1. INTRODUCTIONElectromagnetic inverse scattering problems arise in manyapplications such as radar imaging [1], nondestructive testingand evaluation [2–5], biomedical diagnostics [6–8], geoscienceand remote sensing [9], subsurface inspection [10,11], andmaterial characterization [12]. The growing scientific and in-dustrial interest toward these applications has encouraged thestudy and the development of several inverse scattering tech-niques [13–17]. These approaches are usually classified intostochastic [15,18] or deterministic techniques [19–23]). In thelatter framework, the contrast source (CS) formulation, wherethe problem unknown are the so-called CSs (i.e., the inducedcurrents) and the contrast function, has received great atten-tion because of the more accurate reconstructions and anenhanced robustness with respect to the contrast field formu-lation, even exploiting deterministic minimization tools [20].
However, a critical issue when dealing with CS-based meth-ods lies in the nonuniqueness of the arising inverse sourceproblem [24], which is related to the existence of nonradiatingcomponents (or nonmeasurable, if a finite number of mea-surements is at hand) of the unknown CS. To overcome thisdrawback, Habashy et al. proposed in [25] an iterative algo-rithm first aimed at reconstructing the minimum norm (orradiating components) currents through its conjugate kernelexpansion and then iteratively refining the nonmeasurablecontrast-source components and the corresponding contrastproperties by enforcing the consistency of the their spatialvariations. On the same line-of-reasoning, but resorting to thesingular value decomposition (SVD) of the discretized Green’soperator to compute the radiating components, a nonlinearprocedure was proposed in [26] and further extended in
[27] where the original inverse problem is recast as theminimization of the normalized data misfit function.
Recently, the subspace-based optimization method (SOM)has been proposed as a complement to the existing inversescattering techniques within the CS formulation [28,29]. In thisapproach, the unknown CS is subdivided in a deterministicpart (i.e., the stronger radiating components) and an ambig-uous one, which differs from the nonradiating componentin [26,27] for the presence of the so-called weakly radiatingcomponents [28]. The deterministic currents are computedthrough SVD as in [26,27], whereas the ambiguous ones aredetermined by minimizing the scattering data mismatch quan-tified by the state term and, unlike [26,27], the data term [28]for taking into account the weakly radiating contributions.The SOM shares several positive features of other CS-basedinversion techniques, while it also allows faster convergenceespecially if some a priori information on the noise level isavailable [28]. This has motivated its successful applicationto different microwave imaging problems [28,30–34]. Never-theless, it also presents some drawbacks, briefly summarizedin the following:
• Local minima. A local search in a high-dimensionalspace (when wide domains and high resolutions are at hand)is performed to look for the ambiguous components. Like forevery deterministic optimization algorithm, the estimatedsolution may be occasionally trapped in local minima [35]leading to wrong reconstructions.
• Computational complexity. Since the computation ofthe SVD of the discretized Green’s operator is required to de-termine the deterministic currents, the computational burden
Oliveri et al. Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. A 2057
grows with the dimension of the investigation domain and thespatial resolution [28,31].
• Robustness to noise. When no information on the signal-to-noise ratio (SNR) is available, the choice of the SVD trun-cation index, L, which identifies deterministic and ambiguouscomponents, can become critical in terms of inversion speedand reliability even though the sensitivity of the reconstruc-tion on L has been proven to be not so significant [28].
To overcome these limitations, a new inverse scattering ap-proach based on the integration of a multifocusing strategy,namely the iterative multiscaling approach (IMSA) [36–39],with the SOM [28] is proposed in this paper. More in detail,the IMSA-SOM method implements the following nested pro-cedure: (a) an external IMSA loop devoted to correctly locatewithin the investigation domain the regions of interest (RoIs)where the scatterers are supposed to lie and where adaptivelyrefining the inversion grid to enhance the spatial resolutionand accuracy of the reconstruction [36], (b) a truncated SVDstep aimed at computing the deterministic currents in theRoIs, (c) a conjugate gradient (CG)-based iterative optimiza-tion step aimed at determining the ambiguous currents byminimizing the scattering data mismatching cost function
defined over the IMSA discretization [28]. Such a strategy ismotivated by the following considerations.
1. The IMSA can mitigate the local minima problem [36]since the number of unknowns is kept close to the amountof collectable information from the field measurements[35,40] at each iteration of the multiscaling loop.
START
Initialization
Compute SVD
t=T ?(s)no
Compute Matrices
Filtering
Clustering
Estimate RoIs
Set s=s+1
Stationarity
verified?END
no
yes
Estimate I for v=1,...V from (10)(s)v det
Set t=1
E H G(s)(s)
vinc (s)
Set s=1
yes Computeτ (s)
n(r )
Compute a from (15)(s)v t+1
Compute from (16)ξ(s)
n t+1
Set t=t+1
Fig. 1. (Color online) Flowchart of the proposed IMSA-SOMprocedure.
2058 J. Opt. Soc. Am. A / Vol. 28, No. 10 / October 2011 Oliveri et al.
2. The SVD is computed on reduced-size matrices, alsowhen the spatial resolution within the RoIs is very high [36],since a smaller problem is solved by the SOM at each step ofthe multizooming loop [41].
3. An adaptive choice of the SVD truncation index miti-gates or avoids the issues related to the information on thenoise level, thus indirectly increasing the inversion robustnessto the noise.
The outline of the paper is as follows. The inversion problemis mathematically formulated in Section 2, while the proposedIMSA-SOM procedure is detailed in Section 3. Section 4 is con-cerned with a numerical assessment carried out, also throughsome comparisons with the uniform-grid SOM implementa-tion. Finally, some conclusions are drawn (Section 5).
2. PROBLEM FORMULATIONLet us consider a two-dimensional (2D) inverse scattering pro-blem (z being the longitudinal direction and invariance axis)under transverse magnetic (TMz) incidence. In a lossless andnonmagnetic background with permittivity ε0 and permeabil-ity μ0, a cylindrical scatterer of the arbitrary cross section Ωis successively irradiated by V known monochromatic wavesEincv ðrÞz, v ¼ 1;…; V with time-dependence expðj2πf tÞ
omitted hereinafter. At each incidence v ¼ 1;…; V , the elec-tric field with, Etot
v , and without, Eincv , the scatterer is collected
at M measurement points lying in an observation domain DO
(i.e., rm ∈ DO, m ¼ 1;…; M) external to the investigationdomain D (rm∉D). According to the CS formulation, thefollowing data equation holds true in DO [28]:
Escattv ðrmÞ ¼ k20
ZΩJvðr0ÞGðrm=r0Þdr0;
v ¼ 1;…V ;m ¼ 1;…; M; ð1Þwhere k0 is the free-space wavenumber, JvðrÞ≜τðrÞEtot
v ðrÞ isthe CS, Escatt
v is the scattered field given by Escattv ðrÞ≜
Etotv ðrÞ − Einc
v ðrÞ, Gðr=r0Þ is the 2D free-space Green’s function,τðrÞ≜½εrðrÞ − j
σðrÞ2πf ε0� − 1 is the object function describing the
material properties of the nonmagnetic scatterer, whileεrðrÞ and σðrÞ indicate the relative dielectric permittivity andthe electric conductivity, respectively. Moreover, the stateequation is satisfied within the investigation domain [28]
τðrÞEincv ðrÞ ¼ JvðrÞ − k20τðrÞ
ZΩJvðr0ÞGðr=r0Þdr0;
v ¼ 1;…V ; r ∈ D:
ð2Þ
Starting from the knowledge of the scattering data (i.e., themeasurement of Etot
v in DO and Eincv in both D and DO), CS-
based inverse scattering techniques are aimed at solvingthe scattering Eqs. (1) and (2) to retrieve the distributionof τðrÞ within D.
10-3
10-2
60 50 40 30 20 10 5
Ψto
t [ar
bitr
ary
unit]
SNR [dB]
V=M=24, τ=1.0, Adaptive IMSA-SOM
α=0.5 α=0.7 α=0.9Fig. 3. (Color online) Sensitivity analysis (square cylinder: l ¼ 2:4λ,V ¼ 24, M ¼ 24, τ ¼ 1:0). Behavior ofΨtot versus SNR when applyingthe adaptive IMSA-SOM with different α values.
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(d)
Fig. 4. Sensitivity analysis (square cylinder: l ¼ 2:4λ, V ¼ 24,M ¼ 24, τ ¼ 1:0, α ¼ 0:7). Reconstruction of the dielectric profile(a) when applying the adaptive IMSA-SOM to process (b) noiselessand noisy data with a (c) SNR ¼ 30dB and a (d) SNR ¼ 10dB.
Oliveri et al. Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. A 2059
3. IMSA-SOM INVERSIONIn order to numerically solve the inverse scattering problem athand, the IMSA-SOM scheme works as follows. At each s-stepof the IMSA procedure (s ¼ 1 at the initialization), Eqs. (1) and(2) are discretized according to the coupled dipole method[28]. The investigation domain D is partitioned in N squarecells D
ðsÞn centered at rðsÞn (rðsÞn ∈ D
ðsÞn ⊂D, n ¼ 1;…; N ðsÞ) to
yield the following set of equations [28]:
Escattv ðrvmÞ ¼
XNp¼1
IvðrðsÞp Þgðrvm; rðsÞp Þdr0; v ¼ 1;…V ;m
¼ 1;…; M; ð3Þ
ξðsÞn Eincv ðrðsÞn Þ ¼ IvðrðsÞn Þ − ξðsÞn
XNp¼1
IvðrðsÞp ÞgðrðsÞn ; rðsÞp Þ;
v ¼ 1;…V ;n ¼ 1;…; N ðsÞ; ð4Þ
where IvðrðsÞn Þ ¼ jk0η0ξðsÞn Etotv ðrðsÞn Þ is the contrast displacement
current,
ξðsÞn ¼ −jk0
η0AðsÞn τðrðsÞn Þ; ð5Þ
is the polarization strength, gðr; rðsÞp Þ ¼ 1AðDðsÞ
Fig. 5. (Color online) Sensitivity analysis (square cylinder: l ¼ 2:4λ, V ¼ 24, M ¼ 24, τ ¼ 1:0, α ¼ 0:7). Behavior of Ψtot versus T0 when applyingthe adaptive IMSA-SOM and for different SNRs. (a) Retrieved dielectric profiles when (b) and (d) (T0 ¼ 25) and (c) and (e) (T0 ¼ 25) by processing(b) and (c) noiseless (SNR ¼ ∞) and (d) and (e) noisy data (SNR ¼ 10dB).
10-5
10-4
10-3
10-2
0 0.5 1 1.5 2 2.5
Ψto
t [ar
bitr
ary
unit]
τ
V=M=24 - Adaptive IMSA-SOM (α=αopt, T0=Topt)
SNR=30 [dB]
SNR=10 [dB]
SNR=5 [dB]
Fig. 6. (Color online) Sensitivity analysis (square cylinder: l ¼ 2:4λ,V ¼ 24, M ¼ 24, α ¼ 0:7, T0 ¼ 50). Behavior of Ψtot versus τ fordifferent SNRs.
2060 J. Opt. Soc. Am. A / Vol. 28, No. 10 / October 2011 Oliveri et al.
the nth cell DðsÞn , and η0 is the free-space impedance. Rewriting
Eqs. (3) and (4) in matrix form [28], it results in
IðsÞv ¼ ΞðsÞðfEincv gðsÞ þHðsÞIðsÞv Þ; v ¼ 1;…; V ; ð6Þ
IvðrðsÞN Þ�0, GðsÞ is an M × N matrix whose ðm;nÞ entry isgðrm; rðsÞn Þ, HðsÞ is an N × N matrix whose ðp; nÞ element is
gðrðsÞp ; rðsÞn Þ, and 0 stands for the transpose operator. The SOMis then applied to solve Eqs. (6) and (7). More specifically, theSVD of GðsÞ is first computed as
GðsÞ ¼ UðsÞXðsÞðVðsÞÞH; ð8Þ
where H indicates the transpose and complex conjugate,
X ðsÞ ¼ ½diagðχðsÞ1 ;…; χðsÞM Þ; 0M;…; 0M|fflfflfflfflfflffl{zfflfflfflfflfflffl}N−M
� is the singular value ma-
trix where χðsÞ1 ≥ χðsÞ2 ≥ … ≥ χðsÞM , and 0M is the null-vector ofsizeM , while VðsÞ and UðsÞ are the (right and left) singular vec-tor matrices. By decomposing the singular vector matrices ina deterministic and an ambiguous part (i.e., VðsÞ¼VðsÞ
LðsÞ ∈ ½0; M� being the SVD truncation index and VðsÞp , UðsÞ
p
stands for the p-th columns of VðsÞ, and UðsÞ. By substitutingEq. (8) in Eq. (7) and taking into account the singular vectormatrices decomposition, it is possible to write the following:
where fIdetv gðsÞ is the deterministic part of the currents givenby
Act
ualO
bjec
t
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Re[τ(x,y)]
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y/λ
0
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1.5Re[τ(x,y)]
(c
(
)
d)
Fig. 7. Numerical assessment (inhomogeneous cylinder: l ¼ 2:4λ,V ¼ 24, M ¼ 24, τout ¼ 0:6, τin ¼ 1:2, SNR ¼ 30dB). (a) Actual distri-bution and dielectric profiles retrieved with the IMSA-SOM at(b) s ¼ 1, (c) s ¼ 2, and (d) s ¼ S ¼ 3.
Inhomogeneous Concentric Cylinder (τout=0.6, τin=1.2), SNR=30 dB
IMSA-SOM, s=1
IMSA-SOM, s=2
IMSA-SOM, s=3
BARE-SOM
(b)
Fig. 8. (Color online) Numerical assessment (inhomogeneouscylinder: l ¼ 2:4λ, V ¼ 24, M ¼ 24, τout ¼ 0:6, τin ¼ 1:2, SNR ¼30dB). Behavior of (a) the objective function versus k and (b) thesingular values of the mapping operator GðsÞ.
Oliveri et al. Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. A 2061
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(e () f )
Fig. 9. Numerical assessment (inhomogeneous cylinder: l ¼ 2:4λ, V ¼ 24,M ¼ 24, τout ¼ 0:6, τin ¼ 1:2). Dielectric profiles reconstructed by meansof (a), (c), and (e) as the standard SOM and (b), (d), and (f) as the adaptive IMSA-SOM processing. (a) and (b) are noiseless data and noisy sampleswith (c) and (d) having a SNR ¼ 20dB and (e) and (f) having a SNR ¼ 10dB.
the ambiguous part of IðsÞv can be expressed as the linearcombination of the ðN − LðsÞÞ nonnull ambiguous vectorsthrough the set of unknown auxiliary coefficients aðsÞv
fIambv gðsÞ ¼ VðsÞ
ambaðsÞv : ð11Þ
By exploiting Eqs. (9)–(11) and (7) or Eqs. (6), the data misfit
function defined asΔðsÞdata≜
PVv¼1
�‖GðsÞIðsÞv −Escatt
v ‖2
‖Escattv ‖2
�turns out to be
ΔðsÞdataðaðsÞv ;ΞðsÞÞ ¼
XVv¼1
�‖GðsÞVðsÞ
ambaðsÞv þ GðsÞfIdetv gðsÞ − Escatt
v ‖2
‖Escattv ‖2
�;
ð12Þ
while the state misfit given by ΔðsÞstate≜
PVv¼1�
‖IðsÞv −ΞðsÞðfEincv gðsÞþHðsÞIðsÞv Þ‖2
‖fIdetv gðsÞ‖2
�can be expressed as
ΔðsÞstateðaðsÞv ;ΞðsÞÞ ¼
XVv¼1
�‖ðVðsÞ
amb − ΞðsÞHðsÞVðsÞambÞaðsÞv − ΞðsÞðfEinc
v gðsÞ þHðsÞfIdetv gðsÞÞ þ fIdetv gðsÞ‖2
‖fIdetv gðsÞ‖2
�: ð13Þ
To solve the inversion problem, Eqs. (12) and (13) have to befitted by determining the unknowns aðsÞv and ΞðsÞ through theminimization of the following objective function
F ðsÞðaðsÞv ;ΞðsÞÞ≜ΔðsÞdataðaðsÞv ;ΞðsÞÞ þΔðsÞ
stateðaðsÞv ;ΞðsÞÞ: ð14Þ
Starting from an initial guess ΞðsÞjt¼0 solution obtained byback propagation [28] and setting aðsÞv jt¼0 ¼ 0, the ambiguouscurrent coefficients are iteratively (t being the iteration step)updated:
aðsÞv jtþ1 ¼ aðsÞv jt þ δðsÞv jthðsÞv jt: ð15Þ
hðsÞv jt being in the Polak–Ribiere CG search direction [28]
2062 J. Opt. Soc. Am. A / Vol. 28, No. 10 / October 2011 Oliveri et al.
(hðsÞv jt¼0 ¼ 0), while δðsÞv jt is determined through line minimiza-tion [28]. As for ΞðsÞ, it is updated by modifying its entriesaccording to the following equations:
ξðsÞn jtþ1 ¼(XV
v¼1
½Etotv ðrðsÞn Þjtþ1��½IvðrðsÞn Þjtþ1�
‖fIdetv gðsÞ‖2
)�(XV
v¼1
‖Etotv ðrðsÞn Þjtþ1‖
2
‖fIdetv gðsÞ‖2
); ð16Þ
where IðsÞv jtþ1 ¼ fIdetv gðsÞ þ VðsÞamba
ðsÞv jtþ1 ¼ ½IvðrðsÞ1 Þjtþ1; …;
IvðrðsÞN Þjtþ1�0 and fEtotv gðsÞjtþ1 ¼ fEinc
v gðsÞ þ HðsÞIðsÞv jtþ1 ¼½Etot
v ðrðsÞ1 Þjtþ1; …; Etotv ðrðsÞN Þjtþ1�0. Such an updating process
[Eqs. (15) and (16)] is applied until a user-defined maximumnumber of iterations, T ðsÞ, is reached. The s-th estimate of thedielectric profile and the ambiguous currents are thendetermined by using Eqs. (5) and (11):
~τðrðsÞn Þ ¼ jη0k0
ξðsÞn jt¼T ðsÞ
AðDðsÞn Þ
; n ¼ 1;…; N ðsÞ; ð17Þ
f~Iambv gðsÞ ¼ VðsÞ
ambaðsÞv jt¼T ðsÞ ; v ¼ 1;…; V: ð18Þ
The information on the estimated dielectric distribution with-in the investigation domain coming from Eq. (17) (also indi-cated as acquired information) is then exploited by suitablefiltering and clustering procedures [36] to identify/updatethe RoIs. Successively (s → sþ 1), the discretization grid islocally refined within the RoIs by updating rðsþ1Þ
n (n ¼ 1;…; N)and the SOM is again applied to the new multiresolution re-presentation of D. The IMSA scheme is iterated until a setof stationary conditions on the RoIs holds true [36] (s ¼ S).The distribution f~τgðSÞ, retrieved at the final zooming step,is then assumed as the IMSA-SOM solution.
The whole IMSA-SOM procedure is pictorially sketched inFig. 1, and its implementation can be summarized as follows.
1. Step 0: problem definition. Define D, M , V . For eachview (v ¼ 1;…; V), compute the Escatt
v at the M measurementpoints starting from the measurement of Etot
v and Eincv in DO.
Set N to the number of degrees of freedom (DoF) of the scat-tered field [40], which is related to the extension of D and thevalues of M and V .
2. Step 1: initialization (s ¼ 1). Set a (coarse) grid of cellspositioned at rðs¼1Þ
n to uniformly discretize D.3. Step 2: matrix computation. Measure the samples of the
incident field fEincv gðsÞ at rðsÞn , n ¼ 1;…; N ðsÞ. ComputeHðsÞ and
GðsÞ in Eqs. (6) and (7);4. Step 3: deterministic current components estimation.
Compute the SVD of GðsÞ from Eq. (8). Select LðsÞ and deter-mine fIdetv gðsÞ according to Eq. (10).
5. Step 4: Ambiguous current components and contrastestimation. Minimize Eq. (14) by applying Eqs. (15) and (16)for T ðsÞ iterations. Determine the s-th estimate of the ambig-uous currents and contrast through Eqs. (17) and (18);
6. Step 5: RoIs identification. Filter and cluster theestimated distribution f~τgðsÞ to identify the locations andthe extensions of the RoIs [36].
7. Step 6: convergence check. If the stationary condition[36] on the RoIs is satisfied, terminate. Otherwise (s → sþ 1),enhance the spatial resolution within the RoIs and go toStep 2.
4. NUMERICAL RESULTSThis section has a twofold objective. The former is to providereliable guidelines for the application of the IMSA-SOM. Thelatter is to test its effectiveness in terms of accuracy, compu-tational complexity, and robustness when dealing with var-ious scatterers, measurement setups, and noise conditions.Unless explicitly indicated, the following scenario is assumedas reference: a square investigation domain l ¼ 2:4λ0 in sideilluminated by V ¼ 24 plane TM waves with incidence anglesθv ¼ 2π ðv−1Þ
V(v ¼ 1;…; V). For each illumination, the scattered
Fig. 10. (Color online) Numerical assessment (inhomogeneous cylin-der: l ¼ 2:4λ, V ¼ 24, M ¼ 24, τout ¼ 0:6). Plots of (a) Ψtot, (b) Ψint,and (c) Ψext versus τin for different SNR values.
Oliveri et al. Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. A 2063
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0.9
1.2
1.5Re[τ(x,y)]
-1.2-0.6
0 0.6
1.2
x/λ-1.2
-0.6 0
0.6 1.2
y/λ
0
0.3
0.6
0.9
1.2
1.5Re[τ(x,y)]
(d () e () f )
-1.2-0.6
0 0.6
1.2
x/λ-1.2
-0.6 0
0.6 1.2
y/λ
0
0.3
0.6
0.9
1.2
1.5Re[τ(x,y)]
-1.2-0.6
0 0.6
1.2
x/λ
-1.2-0.6
0 0.6
1.2
y/λ
0
0.3
0.6
0.9
1.2
1.5Re[τ(x,y)]
-1.2-0.6
0 0.6
1.2
x/λ-1.2
-0.6 0
0.6 1.2
y/λ
0
0.3
0.6
0.9
1.2
1.5Re[τ(x,y)]
(g () h () i)
Fig. 11. Numerical assessment (inhomogeneous cylinder: l ¼ 2:4λ, V ¼ 24,M ¼ 24, τout ¼ 0:6, SNR ¼ 20dB). Actual distribution [(a), (d), and (g)]and dielectric profiles retrieved with [(b), (e), and (h)] the SOM and [(c), (f), and (i)] the adaptive IMSA-SOM when [(a), (b), and (c)] τin ¼ 0:0, [(d),(e), and (f)] τin ¼ 0:6, and [g), (h), and (i)] τin ¼ 1:0.
-3 -2 -1 0 1 2 3
x/λ
-3
-2
-1
0
1
2
3
y/λ
10 -3
10 -2
10 -1
10 0
Re[
τ(x,
y)]
-3 -2 -1 0 1 2 3
x/λ
-3
-2
-1
0
1
2
3
y/λ
10 -3
10 -2
10 -1
10 0
Re[
τ(x,
y)]
(a () b)
-1.2
-0.6
0
0.6
1.2
x/λ
-1.2-0.6
0 0.6
1.2
y/λ
0
0.3
0.6
0.9
1.2
1.5
Re[τ(x,y)]
-1.2
-0.6
0
0.6
1.2
x/λ
-1.2-0.6
0 0.6
1.2
y/λ
0
0.3
0.6
0.9
1.2
1.5
Re[τ(x,y)]
(c () d)
Fig. 12. (Color online) Numerical assessment (inhomogeneous cylinder: l ¼ 6λ, V ¼ 60, M ¼ 60, τout ¼ 0:6, τin ¼ 1:2, SNR ¼ 10dB). Dielectricprofiles retrieved with [(a) and (c)] the SOM and [(b) and (d)] the adaptive IMSA-SOM. Reconstructions (a) and (b) are within the investigationdomain (6λ × 6λ), and (c) and (d) are on a smaller region around the scatterer (2:4λ × 2:4λ).
2064 J. Opt. Soc. Am. A / Vol. 28, No. 10 / October 2011 Oliveri et al.
field has been computed at M ¼ 24 points uniformly distrib-uted on a circular observation domain b ¼ 1:8λ0 in radius andlocated at rvm ¼ ðb cos θvm; b sin θvmÞ, m ¼ 1;…; M , v ¼ 1;…; V ,being θvm≜θv þ 2π ðm−1Þ
M. Toward this end, the Richmond’s pro-
cedure [42] has been applied by discretizingDwith aN ¼ 48 ×48 grid of λ0
20 sided cells. As for the inversion, the investigationdomain has been partitioned, at each step of the IMSA, inN ðsÞ ¼ 15 × 15 subdomains [40].
A. Sensitivity AnalysisIn order to analyze the sensitivity of the IMSA-SOM on itscontrol parameters (i.e., LðsÞ and T ðsÞ), a lossless (τ ¼ 1:0)
off-centered [r ¼ ð−0:2λ; 0:2λÞ] square cylinder of side d ¼0:4λ has been considered as a benchmark [Fig. 4(a)]. In thefirst numerical experiment, the IMSA-SOM inversions havebeen performed by fixing T ðsÞ ¼ 100 (s ¼ 1;…; S), Smax ¼ 5,and choosing LðsÞ according to the following strategies.
1. Nonadaptive IMSA-SOM. The SVD truncation index isset to a constant value whatever s:
LðsÞNA ¼ L0 s ¼ 1;…; S: ð19Þ
2. Adaptive IMSA-SOM. The value of LðsÞ evolves duringthe IMSA process to include (at each step) the singular valuesaccounting for a fraction α of the total spectrum energy.Accordingly,
LðsÞA ¼ arg
�minLðsÞ
�PLðsÞm¼1ðχðsÞm ÞPMm¼1ðχðsÞm Þ
− α
;
subject to
PLðsÞm¼1ðχðsÞm ÞPMm¼1ðχðsÞm Þ
− α > 0; s ¼ 1;…; S; ð20Þ
where α ∈ ½0; 1� is an user-defined control threshold.
Figure 2 shows the behavior of the objective function[Eq. (14)] versus the iteration index k≜ðs − 1ÞT ðsÞ þ twhen ap-plying the nonadaptive IMSA-SOM and the adaptive IMSA-SOM with a representative set of values of the correspondingcontrol parameters L0 and α. More specifically, Fig. 2(a) com-pares the two strategies when noisy data are at hand(SNR ¼ 20dB [37]) by setting L0 and choosing the corre-sponding α value such that Lð1Þ
A ¼ Lð1ÞNA. As it can be observed,
the value of the objective function diverges when using thenonadaptive truncation, while it progressively reduces whenproceeding throughout the multiscaling process of the adap-tive strategy. The value α ¼ 0:7 turns out to be an optimalchoice also in view of the robustness of the approach tothe noise, as pointed out by the decreasing behavior of F ðsÞ
whatever the noise level [Fig. 2(b)] and by the plots of the to-tal, internal, and external reconstruction errors defined as
where NR is the number of cells of the investigation domain(R ¼ tot), of the scatterer support (R ¼ int), and of the back-ground region (R ¼ ext), and τðrÞ, ~τðrÞ denoting the actual andretrieved contrast versus the SNR (Fig. 3). As for the latter, itis worth observing that, although the optimal α value (i.e., thevalue of the parameter α for which the reconstruction error isminimum) depends on the SNR (e.g., αopt ¼ 0:9 for noiselessdata while αopt ¼ 0:7 when SNR ¼ 10dB), the setting α ¼ 0:7provides reliable and almost invariant results whatever thenoise level. More in general, the plots ofΨtot in Fig. 3 indicatethat larger values of α provide higher accuracies for lownoises and vice-versa, while intermediate values (e.g.,α ¼ ½0:5; 0:7�) yield almost constant errors whatever theSNR. For completeness and illustrative purposes, Fig. 4 showsthe reconstructions yielded with α ¼ 0:7 and for differentnoisy data.
-3 -2 -1 0 1 2 3
x/λ
-3
-2
-1
0
1
2
3
y/λ
10-4
10-3
10-2
10-1
100
Re[
τ(x,
y)]
(a)
-3 -2 -1 0 1 2 3x/λ
-3
-2
-1
0
1
2
3
y/λ
10-4
10-3
10-2
10-1
100
Re[
τ(x,
y)]
(b)
-3 -2 -1 0 1 2 3x/λ
-3
-2
-1
0
1
2
3
y/λ
10-4
10-3
10-2
10-1
100
Re[
τ(x,
y)]
(c)
Fig. 13. (Color online) Numerical assessment (irregular cylinder:l ¼ 6λ, V ¼ 60, M ¼ 60, τ ¼ 1:0, SNR ¼ 10dB). (a) Actual profile andreconstructions with the (b) SOM and (c) the adaptive IMSA-SOM.
Oliveri et al. Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. A 2065
As for the selection of the number of CG steps in the innerSOM loop, Fig. 5(a) shows the plot of Ψtot versus T0 (beingT ðsÞ ¼ T0, s ¼ 1;…; S) when α ¼ 0:7 and for different noisyconditions. As it can be observed, larger values of T0 givesmaller errors only for low noises (e.g., Ψtot⌋T0¼500 ≈5:8 × 10−4 versus Ψtot⌋T0¼25 ≈ 1:8 × 10−3, noiseless data).Otherwise, a low number of iterations usually correspondsto a reduced Ψtot in heavy noisy conditions (e.g.,Ψtot⌋T0¼500 ≈8:6 × 10−3 versusΨtot⌋T0¼25 ≈ 2:5 × 10−3, SNR ¼ 10dB). This isalso pictorially confirmed by the retrieved profiles displayedin Fig. 5. Starting from these outcomes, the value T0 ¼ 50 hasbeen selected as an optimal trade-off.
To assess the optimality and generality of the control pa-rameters (i.e., α ¼ 0:7 and T0 ¼ 50), a wider set of experi-ments has been carried out and some other representativeresults concerned with the same reference scenario, butdifferent contrasts, τ ∈ ½0:2; 2:2�, are reported in Fig. 6. Theplots of the total errors versus τ and for different SNRs indi-cate that the IMSA-SOM also provides stable performancesfor stronger scatterers (e.g., Ψtot⌋τ¼1:2 ¼ 2:11 × 10−3 versusΨtot⌋τ¼2:0 ¼ 2:65 × 10−3, SNR ¼ 10dB) and an increased accu-racy is obtained when τ < 1 (e.g., Ψtot⌋τ¼0:2 ¼ 7:16 × 10−5
versus Ψtot⌋τ¼1:2 ¼ 1:35 × 10−3, SNR ¼ 30dB). Althoughnonexhaustive, such an analysis suggests that the chosen
parameter setup can provide reliable performances in a wideclass of problems. Of course, further improvements could beyielded by means of an ad hoc parameter selection, whichwould be certainly possible whether some a priori informa-tion on the noise level is at hand.
B. Accuracy and Reliability AssessmentThe next set of experiments is aimed at assessing the perfor-mance of the proposed IMSA-SOM approach, also in compar-ison with the state-of-the-art implementation of the SOM [28](BARE-SOM), when dealing with various scatterer configura-tions. Toward this end, the retrieval of a square nonhomoge-neous scatterer [τin ¼ 1:2, τout ¼ 0:6, see Fig. 7(a)] has beenfirst considered and a set of inversions has been performedvarying the scattering data SNRs.
The dielectric profiles estimated at the successive stepsof the IMSA-SOM show that, as expected, the reconstructionimproves as the multiscaling procedure iterates (s ¼ 1; 2; 3,SNR ¼ 30dB, see Fig. 7). After the initial coarse retrieval[s ¼ 1, see Fig. 7(b)], a step-by-step refinement can be ob-served until the accurate inversion in Fig. 7(d) (s ¼ S ¼ 3)where the contrast discontinuities of the actual objects areclearly distinguishable. This cannot be yielded with theBARE-SOM [Fig. 9(a)] as also pointed out by the behavior
Actual Object
0.9 0.5 0.1
-1.2-0.6
0 0.6
1.2
x/λ
-1.2-0.6
0 0.6
1.2
y/λ
0 0.3 0.6 0.9 1.2 1.5
Re[τ(x,y)]
(a)
0.9 0.5 0.1
-1.2-0.6
0 0.6
1.2
x/λ
-1.2-0.6
0 0.6
1.2
y/λ
0 0.3 0.6 0.9 1.2 1.5
Re[τ(x,y)] 0.9 0.5 0.1
-1.2-0.6
0 0.6
1.2
x/λ-1.2
-0.6 0
0.6 1.2
y/λ
0 0.3 0.6 0.9 1.2 1.5
Re[τ(x,y)]
(b () c)
0.9 0.5 0.1
-1.2-0.6
0 0.6
1.2
x/λ-1.2
-0.6 0
0.6 1.2
y/λ
0 0.3 0.6 0.9 1.2 1.5
Re[τ(x,y)] 0.9 0.5 0.1
-1.2-0.6
0 0.6
1.2
x/λ-1.2
-0.6 0
0.6 1.2
y/λ
0 0.3 0.6 0.9 1.2 1.5
Re[τ(x,y)]
(d () e)
Fig. 14. (Color online) Numerical assessment (irregular cylinder: l ¼ 2:4λ, V ¼ 24,M ¼ 24, τ ¼ 1:0). (a) Actual distribution and dielectric profilesreconstructed with [(b) and (d)] the SOM and [(c) and (e)] the adaptive IMSA-SOM when (b) and (c) have a SNR ¼ 30dB and (d) and (e) have aSNR ¼ 10dB.
2066 J. Opt. Soc. Am. A / Vol. 28, No. 10 / October 2011 Oliveri et al.
of the objective function in Fig. 8(a). As a matter of fact, themultifocusing method reaches smaller and smaller F ðsÞ valuesat each multiresolution step, and the convergence value (i.e.,F ðSÞ) turns out to be significantly below that of the BARE-SOM, which instead stagnates after a few iterations [Fig. 8(a)].Such a result is mainly due to the IMSA capability to acquireinformation through its iterative steps as implicitly confirmedby the behavior of the spectrum of GðsÞ, s ¼ 1;…; S [Fig. 8(b)].While the singular values of the standard SOM are fixedand determined by the measurement points and the inversiongrid, the slope of χðsÞm (m ¼ 1;…; M) changes at each step ofthe IMSA-SOM thanks to the adaptive choice of LðsÞ [seeEq. (20)]. This certainly allows a clearer and easier (i.e., in anunsupervised way and without the need of some knowledgeabout the noise level) separation between the deterministiccomponent and the ambiguous one of the CSs.
The effectiveness and reliability of the IMSA-SOM are alsopointed out from the pictorial comparisons of the multiresolu-tion reconstructions with those from the uniform resolutionimplementation in both noiseless [Figs. 9(a) and 9(b)] andnoisy [Figs. 9(c)–9(f)] conditions. Quantitatively, the improve-ments enabled by the IMSA strategy are clearly highlighted bythe error indices in Table 1. For example, it turns out that thereconstruction error for the IMSA-SOM at SNR ¼ 10dB islower than that of the standard SOM in the noiselesscase (ΨIMSA-SOM
tot ⌋SNR¼10 ¼ 8:58 × 10−3 versus ΨSOMtot ⌋SNR¼∞ ¼
9:06 × 10−3). Indeed, the SOM fails in retrieving the disconti-nuities within the object support in the noiseless case[Fig. 10(a)] as well as in the noisy ones [Figs. 10(b) and 10(c)].Moreover, unlike the IMSA-SOM, some artifacts appear inthe background region as the SNR decreases [Fig. 9(e)] asalso confirmed by the values of Ψext in Table 1(ΨIMSA-SOM
Concerning the inversion time,Δt, the multifocusing proce-dure guarantees a significantly higher computational effi-ciency as it can be inferred from the values in Table 1.Generally, the CPU saving turns out to be of about seven timeswhatever the noise level in the scattering data (e.g.,ΔtSOM⌋SNR¼∞ ¼ 348 s versus ΔtIMSA-SOM⌋SNR¼∞ ¼ 51:9 s) (forfair comparisons, all simulations have been run on a single-core laptop personal computer running at 2:16GHz). Sucha speed-up is mainly due to the smaller problems actuallysolved by the IMSA-SOM at each step, which yield to quickerSVDs as well as faster CG convergences.
To give some additional insights on the enhanced efficiencyof the IMSA-SOM, Fig. 10 compares the total [Fig. 10(a)]internal [Fig. 10(b)] and external [Fig. 10(c)] errors in recon-structing the same two-layer square profile in Fig. 7(a), butvarying the dielectric properties of its core, τin. With referenceto the noiseless case, the outcomes from such experimentscan be summarized as follows: (a) the IMSA enhanced versionalways outperforms the standard implementation whatever
Table 2. Performance Assessment (l � 6λ, V � 60, M � 60, α � 0:7, T0 � 50, SNR � 10 dB):Reconstruction Errors and Computational Indices
SOM IMSA-SOM
Actual Object Ψtot Ψint Ψext Δt (s) Ψtot Ψint Ψext Δt (s)
Fig. 15. (Color online) Numerical assessment (separate cylinders:l ¼ 6λ, V ¼ 60, M ¼ 60, τsquare ¼ 1:2, τL ¼ 0:6, SNR ¼ 10 dB). (a) Ac-tual distribution and dielectric profiles reconstructed with (b) theSOM and (c) the adaptive IMSA-SOM.
Oliveri et al. Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. A 2067
τin, as shown in Fig. 10(a), and quantitatively assessed by thevalues of Ψtot in Table 1, (b) as expected, the smallestreconstruction errors are yielded around the homogeneouscase (i.e., τin ≈ τout ¼ 0:6) due to the simpler nature of the ob-ject at hand [Fig. 10(a)], (c) unlike the state-of-the-art SOM,the IMSA-SOM is able to carefully locate the object supportand to avoid artifacts outside such a region. Indeed, thecorresponding external error turns out to be null for mostτin values [Fig. 10(c)]; (d) whatever the approach, both Ψtot
[Fig. 10(a)] and Ψint [Fig. 10(b)] grow as τin increases overthe homogeneous scenario. Such conclusions hold true also inthe noisy cases as confirmed by the plots in Fig. 10 and theerror indices in Table 1. For illustrative purposes, a set ofrepresentative reconstructions (SNR ¼ 20 dB) is shownin Fig. 11.
To further investigate the reliability of the IMSA-SOM inbetter locating and resolving the scatterers, the following testcases deal with wider investigation domains. Toward the end,D has been enlarged to a square domain of side l ¼ 6:0λ0,while the following configuration has been chosen for the ob-servation domain: V ¼ M ¼ 60 and b ¼ 4:5λ0. As for the dis-cretization of the investigation domain, the following gridshave been adopted: N ¼ 120 × 120 (direct solver), N ðsÞ ¼ 40 ×40 (IMSA-SOM), NSOM ¼ 60 × 60 (SOM).
Dealing with the reconstruction of the scatterer in Fig. 7(a)from noisy data characterized by a SNR of 10dB, the IMSAapproach still confirms its ability to carefully locate the scat-terer support by cleaning the outer region [Fig. 12(a) versusFigure 12(b)] as well as give a higher spatial resolution withinthe scatterer, as shown in the zoom of Fig. 12(c) versusFig. 14(d). Therefore, the error reduces about four times
( ΨSOMtot
ΨIMSA−SOMtot
¼ 3:707, see Table 2). Similar conclusions hold true
for the retrieval of the scatterer with finer details in Fig. 13(a),as well. It is also worth pointing out that the enhanced accu-racy of the IMSA-SOM does not depend on the domain size orSNR, as confirmed by the numerical results obtained for thesame object, but assuming a smaller l ¼ 2:4λ sized domain anddifferent noisy conditions (Fig. 14, Table 2).
The last experiment is concerned with separate scattererswith different permittivities [Fig. 15(a)]. The reconstructionresults, summarized in Fig. 15 and quantified in Table 2, alsoassess in this scenario the reliability and effectiveness of themultifocusing scheme. More specifically, the inversion accu-racy improves in a nonnegligible way (ΨSOM
tot ¼ 6:53 × 10−3 →ΨIMSA−SOM
tot ¼ 2:24 × 10−3), with a half reduction of the externalerror. As for the computational cost dealing with a widerinvestigation region (l ¼ 6λ), the higher computational effi-ciency of IMSA-SOM, previously assessed for the case withl ¼ 2:4λ, turns out even more impressive reaching a speed-up ratio of about ΔtSOM
ΔtIMSA-SOM ≈ 2:2 × 102 (Table 2).
5. CONCLUSIONS AND REMARKSIn this paper, the IMSA-SOM has been proposed within the CSformulation of the inverse scattering problem. The nestediterative method proceeds by identifying and refining theRoIs within the investigation domain (outer loop), then recon-structing the deterministic components of the CSs through thecomputation of the truncated SVD of the scattering operator(spectrum analysis step), and finally retrieving the ambiguouscomponents of the CSs by means of the conjugate gradientminimization of a suitable data misfit function (inner loop).
An adaptive choice of the SVD truncation parameter hasbeen introduced, and the guidelines for an unsupervisedand with limited a priori information application of theproposed approach to the inversion of noisy data has beenpresented.
The numerical investigations have pointed out the follow-ing key features of the multifocusing scheme: (a) the calibra-tion parameters (ðα; T0Þ of the IMSA-SOM can be easilyselected to yield noise robustness as well as fast convergence(Subsection 4.A), (b) thanks to the multizooming strategy, animproved accuracy (compared to the state-of-the-art SOM) isalways reached in reconstructing both simple and complexscatterers (Subsection 4.B), (c) unlike the standard SOM im-plementation, the IMSA-SOM is able to detect fine details ofthe object under investigation as well as to avoid artifacts out-side the scatterers’ regions also when dealing with wide inves-tigation domains and low SNR values, and (d) the IMSA-SOMalways outperforms the Bare-SOM in terms of computationalefficiency.
Future studies, besides the extension of the IMSA-SOM tofull three-dimensional inversions or to different backgroundsand measurement configurations, will deal with a fruitful in-tegration of the available physical information on the scenarioat hand. Moreover, the possible application of the proposedmethodology to subsurface imaging problems as well as bio-medical diagnostics will be investigated as well.
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