and non-parametric models. The former as-sumes a predefined model structure, whereas the latterdoes not take a predefined form, rather than use thedata directly. In the case of error analysis of measureddistributions it is favorable not to assume a functionalform of the distributions involved, since this might implyunjustified altering of the actual measurement. Specificinformation on recent developments in non-parametricdensity estimation is given in the extensive literature re-view by Carroll and Hall [7].Statistical research in recent years has focussed on redu-cing the required information on the error distribution,e.g., Carroll and Hall [7] propose methods that requireonly low order moments of the error distribution.

http://www.ppsc-journal.com © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

0

2

4

6

8

10

12

I II III IV V VI VII VIII

Class

Fre

qu

ency

X

a)

III

IIIIV

VVI

VIIVIII

X Class III

X Class IV

X Class V

X Class VI

0

0,5

1

1,5

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3,5

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4,5

Fre

qu

ency

b)

0

2

4

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10

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I II III IV V VI VII VIII

Class

Fre

qu

ency

Z

c)

Fig: 1: Illustration of the consequences of a measurement error ona distribution. The histogram of the true distribution, X, is shownin part a). The 4 nonempty classes are depicted in differenthatches. Part b) depicts the distribution of each of the four none-mpty X-classes affected by a normally distributed measurementerror, Y. Finally part c) shows the resulting Z distribution witheach column of the histogram being constituted from contribu-tions of different X classes, which are added up from part b).


This work deals with the problematic of non-parametricdensity estimation for measuring distributions in engineer-ing science and specifically particle technology with mea-surement error. Firstly, the basic concept of error convo-lution is described. Deconvolution is achieved by (i) adirect deconvolution using Fast Fourier transformation,and (ii) a moment based method from the statistical litera-ture. Both methods are outlined in more detail and even-tually applied to data arising from an optical cake thick-ness distribution measurement on a laboratory scale bagfilter plant.

2 Distribution of a Characteristic Affected by anError

Each random variable X, Y, and Z should follow a cer-tain distribution with the density functions, �x, �y, and�z, respectively. The relationship given by Eq. (3) yieldsthe following relationship of the density functions withthe asterisk denoting convolution:

�z�u� � ��x � �y��u� ��∞

�∞

�x�n��y�u � n�dn (4)

An important property of the convolution integral isthat it is fulfills the continuity condition for probabilitydensity functions, i.e. the integral over the resultingfunction equals one. Moreover, the function centroidsadd linearly and for a probability function, the functioncentroid equals the expected value.In mathematical terms this reads as follows:�∞

�∞

�z�u�du � 1 (5)

and

E�Z� � E�X� � E�Y� with E�� ∞

�∞

u��u�du (6)

Eq. (6) is especially relevant when the density of Y, i.e.,�y, is zero-biased. If �y is the density of a normally dis-tributed measurement error with a mean zero the expec-tancy, �y, equal to zero, the function centroid of �z isequal to that of �x. If this property is not intrinsicallypresent, it can often be achieved by transformation.

2.1 Example: Convolution of a Uniform and a NormalDistribution

The above theory can be illustrated by taking the exam-ple of the true characteristic, X, being standard uni-formly distributed, i.e., a uniform distribution on the in-terval [0,1]. The error Y obeys a zero-biased normaldistribution with variance r2.

X ∼ U1

Y ∼ N�0� r2�

In Figure 2, the probability density functions �x and �y

are displayed in conjunction with the convolution result�z. The standard deviation r is chosen to be 0.25.One can see that the resulting distribution is centeredon the uniform distribution, but its edges are smearedout. However the expected value (centroid) is not al-tered.

In this special case, the resulting density function canalso be given explicitly as follows:

�z�u� �12

erfu��2

�

2r

� �� 1

2erf

��2

� �1 � u� �2r

� �(7)

3 Implementation

3.1 Estimation of the Density Functions

The density function of the variables Y and Z may eitherbe already available as continuous functions or else theymust be estimated from experimental realizations. Inthe former case, one can proceed directly to the decon-volution described in the next section. In the latter case,the experimental realizations zi and yi can be used to es-timate continuous density functions using kernel densityestimates, as shown below.The direct inversion of Eq. (4) requires continuous dis-tribution density functions, �z and �y. These can be esti-mated from realizations by kernel density estimates inthe form of:


−1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

u, −

φ (u

),−

φx

φy

φz

Fig. 2: Analytic probability density functions for X, Y, and Z.


��u�u� �1

nh

�n

i�1

Kui � u

h

� �(8)

Here, the function K�� denotes the kernel function,which is a density function itself, and h is the bandwidth.The actual choice of the kernel does not affect the esti-mation of �z for a sufficiently large sample size, andhence, the standard normal density is frequently used.The bandwidth is usually chosen by optimization for acriterion that reflects good agreement of the estimateddistribution with actual data.The form of the kernel density estimator Eq. (8), doesnot account for a possible uneven weighting of indivi-dual measurements, and thus, is only applicable toequally weighted samples. In the case of, e.g., mass mea-surements for particle size analysis, the individual sam-ples are weighted, and therefore, a modified kernel den-sity estimator for weighted data must be employed:

��u�u� �1

h�ni�1

wi

�n

i�1

wiKui � u

h

� �(9)

Here, wi is the weight of a certain sample value ui. Forconstant weights wi, Eq. (9) is obviously equivalent toEq. (8).In Figure 3, the densities of the uniform distribution, theuniform convolved with a standard distribution, Eq. (7)and the respective kernel estimates, are displayed. Thesharp edges of the uniform distribution are smoothedout, but the kernel estimate of the convolved distribu-tion can hardly be distinguished from the true function.

3.2 Deconvolution Using Fourier Transformations

In the frequency domain, convolution can be expressedby multiplication. The transform used is the Fouriertransform U�t� � �� eitn � ��n�dnIts inverse transform is given by��u� � �1�U� � e�igu � U�g� dgThe above functions are valid for discretized functionvalues. Therefore, the convolution and deconvolutionare given by:

Uz�t� � Ux�t� � Uy�t� (10)

Ux�t� �Uz�t�Uy�t�

(11)

It is computationally favorable to use the Fast Fouriertransformation (FFT) in the implementation [8]. TheFFT is a version of the discrete Fourier transformationand works on discrete data sets. It assumes that periodicdata is provided, and thus, reflects the end of a data setinto its beginning. Therefore, one must ensure that, e.g.,density functions are padded with a sufficient number ofzeros, so that the FFT only reflects the tails which are al-ready zero into the data set. Moreover, the FFT worksonly on data sets of a length 2m with m being an integer.The use of appropriate filtering of the obtained Fouriercoefficients is crucial for the application of the Fouriertransformation. In Eq. (11), values of the transformederror function, Uy, that are close to zero overamplifyslight errors disproportionately. The filter used here is acutoff filter for the Fourier coefficients in the middlerange, according to Eq. (12)

Ux�t� � Uz�t�Uy�t�

� B�t� (12)

The actual determination of the filter B(t) is not trivialand usually requires both experience of the filter de-signer and highly empirical approaches. Theoretically,an optimal Wiener filter for a certain noise level can bedetermined by following reference [8]. However, appro-priate modeling of the noise in the input functionswould be necessary and such attempts were not fruitfulhere.Instead, the filter is modeled by an analytic function thatcuts out the high, erroneous frequencies. No entire cut-off is used, rather than a Gaussian shaped cutoff func-tion. The spectrum of a convolved distribution with atypical filter function B(t), is displayed in Figure 4. Thefilter is zero in the center of the spectrum, and thus, cutsthe Fourier coefficients which are small in magnitude,and consequently, would introduce a big error in the di-vision. In order to reflect rather big Fourier coefficients,above the threshold B<0.6, no filtering is applied at all,


−1.5 −1 −0.5 0 0.5 1 1.50

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0.8

1

1.2

1.4

u, −

φ (u

),−

φuni,true

φuni,kernel

φconv,true

φconv,kernel

Fig. 3: Density functions for the true uniform and the uniform con-volved with standard distribution, Eq. (7); the respective kernelestimates with the standard normal kernel and bandwidth h =0.025 are also displayed. Note that the convolved curves are eitherdotted or dashed.


i.e., B = 1. The filter can be expressed in mathematicalterms by Eq. (13).

B�t� � exp� t2

r2f

�if B � 0�6 � B � 1 (13)

The Fourier transformation and filtering introduces aphase error and also a scaling error in the deconvolveddistribution. This is corrected for by shifting and scalingthe obtained distribution to meet the conditions of Eqs.(5) and (6).The drawback of using an analytic filter function is thatthe parameter rf must be determined externally. In thecurrent case, this value is decided from an optimization.Carroll et al. [5] give an approach for parametric de-convolution by using the central moments of the dis-tributions involved. Strictly speaking, the parametricdeconvolution is only correct for a symmetric error dis-tribution, �y, i.e., the uneven moments of this distribu-tion must be zero. Here the moment method is onlyused to estimate the filter parameter for the subsequentFourier based deconvolution.The central moment of order k>2 of a distribution isgiven by Eq. (14):

�k � E �U � l�k�

��∞

�∞

�u � l�k��u�du (14)

with l being the mean value of the distribution. The mo-ments of orders 2 to 4 of the distributions, �x, �y, and�z, obey the following relations:

�x�2 � �z�2 � �y�2 (15)

�x�3 � �z�3 (16)

�x�4 � �z�4 � �y�4 � 6 � ��z�2 � �y�2� � �y�2 (17)

Although the exact values of the moments are notknown, the moments of �z and �y can be estimated fromdata. Optimization is carried out for the filter parameterrf of the filter. The objective is to minimize the sum ofthe square errors of the moments calculated for thedeconvolved distribution, �x, and their estimators givenby Eqs. (15) to (17).

3.3 Approximate Deconvolution Using KernelEstimators

In this section, a different method for deconvolving er-ror distributions discussed by Carroll and Hall [7], ispresented. The main advantage of this method com-pared to a full deconvolution as outlined above is thatlow order moments of the error distribution are suffi-cient to capture the error distribution. These low ordermoments are obtained rather easily from experimentaldata. The measured variable Z is used as the input in amodified kernel estimator similar to Eq. (8), which isanalytically combined with the moments of the errordistribution, to give an estimate of the true distribution.The error distribution, Y, is captured by its non-centralmoments, my�k, whereas k denotes the order:

my�k � E �Y�k�

��∞

�∞

uk�y�u�du (18)

Hence, my�1 is the mean value of the distribution. Themeasurements of Z are used in Eq. (19), which intro-duces a slightly modified kernel estimator for equallyweighted samples. Here, K�j� denotes the jth derivativeof the kernel K(·). It is assumed that the kernel, K, is ananalytic function and the jth derivative exists. This as-sumption is fulfilled, e.g., for the Gaussian kernel, whichis used in this work.

��j�z �u� � 1

nhj�1

�n

i�1

K�j� zi � uh

� �(19)

Obviously for j=0, the kernel estimator Eq. (19) is iden-tical to Eq. (8) for the variable Z. The kernel estimator,��j��z�u�, can be calculated for different values of j frommeasured values of zi and again for a chosen bandwidthh.Carroll and Hall [7] give an explicit equation for approx-imating the density �x. Eq. (20) gives the approximationof order two denoted ��x�2. One should note that only themoments up to order two of the error distribution of Yare used here. Hence, only the mean value, my�1, and the2nd order moment, my�2, represent the error distribution.The kernel estimator Eq. (19) is used for j=0, 1, 2.

��x�2�u� � ��z�u� � my�1��1�

z �u� � 12�2m2

y�1 � my�2��2�z �u� (20)


10 20 30 40 5010

−8

10−6

10−4

10−2

100

102

|Φ|2

, |B

(t)

⋅ Φz(t

)|2,

−

10 20 30 40 500

0.2

0.4

0.6

0.8

1

B,

−

t, mm−1

|Φz(t)|

2

|B(t) ⋅ Φz(t)|

2

B(t)

Fig. 4: Part of the Fourier spectrum Uz2 of the actual data exam-ple from the results section and the filtered spectrum, B � Uz2.The respective Gaussian shaped filter with optimized rf is dis-played on the right axis.


This approximate deconvolution procedure is a viableoption when the error distribution is only known rudi-mentarily. One must also note that in contrast to decon-volution using the Fourier method, Eq. (20) gives an apriori approximation. Carroll and Hall [7] also provideapproximations of higher orders, but limitations in dataaccuracy for the application presented in the next sec-tion show that the second order approximation is themost useful practically. It should be noted that usingonly second order moments of the error distribution im-plies omitting more detailed information on this distri-bution, even if it is available. Frequently, the error distri-bution is not known in very much detail, and thus, it israther an advantage that only basic information on thedistribution is required.

4 Results and Discussion

The procedure for deconvolving distributions via theFourier transform is applied to artificially generateddata to demonstrate its use. Here, the true original dis-tribution is known, and therefore, it can be used to as-sess the quality of deconvolution. In Figure 5, the situa-tion for �x as a t-distribution with degrees of freedom,m=6, is displayed. Good agreement of the true functionand the deconvolved distribution is achieved. The de-convolved density becomes negative towards its tails,which is physically meaningless. However, the deconvo-lution has large uncertainties and there is always a tra-deoff between the oscillations in �x and the resolutionof rapid changes in the true function.Another example situation of the convolution of a uni-form distribution with a normal distribution is shown inFigure 6. The deconvolution result that is achieved here

is not in equally good agreement with the true distribu-tion as for the smoother t-distribution. The pooreragreement must partly be ascribed to the kernel estima-tion of the distribution functions, which introduces someadditional smoothing effect. However, the kernel esti-mation of the convolved distribution, �z, is in very goodagreement with the actual function, Eq. (7), and is notdisplayed since no difference could be observed be-tween the true and the estimated functions, respectively.Another more important result is that the Fourier de-convolution is not able to resolve discontinuities in thedensity of X very well. This limitation must be kept inmind when applying this procedure. A possible jump inthe density will be smoothed out and in the extremecase, a jump discontinuity in the cumulative distributionfunction cannot be resolved accurately either.In general, better results for deconvolution are obtainedwhen the error function, �y, is supplied explicitly as aparametric function and not in the form of a realizationon which a kernel density estimation method must beapplied.The experimental data used for evaluating the deconvo-lution procedure arises from an optical cake thicknessmeasurement on a laboratory scale bag filter plant. Theexperimental setup is described in detail by Saleem etal. [9]. In this setup a filter cake surface is measured3-dimensionally using a stereo camera setup. The mea-surement principle is shown schematically in Figure 7 ina projection perpendicular to the cake height. For ameasurement, the two cameras take a picture of the fil-ter media carrying a filter cake. In post processing, thetwo pictures are aligned and a 3-dimensional height fieldof the filter cake surface is calculated. From this heightfield, a reference measurement of the clean filter cloth


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0

1

2

3

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5

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8

φ, m

m−1

u, mm

φz

φy

φx FFT

φx true

Fig. 5: Deconvolution of Z=X+Y with a t-distributed true X and anormal distributed Y. For reasons of comparison, the true t-distri-bution of X is also plotted.

−0.2 0 0.2 0.4 0.6 0.8−1

0

1

2

3

4

5

6

7

8

φ, m

m−1

u, mm

φz

φy

φx FFT

φx kernel

φx true

Fig. 6: Deconvolution of Z=X+Y with a uniformly distributed trueX and a normally distributed Y. For reasons of comparison, thetrue uniform distribution of X and the kernel estimate for �x fromuniformly distributed data are also plotted. Note that the parame-ters here are not equal to the situation shown in Figure 3.


is subtracted to obtain the cake thickness perpendicularto the filter media. During the extensive post processing,mechanistic sources of error are also corrected to a largeextent, which is outlined in detail by Rüther et al. [10].Since the amount of data collected on cake thickness isdifficult to evaluate in the form of height fields, the datais preferentially treated as cake thickness distributionsomitting information of the location of the actual mea-surement. Each reconstructed point of an actual cakethickness measurement is viewed as a sample zi.However, repeated measurements of the clean filtershow that a significant error remains in the results. Ide-ally the repetitions would show a zero cake thicknessthroughout the measured filter cake area. This is not thecase due to limitations of the measurement resolution,and an erroneous height field is returned. Theserepeated measurements of the clean filter are used todescribe the error distribution, i.e., each point of the er-roneous height field is taken as a realization of Y. Sinceno correlation between the actual location of the pointin the height field and its value can be found, the error isindependent on the measurement location.The deconvolution situation for experimental data is de-picted in Figure 8. For this application, the cumulativedistribution function, F, is depicted rather the density, �.These two functions are related via Eq. (21).

F�u� ��u

�∞

��n�dn (21)

Both input distributions, �y and �z, are estimated fromexperimental data via kernel density estimates. The de-convolution using the moment method from Carroll andHall [7] is also displayed. Both methods exhibit ratherstrong oscillations of even the cumulative distribution

function, which becomes physically meaningless when itis negative or goes beyond one. The filter parameter, rf ,for the Fourier method is tuned to fit the parametric de-convolution criteria in Eqs. (15) to (17), in a least squareerror sense.The same deconvolution problem is depicted for Fouriertransform solutions in Figure 9. Here the filter para-meter, rf , is reduced by 25 % from its optimal value ac-cording to the moment estimation. As a consequence,the oscillations in the tails are damped, but the resultingdistribution smoothens out details. It is clear from theoptimization criterion that the tuned distribution is thebetter deconvolution result in the sense of momentagreement, but from the raw image, one tends to


cake thickness

Filter media

Stereo cameras

Filter cake

Fig. 7: Schematic of the stereo optical cake measurement tech-nique.

−0.5 0 0.5 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

F,

−

u, mm

Fz

Fy

Fx FFT

Fx moment

Fig. 8: Cumulative distribution functions for deconvolution of cakethickness data. The kernel estimates of Fy and Fz with the decon-volution results via the Fourier transform method ’FxFFT‘and themoment method ’Fxmoment‘, are displayed.

−0.5 0 0.5 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

F,

−

u, mm

Fz

Fy

Fx σ

f

Fx 0.75⋅σ

f

Fig. 9: Cumulative distribution functions for deconvolution of cakethickness data. The kernel estimates of Fy and Fz and the decon-volved results with the Fourier method Fx are displayed. The samecurve as in Figure 8 with the tuned filter parameter, rf , is dis-played together with one where rf is reduced by 25 %.


decrease the filter parameter slightly further to reduceoscillations.It should be noted that the deconvolution quality largelydepends on the error distribution. Attempts have beenmade to replace the kernel estimate error distribution ofthe experimental data example with a normal distribu-tion via a parametric fit. The deconvolution quality be-came significantly worse, thus indicating that nuances ofthe error distribution that are not reflected in the fittednormal distribution can have a significant impact on thedeconvolution result.With regard to the computational effort for deconvolu-tion, the method using Fast Fourier transforms is foundto have superior performance to the moment deconvo-lution method. Despite the rather simple representationof the approximate deconvolution using moments com-pared to the Fourier approach, one should note that thekernel estimation is the computationally most expensivestep for all data sets. Admittedly, kernel estimation isalso a step for density estimation using Fourier trans-forms, but the estimation using the original Gaussiankernel is much faster than using its derivatives.

5 Conclusions

The problem of treating measured characteristics as dis-tribution functions is discussed. It is shown that the di-rectly measured distributions can differ significantlyfrom the underlying true distributions. In contrast to themeasurement of a nondistributed property, where themeasured mean is a good estimator for the true value gi-ven a zero-biased error, a measured distribution is not agood a priori estimator for the true distribution. A possi-ble measurement error especially affects the tails of ameasured distribution, which are flattened out and canthereby differ significantly from the true distribution.Two methods are described and applied to deconvolvethe measured data from a measurement error, yieldingan estimate for the true distribution.The proposed procedures to treat data are able to de-convolve measured data from a measurement error,although limitations of the resolution due to deconvolu-tion apply. Oscillations are observed especially towardsthe tails of a deconvolved distribution. This is a result ofthe strong influence of the error in the tails of the mea-sured distribution. Hence, the tails of both the measureddistribution and the deconvolved distribution are rela-tively more uncertain than the distributions around theircentroids.The filtering of the Fourier spectra is crucial and an opti-mal filter design is difficult to obtain. This is also indi-cated by the remaining disagreement of the moments

after the filter tuning when based on parametric decon-volution via moments.Approximate moment based deconvolution cannot reflectthe fine details of the error distribution. However, giventhe quality of deconvolution achievable, it might often be aviable method that is relatively easy to implement. A tun-ing possibility also arises for the bandwidth, h, of thekernel estimator, which is not assessed in this work.During evaluating the deconvolution methods for artifi-cial data, parametric error distributions give results ofbetter quality than when the corresponding kernel esti-mates are supplied. However, the opposite is found forthe deconvolution of experimental data, indicating thata parametric fit of the error distribution is not justifiedfor the given application.The methods are applied to experimental cake thicknessmeasurements. Corresponding error distributions areobtained for different experimental parameters and theintercomparability of the deconvolved cake thicknessdistributions affected by different measurement errors isalso improved, which is not elaborated on here.The deconvolution methods are applied to the specificapplication of cake thickness measurements. However,the basic idea and the concepts outlined here are applic-able to a much broader variety of measurement applica-tions, where a distributed characteristic is of interest.

6 Acknowledgements

The authors are grateful for valuable discussions withProf. Stadlober, Department of Statistics and Prof. Stau-dinger, Department of Chemical Apparatus Design,Particle Technology and Combustion, University ofTechnology, Graz, Austria.

7 Nomenclature

Latin symbolsB filter function Fourier transformationF cumulative distribution function –1 inverse Fourier transformationh kernel density estimator bandwidthK kernel functionm true mean value (not distributed)n sample numbert frequency domain variableu general characteristicw sample weightX random variable for the true valuex realization of the true valueY random variable for the measurement error



y realization of the measurement errorZ random variable for the measured valuez realization of the measured value

Greek Symbolsg integration variablejj central moment of jth orderl mean valuemj non-central moment of kth order, (-)U Fourier transform of the density function _� distribution density function, ([u]–1)r standard deviationrf filter parametern integration variable

Indicesi discrete realizationx of the true valuey of the measurement errorz of the measured value

8 References

[1] VDI-3491: Messen von Partikeln; Kennzeichnung vonPartikeldispersionen in Gasen. VDI-Richtlinie, 1980.

[2] ISO-57251: Accuracy (Trueness and Precision) of Mea-surement Methods and Results. ISO International Stan-dard, 1994.

[3] H. Büning, G. Trenker, Nichtparametrische statistischeMethoden. De Gruyter, Berlin 1978.

[4] W. A. Fuller, Measurement Error Models. Wiley Series inProbability and Mathematical Statistics: Applied Prob-ability and Statistics, John Wiley & Sons, New York,1987.

[5] R. J. Carroll, D. Ruppert, L. A. Stefanski, MeasurementError in Nonlinear Models. Chapman & Hall, London,1995.

[6] L. A. Stefanski, Measurement Error Models. J. Am. Stat.Assoc. 2000, 95, 1353–1358.

[7] R. J. Carroll, P. Hall, Low Order Approximation in De-convolution and Regression with Errors in Variables. J.Royal Stat. Soc. B, 2004, 66, 31–46.

[8] W. H. Press, Numerical Recipes in C: The Art of ScientificComputing. 2nd ed., Cambridge University Press, Cam-bridge, 1987.

[9] M. Saleem, G. Krammer, M. Rüther, H. Bischof, OpticalMeasurements of Cake Thickness Distribution and CakeDetachment on Patchily Cleaned Commercial Bag Fil-ters. in Proc. European Conf. on Filtration SeparationTechnology FILTECH, Wiesbaden (Germany) 2005, II-51–II-58.

[10] M. Rüther, M. Saleem, H. Bischof, G. Krammer, In-SituMeasurement of Dust Deposition on Bag Filters UsingStereo Vision and Non-Rigid Registration. AssemblyAutom. 2005, 25, 196–203.


Measurement Error Correction for Distributed Properties – Filter Cake Thickness Measurements

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