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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 54, NO. 4, AUGUST 2007 12

Radiographic Testing of Anomalies in ThickMetal Components: Fitting the Standard

Line-Integral ModelMarc C. Robini , Member, IEEE , and Isabelle E. Magnin , Member, IEEE

Abstract— Radiography is an effective method for nondestruc-tive inspection of thick metal components. However, completediagnosis of internal defects is a difcult task that needs reliablecomputer assistance. In particular, fully 3-D reconstruction wouldbe a very valuable tool for the expert, but it is not so clear whetherdigitized radiographs can be made suitable for reconstruction.We show that this is indeed the case by studying the radiographicimage formation process in the context of primary loop pipinginspection in pressurized water nuclear reactors. In a more gen-eral way, our ndings apply to the testing of thick metal pecimens

with simple, known geometry. Based on justied simplicationsand approximations, we demonstrate that it is possible to processthe raw data to closely t the conventional line-integral projectionmodel. More specically, we provide a full processing procedurethat includes (i) a criterion for subsampling the data without lossof pertinent information, (ii) a novel eld-attening algorithm, and(iii) a calibration method that requires minimal knowledge aboutthe data acquisition parameters. The actual 3-D reconstructionissue is addressed in another paper [1] whose results furthervalidate the present work.

Index Terms— Data models, eld-attening, image processing,nondestructive testing, nuclear imaging, radiography, scattering.

I. INTRODUCTION

A. Motivation and Contribution

O UR study is motivated by a nondestructive evaluation taskthat plays a vital role in the nuclear power plant main-

tenance program of the EDF group, namely the radiographicinspection of aws in casted elbows of the primary loop of pressurized water reactors [2]. These components undergo reg-ular checks, as they are subject to severe temperature and pres-sure conditions that may lead to structural aw formation orworsen existing anomalies. The testing procedure consists inintroducing a gamma-ray source inside the inspected pipe andrecording the outgoing radiation by means of X-ray lms at-tened against the external surface. Because of the large spec-imen thickness (about 7–8 cm), the angle of incidence is lim-

Manuscript received July 12, 2006; revised April 27, 2007. This work wassupported by Électricité de France who also supplied the data and associatedhuman expert know-how. This material was presented in part at the IEEE Nu-clear Science Symposium, SanDiego, CA,November 2007. This work hasbeenperformedin accordance with thescientic trendsof thenationalresearchgroupGDR ISIS of the CNRS.

The authors are with CREATIS (CNRS Research Unit UMR5520 and IN-SERM Research Unit U630), INSA Lyon, 69621 Villeurbanne cedex, France(e-mail: [email protected]).

Digital Object Identier 10.1109/TNS.2007.901219

ited and only ve to seven radiographs are generally availablfor a given defect. Consequently, building up a complete diagnosis includingprecise localization, orientation,andshape characterization of aws is a difcult task that needs reliable computer assistance. The issue at stake here is to demonstrate thfeasibility of reconstructing the 3-D shape of voids (such as gaholes,airlocks, shrinkagecavities,or cracks) by considering reatest data gathered under conditions similar to those met on sit

This turns out to be a great challenge since dealing with a fe2-D projections collected under a limited angle of incidence inot the only difculty: rst, due to scattered radiation predominance, and because the aw size is small with respect to thspecimen thickness, both thesignal-to-noiseratio (SNR) andthcontrast are low; second, thescattered radiationand, in ourcasethe digitization apparatus produce unknown, smooth gradientof luminosity which contaminate the attenuation data; third, thprojections are not calibrated in the sense that the aw informtion signals are weighted by unknown constants.

Let , , be a set of 2-Ddigitized projections available for reconstruction, where denotes a rectangular pixel lattice. Manyauthors (see, e.g., [3]–[6for closely related problems) resort to the standard direct mod

(1)

where the eld , dened over a 3-D voxellattice , characterizes the shape of the defects, is a lineamap from to , and is Gaussian white noise withmeanzeroand variance (whichwedenote bywhether is a constant eld or not). However, to our knowedge, no attempt was made to rigorously justify this model fothe testing of thick metal components in the context of a reaindustrial application. This task constitutes the main focus othis paper; the actual 3-D reconstruction issue is discussed i[1]. Assuming that the inspected object geometry is known, wshow that, at the expense of some model-driven processing othe data, one can indeed come up with a set of projections of thform

(2)

for someconstant closeto one and independent of , where theeld is directly connected to the attenuation of a monochromatic radiation by the “negative” of the aws (i.e., by metal ob jects taking the shape of the defects). These projections t weintomodel (1) for small size awssuchas those wewish tochar

acterize.0018-9499/$25.00 © 2007 IEEE



1286 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 54, NO. 4, AUGUST 2007

B. Overview

In Section II, we provide a brief description of the experi-mental setup — the real data presented therein will be usedthroughout the rest of the paper. Section III is devoted to theanalysis of the image formation process, except for scatteredradiation which is studied in Section IV. We show that the lm

exposure satis es

(3)

where is the exposure time, denotes the irradiance pro-duced by scattered photons, stands for the primary irra-diance that would be observed in the absence of defect, and

is the aw information signal. In addition, we establish thatthe common logarithm of the available digital radiographs, say

, is related to through an expression of the form

where is an unknown, smoothly varying eld resulting fromdigitization and , are unknown positive constants. The pri-mary irradiance in (3) can be easily estimated, but the scat-tered component is a real nuisance for extracting . How-ever, it is intuitively reasonable to assume that is not sensi-tive to moderate size aws. Indeed, starting with a single-scatterapproximation, it is shown in Section IV that the term

can be approximated by the spatial distributionof the build-up factor that would be observed if the inspectedobject were awless.

Section V discusses how to obtain projection data of type (2)from our model:

In the prospect of reducing computational costs, we rst demon-strate that the digitized radiographs can be subsampled withoutloss of useful aw information. Strictly speaking, we give anupper bound on the subsampling factor that permits one to re-tain the aw information located in the frequency regions wherethe Fourier domain SNR is greater than one. Next, we pro-pose a novel eld- attening algorithm to remove the unknown,smooth component . This is a quite dif-cult task, as the support of is not known a priori and asthe spectral ranges of and signi cantly overlap. Our al-gorithm performs well even for very low SNR data; it leads to

projections of the form for some set of constants, where is a nonstationary, zero-mean,Gaussian white noise eld. The last stage is to get rid of the

’s up to a multiplicative constant preferably close to one.For this purpose, we develop a calibration procedure that doesnot depend on lm characteristics, exposure time, or source ac-tivity. This ultimately leads to projections of type (2), as con-rmed by comparing simulations to experimental results. Thefull processing procedure is summarized in Section VI, wherewe also give concluding remarks about how to obtain model (1)from (2).

II. EXPERIMENTAL SETUP

Gamma-ray data have been collected under the constraintsimposed by on-site testing conditions. The inspected specimens

Fig. 1. Data acquisition geometry.

are nitrogen-strengthened austenitic stainless steel castings withuniform thickness of 7 or 8 cm including arti cial aws with

known shape, dimensions, and localization. In accordance withFig. 1, the source positions are in a plane parallelto the base of the specimen. The lm cassettes are taped on thespecimen and the source- lm distance is xed by the externaldiameter of the pipes to be inspected. Within the source plane,

lie on a circle whose center and whose diam-eter are chosen to ensure an upper thickness limit of 110% of thespecimen thickness in the projection regions of interest. There-fore, the angle of incidence is less than 25 degrees. Each lmcassette contains a 2 mm lead lter and is backed with a 3 mmlead screen for protection against backscatter. In addition, thelms are sandwiched between 0.2 mm lead intensifying screens

whose main effect is to convert part of the ionizing radiation intoelectrons that assist in the formation of the latent image. Thesources are 3 mm 3 mm iridium-192 cylindrical pellets withstrength close to 100 Ci and the exposure times varies between 2and 5 hours as a function of source position and specimen thick-ness. Note that on-site inspections are scheduled during shut-down periods that are long enough to assume constant temper-ature and pressure conditions.

After lm development, digitization is achieved by placinglms on an illuminated screen and recording the transmittedirradiance using a CCD array camera. This system allowsone to obtain 5 cm square projections with spatial resolution

, some examples of which are given in Fig. 2. Theradiographs in Figs. 2(a) –(d) show an open notch, fabricatedby electro-erosion, simulating a crack with a large depth/width



ROBINI AND MAGNIN: RADIOGRAPHIC TESTING OF ANOMALIES IN THICK METAL COMPONENTS 1287

Fig. 2. Digital radiograph examples: (a) –(d) 2 mm 2 mm electro-eroded notch; (e) –(h) 2 mm diameter drilled hole.

ratio. Figs. 2(e) –(h) are views of a 2 mm diameter hole parallelto the base of the specimen at a depth of 10 mm on the sourceside. Note that the smooth background intensity variations arenot representative of metal thickness, as the incident irradianceproduced by the illuminator of the digitization apparatus is notconstant. This could be avoided by using a laser digitizer, butthere was not one available at the time of the experiments (thedata acquisition process was outside our control).

III. IMAGE FORMATION MODEL

The data model we develop here is dedicated to the exper-

imental setup described in Section II, but it can be easily cus-tomized to other radiographic inspection systems for thick metalcomponents. The results presented in Sections III-A and III-Bare to be interpreted as statistical averages; deviations from thismean behavior is the subject of Section III-C.

A. The Radiant Image

When the detector is an X-ray lm, the information to becaptured is the energy uence that exposed the emulsion. Thisquantity, called exposure and denoted by , is obtained by in-tegrating the irradiance (i.e., the energy uence rate) over theexposure period. For iridium-192 sources, the radioactive decayover a several hours period is negligible and the relevant formsof interactions of photons with matter are photoelectric absorp-tion and scattering. Hence, for any point in the detector plane

(4)

where is the exposure time and and denote the en-ergy uence rates respectively produced by primary (i.e., un-scattered) and scattered photons. The remainder of this subsec-tion is devoted to the characterization of the primary component(scattered radiation is studied in Section IV).

Consider the geometry shown in Fig. 3. For any point inthe source , is the distance from to the detector plane

and, given any point , denotes the distancefrom to . Let be the total linear attenuation coef -cient at point for energy ( is Planck ’s constant) and let

Fig. 3. Geometry used for modeling the primary irradiance.

be the line integral running from to . The pri-mary irradiance can be written as

where is the volume of the source, is the source activity, anddenotes the probability that a disintegration is accompanied

by the emission of a photon of energy . Our experimentalconditions allows us to make useful simpli cations.

• First, letting (respectively, ) be the total linear atten-uation coef cient of the material composing the inspectedobject (respectively, the lter), we have




where , and respectively de-note the lengths of intersection between line segmentand the awless specimen, the aw(s) and the lter. In ourcase, and , where and arethe total linear attenuation coef cients of iron and lead.

• Second, because the source dimensions are very small

compared to the source- lm distance, we can setandover the projection region of interest . To the same orderof approximation, since the inspected specimen and thelter have uniform thickness,and .

Taking this into account, we obtain

(5)

with

The limited angle of incidence opens the way to another sim-plication: for any possible source position, we can nd and

such that

(6)

In other words, from the primary irradiance standpoint, thereal source behaves like a monochromatic source with activity

and frequency . Finding and is an easy to solvenonlinear optimization task which can be performed regardlessof the shape of the defects [2]. For instance, Fig. 4 displaysthe results associated with the testing conditions describedin Section II in the least favorable situation (i.e., when themean angle of incidence over is 25 degrees). The estimatedquantities and the corresponding relative error

are plotted versus . For small

defects (e.g., ), is of the order of and thenumeric values of and stay close to those obtained fora awless specimen (i.e., for ). This shows that theproposed approximation is sharp and that the speci cation of the equivalent sources can be accomplished on the sole basis of the specimen geometry.

From (4), (5) and (6), the exposure can be described by

(7)

where

(8)

Fig. 4. (a) Equivalent monochromatic source characteristics and (b) associatedrelative error versus maximum path length through defect cavities.

conveys the pertinent aw information, and where denotesthe primary irradiance that would be “observed ” in the absenceof defect:

where

(9)

( is the ratio of the mean number of primary photonsintercepted by the detector element to the number of pho-tons emitted by the equivalent monochromatic source.)

B. Detection and Digitization

The optical density, , of a developed X-ray lm depends

linearly on exposure up to (see, e.g., [7] –[9]). Theexposure time is typically determined so that this relationshipholds within the projection region of interest. Thus,

(10)

where the constant measures lm sensitivity and wheredenotes the fog density resulting from the in-

herent density of the base of the lm together with the presenceof self-developable grains in the emulsion. In fact, because of the blurring due to the scattering of electrons in the emulsion,it would be more accurate to write




where stands for the 2-D convolution operator and is alow-pass lter impulse response. However, this blurring effectis negligible compared to the geometric blurring due to nitesource size (the former is of the order of 0.1 mm and the latteris of the order of 1 mm).

As regards digitization, the CCD camera imperfections can

be safely overlooked, but the intensity variations produced bythe illuminator have to be taken into account. Hence, the avail-able numerical data consist of projections

of the form

(11)

where is an unknown, smoothly varying eld, anddenotes the density measured over the square domain

(12)

whose size is the spatial resolution of the digitization system.

C. Noise Characteristics

Let be the mean number of photons per unit area andunit time (i.e., the uence rate) impinging on the detector.During the exposure period, photon detection can be describedby the spatial random process

where is the Dirac delta function, is the total number of detected photons, and the 2-D vectors are independent andidentically distributed random variables with probability den-sity function . Restricting ourselves to themonochromatic case, the derivation of a noise model for ourradiographic images starts with the characterization of the non-stationary discrete random eld

(13)

where is given in (12). Let be the quantum ef ciencyof the detector (i.e., the probability that an incident photoncontributes to latent image formation) and let be the total

number of photons incident on the emulsion. Clearly, isa binomial random variable with parameters and . Sincecan be assimilated to a poisson random variable with mean

, which we denote by , wededuce that . It follows that

with

In addition, it is not dif cult to check that (13) is uncorrelated:

where is the Kronecker delta symbol.The above description can be further simpli ed when using

very ne grain lms. For an average density of about 2 –3 anda spatial resolution, the number of developedgrains in the volume of emulsion under is of the order of

(which is the reason whygranularity is not considered here).Also, for the principal energy spectrum lines of iridium-192(i.e., between 0.3 MeV and 0.6 MeV) the average number of halide grains that are made developable for each quantum ab-sorbed ranges between 20 and 40. Therefore, is largeenough to fully justify a Gaussian approximation

where . Since is a linear function of in the range of linearity of the density-exposure re-

lation, the same kind of approximation holds for , butwith correlated. Indeed, two-dimensionally, the set of grainsthat are rendered developable for one quantum absorbed is con-ned to a small disc whose diameter, of the order of 0.1 mm, isgreater than the spatial resolution. Because is a linearfunction of with mean behavior described by (10), wededuce that

(14)

for some positive constant . It should be stressed that the mag-nitude of geometric blurring is large, as on-site testing con-ditions impose small source- lm distances. Consequently, thecut-off frequency of the power spectrum of density uctuationsis greater than the useful bandwidth of the mean density distri-bution . Hence we can actually assume that

in (14).

IV. S CATTERED RADIATION

As suggested by (7), scattered radiation does not contribute tovaluable aw information, but rather reduces the contrast of theradiographs. Moreover, scattering is the dominant interaction insteel and other metallic materials for photons in the iridium-192spectrum. For large specimen thickness, most photons undergoseveral scattering events before reaching the detector or beingdestroyed by photoelectric absorption. In this section, we showthat, in comparison with the primary component, the aw in-formation content in the scattered component is so poor as tobe disregarded. In a rst analysis, we assume that each detectedphoton experiences at most one scattering event; this suggests auseful approximation that is validated by Monte Carlo simula-tion.

Let us focus on the simple transmission imaging geometryshown in Fig. 5, where and are vectors in the detector plane

with equation . Without loss of generality, we assumethat the scattering medium is between the planes and

. Consider a thin pencil of rays of frequency and cross-

sectional area , and let be the global attenuationfactor along line segment , that is, .




A volume element situated along the pencil at adistance from receives the primary uence rate

where is the incident uence rate, and the uence rate perunit solid angle scattered by is

(15)

where is the spatial distribution of atom density and whereand are respectively the differential

Rayleigh and Compton scattering cross sections per atom [10](both quantities depend on and ). The number of pho-tons scattered in the direction of a detector element of arealocated at is obtained by multiplying (15) with the differ-ential solid angle , where

Moreover, these photons areattenuated before they emergefromthe inspected object. Therefore, under the single-scatter approx-imation, the amount of energy scattered by and received by

per unit time is

(16

where and where denotes the frequencyof Compton photons de ected through angle , that is,

with ( is the electronrest-mass and is the velocity of light). Then, if we assumea distant point source, integrating (16) over and gives thetotal amount of scattered energy received by per unit time,that is, . It comes up that can be connectedto the primary irradiance via an inhomogeneous Fredholmequation of the rst kind:

(17)

where and where thekernel is given by

(18)

Let us apply this result to the arrangement shown in Fig. 6(a),which is closely representative of the acquisition geometry de-scribed in Section II. The inspected iron specimen, with uni-form thickness of 7 cm, includes a spherical defect of radius5 mm (the spherical shape assumption simpli es computations).Clearly, if the distance from to is large enough (e.g., of theorder of the sample thickness), then for all

, where denotes the shift-invariant point spread function ob-

Fig. 5. Simple transmission imaging arrangement in which a scatteringmedium (the inspected object) is irradiated by a thin pencil of rays.

tained from (18) in the case of a awless specimen. In fact, themost marked difference between and is observed for. These kernels are displayed in Fig. 6(b) for the two extreme

spectrum lines of iridium-192 with transition probability greaterthan 0.05% (i.e., MeV and MeV). Inboth cases, and are close enough to con-clude that is a suitable approximation forall and for all . It follows that (17) is well described by aconvolution whose modulation transfer function is depicted inFig. 6(c) for MeV and MeV. The cor-responding 20 dB cut-off frequencies (about 0.11 mm and0.06 mm ) indicate that the unscattered defect signal is atten-uated by a factor of at least 10 if one of its local dimensions issmaller than 9 mm, which is the case in practice.

The preceding observations show that, under the single-scatter approximation, for any spectrum line of iridium-192,the aw information to be found in the scattered componentis substantially smoothed out and may therefore be neglected.Since the real (i.e., multiple-scatter) point spread function isbroader than , there is reasonable evidence that ,where denotes the scattered irradiance in the absence of defect. To substantiate this approximation, we simulated theexposure producing the radiograph in Fig. 2(a) by Monte Carlocalculation. Without going into details, we considered a 230

230 detector array in contact with a 2 mm lead lter and

we computed the trajectories of photons. The resultingprimary and scattered components are shown in Fig. 7(a) andFig. 7(b), respectively, while Fig. 7(c) compares the associatedrow-means over the regions delimited by the dashed lines. Themagnitude of the little bump in the scatter pro le is ten timessmaller than the magnitude of the primary aw signal. Usingthis ratio, we can estimate the value of the peak signal-to-noiseratio (PSNR) of the scattered aw signal in the real radiograph,which is about 3.5 dB PSNR , whereis the peak-to-peak value of the considered signal and is thestandard deviation of the noise). This example suggests that, asa rst approximation, the eld




Fig. 6. Single-scatter approximation: (a) schematic of transmission imaginggeometry; (b) point spread functions from (18) (dashed lines)and theirshift-invariant approximations 0 (solid lines) for MeVand MeV; (c) modulation transfer functions of the shift-invariantapproximations in (b).

in (7) can be replaced by , that is, by the spa-tial distribution of the build-up factor corresponding to the aw-less specimen. This approximation is more accurate in real sit-uations, since the real defects of interest produce signals withlower SNR. However, should it be questionable whether thescattered aw signal can be ignored, we take advantage of thefact that is substantially smoother than .

V. FROM RAW RADIOGRAPHS TO LINE -INTEGRAL PROJECTIONS

Let us review the present situation. From (7), and consideringthe conclusion of Section IV, we have

Fig. 7. Monte Carlo simulation of the radiograph shown in Fig. 2(a): (a) pri-mary exposure ; (b) scattered exposure ; (c) means of the rows lo-cated between the dashed lines.

where is the amount of energy released bythe equivalent monochromatic source during exposure, isgiven by (9), and the notation ( , , or )stands for . Hence, from (11) and (14), thecommon logarithm of a digitized radiograph can be describedby

(19)

where is a smooth distribution representing the drift intro-duced by and by the digitization system, and where

. In this model, and are unknownelds, and can be estimated based on the knowledge of the specimen geometry, and , and are unknown scalars.

This section is devoted to the elaboration of processing tech-niques leading to a set of reduced size pro-

jections of the form(20)

where is a constant close to one and independent of , andwhere is some noise component to be characterized. We rstshow in Section V-A that the original data (19) can be subsam-pled without loss of aw information. Then, in Section V-B, wepropose a novel eld- attening algorithm which allows us to ob-tain projections of the form (20), but with depending on . Theremoval of this residual dependency is discussed in Section V-C.

A. Subsampling

The useful bandwidth of the aw signal is very limited be-cause of geometric blurring as well as the presence of noise. We




can therefore reasonably assume that the available digital radio-graphs can be subsampled without risk. But how far can we go?Up to a multiplicative constant, dividing (19) by leads to thesuperposition of a very low frequency drift with the aw infor-mation component corrupted by some noise .Reasoning in the continuous domain and temporarily assuming

that is constant, the Fourier domain signal-to-noise ratiocan be de ned by , whereand where and denote the power spectral densi-

ties of and , respectively. Since is a transient signal,we have , where is the area of thesupport of and stands for the 2-D Fourier transform of

. Besides, because the noise cut-off frequency is located be-yond the bandwidth of , we can model by a band-limitedwhite noise with density , where

if , 0 otherwise. Hence, set-ting , we obtain

for all in the Nyquist domain.The information located in the frequency domain

is of no use for 3-D reconstruc-tion. Consequently, if there exists a positive integer such that

(21)

then the corresponding digitized radiograph can be subsampledby a factor of . In order to safely estimate , we considerthe very unfavorable (arti cial) situation where the gamma-raysource and the defect are cuboids with edges parallel to the axesof a rectangular coordinate system whose -plane coincideswith the detector plane. Let be the area of the -base of the source and let be the maximum angle of incidence (25degrees in our case). Under approximation (8), it can be shown[2] that

with

where and respectively denote the minimum and max-

imum -coordinates of the defect and is the distance fromthe source to the detector plane. It follows that

(22)

is a suf cient condition for (21) to hold. Note that, as we mightexpect, the upper bound is an increasing function of thesource dimensions, the sampling rate and the noise standarddeviation. As an example, is depicted in Fig. 8 as afunction of and for the testing conditions associatedwith radiographs (a) –(d) in Fig. 2. The corresponding pairs of values for and are located between the level curves

and , which showsthat subsampling by a factor of 4 does not affect the aw

Fig. 8. Upper bound (22) for the logarithm to base 2 of the subsamplingfactor as a function of the area of the support of 7 and of the minimum noisestandard deviation. The “2 ” marks correspond to radiographs (a) –(d) in Fig. 2.

information content (similar observations were made for radio-

graphs (e) –(h) in Fig. 2). In practice, subsampling is achievedby keeping the low resolution residual stemming from subbanddecomposition of the original data (we use Johnston ’s lters[11]). This operation leaves description (19) unchanged, exceptfor the noise standard deviation which must be divided by thesubsampling factor.

B. Field-Flattening

Our goal here is to get rid of the smooth eld in (19) so asto give access to . We start from the data obtained by dividing

by , that is, projections of the form

(23)

where , , andwith (the factor of is due tosubsampling). Since metal thickness does not vary signi cantlyover the projection region of interest, is nearly constant out-side the support of and its immediate neighborhood(for instance for thesimulation results depicted in Fig. 7,

if ). Consequently, approximatelyreduces to a constant, say , outside , whereaswithin . With this in mind, the eld- attening operation is toproduce an estimate for in order to obtain projections pro-

portional to according to

(24)

where . This is a non-trivialestimation task, not only because the spectra of the unknownelds and signi cantly overlap, but also becausecannot be assumed to be known in the prospect of automaticprocessing. In particular, spatial low-pass ltering is ineffectiveand standard polynomial tting techniques produce biased es-timations. Alternatively, a sophisticated method that apparentlyyields good results has been proposed in [12] and [13], but theresulting algorithm is cumbersome and has many parameters,some of which are dif cult to set. For these reasons, we proposea new eld- attening approach that jointly estimates and




together with a continuous-valued representation of . Thecorresponding algorithm is fairly simple to implement andinvolves few parameters that can be set without dif culty.

Let be the regular grid supporting .Since is the sampled version of a smooth surface, we identifyit with the bicubic spline interpolant of a reduced size represen-

tation de ned on a coarse grid withnode spacing ; we shall use the notation forshort. Assume for the moment that is known and let bethe eld de ned by

if otherwise

where the constant assures that . Let usput

and

where stands for the 2-D discrete convolution operator overand where is a Gaussian function with standard deviationsatisfying . Then, can be approxi-

mated by with

(25)

where the regularization term is the discrete version of thethin plate spline functional (see, e.g., [14], [15]) and the free pa-rameter adjusts the degree of stabilization of the solution.In practice, this minimization task can be solved ef ciently bymeans of fast surface approximation algorithms such as the onesdescribed in [16], [17] or [18]. We also have with

(26)

In the case where is unknown, these observations suggestto design an iterative algorithm that jointly evaluates andthrough alternate updating of the associated estimates togetherwith a hidden eld representing the accumulated knowledgeabout . Let us respectively denote the estimates for , and

at step by , and . Appealing to de nitions (25) and(26), and given , the proposed algorithm is of the fol-lowing form:

(27)

The r ole o f the c onstant is t o guarantee t hatand the updating mechanism is

designed to relax the hidden eld at locations that are likely

to be within . The precise speci cation of relies onthe de nition of the local noise standard deviation estimate

together with an additional parameter that expresses

the noise standard deviation tolerance outside . De-noting the total number of iterations of the algorithm by ,we wish to have for all

. To see how this can be achieved, note that we alwayshave either or because of the bias produced bythe aw signal together with the fact that is a very smoothdiscrete surface. Hence, means that

is likely to be in so that should be relaxedat this location. Moreover, in such a situation, the greater the dif-ference , the more should beloosened. There are, of course, many ways to get this behavior.We suggest the following point-wise updating scheme:

if otherwise

where .The algorithm described above involves four parameters,

namely the node spacing of the coarse grid , the regular-ization parameter in (25), the positive real xing the stopcriterion, and the noise standard deviation tolerance . Thechoice of these parameters does not pose any special problemunder testing conditions similar to those described in Section II.

For instance, a 20 20 grid is suf cient to capture thevariations of the drift . Also, because of prior smoothing by

, the solutions we are looking for via (25) are very close tothe ’s at those locations where ; consequently,setting is enough for the resulting surfaces to t thethin plate constraint tightly (note that can always be selectedby means of generalized cross-validation [19] or by using theL-curve method [20] at the expense of a substantial increasein computational load). Finally, choosing givessatisfactory precision and setting permits one tocope with low contrast aw information.

As a rst example, let us consider the case of projection

(d) in Fig. 2. In accordance with the results of Section V-A,the associated input projection (23) shown in Fig. 9(a)was computed after subsampling by a factor of 4. Usingthe proposed set of parameter values, our algorithm took 90iterations to produce the estimates for and respectivelydepicted in Figs. 9(b) and (c). The absolute value of the re-sulting projection (24) is displayed in Fig. 9(d). A fewinsigni cant irregularities caused by lm manipulation arevisible. Aside from these, the estimated hidden eld provides afaithful representation of that can be used for computing a3-D region of interest for reconstruction. The second exampledepicted in Fig. 10 concerns projection (c) in Fig. 2; the resultsdisplayed in Figs. 10(b) and (c) were obtained in 63 iterations.Fig. 10(d) superimposes the intensity pro les of the inputprojection and the estimate for along the indicated dashed




Fig. 9. Application of the eld- attening algorithm to projection (d) in Fig. 2: (a) associated input projection (23); (b) estimate for ; (c) estimate for ; (d)absolute value of the resulting projection (24).

Fig. 10. Application of the eld- attening algorithm to projection (c) in Fig. 2: (a) associated input projection (23); (b) estimate for ; (c) absolute value of the resulting projection (24); (d) intensity pro les of (a) and (b) along the dashed lines.

lines. It shows that the algorithm behaves well in very lowconstrast situations.

C. Calibration

The eld- attening process leads to a setof projections of the form

(28)

where . While the lm sensitivity measure obvi-ously doesnotdepend on , itis not soclearthat isthesame foreach projection. In fact, is de ned by the quantum ef ciencyof the detector and by the mean number of halide grains that aremade developable per absorbed quantum. Strictly speaking, be-cause of the latter dependency, is a function of both incidenceangle and metal thickness and hence varies with . However,within our framework, the angle of incidence and the specimenthickness latitude are limited enough to disregard the variationsin between any two projections of a same set.

We want to remove the unknown factor in (28) up tosome multiplicative constant independent of and close to one.Recall that our eld- attening algorithm provides an estimatefor the noise standard deviation in (23) outside . Morespeci cally, for each , we have the value of

(29)

where stands for the mean build-up factor over the supportof projection . Since can be estimated via Monte Carlo

simulation in a reasonable amount of time, we choose to mul-tiply each by , which leads to projection data of

the form plus some noise. We are then left withthe problem of estimating . Because we are not actually

looking for a very sharp estimate, we can resort to mean be-havior approximations. Let us drop the index for a moment.According to (14), the standard deviation of is approx-imately equal to and thuswhere and respectively stand for the mean standarddeviation of the density and the mean exposure over .Hence, from (10),

where is the average lm density xed by testing speci ca-tions (about 2.5 in our case). Substituting into (29) gives

(30)

At the same time, relations (11) and (23) show that is also anestimate for the mean standard deviation of the uctuations of

outside . Therefore

(31)

where . It follows from (30) and(31) that

(32)

The value of can be estimated from either (29) or (32).Setting the corresponding expressions equal to each other, weobtain

(33)




Fig. 11. Processing of projections (a) –(d) in Fig. 2: (a) –(d) nal outputs; (e) –(h) representation of the corresponding simulated aw signals using the same graylevel mappings (the background intensity value is zero).

Fig. 12. Processing of projections (e) –(h) in Fig. 2: (a) –(d) nal outputs; (e) –(h) representation of the corresponding simulated aw signals using the same graylevel mappings.

In the end, multiplying each (28) by pro-duces data of the required form, that is, a set

of projections such that

(34)

where is close to one and . To con rm thisresult, Figs. 11(a) –(d) and 12(a) –(d) show the outputs of thefull processing procedure for the raw data in Fig. 2. The cor-responding simulations of (see (8)) are respectively represented in Figs. 11(e) –(h) and12(e) –(h) using the same gray level mappings for suitablecomparison. Apart from noise, each processed projection isvery close to its simulated counterpart, which gives strongcredit to our image formation model as well as to the proposedprocessing techniques. Note that if the metal thickness exhibitsmoderate variations over the projection regions of interest, then

so do and , and it follows that the noise componentin (34) is approximately stationary outside . Besides,although is unknown, the mean noise variance outsidecan be easily estimated by appealing to the description of provided by the eld- attening algorithm.

VI. S UMMARY AND CONCLUSION

We have proposed a complete radiographic image formationmodel together with a detailed processing procedure to set thebasis for 3-D reconstruction of defects in thick metal compo-nents. The raw data consists of a set of projections whose common logarithm has been shown to sat-isfy

where is Gaussian white noise with mean zero and variance(we refer to the very beginning of Section V




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