degree of nonlinearity and a new solution procedure in scalar two-dimensional inverse scattering...

1832 J. Opt. Soc. Am. A/Vol. 18, No. 8 /August 2001 Bucci et al.

Degree of nonlinearity and a new solutionprocedure in scalar

two-dimensional inverse scattering problems

Ovidio M. Bucci, Nicola Cardace, Lorenzo Crocco, and Tommaso Isernia

Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni, Universita di Napoli ‘‘Federico II,’’Via Claudio 21, 80125, Napoli, Italy

Received May 31, 2000; accepted January 10, 2001; revised manuscript received February 2, 2001

Within the framework of inverse scattering problems, the quantifying of the degree of nonlinearity of the prob-lem at hand provides an interesting possibility for evaluating the validity range of the Born series and forquantifying the difficulty of both forward and inverse problems. With reference to the two-dimensional scalarproblem, new tools are proposed that allow the determination of the degree of nonlinearity in scattering prob-lems when the maximum value, dimensions, and spatial-frequency content of the unknown permittivity arechanged at the same time. As such, the proposed tools make it possible to identify useful guidelines for thesolution of both forward and inverse problems and suggest an effective solution procedure for the latter. Nu-merical examples are reported to confirm the usefulness of the tools introduced and of the procedure proposed.© 2001 Optical Society of America

OCIS codes: 100.3190, 290.3200, 100.6950.

1. INTRODUCTION AND STATEMENT OFTHE PROBLEMMany studies have been performed to ascertain the rangeof validity of the classical Born approximation with re-spect to the scatterer size and the maximum value of itsabsolute permittivity. In particular, an upper bound hasbeen reported1–3 for the range of validity of such an ap-proximation in terms of the product between the maxi-mum electrical dimension of the object and the maximumvalue of its contrast. Such a product has also been usedto quantify the enhanced reconstruction capabilities ofnonlinearized inversion approaches, such as in Refs. 4–8.Although they do not perform any approximation on thescattering equations, the latter have limited reconstruc-tion capabilities because of the possible occurrence offalse solutions.

In this paper, with reference to the two-dimensional(2-D) scalar case, new tools for the analysis of the abovepoint are proposed. These make it possible to evaluatethe range of validity of the Born approximation in termsof the scatterer size (and shape) as well as the maximumvalue and spatial frequency content of the unknown con-trast. The proposed tools also allow the degree of nonlin-earity of the scattering problem at hand to be evaluated,thus furnishing useful guidelines for the solution of bothforward and inverse scattering problems when the Bornapproximation is no longer valid. In particular, a new so-lution procedure is proposed that proves more favorablein accurately solving the inverse problem as comparedwith previous approaches.

Let us assume that the scatterer is enclosed in a com-pact support V, and let the z axis be the invariance direc-tion. The incident waves are a given collection of TM po-larized plane waves. In such a situation, (from the

0740-3232/2001/081832-12$15.00 ©

Maxwell equations) the scattering problem can be ex-pressed in a compact scalar fashion as9

¹2Ev 1 b2~1 1 x!Ev 5 0, ;r P V

¹2ESv 1 b2ES

v 5 0, ;r ¹ V

Ev 5 E incv 1 ES

v ,

where b 5 v(ebmb)1/2 is the background wave numberand, for each different illumination v, E inc

v is the incidentfield, Ev is the total electric field, and ES

v is the scatteredelectric field, which is supposed to be known on an obser-vation curve G external to V. The complex function x(r)is the so-called contrast function between the (possiblycomplex) permittivity of the scatterer e(r) and that ofthe homogeneous background eb , and it is defined asx(r) 5 @e(r)/eb# 2 1.

To better emphasize the relationship between the un-known x and the data ES

v of the inverse problem, thescattering equations can also be written in the integralform

ESv~r!

5 EV

g~r, r8!x~r8!Ev~r8!dr8,Ae~xEv!, r P G, (1a)

Ev~r! 2 E incv ~r!

5 EV

g~r, r8!x~r8!Ev~r8!dr8,Ai~xEv!, r P V, (1b)

where g(r, r8) 5 2( jb2/4)H0(2)(bur 2 r8u) is the Green’s

function at the prescribed frequency; H0(2)( • ) is the

second-kind zero-order Hankel function [an exp( jvt) timedependence is assumed; j denotes the imaginary unit].

2001 Optical Society of America

Bucci et al. Vol. 18, No. 8 /August 2001 /J. Opt. Soc. Am. A 1833

Operators Ai and Ae represent the internal and externalradiation operators, respectively. In particular, they arelinear in xEv or, equivalently, bilinear in Ev and x.

By rearranging Eq. (1b), one obtains

Ev 5 ~I 2 Ai X !21E incv , (2)

wherein I is the identity operator and X( • ) is the diag-onal operator that gives the product x times (•). If

iAi Xi , 1, (3)

then the inverse operator in Eq. (2) can be expanded in aNeumann series 10 around the point x 5 0, i.e.,

~I 2 Ai X !21 5 I 1 Ai X 1 ~Ai X !2 1 ... 1 ~Ai X !n

1 ... . (4)

As is well known, the Born approximation considers onlythe first term of the series (4), which is justified whenever

iAi Xi ! 1. (5)

It is therefore interesting to try to evaluate the norm ofthe operator Ai X as a function of the main characteristicsof the scatterer. Besides giving information about the va-lidity of the Born approximation, the norm also can giveuseful information about the applicability of the Born se-ries, the number of terms required to achieve a given ap-proximation accuracy, and therefore the overall degree ofnonlinearity (as a function of x) of the given scatteringproblem.

Expansion (4) holds true when the (sufficient) condition(3) is satisfied. However, the norm iAi Xi also proves im-portant to iterative methods of solving Eq. (2) even whencondition (3) is not fulfilled (see, for example, Ref. 11). Inparticular, the larger iAi Xi is, the larger are the numberof iterations required to achieve a satisfactory conver-gence. Therefore, as each iteration implies a further ap-plication of the operator Ai X, iAi Xi plays a key role inestablishing the computational burden of the scatteringproblem at hand regardless of the validity of condition (3).

It is worth noting that the degree of nonlinearity has astraight influence on the solution of inverse problemswith actual (and henceforth error-affected) data. In thiscase, in order to tackle its intrinsic ill-posedness, the in-verse problem is conveniently reduced to the minimiza-tion of properly defined cost functionals12 whose globalminimum represents the generalized solution of the prob-lem. Now, any increase in the degree of nonlinearity ofthe problem, and hence in the nonlinearity of the costfunctional, is likely to increase the number of possible lo-cal minima. As the latter could trap any nonglobal opti-mization procedure into false solutions, and performancesof global optimization procedures are still limited to fewtens of unknowns, iAi Xi also represents a measure of thedifficulty encountered in such methods in solving inversescattering problems. Therefore in the following we focuson factors affecting iAi Xi . We stress that even assum-ing the availability of a computationally effective (global)optimization scheme, the comprehension of factors affect-ing nonlinearity, and henceforth the possible occurrenceof false solutions, would allow a considerable speed-up ofthe overall solution scheme.

The paper is organized as follows. First, in Section 2,upper and lower bounds to the norm of Ai X in the spaceof square integrable functions are derived. These boundstake into account the scatterer size and shape, as well asthe harmonic content of the unknown contrast. As such,they make it possible to understand and quantify the dif-ferent roles played by the spatial harmonic behavior ofthe contrast. Moreover, the role of possible losses in thebackground medium, which reduce the degree of nonlin-earity, can also be quantified.

Exploiting these results, we present a new solution pro-cedure for the inverse scattering problem in Section 3.Such a procedure makes it possible to undertake a pre-liminary (offline) estimation of the mean value of the con-trast to be retrieved and then exploits such an estimationas a starting guess for the actual inversion of data. Nu-merical results presented in Section 4 fully confirm the ef-fectiveness of such a procedure. Conclusions and sugges-tions for future work are contained in Section 5.

2. BOUNDS TO THE NORM OF Ai XA. An Upper Bound to iAi XiLet us now introduce the linear operatorK 5 Ai X:L2(V) → L2(V), defined as

K(e~r!),EV

k~r, r8!e~r8!dr8 r, r8 P V, (6)

wherein k(r, r8) 5 g(r, r8)x(r8). As g(r, r8) is squareintegrable, any element belonging to the range of K is in-deed in L2(V), provided that x is bounded. Indeed ap-plying the Schwarz inequality, one then obtains

iAi X~e !i2 , S EEV

uk~r, r8!u2drdr8D S EV

ue~r8!u2dr8D5 ikiL2~V3V!

2• ieiL2~V!

2 . (7)

Thus by exploiting the definition of the L2 norm for a lin-ear continuous operator,10 we get

iAi Xi2 , ikiL2~V3V!2

5 EV

drEV

dr8u g~r 2 r8!u2ux~r8!u2.

(8)

As the support of x is contained in V, the internal inte-gral can be extended to 6 infinity. Exploiting the convo-lution theorem, Eq. (8) can then be rewritten as:

iAi Xi2 , EV

drE2`

1`

G2~u!X2~u!exp~ ju • r!du, (9)

wherein u 5 (u, v), and G2(u) and X2(u) represent theFourier transforms (in the spatial frequency domain) ofu g(r)u2 and ux(r)u2, respectively. If the integration orderis inverted in Eq. (9), such an expression can be furthermodified as

iAi Xi2 , E2`

1`

G2~u!X2~u!Fs~u!du, (10)

wherein


Fs~u! 5 EV

exp~ ju • r!dr (11)

depends only on the shape of the domain V. Equation(10) then gives an upper bound to iAi Xi2, which dependson how the (spatial) spectral contents of u g(r)u2 andux(r)u2, as well as of Fs(u), combine with each other. Itis interesting to note that objects having the same area(Fs(0)) and the same maximum value of permittivity maygive rise to completely different bounds, depending on theactual shape and harmonic content of the contrast func-tion.

To further analyze this point, we need to evaluate thespectrum of u g(r)u2. The Fourier transform of u g(r)u2

can be rewritten as

G2~u, v ! 5 E0

2pE0

1`

uH0~2 !~br!u2

3 exp@2jr~u cos u 1 v sin u!#rdrdu. (12)

After some manipulations (see Appendix A), the desiredFourier transform for a lossless background medium canbe obtained by the following:

If u2 1 v2 , 4b2,

G2~u, v ! 58

pb2

1

~2j 2 j2!1/2 arctanF ~2j 2 j2!1/2

jG ,

(13)

whereas, if u2 1 v2 . 4b2,

G2~u, v ! 54

pb2

1

~j2 2 2j!1/2 lnF j 1 ~j2 2 2j!1/2

j 2 ~j2 2 2j!1/2G (14)

and j 5 (u2 1 v2)/2b2. As G2(u, v) is radially symmet-ric, only its cut along the v 5 0 axis (for u > 0) has beendepicted in Fig. 1.

It is also interesting to discuss the case of a scattererembedded in a homogenous lossy medium, as happenswith well-to-well tomography (with the object buried indepth). In this case, because Eq. (A3) (see Appendix A) isno longer true, results (13) and (14) do not hold. It is,however, possible to numerically evaluate Eq. (12), rely-ing on the exponential decay of H0

(2)(br) as r → ` when bis complex. It is also possible to evaluate G2(u, v) in aclosed form for u 5 v 5 0 (see Appendix B), which reads

Fig. 1. (Radially symmetric) spectrum G2(u, v) along thev 5 0 axis, for u > 0.

G2~0, 0 ! 5 2plog~b* ! 2 log~b!

b2 2 ~b* !2 . (15)

Figure 2 shows a comparison between differentG2(u, v) obtained by changing the amount of losses in thebackground. As expected, owing to the presence of acomplex b, the spectrum does not show any singularity inu 5 v 5 0. Moreover, the higher the imaginary part ofb, the smaller G2(u, v) is in magnitude.

B. A Lower Bound to iAi XiAs the norm of an operator is equal to the norm of itsadjoint,10 and

~Ai X !1 5 X1Ai1 5 X* Ai

1, (16)

wherein ( • )1 denotes the adjoint operator and X* is theoperator of multiplication by x* , one also obtains

iAi Xi 5 iX* Ai1i 5 sup

iei51

iX* Ai1ei

iei>

iX* Ai1~v0!i

iv0i,

(17)

wherein v0 is the first right singular function of the sin-gular system10 $uk , sk , vk% of Ai . Note explicitly that,because of the above-mentioned properties of g(r, r8), Aiis a compact operator.10

When we exploit the singular-system properties, it fol-lows that

iAi Xi > iX* s0u0i 5 s0iX* u0i , (18)

wherein s0 is the first singular value of Ai1. Therefore

one needs to know s0 and u0 (which depends on the do-main being tested) in order to evaluate the lower bound(18). Even assuming a circular domain, we were not ableto compute a closed-form expression for u0 as a functionof the spatial coordinates and of the parameter bRMAX ofthe domain being tested. It is, however, possible to com-pute such a function and the corresponding singularvalue by using the singular value decomposition of thediscretized operator.

The behavior of such a function, which turns out to be afunction of the radial coordinate only, is reported in Fig. 3for different values of bRMAX . It is worth noting thatthrough use of an appropriate normalization factor de-pending on bRMAX , the different curves collapse into auniversal one for both the real and the imaginary parts.

Fig. 2. Comparison between the spectra of G2(u, 0) for differentvalues of the losses in the background medium.


As a consequence, the singular function u0 can be ex-pressed in closed form as a function of bRMAX as

u0~br, bRMAX! 5 u0r~br !~bRMAX!23/2

2 ju0i~br !~bRMAX!21/2, (19)

wherein u0r(br) and u0i(br) are two universal (i.e., inde-pendent of bRMAX) functions, depicted in Fig. 4.

It is worth noting that both u0r(br) and u0i(br) exhibitan oscillatory behavior in the variable br with period 2p,so they share the same properties as H0

(2)(br). This is aninteresting feature, as the results we have reported inSubsection 2.A were strictly related to the spectral behav-ior of the Hankel function.

When we consider circular domains with different ra-dii, the numerical evaluation of the first singular value s0shows that it also exhibits a definite dependence uponbRMAX . In fact, we have

s0~bRMAX!

bRMAX5

s0~bRMAX8 !

bRMAX8, (20)

i.e., s0 grows linearly with the electrical dimension of the(circular) domain at hand. Let A be the constant suchthat s0(bRMAX) 5 AbRMAX (numerical computation pro-vides A > 0.546). By further rearranging expression

Fig. 3. Comparison between the different singular functionsu0(br, bRMAX) for different values of bRMAX . Real and imagi-nary parts are respectively shown in parts (a) and (b).

(18), taking into account the definition of the norm andthe unitarity of the Fourier transform, one can easilyprove that

iAi Xi2 . A2b2RMAX2 E

2`

1`

U2~u, bRMAX!X2~u!du,

(21)

wherein U2(u, bRMAX) 5 F(uu0u2).Further information can be gained from expression (21)by noting that

s02U2~u, bRMAX! 5 A2F U2r~u!

bRMAX1 U2i~u! • bRMAXG ,

(22)

wherein U2r(u) 5 F (uu0ru2) and U2i(u) 5 F (uu0iu2),evaluated for a very large value of bRMAX , are depicted inFig. 5.

We should note that the lower bound (21) does not ex-hibit an explicit dependence upon the shape of the scat-terer, which is in fact embedded in X2(u). It is, however,still possible to reintroduce explicitly such dependence byrearranging Eq. (18). In fact, one can easily prove that

iAi Xi2 . A2b2RMAXE2`

1`

3 @U2~u, bRMAX! * X2~u!#FS~u!du. (23)

Fig. 4. Universal curves u0r(br) and u0i(br), as defined in Eq.(20).


C. DiscussionFrom Eqs. (13) and (14) it appears that the spectrumG2(u) is a decreasing function of the spectral variables, inagreement with the known smoothing properties of ellip-tic operators.13 The main contribution in the losslesscase is due to its singularity at the origin. In addition,Fs(u) is also a decreasing function of the radial coordi-nate in the spectral domain. Provided the scatterer issmall with respect to the wavelength (i.e., in the low-frequency limit), Fs(u) varies gently with respect toG2(u), so that the latter plays a dominant role around theorigin. On the other hand, in the high-frequency limit(i.e., for large scatterers with respect to the wavelength)Fs(u) cannot be neglected in evaluating the upper bound(10), as it shows strong filtering properties.

In any case, given the decreasing behavior of bothG2(u) and Fs(u) with the radial coordinate, it can be de-duced that only the low-spatial-frequency part of thespectrum X2(u) can make a contribution to the upperbound (10), as its high-spatial-frequency part is filteredout by G2(u)Fs(u). In a similar fashion, due to the be-havior of U2r(u) and U2i(u) (see Fig. 5), the low-frequency part of X2(u) is the main factor in determiningthe lower bound to iAi Xi . Therefore the more signifi-cant the low-frequency content of X2(u), the higher thebounds to iAi Xi , with a subsequent increase in the non-linearity of the scattering problem at hand. It is inter-esting to understand under which conditions the spec-trum X2(u) proves small at low spatial frequencies. For

Fig. 5. Sketch of the (numerically computed) U2r(u, 0) andU2i(u, 0), as defined in Eq. (22).

this purpose, note that the spectrum X2(u) can be evalu-ated by applying the properties of the convolution prod-uct, thus

X2~u! 5 F@x • x* # 5 X~u! * @X~2u!#* , (24)

wherein X(u) denotes the Fourier transform of the con-trast function x. From Eq. (24) it appears that X(u) canbe easily computed from the knowledge of the spectrum ofthe contrast function.

The above arguments lead to interesting results thatfully agree with numerical experience. First, in both thehigh- and low-frequency regime, the mean value of thecontrast plays a major role in determining bounds (10)and (21). In fact, two objects with the same shape andthe same maximum contrast value, which is constant inone case and oscillating with a zero-mean value in theother, give rise to functions X2(u), with values differingin the origin by a factor of two or four, depending onwhether the oscillations take place in one or two dimen-sions. As a consequence, the oscillating case, despite thefact it has an overall double excursion, gives rise to alower degree of nonlinearity and is thus simpler to re-trieve in the inverse problem.

For a second interesting result regarding purely oscil-lating (i.e., zero-mean) profiles, let us first recall someknown results. Provided there are no primary sourcesand measurement probes in the reactive zone of the scat-terer (see Ref. 14 for more details), in the Born approxi-mation Fourier components of up to 2k can be recoveredin the spectral domain. This corresponds exactly to a l/4sampling in the spatial domain, so that one expects atmost to retrieve spatial variations of that order, and itmakes no sense to look for faster variations, as they arefiltered out by the scattering phenomena. However, nospatial filtering occurs with lower-order Fourier compo-nents, which correspond to visible profiles. If we assumethat similar filtering effects hold true even in the nonlin-ear case, it makes sense to compare two purely oscillatingcontrasts. In particular, let us consider two profiles hav-ing the same amplitude and boundary, but different oscil-lation frequencies (both less than 2k). They will give riseto the same value of X2(0) but to two different functionsX2(u) (see Fig. 6). Given the decaying behavior of bothG2(u) and U2(u), the function that oscillates faster givesrise to a lower value of the upper bound (10) and of thelower bound (21) as well. However, as even little differ-ences in iAi Xi may give rise to considerable differencesin the number of terms to be retained in expansion (4) (orin the modified Born series11 in case iAi Xi . 1), thismeans that the contrast oscillating more rapidly givesrise to a lower degree of nonlinearity in the relationshipbetween the profile and the internal field. In inverseproblems, this suggests that the latter should be easier toreconstruct than the slowly oscillating contrast. As al-ready stressed, this kind of comparison only makes senseamongst visible profiles.

In Fig. 7 the (numerically computed) actual norm of os-cillating contrasts that have different periods are re-ported together with the corresponding upper and lowerbounds. Two different cases have been considered: Fig-ure 7(a) corresponds to bRMAX 5 p, while Fig. 7(b) corre-


sponds to bRMAX 5 2p. It has been possible to appreci-ate how the dependence on the (spatial) frequency iscorrectly predicted by the bounds, and in particular, thelower bound turns out to be quite accurate. It is inter-

Fig. 6. Comparison between the X2(u, v) spectra for the threedifferent profiles: (a) constant profile (solid curve), (b)cos (x) cos ( y) profile (dashed curve), (c) cos (2x) cos ( y) (dottedcurve). The cut along the v 5 0 axis of the three spectra is re-ported.

Fig. 7. Comparison between the (numerically computed) actualnorm of oscillating contrasts having different periods [see Eq.(30)] and the estimated bounds, for different electrical dimen-sions of the scatterer. (a), bRMAX 5 p; (b) bRMAX 5 2p. Notethat in (a) the norm is almost coincident with the estimatedlower bound.

esting to note that, because of the decay of bothFS(u)G2(u) and U2(u), the reduction in nonlinearitysaturates when the number of complete oscillations in-creases. This circumstance suggests that our conclusionabout the reduction of nonlinearity with increasing oscil-lations only holds true for the first few harmonics.

When a lossy background is concerned, which is thecase when a (sufficient deeply) buried object is beingsought, the upper bound (10) shows that iAi Xi2 will belower if compared with the corresponding lossless case.Therefore the Born approximation will have an extendedvalidity range. This result is partially expected, as themultiple-scattering effects between the different parts ofthe scatterer, which are the main contributions to theoverall nonlinearity, are certainly reduced because oflosses. However, the upper bound (10) and Eq. (16) allowa quantitative estimation of the amount of reduction andtherefore of the applicability of the Born expansion in thedifferent cases. In inverse problems, two contrasting fac-tors are to be considered. On the one hand, when dealingwith a reduced nonlinearity it is possible to deal withfalse-solution-free problems in an enlarged set of cases.In particular, the higher the losses, the lower the upperbound for iAi Xi2. On the other hand, the presence oflosses also significantly reduces the essential dimensionof the space of data,12 which proves to be a function ofboth the measurement distance and the losses. There-fore even if one is more confident in approaching the in-verse problem as a linear one, the latter becomes severelyill-conditioned as far as continuity in data is concerned.

3. AN EFFECTIVE SOLUTION PROCEDUREFOR THE INVERSE SCATTERINGPROBLEMThe degree of nonlinearity, and henceforth the difficultyfaced by (iterative) approaches to inverse scattering prob-lems, is usually attributed to electrical dimensions andmaximum permittivity of the contrast.1–3

From what has been reported above, we learned thatthe degree of nonlinearity in inverse scattering problemsalso depends on the shape and harmonic content of thecontrast, particularly in case of electrically small objects.Moreover, irrespective of the electrical dimension, themean value of the contrast plays a crucial role in deter-mining X2(0), and therefore the norm of the operatorAi X. As an example, in the case of a positive definite realcontrast, the mean value contributes by more than 2/3 toX2(0) and therefore, in a first instance, by more than 80%to iAi Xi .

This circumstance suggests a new processing procedurefor the inverse problem. In fact, let us separate the con-trast into its mean value and oscillating parts, i.e.,

x 5 x0 1 Dx. (25)

Substituting Eq. (25) into expansion (4) (or the modi-fied Born series in case iAi Xi . 1), the Born series be-comes


E 5 Ei 1 Aix0~Ei! 1 AiDx~Ei! 1 Aix0@Aix0~Ei!#

1 Aix0@AiDx~Ei!# 1 AiDx@AiDx~Ei!# 1 ...,

(26)

wherein, for the sake of simplicity, the expansion hasbeen truncated to the second-order term and operatorsAix0 and AiDx are operator Ai X for x 5 x0 andx 5 Dx, respectively. Now, if x0 is known through pre-vious knowledge or estimation, Eq. (26) can be rearrangedas a series in AiDx. As the norm of the latter is consider-ably lower (in the above hypothesis) than that of Aix0 ,much fewer terms are required in such a series than inthe original one, simplifying the overall solution of boththe forward problem and the inverse problem. Such pre-liminary knowledge is not usually available and must beestimated from the data themselves or from other kinds ofdata.

When multifrequency data are available, the lowest-frequency results provide the estimate required, and theabove analysis helps to understand the effectiveness offrequency-hopping schemes.15,16 In particular, it isworth noting that the success of these procedures is re-lated not only to the relative strength of low-order andhigh-order harmonics, as shown in Ref. 16, but also to thenorms iAix0i and iAiDxi .

In the single-frequency case we are considering, themean value of the contrast must be directly estimatedfrom the data available. For a given configuration, thepreliminary estimation of the mean value of the unknownpermittivity is performed by searching for the globalminimum of the cost functional:

F~x t! 5 (v

iEMv 2 Etrial

v~ x t!i2,

xMIN < x t < xMAX , (27)

wherein EMv is the value of the scattered field corre-

sponding to the vth view and Etrialv(x t) is the field scat-

tered from a constant contrast profile x t under the sameincidence conditions. In our experience, global minimi-zation of expression (27) generally furnishes a fairly accu-rate estimation of the mean value.

Once the mean contrast has been estimated, it is usedas a starting guess for the full inversion, which we per-form by adopting the approach of Ref. 5 in its final form.

Note that a similar two-step solution procedure wasproposed on a purely heuristic basis in Ref. 4. Our ap-proach, however, is different for at least three differentreasons. First, it has a firm basis in the theoreticalanalysis of the degree of nonlinearity. Second, a global(rather than local) optimization procedure is adopted forestimating the mean value. As this estimation could alsobe affected by false solutions, such a choice allows in-creased robustness in evaluating the fundamental har-monic. We have verified that this is indeed the case forseveral profiles. Third, the inversion approach used inthe second step (see Ref. 5) is quite different from thatused in Ref. 4.

4. DISCUSSION AND NUMERICALEXAMPLESIn the following, we validate through numerical examplesthe analysis of Section 2 (and its relevance in the solutionof the inverse problem) as well as the effectiveness of thesolution procedure developed in Section 3. For this pur-pose, the numerical analysis is subdivided into threeparts.

A. Comparison between Constant and OscillatingProfilesTo show how the spatial behavior of the contrast functionaffects nonlinearity and hence the difficulty in solving theinverse scattering problem, let us compare, in the freespace case, two profiles with the same dimensions andshape but different spatial-frequency content. As a firststep, let us analyze our reconstruction capabilities by con-sidering two contrast profiles that have the same maxi-mum value of the contrast and the same extension andthat are respectively constant and cosinusoidally varying(with a zero mean). The approach of Ref. 5 has beenadopted in the inversion, using the background as a start-ing guess and expanding the unknown contrast into 2-Dtruncated Fourier series.

By considering a scatterer of size l 3 l, we were ableto successfully retrieve a one-dimensional cosinusoidallyvarying (along the x direction) complex profile

x~x, y ! 5 xMAX~1 2 0, 1 j !cosS 2px

lD , (28)

whose amplitude was as large as xMAX 5 2.4. Moreover,the reconstruction of a 2-D cosinusoidally varying con-trast distribution (along the x and y directions),

x~x, y ! 5 xMAX~1 2 0, 1 j !cosS 2px

lD cosS 2p

y

lD , (29)

was possible up to an amplitude as large as xMAX 5 3.2,while it was only possible to retrieve constant profiles upto a maximum amplitude of xMAX 5 1.1. Note that re-sults are correctly predicted by the first result of Section2.C.

In each case, 3 3 3 Fourier harmonics of the contrastwere looked for, 12 plane waves were considered as inci-dent fields, and 12 receivers placed on a circle 3 l in ra-dius surrounding the scatterer were used. The real partsof actual and reconstructed profiles are shown in Figs.8(a)–8(d).

B. Comparison of Oscillating ProfilesThe second result of Section 2.C also suggests that the dif-ficulty of a given inverse scattering problem does not sim-ply depend on the energy of the unknown contrast, butalso on its spectral distribution. This result is fully con-firmed by numerical experience. For example, the profile

x~x, y ! 5 xMAX~1 2 0, 1 j !cosS 2pnx

lD cosS 2p

my

lD(30)

(with n 5 2 and m 5 2) has the same energy as Eq. (29)but, according to Section 2, an even lower iAixi due to the


Fig. 8. Real part of the actual and reconstructed cosinusoidally oscillating profiles, as defined in Eq. (30). (a) and (b) n 5 1, m 5 0(xMAX52.4); (c) and (d) n 5 1, m 5 1 (xMAX 5 3.2); (e) and (f) n 5 2, m 5 0 (xMAX 5 4.0); (g) and (h) n 5 2, m 5 2 (xMAX 5 4.5).

different behavior of X2(u) (see Fig. 6). As a conse-quence, it should be easier to reconstruct than Eq. (29).

In fact, we were able to successfully retrieve unknowncontrast functions of this kind by enlarging the amplitudeof the oscillations up to xMAX 5 4.5; a smaller value,xMAX 5 4.0, was instead obtained with reference to Eq.

(30) with n 5 2 and m 5 0, corresponding to double-frequency oscillations along the x axis only.

Also in this case the approach of Ref. 5 was adopted.In particular, 5 3 5 Fourier harmonics of the contrastfunction were looked for, and the same number of incidentfields as well as receivers of the previous case were con-


sidered. In this example, the background permittivityhas also taken as the starting guess for the inversion pro-cedure. The reference profiles and the reconstructed pro-file are shown in Fig. 8(e)–8(h).

The above examples confirm that if the low-frequencycontent of the spatial-frequency spectrum of the contrastfunction is reduced, then the nonlinearity of the problemis also reduced. Hence, the inversion procedure is morelikely to retrieve the actual solution of the problem, thusovercoming the false-solution problem.5

For a rapid comparison, all of the above results havebeen collected in Table 1.

C. Effectiveness of the Procedure ProposedThe procedure proposed in Section 3 has proved to be ex-tremely effective. As a matter of fact, a large number ofnumerical examples show that it is possible to achieve ac-curate reconstructions in many different cases whereusual techniques do not lead to the actual solution.

To illustrate such effectiveness, we present below twodifferent examples pertaining to lossless scatterers. In allcases, a 10% additive noise was introduced to corrupt thesynthetic data in order to simulate realistic measurementconditions.

In the first example [see Fig. 9(a)], wherein the un-known object consists of a cylindrical cavity partiallyfilled with a cylindrical dielectric, 20 different views and20 measurements for each view were used. By using theinversion approach of Ref. 5 and taking the backgroundas a starting guess, it is not possible to achieve any sig-nificant reconstruction [see Fig. 9(b)]. In the proposedapproach, the mean value of the permittivity is first esti-mated (with an estimation error as low as 5.2%). Then,using Ref. 5 and looking for 9 3 9 Fourier harmonics, weachieve the reconstruction of Fig. 9(c) after 393 iterations.As only low-pass reconstructions of actual profiles can beachieved given the lack of proper edge-preserving regular-ization techniques,17 the result is more than satisfactory,particularly in view of the high value of the product(maximum contrast) 3 (electrical dimensions). The cor-responding normalized mean square error is 0.095.

Similar comments apply to the second example[wherein the unknown object is made up of two cylinders;see Fig. 10(a)] as well as to many others. Usual strate-gies (such as Ref. 5) become trapped in local minima,yielding false solutions [see Fig. 10(b)], whereas the pro-posed two-step procedure allows satisfactory reconstruc-tions [see Fig. 10(c), obtained after 187 iterations, with a

Table 1. Comparison between MaximumContrasts with Reference to the HarmonicContent of the Oscillating Contrast Profile

Harmonic ContentEq. (30)

n, m xMAX

Number of Unknowns(Contrast Fourier Harmonics)

n 5 0, m 5 0 1.1 3 3 3n 5 1, m 5 0 2.4 3 3 3n 5 1, m 5 1 4.2 3 3 3n 5 2, m 5 0 4.0 5 3 5n 5 2, m 5 2 4.5 5 3 5

mean square error of 0.203]. Ten different views and tendifferent measures for each view have been used in thissecond example.

Fig. 9. (a) Case 1, real part of the actual profile; (b), result ob-tained with inversion approach in Ref. 5; (c), reconstructed pro-file exploiting the proposed procedure.


Although the above examples show that the procedureproposed largely extends the class of profiles that can bereliably reconstructed, further work is required in order

Fig. 10. (a) Case 2, real part of the actual profile; (b) result ob-tained with inversion approach in Ref. 5; (c) reconstructed profileexploiting the proposed procedure.

to trust the results of the estimation performed in thefirst step and to understand the consequent ultimate limi-tations of the approach proposed.

5. CONCLUSIONSIn this paper, new upper and lower bounds to the norm ofthe operator Ai X have been derived. As this quantitydictates the degree of nonlinearity in the relationship be-tween the contrast and scattered fields, the relationshipsobtained may be particularly helpful in quantifying thedifficulty of iterative approaches to both forward and in-verse scattering problems.

In forward problems, Eqs. (10) and (18) make it pos-sible to evaluate bounds to iAi Xi as a function of size andshape of the scatterer, as well as the maximum value andspatial-frequency content of its permittivity profile. Thisgives indications of the possibility of quickly and accu-rately evaluating the formal solution (2) by adopting aproper series expansion. In particular, in the case inwhich iAi Xi , 1, the knowledge of the norm iAi Xi mayhelp when choosing the number of terms in the Neumannseries (4). In the large permittivity case, i.e., when con-dition (5) does not hold, the modified Born series proposedin Ref. 11 may be adopted.

In inverse problems, given the fact that the bounds de-rived give explicit and analytic evidence dependencies onall of the above factors, we can understand the role theyplay in the fulfillment of the Born approximation, andthis fact allows us to discuss how the difficulties in itera-tive approaches to the inverse problem change whenmodifying the above factors at the same time.

The theoretical and numerical analysis performedshows that in addition to the scatterer’s spatial extensionand the maximum magnitude of its contrast, the spatialbehavior of the permittivity profile also plays a key role inthe solution of the inverse scattering problem. In par-ticular, when the contrast has a nonnegligible meanvalue, the degree of nonlinearity grows considerably.

To reduce the degree of nonlinearity, both software pro-cedures and hardware solutions (such as immersing theunknown body in a water bath) are available. In fact, abackground medium different from free space and whosepermittivity value is close to the expected mean value (asthe one in Ref. 18) could be of substantial help.

However, doing this also increases the electrical dimen-sions of the scatterer. While increasing the essential di-mension of the space of data (if eb . e0), this circum-stance also increases iAi Xi so that a trade-off does existbetween the reduction in nonlinearity due to a reductionin the fundamental harmonic and an increase in nonlin-earity given the enlarged electrical dimensions.

As a software solution, the fundamental harmonic x0must be estimated in order to hopefully deal with an in-verse problem, wherein only the oscillating part Dx is un-known. This is related to what occurs in the so-calledfrequency-hopping schemes,15,16 and the analysis per-formed helps to understand their effectiveness. A differ-ent solution procedure, which relies on monochromaticdata, has been developed, discussed, and validatedherein. Numerical examples confirm the ability of the


scheme proposed to accurately retrieve unknown profilesthat cannot be reconstructed through the usual schemes.

APPENDIX AThe Fourier transform of u g(r)u2, Eq. (12) can be rewrit-ten as

G2~u, v ! 5 E0

1`E0

2p

uH0~2 !~br!u2 exp@2jr~u2 1 v2!1/2

3 cos~u 2 f!#rdrdu, (A1)

wherein

cos f 5u

~u2 1 v2!1/2 , sin f 5v

~u2 1 v2!1/2 .

By means of such a substitution, we exploit the definitionof the Bessel function in terms of exponential functionsand the relationship between the absolute value of theHankel function and the Kelvin function for realarguments19:

2pJ0@r~u2 1 v2!1/2# 5 E0

2p

exp@2jr~u2 1 v2!1/2

3 cos~u 2 f!#du, (A2)

uH0~2 !~br!u2 5

8

p2 E0

1`

K0@2br • sinh~t !#dt,

(A3)

wherein sinh( • ) denotes the hyperbolic sine function.Therefore Eq. (A1) can be rewritten as

G2~u, v ! 516

pE

0

1`E0

1`

K0@2br • sinh~t !#

3 J0@r~u2 1 v2!1/2#rdrdt. (A4)

Since19

E0

1`

K0@2br • sinh~t !#J0@r~u2 1 v2!1/2#rdr

51

@2b • sinh~t !#2 1 u2 1 v2 , (A5)

G2~u, v ! 516

pE

0

1` 1

@2b • sinh~t !#2 1 u2 1 v2 dt

58

b2pE

0

1`

@cosh~2t ! 1 ~j 2 1 !#21dt, (A6)

wherein j 5 (u2 1 v2)/2b2; then, if (u2 1 v2)1/2 , 2b,one can obtain

G2~u, v ! 58

b2p@1 2 ~j 2 1 !2#21/2

3 arctanH @1 2 ~j 2 1 !2#1/2

jJ , (A7)

while, if (u2 1 v2)1/2 . 2b, one can obtain

G2~u, v ! 54

b2p@~j 2 1 !2 2 1#21/2

3 lnH j 1 @~j 2 1 !2 2 1#1/2

j 2 @~j 2 1 !2 2 1#1/2J . (A8)

APPENDIX BTo evaluate the spectrum G2(u, v) in the case of lossy me-dium (modelled with a complex b), we can recall the con-volution product properties, i.e.,

G2~u, v ! 5 G~u, v !* @G~2u, 2 v !#* , (B1)

wherein G( • ) 5 F @ g( • )#.Exploiting the Helmotz equation in the wave number

domain, the spectrum G(u, v) can be expressed as

G~u, v ! 51

b2 2 ~u2 1 v2!, (B2)

where b 5 v@e0(eb 2 jsb /ve0)m0#1/2, and Im@b# , 0.Note that Eq. (B2) would have singularities in the losslesscase. By combining the spectrum of Eq. (B2) as in Eq.(B1) one can then obtain the desired spectrum G2(u, v).By computing the convolution for u 5 v 5 0 via decom-position in simpler fractions,20 one obtains

G2~0, 0 ! 5 2p •

log~b* ! 2 log~b!

b2 2 ~b* !2 . (B3)

The authors can be reached at the address on the titlepage or by email: [email protected], [email protected],[email protected], and isernia@unina@it.

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degree of nonlinearity and a new solution procedure in scalar two-dimensional inverse scattering...

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