Detection and localization of a slab by a linearlike δ approach

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R. Solimene and R. Pierri Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. A 2661

Detection and localization of a slab by a linearlike� approach

Raffaele Solimene and Rocco Pierri*

Dipartimento di Ingegneria dell’Informazione, Seconda Università di Napoli, Via Roma 29, I-81031, Aversa, Italy*Corresponding author: [email protected]

Received June 19, 2006; revised March 7, 2007; accepted April 12, 2007;posted May 7, 2007 (Doc. ID 72135); published July 30, 2007

The problem of the detection and localization of a dielectric slab under a multifrequency plane-wave illumina-tion is addressed. A linear inversion scheme, based on �-functions, whose support represents the positions ofthe two slab interfaces, and on the truncated singular-value decomposition is exploited. The closed-form deri-vation of the amplitude of the reconstruction at the slab interface positions when the inversion scheme acts onexact model data allows us to analyze by means of analytical results the role played by the parameters of theslab, the frequency bandwidth, and the noise on the achievable probability of detection. © 2007 Optical Soci-ety of America

OCIS codes: 290.3200, 100.3190, 100.3010.

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. INTRODUCTIONne-dimensional inverse-scattering [1] problems arise inany fields of applied science. Just to quote a few ex-

mples, they can be encountered in meteorological explo-ation [2], in the design of optical waveguides [3], and inivil engineering applications [4]. Many solution strate-ies, which rely on either exact or approximate modeling,an be found in the literature.

A first class of inversion techniques consists of thosehat transform the reduced Helmholtz equation, perti-ent to the one-dimensional inverse-scattering problem,o a Schrödinger-like equation so that the solution can beound via the Gel’fand–Levitan method. These techniquesllow us to reach analytical solutions but in principle re-uire that the data be collected over an infinite band-idth in the wavenumber domain and allow the closed-

orm solution to be obtained only for particular profiles5].

Exact models lead to a nonlinear and ill-posed inver-ion [6], as in the case of the Riccati equation [7], or toterative minimization procedures of suitable cost func-ionals accounting for the mismatch between the scat-ered field data and those predicted by the model [8].owever, due to the nonlinearity, the inversion procedure

an arrest within a local minimum of the cost functionali.e., the so-called false solutions) [9].

The problem is drastically simplified if the scatteringquations are linearized [4,10–12]. Linear inversionchemes based on the so-called Born approximation [4], orven on the Rytov approximation [12], are widely em-loyed. This is because linear inversion schemes allow uso avoid the local minima problems, are computationallyfficient, and can recover the stability against noise by auitable regularization procedure. However, quantitativeeconstructions are obtained only for a restricted class ofrofiles that are slowly varying [10] or meet the conditionf a weak scatterer [13]. To overcome this shortcoming,igh-order inversion schemes, which retain another term

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f the Born series beyond the linear one, have proved use-ul [14,15]. However, these techniques also require that aonlinear inversion be performed and thus in general suf-er from the above-mentioned drawbacks due to the non-inearity.

In this paper the problem of detecting a one-imensional inhomogeneity is addressed. In particular,e are interested in detecting the discontinuities of a di-lectric profile. The applications we have in mind concernhe problem of detecting cracks occurring in masonrytructures or the detection of a subsurface void layer. Tohis end, as a preliminary step toward more realistic situ-tions, we consider the canonical case of a lossless three-ayered medium where the slab to be detected is represen-end by the second layer. The scattered field collectednder a multifrequency plane-wave illumination over a fi-ite bandwidth constitutes the data of the problem. Inarticular, the emphasis is on establishing the role playedy the adopted frequency bandwidth, the thickness andhe dielectric permittivity of the inhomogeneity, and theoise in detecting the slab.In order to carry out such a task, we adopt a linear in-

ersion scheme. More specifically, the slab reflection coef-cient is first approximated by neglecting the mutual re-ection occurring between the two slab interfaces.oreover, by following the idea outlined in [16], where a

ingular distribution supported over the surface of an ex-ended scattering object is used to represent its shape, weepresent the unknown positions of the two interfaces ashe support of two �-functions [17]. This allows us to casthe problem as the inversion of a linear integral operatorcting on a distributional set. Furthermore, results re-orted in [18] allow us to carry out the reconstruction byeans of a standard truncated singular-value decomposi-

ion (TSVD) inversion scheme [19]. The problem of detec-ion is then set as that of finding the probability that theeconstruction of the two interfaces might be above axed decision threshold [20].

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2662 J. Opt. Soc. Am. A/Vol. 24, No. 9 /September 2007 R. Solimene and R. Pierri

As outlined above, a linear formulation of the inverseroblem allows us to achieve a number of advantages athe expense, almost always, of a large model error. How-ver, here, the linear formulation is adopted only to de-elop a linear inversion algorithm. Indeed, we found, in alosed form, the amplitude of the reconstruction returnedy the linear inversion scheme at the two slab interfaceshen exact model data are exploited. This allows us toighlight the role played by the various parameters onhe probability of detection and ensures that the analysiss not restricted to the cases for which the linear modelan be applied.

Finally, the closed-form results are checked against thease of losses present in the background or in the slab toe detected and for the case of a void layer buried in aalf-space medium.The paper is organized as follows: In Section 2, the

roblem is formulated, the linear model of the electro-agnetic scattering that we rely on is presented, and theSVD inversion scheme is briefly outlined. In Section 3,e report the statistical features of the reconstructionhen the data are corrupted by an additive white Gauss-

an noise. In Section 4, an analytical estimate of the prob-bility of detection is derived, and the roles of the slab’seatures and of the parameters of the configuration areiscussed. In Section 5, the cases of lossy background me-ium and of lossy slab are considered. In Section 6, thease of a void layer buried in a half-space host medium isnalyzed. Finally, conclusions are drawn in Section 7.

. FORMULATION OF THE PROBLEM ANDHE LINEAR INVERSION ALGORITHMe consider the scattering configuration represented by

he three-layered medium depicted in Fig. 1. In particu-ar, the first and third layers are homogeneous and char-cterized by a dielectric permittivity �1 and �3, respec-ively. The second layer is, in general, not homogeneous,ut its dielectric permittivity, denoted as �2�z�, is assumedo be a WKB profile [21]. All the layers are nonmagnetic—hat is, the magnetic permeability is everywhere that of

Fig. 1. (Color online) Geometry of the problem.

he vacuum �0—and lossless. The presence of the losses isddressed later in Section 5.We first want to see whether, under the incidence of

EM plane waves along the z axis at different frequen-ies, a slab inhomogeneity is present and then possibly toetermine the positions z1 and z2 of its interfaces startingrom the knowledge of the scattered field ES gathered athe different frequencies.

The scattered field at the observation point z=0 can beritten as

ES�� = ��,�1,�2,�3,z1,z2�exp�− j2k1z1�, �1�

here � is the angular frequency ranging within theorking bandwidth ��min ,�max�, k1=��1�1 is the wave-umber in the first layer, and �� ,�1 ,�2 ,�3 ,z1 ,z2� is theresnel reflection coefficient at the first slab interface.To introduce the linear inversion algorithm, let us ex-

ress the reflection coefficient ��·� in terms of the local re-ection coefficients of the slab (see Fig. 1) [22]:

��,�1,�2,�3,z1,z2� =�12 + �23 exp�− 2jk1��2,z1,z2�/��1�

1 + �12�23 exp�− 2jk1��2,z1,z2�/��1�,

�2�

here

�12 =��1/�2�z1� − 1

��1/�2�z1� + 1, �3�

�23 =��2�z2�/�3 − 1

��2�z2�/�3 + 1, �4�

��2,z1,z2� =�z1

z2

��2�z�dz. �5�

Equation (2) is exact when �2 is constant. It is an ap-roximation for the reflection coefficent when the secondayer is inhomogeneous and holds as long as the WKB ap-roximation works (see [23] for details concerning itserivation in the latter case).Equation (2) can be conveniently rewritten under a se-

ies expression as [22]

��,�1,�2,�3,z1,z2� = �12 + �1 − �122 ��23 �

n=0

�

�− �12�n�23n

�exp�− j2�n + 1�k1��2,z1,z2�/��1�.

�6�

Now, we approximate the reflection coefficient in Eq. (6)s

��,�1,�2,�3,z1,z2� � �12 + �1 − �122 ��23

�exp�− j2k1��2,z1,z2�/��1�, �7�

hat is, by arresting the series in Eq. (6) up to the seconderm. From a physical point of view, the approximation inq. (7) consists in neglecting the multiple reflections be-

ween the slab (i.e., the second layer) interfaces, and itorks when � � 1. Accordingly, by substituting for �
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n Eq. (1), the scattered field can be rewritten as

Es�k1� � �12 exp�− j2k1z1� + �1 − �212 ��23 exp�− j2k1z2�,

�8�

here z2=z1+��2 ,z1 ,z2� /��1, which becomes z2=z1

d��2 /�1, d=z2−z1 when the slab is homogeneous. Fi-ally, by assuming that the slab is located within an in-estigation domain given by D= �zmin ,zmax� and exploitingwo Dirac �-functions supported just over z1 and z2 to ac-ount for the slab interface locations [17], Eq. (8) can beecast as follows:

Es�k1� =�zmin

zmax

f�z�exp�− j2k1z�dz, �9�

here f�z�=�21��z−z1�+ �1−�122 ��23��z− z2�.

According to the linear integral equation (9), the prob-em is formulated as one of determining the unknown dis-ribution f�z� starting from the knowledge of the scatteredeld within the adopted frequency band. Thanks to theesults reported in [18], the reconstruction, that is, the in-ersion of the operator in Eq. (9), can be performed byeans of its singular-value decomposition (SVD)

un ,n ,�n�n=0� , where the uns and ns are the singular

unctions spanning the unknown and the data sets, re-pectively, and the �ns are the singular values [24]. More-ver, to curtail the effect of the noise on the reconstruc-ion, the required regularization can be achieved byruncating the SVD expansion in correspondence to aoise-dependent index NT (TSVD inversion scheme) [19].The operator in Eq. (9) is a Fourier transform acting

ver a compact support; hence its singular system is wellnown and can be easily expressed in terms of the so-alled prolate spheroidal wave functions (PSWFs) [25]however, to make the paper self-contained, we report itserivation in Appedix A). More specifically, the singularystem is related to the PSWF �n�c , · �, with spatial-andwidth product c= �k1max−k1min��zmax−zmin� /2,k1min ,k1max� being the wavenumber bandwidth, and theingular values are �n=� n�c��, with n�c� being the nthigenvalue corresponding to the nth PSWF [25].

The inversion scheme point-spread function is given by19] (see Appendix A for the mathematical expression ofhe uns):

h�z − z� = �n=0

NT

un*�z�un�z� = exp�j2k1av�z

− z��n=0

NT �n�z − zav��n�z − zav�

n, �10�

here * denotes the complex conjugation and k1av�k1max+k1min� /2 and zav= �zmin+zmax� /2.The point-spread function is a bandpass function whose

nvelope, that is, the summation on the right-hand side ofq. (10), is a sinclike function having the main beamidth and its amplitude both dependent on the truncation

ndex NT, which in turn depends on the tolerable level ofoise. Indeed, when c is sufficiently greater than 1, theSWF eigenvalues ns, and hence the singular values, ex-ibit a steplike behavior, with the s being almost equal
n
o 1 before the knee occurring roughly in correspondenceo the index 2c /� [25]. In this case, the truncation indexan be chosen, almost independently of the noise, in cor-espondence to such a knee, leading the point-spreadunction to be roughly given by [26]

h�z − z� � exp�2jk1av�z − z��

�sinc��z − z��, �11�

here �= �k1max−k1min�. This is the situation we consider.Accordingly, the outcome of the inversion scheme will

e

f�z� =�

�exp�2jk1avz��12 exp�− 2jk1avz1�sinc��z − z1��

+ �1 − �122 ��23�

n=0

�

�− �12�n�23n exp− 2jk1av�z1 + �n

+ 1��/��1��sinc��z − z1 − �n + 1��/��1�� . �12�

The first two terms of the series in Eq. (12) are domi-ant when �12�231. However, when this condition isot met, the multiple reflections between the two inter-aces become significant. Hence, in order to study the ca-ability of the inversion scheme to detect the slab, theontribution of these additional terms to the recontruc-ion also have to be accounted for.

In the following study, to assess the capability of the in-ersion scheme to detect the slab, we will employ Eq. (12).his makes the analysis also encompass the multiple re-ections effect, and hence it is not limited to the cases dic-ated by the validity of the linear model.

. STATISTICAL FEATURES OF THEECONSTRUCTION

o analyze how the probability of false alarm PFA (i.e., therobability that a phantom interface is detected) and therobability of detection PD depend on the noise level, onhe slab’s features, and on the parameters of the configu-ation, in this section we report the statistical features ofhe recontruction.

We consider the scattered field data corrupted by anncorrelated additive white Gaussian noise �0�k1� havingero mean and variance equal to �0

2. Thus, the reconstruc-ion will be affected by a noise term given by

��z� = �n=0

NT ��,n�

�nun�z�. �13�

According to the mathematical features of the singularystem, ��z� is a bandpass random process still having aero mean and autocorrelation equal to

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R��z,z�� = E��z�,�*�z�� = �02 exp�2jk1av�z

− z��n=0

NT �n�z − zav��n�z� − zav�

n�n2 , �14�

here E�· , · � denotes the statistical average (see Appen-ix B). In particular, for a large spatial-bandwidth prod-ct c, the autocorrelation is approximated by [see Eqs.10) and (11)]

R��z − z�� exp�2jk1av�z − z��0

2�

�2 sinc��z − z��.

�15�

Hence, the noise process in the reconstruction is wide-ense stationary, and its variance is given by

var� =�0

2�

�2 . �16�

he noisy reconstruction is then a random process giveny

f�z� + ��z� = exp�j2k1avz�Af�z�exp�j�f�z��

+ A��z�exp�j��z��, �17�

here Af�z� and �f�z� are the amplitude and the phase ofhe complex envelope of f�z� and A��z� and ��z� are themplitude and the phase of the noise ��z�.Even though in the lossless case the unknown has a

eal density, we will consider the amplitude of f�z� whiletudying the achievable probability of detection, sinceaking the real part of the reconstruction leads us to dealith a more oscillating function, which makes it more dif-cult to distinguish the “peaks” of the reconstruction cor-esponding to the interfaces from those due to the oscil-ating nature of the reconstruction itself. In particular, forach position z within the investigation domain, the noisyeconstruction amplitude and the noise amplitude areharacterized by the following probability density func-ions (PDFs) [27]:

pf�r� =r

var�

exp− �r2 + Af

2�

2 var�

I0�rAf/var��, �18�

hich is the Rician PDF, and

p��r� =r

var�

exp− r2

2 var�

, �19�

hich is the Rayleigh PDF. In Eq. (18), I0�·� is the modi-

�1 + 2�12�23 cos�2k1

ed Bessel function of zero order and r represents the am-litude of the reconstruction.Hence, if a decision threshold Ath is fixed, then the PFA

nd PD are given by [20]

PFA =�Ath

�

p��r�dr = exp�−Ath

2

2 var�� , �20�

PD =�Ath

�

pf�r�dr = Q� Af

�var�

,Ath

�var�� , �21�

here Q�· , · � is Marcum’s Q function [27]. More specifi-ally, we will fix the decision threshold depending on thecceptable PFA, that, is by employing the results in Eqs.16) and (20):

Ath =�0

�� ln�1/PFA�, �22�

nd we will assume a successful detection when the re-onstruction amplitude is above such a threshold.

. DETECTION OF THE SLABn order to evaluate the capability of the inversion schemeo detect the slab, the amplitude of the reconstruction haso be estimated. In particular, as we are interested in de-ecting the slab’s interfaces, we refer to the amplitude off�z�, Af�z1�, and Af�z2� achieved in correspondence to zz1 and z= z2. The analytical expression of Af in suchoints can be obtained by summing the series in Eq. (12)or z=z1 and z= z2. In particular, after some manipula-ions of Eq. (12), this can be achieved by resorting to theormulas listed in [28] (see Appendix C for details), thusbtaining

Ref�z1�� =�

��12 +

1 − �122

2��12�/��1

��tan−1� − �12�23 sin�2k1 min�/��1�

1 + �12�23 cos�2k1 min�/��1��− tan−1� − �12�23 sin�2k1 max�/��1�

1 + �12�23 cos�2k1 max�/��1�� ,

�23�

Imf�z1�� =1 − �12

2

2��12�/��1�ln� 1

�1 + 2�12�23 cos�2k1 min�/��1� + �122 �23

2 �− ln� 1

2 2 � , �24�
max�/��1� + �12�23

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here Re�·� and Im�·� take the real and imaginary partsf their argument, respectively, and

f�z1� = Af�z1� = �Ref�z1��2 + Imf�z1��2. �25�

Similarly,

f�z2� = Af�z2� = �Ref�z2��2 + Imf�z2��2, �26�

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Ref�z2�� =�

�cos�2k1av�/��1�sinc��/��1��12 +

�

��1

− �122 ��23 +

�1 − �122 ��23

2��/��1

��tan−1� − �12�23 sin�2k1 max�/��1�

1 + �12�23 cos�2k1 max�/��1��− tan−1� − �12�23 sin�2k1 min�/��1�

1 + �12�23 cos�2k1 min�/��1�� ,

�27�

Imf�z2�� =�

�sin�2k1av�/��1�sinc��/��1��12 −

�1 − �122 ��23

2��/��1�ln� 1

�1 + 2�12�23 cos�2k1 min�/��1� + �122 �23

2 �− ln� 1

�1 + 2�12�23 cos�2k1 max�/��1� + �122 �23

2 � . �28�

As an example, we consider the case of a homogeneouslab embedded in free space (therefore �1=�3=�0, �0 beinghe dielectric permittivity of the vacuum) and comparehe behavior of Af�z1� and of Af�z2� with the maximum off�z� returned by Eq. (12). Such a comparison is reportedn Fig. 2. As can be seen, Af�z1� is very close to the maxi-

um of f�z�, and hence we conclude that the probability ofetection can be estimated as

PD = Q�Af�z1�

�var�

,Ath

�var�� . �29�

The expression of PD given by Eq. (29) is what we wereearching for, as it allows us to assess the roles played by

ig. 2. Comparison among maxf�z� (solid curve), Af�z1� (dashedurve), and Af�z2� (dotted-dashed curve): k1min=9� m−1, k1max12� m−1, z1=0.2 m, D= �0.05,1.5� m. [A] Varying thickness dnd fixed � =4� . [B] Varying � and fixed d=0.1 m.

he various parameters on the achievable PD. Indeed, itllows us to estimate the achievable PD for a certain slabnce the parameters of the configuration are given, oronversely, for a certain slab one can determine the pa-ameters of the configuration that make the slab detect-ble. In particular, from Fig. 2 it can be seen that Af�z1� isractically coincident with maxf�z� when the slab’s thick-ess d��1 /�2� /� or when d�2��1 /�2� /�. Indeed,hen d��1 /�2� /�, the reflection coefficient ��k1� can be

onsidered almost constant while evaluating the coeffi-ients of the TSVD expansion �ES ,n�. Hence, the recon-truction returned by the inversion scheme is roughlyiven by

f�z� � Af�z1�exp�j�f�exp�j2k1avz�sinc��z − z1��. �30�

On the other hand, when d�2��1 /�2� /�, the mainobes of the sinc terms appearing in Eq. (12) are not over-apped. Accordingly, the reconstruction amplitudechieves its maximum in z1. Moreover, in this last case

Af�z1� →�12�

�, �31�

hat is, Af�z1� tends to the amplitude of the first term ofhe right-hand side of Eq. (23) particularized to the ex-mple we are considering.When neither d��1 /�2� /� nor d�2��1 /�2� /� are

erified, the maximum of f�z� does not necessarily occurn z1. Accordingly, Eq. (29) will return a lower bound forD.This is confirmed by Fig. 2 and furthermore by Fig. 3,

here the reconstructions corresponding to three differ-nt values of the thickness corresponding in turn to thehree situations described above are shown. In Fig. 4 wehow the comparison between the P returned by Eq. (29)
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nd that obtained by numerical simulations achieved byonsidering 1000 different realizations of the noise andstimating PD as the ratio between the number of timeshe maximum of the reconstruction amplitude is over theecision threshold Ath and the total number of trials. AFA=10−5 is assumed. As can be seen, the agreement isery good. It is obvious that PD increases as the noise �0

2

ecreases, since this leads to exploiting a lower decisionhreshold Ath. Furthermore, an abrupt decay of PD is ob-erved when

d = n�/k1av��1/�2. �32�

As expected, this reflects the fact that, when the slab’shickness is an integer number of the internal half-avelength, the slab is more difficult to image. This be-avior is clearly due to the periodicity of ��k1�. In fact, thealues of �2 and d given by Eq. (32) are those for whichhe scattered field is “observed” over an interval centeredn its amplitude minimum. Accordingly, this entails a de-ay of the reconstruction amplitude maximum as well asf the probability of detection PD. However, not all thebrupt decays prediceted by Eq. (32) are observed. This isecause when d and/or �2 increases, the period of ��k1� de-reases and hence the scattered field amplitude departsore rapidly from its minimum, or in other words the

cattered field is observerd over a greater part of its pe-iod. This leads the decay of the reconstruction amplitudeo be less pronounced, and when evaluating PD, this de-ay can disappear completely. In particular, when d2��1 /�2� /�, Eq. (31) tends to be valid and PD becomes

nly weakly dependent on d. In this last case, the peakorresponding to the second interface could also be seen inhe reconstruction. In particular, its amplitude tends to

Af�z2� →�1 − �12

2 ��12�

�, �33�

hat is, to the second term on the right-hand side of Eq.26). This leads to the following estimate for the corre-ponding P :

ig. 3. Maximum of f�z� does not necessarily occur in z1: k1min9�, k1max=12�, z1=0.2, D= �0.05,1.5� m, �2=9�0, and three dif-

erent values of d.

D

PD2 = Q�Af�z2�

�var�

,Ath

�var�� . �34�

However, Af�z1� increases as the dielectric discontinuityncreases, and hence so does its PD, whereas Af�z2� in-reases until its stationary point ��12=1/�3�, beyondhich a decay is observed. In other words, even in the

ossless case, it becomes increasingly difficult to detecthe second interface as the jump in the dielectric permit-ivity increases. Hence, when d�2��1 /�2� /�, the secondnterface could be discerned in the reconstruction, but theccurrence of such a circumstance is also dependent onhe noise level, and one is actually aware of that only af-er the reconstruction.

Furthermore, as in this case, the ratio between the am-litude of the first and second terms of Eq. (12) [or,quivalently, the ratio between Eqs. (31) and (33)] is 1�12

2 , so when the jump in the dielectric permittivity in-reases, the interference between the sidelobes of the firsterm of the reconstruction in Eq. (12) and the second termentered in z2 could lead the second interface to berongly localized or at worst to be “masked.” In this

ense, Eq. (34) is rather optimistic. Equation (34) is valid,nd a correct localization of the second interface is

ig. 4. Comparison between PD returned by Eq. (29) (solidurves) and numerical simulations (curves with circles): PFA10−5, k1min=9� m−1, k1max=12� m−1, z1=0.2 m, D= �0.05,1.5� m.

Top two panels) �0=10−1; (bottom two panels) �0=10−0.5. [A]arying thickness d and fixed �2=4�0. [B] Varying �2 and fixed=0.1 m.

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chieved if a ratio of 0.13 between the sidelobes of therst term and Af�z2� is restored, that is, if

d ��1

�2� 1

0.13��1 − �122 �

+�

2�� . �35�

In other words, Eq. (35) indicates that the main lobe ofhe second term of the reconstruction overlaps with theidelobes of the first term, which have an amplitude notreater than 0.13 times the amplitude of the reconstruc-ion in z2 as returned by Eq. (33). An illustration of whatas been said above is shown in Fig. 5, where the normal-

zed amplitude of the reconstruction returned by Eq. (12)s reported for different values of �2r (i.e., the relative di-lectric pemittivity of the second layer) and for a fre-uency band �15,60� GHz. In panel [A] the slab’s thick-ess d has been chosen equal to 2��1 /�2� /�. In theseircumstances, having chosen the first interface located at1=0.5 cm, in all the cases, the second interface should beocated at z2=1.16 cm. Instead, in panel [B], d has beenhosen according to Eq. (35).

We end this section by observing that, when d2��1 /�2� /�, the reconstruction allows us to obtain an

ig. 5. Role of �2r in the localization of the interfaces. The ar-ows indicate the positions the interfaces should have in the re-onstruction if properly localized. D= �0,3� cm. [A] d

1 2

stimation of the slab’s parameters. Indeed, from the re-onstruction, z1, z2, and Af�z1� can be derived. Then, byxploiting Eq. (31), an estimation for �2 can be obtained.inally, by employing the estimation of z1 and �2 in z2, onean deduce the slab’s thickness d [remember that d= �z2

z1��1 /�2]. Some examples illustrating this procedurere reported in Table 1, where the effect of noise also isccounted for. In particular, in each case, the estimationshown there are obtained by averaging the results corre-ponding to ten different reconstructions. As can be seen,he procedure works well, mainly as far as the estimationf z1 and d are concerned.

. LOSSY CASErevious results have been derived by considering no

osses, neither in the background medium nor in the slab.n this section we analyze how Eq. (29) can be modified toccount for losses. As an example, we still consider thease of a homogeneous slab embedded in a homogeneousackground medium. We start by considering losses inhe background; that is, we assume �1eq=�1− j�1 /�, with1 being the electric conductivity. In this case, both theavenumber k1loss and the local reflection coefficient12loss are complex functions of �. Hence, Eq. (25) does notpply. However, losses can be at least partially accountedor if the small loss condition is assumed. Under this as-umption, �12loss is considered the same as the losslessase �12 and k1loss�k1− j�1, with �1=�1��0 /�1 /2. There-ore, upon performing the same reasoning as in the pre-ious case, it is straightforward to estimate the amplitudef the reconstruction in z1 as

Afloss1�z1� = exp�− 2�1z1�Af�z1�, �36�

ith Af�z1� given by Eq. (25), and hence to estimate PD as

PD = Q�Afloss1�z1�

�var�

,Ath

�var�� . �37�

A comparison between numerical results (obtained asn the previous example) and P given by Eq. (37) is

Table 1. Estimation of the Slab’s Parameters

1min=9� m−1, k1max=16� m−1, D= �0.05,1.5� mActual Parameters: Estimated Parameters:(z1 (cm), d (cm), �2r) (z1 (cm), d (cm), �2r)

(10, 15, 4) (9.89, 14.93, 4.05), �0=10−1

(10, 15, 4) (9.12, 14.84, 4.21), �0=10−0.5

(10, 15, 8) (10.09, 15.24, 7.66), �0=10−1

(10, 15, 8) (9.99, 14.52, 8.14), �0=10−0.5

D
2�� /� � /�. [B] d chosen according to Eq. (35).

stac

ikce

wai

tEpbwtltmsb

6Ir

F=�c


hown in Fig. 6. As expected, PD decreases with respect tohe lossless case. Furthermore, as is obvious, Eq. (37) isccurate as long as the regime of small losses is appli-able; otherwise, Eq. (37) underestimates PD.

We now consider the case of losses within the slab. Alson this case, under the small loss approximation, that is,

2loss�k2− j�2, with �2=�2��0 /�2 /2, an estimate for PDan be derived. More specifically, the slab’s reflection co-fficient can be approximated as

� = �12 − �1 − �122 ��12 �

n=0

�

��12�2n exp�− j2�n + 1�

� k1��2/�1d�exp�− 2�2�n + 1�d�, �38�

here Eq. (6) has been particularized to the example were considering. By repeating the same passages shownn Appendix B, we obtain
tom panel) �1=10 S/m.
Ref�z1�� =�

��12 −

1 − �122

2�d�12��2/�1�tan−1� �12

2 exp�− 2�2d�sin�2k1 maxd��2/�1�

1 − �122 exp�− 2�2d�cos�2k1 maxd��2/�1��

− tan−1� �122 exp�− 2�2d�sin�2k1 mind��2/�1�

1 − �122 exp�− 2�2d�cos�2k1 mind��2/�1�� , �39�

Imf�z1�� =1 − �12

2

2�d�12��2/�1�ln� 1

�1 − 2�122 exp�− 2�2d�cos�2k1 mind��2/�1� + �12

4 exp�− 4�2d��− ln� 1

�1 − 2�122 exp�− 2�2d�cos�2k1 maxd��2/�1� + �12

4 exp�− 4�2d�� , �40�

Fz(m�

Afloss2�z1� = �Ref�z1��2 + Imf�z1��2. �41�

Accordingly, PD is estimated as

PD = Q�Afloss2�z1�

�var�

,Ath

�var�� . �42�

Figure 7 shows the comparison between the PD ob-ained numerically as said above and the one returned byq. (42). As can be seen, the agreement is very good. Inarticular, for the case of low conductivity (top panel), theehavior of PD resembles that of the lossless case,hereas when �2 increases, the abrupt decays observed in

he previous section disappear. This is clearly due to theosses inside the slab, which make negligible the interac-ions between the two slab interfaces. Of course, this alsoeans that it becomes increasingly difficult to detect the

econd interface when the losses increase. This can alsoe seen by Eq. (26) particularized to the example at hand.

. CASE OF A BURIED VOID LAYERn this section, we consider a further example to check theesults derived previously for a situation relevant to sub-

ig. 6. Role of background losses: k1min=9� m−1, k1max12� m−1, z1=0.1 m, �1=�0, �2=4�0, D= �0.05,1.5� m, PFA=10−4,0=10−1. (Solid curves) PD returned by Eq. (37); (lines withircles) numerical simulations. (Top panel) �1=10−1.4 S/m; (bot-

−1

ig. 7. Role of the slab losses: k1min=9� m−1, k1max=12� m−1,1=0.1 m, �1=�0, �2=4�0, D= �0.05,1.5� m, PFA=10−5, �0=10−1.Solid curves) PD returned by Eq. (42); (curves with circles) nu-

erical simulations. (Top panel) �2=10−2 S/m; (bottom panel)=10−1 S/m.
2

svmtifftf�cHttts

witEpgo

w

c8o

m

wtmntmc

7Iet�pg

sirttfTctFhl

bh

aaapfww[

ATIeh

wt

FmEDe


urface imaging. In particular, we address the case of aoid layer buried in a half-space medium. In this case, theedium has four layers. The first one is assumed to be

he free space, hence its dielectric permittivity is �0, ands separated by the other three layers by a planar inter-ace we call the air/soil interface at z=0. The second andourth layers are representative of the soil, and we denoteheir permittivity by �1. Finally, the third layer extendsrom z1 to z2, it is void, and its dielectric permittivity is2=�0. For any losses in the soil, �1 denotes the electriconductivity. Also, in this case Eq. (25) does not apply.owever, an estimate for PD can be obtained by adopting

he small loss approximation and by neglecting the mu-ual reflections between the void slab and the air/soil in-erface. Under these assumptions, the scattered field ob-erved at the air/soil interface can be written as

ES�k1� = �1 − �as2 �exp�− 2j�k1 − j�1�z1��, �43�

here �as is the local reflection coefficient at the air/soilnterface, �1 is as defined in the previous section, and � ishe Fresnel reflection coefficient of the void slab given byq. (6). Now, by repeating the reasoning achieved in therevious section for the case of losses present in the back-round, one finds the following estimate for the amplitudef the reconstruction in z1:

Af void�z1� = �1 − �as2 �exp�− 2�1z1�Af�z1�, �44�

hich leads to the estimate

PD = Q�Af void�z1�

�var�

,Ath

�var�� . �45�

Also, in this case the theoretical estimate (45) is verylose to the numerical simulations. This is shown in Fig., where some examples corresponding to different valuesf losses, of �1, and of noise are shown.

When the losses increase, as in the previous case, esti-ate (45) tends to underestimate PD. Furthermore, even

ig. 8. Case of a buried void layer. Comparison between the nu-erical evaluated PD (curve with circles) and the one returned byq. (45) (solid curve): fmin=675 MHz, fmax=900 MHz, z1=0.1 m,= �0.05,1.5� m, PFA=10−5. (Panels on the left side) �1=4�0; (pan-

ls on the right side) � =8� .
1 0
hen z1 decreases, the analytical estimate deviates fromhe numerical simulations. This is because in Eq. (45) theultiple reflections with the air/soil interface have beeneglected. However, estimate (45) still remains close tohe numerical results for very low z1; we performed nu-erical simulations (not shown here) up to z1=0.01 m,

onfirming it.

. CONCLUSIONn this paper, the problem of detecting and localizing a di-lectric slab under a multifrequency plane-wave illumina-ion has been addressed. A linear model based on-functions has been introduced, which has allowed us toerform the reconstruction by means of the truncated sin-ular value decomposition (TSVD) inversion scheme.

A closed-form derivation for the amplitude of the recon-truction corresponding to the slab interfaces when thenversion scheme acts on exact model data has been de-ived. This has allowed us to discuss the roles played byhe parameters of the slab, the frequency bandwidth, andhe noise on the achievable probability of detection evenor situations where the linear model was not applicable.he case of a lossless three-layered medium has been firstonsidered. Afterward, the analysis has been extended tohe case of losses present in the background or in the slab.urthermore, the case of a void layer buried in a lossyalf-space (i.e., a four-layered medium) has also been ana-

yzed.A procedure to recover the parameters of the slab when

oth the interfaces are detectable in the reconstructionas been presented and tested against noisy data.As a final remark, we point out that the numerical

nalysis has been developed, for simplicity, for the case ofhomogeneous slab. However, the analytical results have

lso been derived for the case of an inhomogeneous slab,rovided that the WKB approximation is verified. There-ore, the study can be, with minor effort, modified to dealith the case of a more general three-layered medium [4]ith the slab having a WKB-varying pemittivity profile

21,23].

PPENDIX A: THE SINGULAR SYSTEM OFHE OPERATOR IN EQ. (9)

n this appendix we derive the singular system of the op-rator in Eq. (9). To this end, the following eigenproblemas to be solved:

un�n2 = L†Lun, �A1�

here L† denotes the adjoint of L [24] and L is the opera-or in Eq. (9). Equation (A1) can be explicitly written as

un�z��n2 =�

zmin

zmax

un�z��exp�2jk1av�z − z��sin��z − z��

z − z�dz�,

�A2�

wm+w

wc

wPs

aa

ATg

wE

w

�b

Tt

AAIc

f

i


here k1av= �k1min+k1max� /2 and �=k1max−k1min. Now, byaking the change of variable z=zav+s, with zav= �zminzmax� /2, �= �zmax−zmin� /2, and s��, Eq. (A2) can be re-ritten as

un�s��n2 =�

−�

�

un�s��sin��s − s��

s − s�ds�, �A3�

ith un�s�=un�zav+s�exp�−2jk1av�zav+s��. Thus, the unsan be expressed in terms of the PSWFs [25] as

uz�z� = ��z − zav,c�exp�2jk1av�z − zav��/� n�c�, �A4�

here n is the eigenvalue corresponding to the nthSWF and c=�� is the spatial-bandwidth product. Theingular values are then given by

�n = � n�, �A5�

nd the singular functions expanding the scattered fieldre given by

n = 1/�nLun. �A6�

PPENDIX B: DERIVATION OF EQ. (14)he autocorrelation of the noise in the reconstruction isiven by

R��z,z�� = E��z�,�*�z��, �B1�

here E�·� denotes the stochastic average. By exploitingq. (13), Eq. (B1) can be rewritten as

R��z,z�� = �n=0

NT

�m=0

NT un�z�um* �z��

�n�mE��0,vn�,��0,vm�*�.

�B2�

Now, as the noise on the data �0 is assumed to behite, that is, R�0

�k1 ,k1��=�02��k1−k1��, then

E��0,vn�,��0,vm�*� = �02�nm, �B3�

nm being the Kronecker function. Accordingly, Eq. (B2)
ecomes p
�z − z1 −

R��z,z�� = �n=0

NT

�02un�z�un

*�z��

�n�n. �B4�

hen Eq. (14) is obtained by exploiting the expression ofhe singular functions uns as given in Appendix A.

PPENDIX C: CALCULATION OF Af„z1… AND

f„z2…

n this appendix we provide the mathematical details con-erning the derivation of Eqs (25) and (26).

Let us separately write the real and imaginary parts of�z�. According to Eq. (12), this leads to

Ref�z�� =�

��12 cos�2k1av�z − z1��sinc��z − z1�� + �1

− �122 ��23�

n=0

�

�− �12�n�23n cos2k1av�z − z1 − �n

+ 1��/��1��sin��z − z1 − �n + 1��/��1��

��z − z1 − �n + 1��/��1� ,

�C1�

Imf�z�� =�

��12 sin�2k1av�z − z1��sinc��z − z1�� + �1

− �122 ��23�

n=0

�

�− �12�n�23n sin2k1av�z − z1 − �n

+ 1��/��1��sin��z − z1 − �n + 1��/��1��

��z − z1 − �n + 1��/��1� .

�C2�

By exploiting the relationship

sin � cos � =sin�� + �� + sin�� − ��

2,

sin � sin � =cos�� − �� − cos�� + ��

2

n Eqs. (C1) and (C2), respectively, the real and imaginary
arts can be rewritten as
Ref�z�� =�

��12 cos�2k1av�z − z1��sinc��z − z1�� + �1 − �122 �

�23

2��n=0

�

�− �12�n�23n

�sin2k1 max�z − z1 − �n + 1��/��1�� − sin2k1 min�z − z1 − �n + 1��/��1��

�z − z1 − �n + 1��/��1� , �C3�

Imf�z�� =�

��12 sin�2k1av�z − z1��sinc��z − z1�� + �1 − �122 �

�23

2��n=0

�

�− �12�n�23n

�cos2k1 min�z − z1 − �n + 1��/��1�� − cos2k1 max�z − z1 − �n + 1��/��1��

� . �C4�
�n + 1��/ �1�

w

tp

wsit(

uo

wlAa

R

1


Now, we evaluate Eqs. (C3) and (C4) in z1:

Ref�z1�� =�

��12 + �1 − �122 �

�23

2��n=0

�

�− �12�n�23n

�sin2k1 max��n + 1��/��1��

�n + 1��/��1

−sin2k1 min��n + 1��/��1��

�n + 1��/��1 , �C5�

Imf�z1�� =�

��1 − �122 �

�23

2��n=0

�

�− �12�n�23n

�cos2k1 max��n + 1��/��1��

�n + 1��/��1

−cos2k1 min��n + 1��/��1��

�n + 1��/��1 , �C6�

hich can be rearranged in the following form:

Ref�z1�� =�

��12 −�1 − �12

2 �

2��12�/��1��

n=1

�

�− �12�n�23n

�sin2k1 maxn�/��1�

n

− �n=1

�

�− �12�n�23n

sin2k1 minn�/��1�

n � ,

�C7�

Imf�z1�� =�

�� − �1 − �122 �

2��12�/��1��

n=1

�

�− �12�n�23n

�cos2k1 maxn�/��1�

n

− �n=1

�

�− �12�n�23n

cos2k1 minn�/��1�

n � .

�C8�

Equations (C7) and (C8) are now in a form for whichhe sum of the series is known. In particular, we can ex-loit the following results reported in[28]:

�n=1

� pn sin nx

n= tan−1� p sin x

1 − p cos x� , �C9�

�n=1

� pn cos nx

n= ln� 1

�1 − 2p cos x + p2� , �C10�

ith p2�1. In particular, by exploiting Eq. (C9) and con-idering p=−�12�23 and x=2k1max� /��1 in the first seriesn Eq. (C7) and x=2k1min� /��1 in the second series, we ob-ain Eq. (23). Analogously, by employing Eq. (C10) in Eq.C8), we obtain Eq. (24) and hence Eq. (25).

Similarly, Eq. (26) can be derived. More specifically,pon evaluating Eqs. (C3) and (C4) in z2=z1+� /��1, webtain

Ref�z2�� =�

��12 cos�2k1av�/��1�sinc��/��1�

+ �1 − �122 ��23 + �1 − �12

2 ��23

2�

��n=1

�

�− �12�n�23n

sin2k1 maxn�/��1�

n�/��1

− �n=1

�

�− �12�n�23n

sin2k1 minn�/��1�

n�/��1� ,

�C11�

Imf�z2�� =�

��12 sin�2k1av�/��1 sinc��/��1��

+ �1 − �122 �

�23

2��

n=1

�

�− �12�n�23n

�cos2k1 maxn�/��1�

n�/��1

− �n=1

�

�− �12�n�23n

�cos2k1 minn�/��1�

n + 1�/��1� , �C12�

here the terms of index n=0 have been computed asimit values of the generic n-dependent terms as n→0.gain, by exploiting Eqs. (C9) and (C10), Eqs. (27), (28),nd (26) are obtained.

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Detection and localization of a slab by a linearlike δ approach

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Transcript of Detection and localization of a slab by a linearlike δ approach