This article was downloaded by: [Universitat Politècnica de València]On: 24 October 2014, At: 16:09Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Research in Nondestructive EvaluationPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/urnd20

Restoration of the Spread Function inNondestructive Testing: Theory andExperimentsLina Teper a b , Dov Ingman a & Phineas Dickstein a ca Quality Assurance and Reliability, Faculty of Industrial Engineering,Technion–Israel Institute of Technology , Haifa , Israelb RAFAEL Reliability Center , Haifa , Israelc Soreq Nuclear Research Center , Yavne , IsraelPublished online: 10 Jan 2012.

To cite this article: Lina Teper , Dov Ingman & Phineas Dickstein (2012) Restoration of the SpreadFunction in Nondestructive Testing: Theory and Experiments, Research in Nondestructive Evaluation,23:1, 1-16, DOI: 10.1080/09349847.2011.622067

To link to this article: http://dx.doi.org/10.1080/09349847.2011.622067

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

RESTORATION OF THE SPREAD FUNCTION IN NONDESTRUCTIVETESTING: THEORY AND EXPERIMENTS

Lina Teper,1,2 Dov Ingman,1 and Phineas Dickstein1,3

1Quality Assurance and Reliability, Faculty of Industrial Engineering,Technion–Israel Institute of Technology, Haifa, Israel2RAFAEL Reliability Center, Haifa, Israel3Soreq Nuclear Research Center, Yavne, Israel

Measurement systems can be regarded as input–output systems represented by a character-istic instrument transfer function. The characterization of the experimental setup enables aquantitative evaluation of its sensitivity and resolution, a comparison of relative performanceamong several systems, and inverse data restoration through deconvolution procedures.The direct inversion introduces either ill-conditioned sets of equations or singularities.One approach is to regularize the instabilities through minimum-entropy considerations.Another approach is to develop alternative methodologies to by-pass the singularity prob-lems. Following the theoretical work, experiments were planed and conducted so as to char-acterize the transfer function of measurement systems in the areas of ultrasonic and industrialradiography. The experimental ultrasonic and radiography results and the applications of themethodologies for the characterization of an ultrasonic measurement system are presentedand discussed.

Keywords: convolution, instrument function, line spread function, modulation transfer function,nondestructive testing, radiography, ultrasonic

1. INTRODUCTION

The results of laboratory experiments may be regarded as the outcome ofa convolution of the ideal results, i.e., those expected from a model based onphysical principles, and a transfer function which characterizes the measure-ment setup.

Theoretically, an absolutely perfect measurement system would yield the‘‘ideal’’ results, but actually, the nonperfect instrument functions provide adegraded or smeared outcome (see Fig. 1).

The instrument function relates to the whole measurement setup used inthe experiment regardless of its composition and fine structure. It may beconsidered as a ‘‘black box’’ whose output is determined by a characteristicfunction (see Fig. 2).

Address correspondence to Dr. Phineas Dickstein, Soreq Nuclear Research Center, Yavne 81800,Israel. E-mail: dickstein@soreq.gov.il

Research in Nondestructive Evaluation, 23: 1–16, 2012

Copyright # American Society for Nondestructive Testing

ISSN: 0934-9847 print=1432-2110 online

DOI: 10.1080/09349847.2011.622067

1

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

The semi-empirical determination of the transfer function of a measure-ment system is primarily essential in the following cases:

1. An ideal transfer function which does not degrade ideal signals is the dfunction (a convolution with a d function does not affect the input data).Consequently, a system whose transfer function resembles a d function isdefinitely superior to a measurement setup of which the transfer functionis wider. Thus, a comparison between the characteristic transfer functionof different measurement systems provides a mean of leveling measure-ment system according to their performance.

2. The determination of the transfer function of a measurement setup is alsoessential for calibration purposes. Suppose a measurement system is to bewidely used for the examination of a large number of specimens. Theeffort needed for deciphering the transfer function is then of outmostimportance since the ‘‘ideal’’ data can then be revealed through a decon-volution procedure which may not be that complicated if conducted inthe frequency domain.

Thus the ‘‘forward problem’’ is to determine the transfer function. This isaccomplished semi-empirically by means of well-defined reference speci-mens that serve as ‘‘the learning group.’’ Once the transfer function is deter-mined and verified experimentally, it is used for the ‘‘inverse problem,’’where the ‘‘ideal’’ input is resolved out of its experimental image througha deconvolution procedure. It should be emphasized that the system is eval-uated and characterized as a given apparatus (black box) regardless the verydetails of its compartments. The goal is to develop a simple methodology forits characterization through the transfer function. We are not dealing with thecharacterization of defects; rather, the transfer function of the measurement

FIGURE 2. The complete ultrasonic setup is regarded as a single black-box.

FIGURE 1. The ideal image is convolved by the instrument function to result in a non-ideal image.

2 L. TEPER ET AL.

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

system is sought, so as to be engaged in a deconvolution procedure. Formathematical simplicity, the instrument function is assumed to be linearand shift-invariant. Under this assumption, the simplest shape of slits are tobe used in the characterization process, and since we are dealing with theLine Spread Function (LSF), which can be regarded as a ‘‘zero width rec-tangular slit,’’ it is convenient and appropriate to use rectangular slits inthe experimental work. In analogy, the Point Spread Function (PSF) couldbe sought through circle-shape slits of varying diameters.

Theoretically, the experimental results fromwell-defined reference samples,together with a model of the ideal response, should enable a semi-empiricaldetermination of the instrument function via a deconvolution procedure,especially if carried out in the frequency domain, where the de-convolutionoperation becomes an algebraic division. However, this procedure is often verycomplicated from a mathematical viewpoint, and alternative methods to deter-mine the instrument function or its features have to be applied.

An approach described by Teper et al. [1] attempts to optimize therestoration of the LSF functions in nondestructive testing (NDT) by either simpli-fying the restoration procedure and by accounting for a-priori statistical infor-mation. This approach extended the entropy concept of Burshtein [2] in regardto the LSF. However, this article concentrates on the quantitative characteriza-tion of an ultrasonic measurement system, in terms of its instrument function.

Some aspects of this approach were applied in studies dealing withradiation-gauges, radiography systems, industrial tomography, and scanningtunneling microscopes, as referenced by Notea [3]. Kenny et al. [4] appliedsimilar principles to an ultrasonic testing system, but concentrated on thecharacterization of different ultrasonic transducers for a set configurationof test equipment. This work though, characterizes the performance of anentire ultrasonic measurement system in terms of its instrument function.Several methods to determine experimentally the LSF of ultrasonic measure-ment systems were described by Dickstein et al. [5]. These included themoment analysis and the derivative methods. However, these latter methodscannot provide the response of the measurement system to an infinitelynarrow slit, which is, by definition, the LSF of the measurement system.

2. THE INSTRUMENT FUNCTION

The ultrasonic measurement system is regarded as an input–output sys-tem which can be modeled by a transfer function. This function correspondsto the entire measurement setup as a single unit without regard to its detailedstructure.

Consider the common problem in NDE of determining the shape of acrack-like defect. It is essential that the ultrasonic system output (or somequantity that can be derived from the output) be a close representation ofthe system input, i.e., the crack shape. Any degradation of the image by

RESTORATION OF THE SPREAD FUNCTION 3

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

the measurement system would be due to non-idealities in the systeminstrument function. Note that the shape of the crack is not necessarily theideal image. In many cases there exists a nonlinear and local-dependentoperator that is transferring the shape of the crack to the ideal image [3].

For mathematical simplicity, the instrument function is assumed to belinear and shift-invariant [6], and dependent only on the experimental setup,not the defect. It is also assumed to be normalized, so that its integral (0thmoment) is equal to one unit of area. For this study, the instrument functionis assumed to be symmetric, although an extension can bemade to accommo-date a lack of symmetry. In accordance with these assumptions, the experi-mental output of the measurement system Om(r) is expressed as theconvolution of the ideal output, Oi(r), i.e., the output expected from a‘‘perfect’’ system, and the normalized instrument function L(r):

OmðrÞ ¼ LðrÞ � OiðrÞ; ð1Þ

where r is a position-vector in the spatial-domain, and the sign � denotes theconvolution operation.

If the measurements are dependent on a single spatial variable, x, Eq. (1)can be simplified to read:

OmðxÞ ¼ LðxÞ � OiðxÞ ¼Z 1

�1Lðx � x 0Þ �Oiðx 0Þdx 0: ð2Þ

Theoretically, L(x) can be derived from Eq. (2) by a deconvolution of the mea-sured system outputOm(X) by the ideal outputOi(x) known to correspond to areference input. In practice, the intricacies of the deconvolution procedureare prohibitively complicated and often very sensitive to random noise.

Alternatively, the instrument function can be determined empirically. Insome cases this procedure consists of assuming the general functional formof L(x) and conducting experiments on reference samples to determine thefree parameters. The most commonly used functions to describe L(x) areexponentials [7–10], Gaussian profiles, square-impulse functions [11] anda Cauchy type function [12].

Note that for an ideal measurement system, Om(x)¼Oi(x). This corre-sponds to an identity system with an instrument function described by d(x).Thus, the resemblance of L(x) to d(x) is indicative of the system’s quality.As an illustration, consider an ultrasonic B-scan data from reference rec-tangular slits machined in a plate (Fig. 3).

The output of a ‘‘perfect’’ measurement system should be a rectangularshape described by

Oiða; xÞ ¼ A � h x þ a

2

� �� h x� a

2

� �h i; ð3Þ

4 L. TEPER ET AL.

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

where a is the width of the slit centered at x¼ 0, h is the heavyside function,and A is a scaling factor determined by both the measurement system and theattenuation of the ultrasonic beam.

Since the measurement system is imperfect, the shape of the output isdescribed by

Omða; xÞ ¼ LðxÞ � A � h x þ a

2

� �� h x� a

2

� �h i: ð4Þ

For the ideal case where L(x) is so narrow that it approachesd(x), Eqs. (3) and(4) become identical.

3. THE EXTRAPOLATION METHOD TO DETERMINATE L(X)

In the limit that a slit’s width becomes infinitely small, Eq. (4) can besimplified to yield the theoretically measured profile of the reflected signal

lima!0

Omða; xÞ ¼ A

Z x 0¼xþa=2

x 0¼x�a=2

Lðx 0Þdx 0 ¼ A � LðxÞ � a: ð5Þ

Since L(x) is normalized,

lima!0

Z 1

�1Omða; xÞ � dx ¼ A � a �

Z 1

�1LðxÞ � dx ¼ A � a � 1; ð6Þ

so that

LðxÞ ¼ lima!0Omða; xÞlima!0

R1�1Omða; xÞ � dx

� ONð0; xÞ: ð7Þ

The notation ON(a,x) is introduced in Eq. (7) to describe the normalizedmeasured profile obtained from a slit of width a. An extrapolation procedurecan now be used to derive L(x) from the normalized measured profiles ofseveral reference slits, as described before.

FIGURE 3. The B-scan geometry. (Figure appears in color online.)

RESTORATION OF THE SPREAD FUNCTION 5

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

4. DIRECT EXTRAPOLATION

The direct extrapolation procedure is carried out by the following steps. Asequence of x values denoted fxi, i¼ 1,2,. . .Ng is selected, whose range spansthe interval over which nonzero values of Om(a,x) were recorded. For eachvalue of xi, an interpolating polynomial is obtained describing ON(a,x) as afunction of a. For each value of xi, the extrapolated value of ON(a,x) isobtained by inserting a¼ 0 into the interpolating polynomial. The resultingvalues of ON(a,x) are plotted against xi, yielding the approximate profile ofON(0,x), equal to the instrument function L(x) of the measurement system.

To simplify any further calculations, a smooth analytic function may bedetermined to approximate L(x).

5. INDIRECT EXTRAPOLATION

In this technique it is assumed that the normalized profiles ON(a,x) of allthe slits can be approximated by the same general functional form, fi, withdifferent values of free parameters pi for each slit:

ONða; xÞ � f1½p1ðaÞ; p2ðaÞ; . . .pmðaÞ; x�; ð8Þ

where the free parameters pi, i¼ 1,m depend only on the slit width a.The steps of the indirect extrapolation method are then as follows:

For each of the m free parameters, the values of pi(a) are plotted against a.For each of the m free parameters, a numerical routine is used to obtain an

interpolating polynomial which describes the dependence of pi on a.For each of the m free parameters, the value of pi(0) is obtained by inserting

a¼ 0 in the interpolating polynomial.

The instrument function is now given by

ONð0; xÞ � f1½p1ð0Þ; p2ð0Þ; . . .pmð0Þ; x�: ð9Þ

It is noted that this technique benefits from a-priori knowledge regarding theshape of L(x), such that only the values of m free parameters need be sought.

6. EXPERIMENT

The reference items include four aluminum plates (Fig. 4): A and B with11 rectangular slits, C and D with 8 rectangular slits. The nominal widthsof the slits for A and B were: 0.5, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0,and 10.0mm. The depth of the slits in the plate A was 5mm and in the plateB �10mm. The nominal depth of the slits C and D were: 1.0, 3.0, 5.0, 7.0,

6 L. TEPER ET AL.

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

9.0, 11.0, 13.0, and 15.0mm. The width of the slits in the plate C was 1mmand in the plate D �5mm.

Scanning of the items was performed by ultrasonic probe with 0.1mmstep. The probe produced by Harisonic (Catalog number: CM-1004-S). Theultrasonic probe had a diameter 0.25’’, focal length equal to 1.5’’, andnominal central frequency of 10MHz (Fig. 5).

The probe was driven by an ultrasonic analyzer, Panametrics model5601A=TT, and the signal was monitored by a TDS 320, Tektronix oscillo-scope. A sample of the resulting image is demonstrated in Fig. 6.

FIGURE 4. Aluminum plates with rectangular slits. (Figure appears in color online.)

FIGURE 5. Ultrasonic probe.

FIGURE 6. Resulting image from the ultrasonic scan.

RESTORATION OF THE SPREAD FUNCTION 7

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

The raw data as measured by the experimental system is provided inFig. 7. The data was then subjected to further analyses and processing.

7. DATA PROCESSING AND RESULTS

From the measured values, a digitized form of the function Om(a,x) wasconstructed to each of the 11 slits of both plates (A and B) and to each of the8 slits of plates (C and D). For each experimental set of values a smoothsymmetric function was fitted. To achieve the best match, the optimal formof the function was found:

Omða; xÞ � p1ðaÞ � expð�p2ðaÞ � x � p0ðaÞj jp3ðaÞÞ þ p4ðaÞ: ð10Þ

The pi are fitting parameters. The fitting procedure was accomplished usingMatlab software. The program for automatic data processing was written inMatlab. The internal Matlab function nlinfit was used for the Pi parametersestimation. This function implements algorithm of nonlinear least squaresregression. An illustration of the fitting is presented in Fig. 8.

The results of the fitting parameters for the slits of plate A are listed inTable 1.

Direct extrapolation procedure was used to derive L(x). First, a sequenceof x values was selected: x¼f0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4g. Foreach value of x, an interpolating polynomial ON (a,x) was plotted accordingto expression (10) as function of a. Fitting parameters from Table 1 wereapplied corresponding to each a value. The values obtained for ON (a,x)are plotted in Fig. 9.

We note that for the larger slits, the values of ON (a,x) coincide andshow monotonic decline. These slits are wider than the effective width of

FIGURE 7. Measured data for plate A. (Figure appears in color online.)

8 L. TEPER ET AL.

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

FIGURE 8. Result of the fitting procedure for the plate A, slit #5 (A5). (Figure appears in color online.)

TABLE 1 Fitting Parameters for the Plate A

Slit label Slit width a [mm] p1 p2 p3 p4 p0

A11 0.5 29.72 9.55E-01 1.23 6.35 �0.06A10 1 45.70 8.70E-01 1.33 6.00 �0.01A9 2 72.23 5.53E-01 1.75 6.25 �0.07A8 3 84.31 3.38E-01 2.11 6.70 �0.01A7 4 79.13 9.08E-02 2.84 6.70 �0.01A6 5 75.81 2.53E-02 3.79 7.08 0.08A5 6 36.94 1.37E-02 3.82 5.39 0.03A4 7 34.72 4.90E-03 4.17 4.61 0.01A3 8 34.30 1.95E-04 6.00 4.73 �0.05A2 9 30.79 2.63E-05 6.82 4.49 0.01A1 10 14.00 7.25E-07 8.61 4.28 0.06

FIGURE 9. ON(a,x) for plate A. (Figure appears in color online.)

RESTORATION OF THE SPREAD FUNCTION 9

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

the ultrasonic beam, so regardless of the width of the slits, the same energy isbounced back to the ultrasonic probe. The monotonic decline is due to thenormalization procedure, which depends on the ratio between the bouncingenergy and the width of the slits, the wider the slit, the smaller the ratio.

The values of ON(0,x) were obtained through extrapolation of the ON

(a,x) functions. For example, ON(0,1.4) equals 0.12.On the next step, the values of ON(0,x) were plotted against x, resulting

in the graph shown in Fig. 10. This plot is the response of the system to aninfinitesimal narrow slit, i.e., the LSF of the experimental system. Figure 10demonstrates the LSF function for positive values of x only. The completeLSF function is symmetric about the vertical axis.

The Instrument Function of the ultrasonic measurement system can bedescribed by

LðxÞ ¼ 0:513 � expð�ð0:933 � xj jÞ1:111Þ; ð11Þ

with a correlation of R2¼ 0.9997.The instrument function should be, by definition, of area that equals

unity. The area A of L(x) from Eq. (11) is calculated by means of the followingequation:

A ¼Z 1

�10:513 � expð�ð0:933 � xj jÞ1:111Þdx � 1: ð12Þ

FIGURE 10. Instrument function L(x) for the measurement system (plate A). (Figure appears in coloronline.)

10 L. TEPER ET AL.

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

At this point, then, L(x) represents the response of the experimental system toan infinitesimally narrow slit whose area is unity, which is actually theresponse to a d function. This is the sought instrument function of the system.

8. CONVOLUTION

Now that the transfer function has been defined by Eq. (11), the responseto a known slit can be calculated through a convolution procedure.

The convolution integral in general form is presented in the followingequation:

DmðxÞ ¼Z 1

�1DiðfÞLSFðx � fÞdf: ð13Þ

Inserting the instrument function of Eq. (11) into Eq. (13) yields

Dmðx; a;hÞ ¼ 0:513h

Z a2

�a2

expð�ð0:933 x � tj j1:111Þdt ; ð14Þ

where a is the width of the slit centered at x¼ 0, and h is the depth of the slit.Figures 11–13 demonstrate the response to three different slits, both

measured by the ultrasonic experimental system and calculated throughconvolution.

Note that while the general shapes of the calculated and experimentalresponses are similar, the amplitudes are different, so that the experimentalresponses had to be scaled-down in order to be plotted on the same figure.

FIGURE 11. Convolution (blue) versus measured (red) for slit A10. (Figure appears in color online.)

RESTORATION OF THE SPREAD FUNCTION 11

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

It turns out that for all the slits that are narrow than the effective width of theultrasonic beam, the scaling factor is 20 (up to width 5: A6:A11). For thoseslits that are larger than the effective width of the ultrasonic beam (A1:A5),the scaling factor becomes smaller, and varies with the width of the slits(nonconstant).

In conclusion, in addition to the LSF, a characteristic scaling factor is tobe taken into account, which is dependent on the geometrical dimensions ofthe slits.

A comparison between the areas of the calculated and experimentalresponses of the slits reveals, as well, the existence of the scaling factor which

FIGURE 12. Convolution (blue) versus measured (red) for slit A9. (Figure appears in color online.)

FIGURE 13. Convolution (blue) versus Measured (red) for slit A5. (Figure appears in color online.)

12 L. TEPER ET AL.

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

depends on the relative dimensions of the slits and the effective width of theultrasonic beam. For narrow slit, the scaling factor is linear and slightly increas-ing, with and average of about 20, while for slits larger than the ultrasonicbeam, this factor varies significantly. This pattern is demonstrated in Fig. 14.

It has been found that the slits that provide the most pronounced contri-bution to the derivation of the transfer function are those slits that are narrowenough, compared to the effective width of the ultrasonic beam.

Figure 15 shows the area scaling factors obtained for the slits of plate A(same depths, different widths) and those of plate C (same widths, differentdepths). The most pronounced detail in this figure is that for both plates, asimilar scaling factor was obtained for slits of the same area (depth � width).

Based on the ultrasonic experimental results, the observed scaling factorshould be accounted for, so that Eq. (1) is modified to read

OmðrÞ ¼ LðrÞ �OiðrÞ � SðgeometryÞ; ð15Þ

where S stands for the scaling factor.

FIGURE 14. Ratio between ideal to measured signals areas. (Figure appears in color online.)

FIGURE 15. Area ratio as function of width and height. (Figure appears in color online.)

RESTORATION OF THE SPREAD FUNCTION 13

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

9. RADIOGRAPHIC INSTRUMENT FUNCTION

To verify and validate the theory described above, a similar experimentwas conducted for a radiographic measurement system.

The concept of the LSF in radiography has been established in numerousstudies. The geometry scaling factor does not seem to influence the radiography

FIGURE 16. Radiography images of the slits of plate A. (Figure appears in color online.)

FIGURE 17. The experimental LSF of the radiography setup. (Figure appears in color online.)

14 L. TEPER ET AL.

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

images, so that the theory can be used towards a quick and direct calculation ofthe LSF.

The same approach as for the ultrasonic measurement system wasapplied to a radiography experimental setup. The radiography images arepresented in Fig. 16.

Following the some extrapolation procedures as discussed above, the LSFof the radiography setup was derived. It is plotted in Fig. 17.

The instrument function of the radiography measurement system can bedescribed by

LðxÞ ¼ 0:87 � expð�ð1:34 � xj jÞ2:2Þ: ð16Þ

Figure 18 demonstrates the correlation between the calculated and experi-mental areas of the slit images. As anticipated for radiography setups, thereis no apparent geometric effect.

The extrapolation theory described in this study can then be applied toboth radiography and ultrasonic experimental setups. While in radiographythe derived LSF can be used for image restoration, ultrasonic systems needto take into account an additional geometric factor as well, though the LSFdoes allow determining the shape of the images.

10. SUMMARY AND CONCLUSIONS

Characterization of the transfer function of experimental setups enables acomparison between several such measurement systems and the choice ofthe one which fits best to the task.

FIGURE 18. Correlation between measured and calculated areas of the slits (normalized). (Figure appearsin color online.)

RESTORATION OF THE SPREAD FUNCTION 15

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014

The interpolation method described in the work provides a mean for theexperimentally-based calculation of the characteristic transfer function of themeasurement system.

The interpolation method is based upon the scanning of reference slits ofknown dimensions. These very slits are to be used for the derivation of thetransfer function of the different systems under evaluation.

Knowledge of the transfer function enables not only the comparisonbetween different experimental setups, but the inverse deciphering of slitsdimensions through a deconvolution procedure involving the experimentalprofile of the slits and the characteristic LSF.

It has been found that the slits that provide the most pronounced contri-bution to the derivation of the transfer function are those slits that are narrowenough, compared to the effective width of the ultrasonic beam.

The extrapolation method was applied in this study to both ultrasonicand radiography measurement systems. While in radiography the derivedLSF can be used for image restoration, it turned out that for ultrasonic systemsan additional scaling factor should be accounted for. This factor depends onthe dimensions of the reference slits and the effective width of the ultrasonicbeam. However, the ultrasonic LSF does allow determining the shape of theimages.

REFERENCES

1. L. Teper, D. Ingman, P. Dickstein, and J. Suzdalnitsky. Meas. Sci. Technol. 16:775–789 (2005).2. P. D. Burstein. LSF restoration by Neural Networks. Research Thesis. Technion, Israel Institute of

Technology (1993).3. A. Notea. NDT International 21:379–384 (1988).4. P. G. Kenny, J. J. Gruber, and J. J. Smith. Mat. Eval. 45:99–104 (1986).5. P. A. Dickstein, A. N. Sinclair, Y. Bushlin, and D. Ingman. Journal of Research in Nondestructive

Testing 2:29–43 (1990).6. A. Papoulis. Systems and Transforms with Applications in Optics; McGraw-Hill, New York, 1968.7. Y. Segal and F. Trichter. NDT International 21:11–16 (1988).8. E. Segal, A. Notea, and Y. Segal. Mat. Eval. 40:1268–1272 (1982).9. A. Fishman, U. Feldman, A. Notea, and Y. Segal. Mat. Eval. 41:1201–1207 (1983).10. Y. Segal, P. Dickstein, D. Ingman, and E. Segal. Technion Report TNED-R=674 (1985).11. R. Halmshaw. Industrial Radiology: Theory and Practice; Applied Science Publications, London

(1982).12. A. A. Harms. At. Energy Rev. 15:143 (1977).

16 L. TEPER ET AL.

Dow

nloa

ded

by [

Uni

vers

itat P

olitè

cnic

a de

Val

ènci

a] a

t 16:

09 2

4 O

ctob

er 2

014