Virtual Experiments via LSM for Quantitative 2-D Inverse Scattering: Challenges and Opportunities

In this article, we investigate the virtual experiments (VE) setting by means of analytical solution of the far-field equation, that is the equation underlying the linear sampling method. In particular, an analytical expression for the internal field is derived in case of penetrable circular cylinders, which provides a straightforward and more comprehensive overview of the VE settings with particular hints on its reliability and validity range when applied for quantitative imaging. Numerical examples are reported to show the analytic findings, also in case of noncircular targets. The reported analysis can open the way to new possibilities in VE design for electromagnetic inverse scattering, microwave imaging, and related focusing problems.

Following this way, in a body of papers over the last decade, a new paradigm for solving the 2-D inverse scattering problem has been proposed.Such a paradigm stems for recasting the original scattering experiments into new virtual, software defined, ones without requiring any additional measurements.This can be actually pursued by means of a proper design equation which is able to condition the spatial behavior of the internal field or of the contrast source (i.e., the considered auxiliary unknown of the problem).Then, such a relevant conditioning can be conveniently exploited to introduce new inversion strategies.Some inversion approaches are discussed in [8], [9], [10], [11], and [12] and rely on the design of virtual experiments (VE) enforcing circular symmetric contrast sources (or total fields) for a set of points, called pivot points, through the solution of the far-field equation (FFE) [13].VE-based inversion approaches show a great capability in imaging targets which do not belong to the so-called weak scattering approximations based on born approximation (BA) [14], while addressing the problem in a simple, effective, and computationally efficient way.On the other hand, their performance gets worse when targets' dimension and dielectric contrast become larger and larger.This circumstance has been partially analyzed in [15] through a simple numerical analysis, wherein an applicability range of about three times larger than BA has been found.
In order to better understand the reasons of the performance's worsening, in this article, we study analytically the FFE in case of penetrable circular cylinders under the 2-D transverse magnetic (TM) formulation to derive a direct expression for the internal field (or contrast source).This analytical expression, which depends on the target's contrast and on the considered pivot point, provides straightforward outcomes which fully agree with results of previous works on linear sampling method (LSM) and FFE interpretation [16], [17], [18].More interestingly, the analytic expression allows also to understand why the VE-based inversion methods get worse performance beyond a given range of validity and may suggest some counteracting strategies to avoid their performance degradation.
The article is structured as follows.In Section II, the mathematical formulations of the problem and the FFE solution are introduced for the canonical case of a dielectric circular cylinder.In Section III, the analytic expression of the virtual total field is derived, while in Section IV, the main outcomes are discussed.Then, in Section V, a numerical analysis is performed to validate the previously derived analytic findings, while Section VI is instead focused on their exploitation in the framework of a VE-based linear inverse recovery approach.Discussion and Conclusions follow.Throughout the paper, the time harmonic factor ex p( jωt) is assumed and dropped.

II. ANALYTIC SOLUTION OF THE FFE FOR CIRCULAR DIELECTRIC CYLINDER
Let us consider an infinitely long and lossless circular cylinder of radius a along the z axis with complex relative permittivity ϵ s .A TM incidence by plane waves with incoming direction θ i ∈ [0, 2π] is assumed for the following analysis.The corresponding scattered field is collected in the far-field zone of the cylinder over a circular observation domain of radius R o at different angular observation points θ o ∈ [0, 2π].Under the above assumptions, the scattered field can be factored as where E ∞ s is the far-field pattern, k 0 = ω √ ϵ 0 µ 0 is the background wavenumber (free space is assumed herein for the sake of simplicity), ω is the angular frequency, ϵ 0 and µ 0 are the free-space dielectric permittivity and magnetic permeability, respectively.
According to [19], the scattered field pattern E ∞ s and the transmitted field E t , outside and inside the cylinder, respectively, can be analytically expressed through the following modal expansions wherein r = (r cos θ r , r sin θ r ) is the position vector, and the expansion coefficients are given by and In the expressions (4) and (5), being χ the contrast function defined as the difference between the complex relative permittivity of the cylinder and the one of the background medium [2].Moreover, H (2)   n is the nth-order second kind Hankel function, J n is the Bessel functions of nth-order and first kind, and the superscript ′ denotes the derivative with respect to the argument.It is important to note that, as only a finite number of terms in (2) and (3) plays a role in the composition of the total field, the series can be truncated according to the rule of thumb N 0 ≈ k 0 a for (2) and N s ≈ k s a for (3), respectively.
In this framework, the LSM amounts to sense the investigation domain by sampling it into a set of points r s = (r s cos φ s , r s sin φ s ) and by solving the following equation wherein F[•] is the so-called far-field operator, ι(θ i , •) the problem unknown, and f (θ o , r s ) the far-field pattern of a line source placed in the sampling points r s .The far-field operator is the angular pattern of the scattered field when collected under a multiview-multistatic configuration as defined in (2), while the right-hand side of (6) reads as Since F is a compact operator, solution of ( 6) must be pursued by means of a proper regularization strategy [13].In this respect, singular value decomposition (SVD) and Tikhonov regularization can be exploited to get stable and reliable results [8].Then, the support of the scatterer can be effectively retrieved by associating to the target's support those sampling points wherein the solution norm ||ι|| L 2 attains the lowest values with respect to its overall dynamic in the investigated region [13].
On the other hand, in case of circular cylinder, an analytical solution of the FFE can be achieved expanding the problem unknown in Fourier series.By doing so, as shown in [20], the latter can be expressed as In [13], solution (8) has been proposed to solve the inverse obstacle problem since its energy, when plotted over the sampled domain, is able to image the scatterer's support.

III. ANALYTIC EXPRESSION OF THE VIRTUAL TRANSMITTED FIELD
The VE paradigm is based on the simple observation that, for a fixed contrast function, the relationship between the total field transmitted into the target and the incident field holds linear.This circumstance entails that the original scattering experiments can be rearranged in a possibly more convenient form by means of a linear superposition of the incident fields [8], [9], [10], [11].As a result, properly designed VE can give rise to total fields (or induced currents) exhibiting some useful properties which can be conveniently exploited in the recovery stage.In particular, the idea to design-specific VE enforcing total fields which are focused or exhibit a circular symmetry in the neighborhood of a "pivot" point has paved the way to a number of effective solution approaches for quantitative inverse scattering [8], [9], [10], [11].It is also worth to underline that some other possibilities can be exploited to design VE, such as, for example, the factorization method (FM) [21] and the orthogonality sampling method (OSM) [22], but these latter are out of the scope of the present paper.
From a mathematical point of view, a virtual incident field E i can be simply obtained through a linear superposition of the original incident fields E i through the FFE solution evaluated in a pivot point r p belonging to the support of the target, i.e., Then, the virtual total field E t transmitted inside the cylinder can be expressed as superposition of the original ones E t by means of the same coefficients ι(θ i , r p ), i.e., In order to investigate the expression of the virtual total field, let us substitute ι(θ i , r p ) given by ( 8) into (10).Then, by changing the order of the integral and the sum, and substituting the expression of the original total field in (3), one gets Finally, by exploiting the orthogonality relation between the above exponential functions, the following expression can be obtained wherein b m = (a m /c m ).Interestingly, when r p = (0, 0), the only term which survives in the expression (12) is the one for m = 0 as the J m (0) = 0 for m ̸ = 0.Then, the VE total field induced in the cylinder is exactly a Bessel function of zero order apart from the constant b 0 .
On the other hand, when r p ̸ = (0, 0), one has to consider the behavior of the expansion coefficients.In particular, if the coefficients b m are constant with respect to the order m (at least for the first relevant terms in the sum), the above expression can be further simplified by exploiting the Graf's addition theorem for the Bessel functions [23], i.e., The above expression states that the virtual total field can be approximated (apart from an unessential constant factor) by a 0th-order Bessel function.Then, the virtual total field exhibits a circular symmetry and is also focused, but not simply with respect to the pivot point r p , as one would expect from the cylindrical pattern enforced by the right-hand-term of the FFE.Indeed, a rigid translation of the focusing point, i.e., the point where the 0th-order Bessel function attains its maximum, is introduced due to the scale factor Finally, if the coefficients b m are not constant with respect to the order m (at least for the first relevant terms of the sum), a total field distribution resembling a Bessel function is not guaranteed, as it will be shown in Sections IV and V.

IV. EXPECTED PROPERTIES OF THE VIRTUAL TRANSMITTED FIELD
The approximation (13) derived in the previous section allows to gain a deep understanding of the expected spatial distribution of the virtual transmitted field.
First of all, the larger the contrast, the larger the shift of the actual focusing point from the relevant pivot point r p .This implies that the use of the FFE to design circularly symmetric VE is expected to work worsen and worsen for increasing optical density of the target.In particular, for any arbitrary contrast value and electrical dimension of the scatterer, the VE internal fields corresponding to the outermost pivot points are affected by a larger shift than the internal ones.This entails that, for low contrast extended targets, the shift of the focusing point is also expected to be larger and larger for the outermost pivot points.Conversely, there are only two circumstances for which the shift is irrelevant.For χ = 0 the shift would be exactly null, however, this circumstance is not in order since the scattered field would be null.Instead, more interestingly, the shift is null also for the pivot point pertaining to the cylinder's center.Another important role for the internal field behavior is played by the expansion coefficients b m .Indeed, as mentioned above, if the coefficients b m are constant with respect to m, the Graf's theorem can be invoked and the internal field would be exactly a Bessel function given by (13).By relaxing such an hypothesis, even when the coefficients b m are all almost constant with m (at least up to the truncation index in the relevant series), the sum gives rise to a total field that can be approximated by (13) (within a given error).On the other hand, when the coefficients differ too much one from each other, the recombined total field will not resemble the Bessel function stated by the Graf's theorem.
In order to study the behavior of the expansion coefficients with respect to the dimension and dielectric constant of the cylinder, and their impact on the circular symmetry of the virtual internal field, we derive an explicit expression of such coefficients, which is reported in Appendix, see eqs.(A.1)-(A.3).Then, we plot the coefficients for different radius a and dielectric constant ϵ s .As it can be seen in Fig. 1, the coefficients exhibit a finite value up to a given order m, which corresponds to the last relevant term in (12), while they grow to infinity beyond a given order m, that is when the integral in (A.3) goes to zero.For example, for the radius 0.25λ b , only the first two coefficients are relevant in series (12) and are almost equal one each other, even in case of largest contrast, see Fig. 1(a).When the radius increases, the dependence of the coefficients as a function of m becomes more complex for increasing values of the dielectric constant, see Fig. 1(b)-(d).Such an analysis suggests that for increasing value of the product k s a of the cylinder, the internal field cannot show a perfect circular symmetry with respect to the considered pivot point since its expression differs from the 0th-order Bessel function stated by the Graf's theorem.

V. NUMERICAL ASSESSMENT
In this section, we show some numerical tests to better understand and discuss the findings of the previous Section.In particular, the main aim is to compare the actual VE Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.transmitted field (12) and the Bessel function J 0 , when both defined with respect to the same pivot point r p .The benchmark scenario consists of a lossless circle of radius a = 0.5λ 0 and different dielectric constants ϵ s , embedded in free space within a square domain of side L = 2.2λ 0 , which is discretized into 80 × 80 cells.Without loss of generality, we assume θ m = θ i .According to the Nyquist criterion underlying the degrees of freedom of scattered field suggested in [24], N = M = 16 incident directions and measurements have been considered.The scattered field data has been generated by means of analytical formulas given in [19].Fig. 2 shows the amplitudes of the normalized virtual transmitted field designed by means (12) and the related Bessel function J 0 , respectively.As it can be seen, the larger the relative permittivity ϵ s , the larger the shift of the total field with respect to the pivot point r p and the larger the deformation of the cylindrical wavefront for the internal field.Even, for ϵ s = 3.5 an undesired sidelobe appears in the internal field destroying completely the well-focused and cylindrical symmetry of the field pattern.This circumstance does not hold true when the pivot point is located at the center of the cylinder as analytically derived in the previous Section.Indeed, when the r p = 0 no shift of the field's maximum is observed, neither wavefront deformation, but a perfect circularity of the field distribution is ensured (the relevant plots are not shown for the sake of brevity).To quantitatively appraise the difference in term of circular pattern of the internal field, a synthetic parameter has been adopted.It is defined as the L 2 -norm of the angular derivative ∂ φ r of the virtual total field, that is wherein φ r is the angular coordinate of a local polar reference system centered in r p and (r , r p ) identifies the minimum circle centered on r p inscribed in the target.The larger the value of symcir c, the worse the circular symmetry around the considered pivot point.As it can be seen from Table I, the higher the relative permittivity ϵ s , the worse the cylindrical symmetry of the field in terms of the adopted metric.Only in case of ϵ s = 1.5, the symcir c of the virtual field is comparable to the one of the Bessel function.
In order to show that the above findings hold true also for targets other than the circular ones, a kite and a ring shaped targets have been considered.The targets are embedded in a square domain with size of 2λ , discretized with 80 × 80 number of cells.The leading dimension of both targets is 1λ .To probe the targets, 21 plane waves and 21 measurement points are considered, the latter located uniformly over the circle with radius R = 10λ .The synthetic scattered fields  have been numerically evaluated by using a full-wave forward solver based on the method of moments [25] and organized into a complex matrix (with size of 21 × 21), while the FFE equation has been solved numerically by adopting the Tikhonov regularization [8].Results are shown in Figs. 3  and 4.Even if the targets are not circular cylinders and some possible deformations of the internal fields can be due to the geometrical shape, the trend observed in Fig. 2 for the circular cylinder is still confirmed.

VI. EXPLOITATION OF THE OUTCOMES THROUGH A LINEAR INVERSION APPROACH
In this section, the inversion results within the framework of a VE based linear inversion approach [8] are reported in order to assess the outcomes of the derived analysis.The reconstructions of the cylinder and kite targets are discussed and shown with and without the shift compensation of the pivot points foreseen by (13).Of course, it is possible to compensate this shift when the target contrast ϵ s is approximately known.In the following, ϵ s is assumed known, in order to check if the shift compensation can have an impact on the final reconstruction.
The region of interest has been probed by means of N = 19 angular equally spaced plane waves and the scattered field is collected at M = 19 points all around the investigation domain at a distance of 4λ from the center of the reference system.Finally, the scattered field has been corrupted by additive white Gaussian noise with a SNR = 25 dB.The reader is referred to [8] for details about the linear reconstruction approach.A standard truncated SVD has been selected to solve the relevant ill-posed linear problem, wherein the cutoff value in the reconstruction formula is chosen according to the Picard's plot [26].
The first test bench is a perfect lossless cylindrical target with radius 0.6λ and dielectric constant 1.75, see Fig. 5(a).In Fig. 5(b)-(e), the set of pivot points without and with shift compensation and the relevant reconstruction of the target are reported, respectively.For the sake of brevity, the imaginary part is not reported being negligible with respect to the real one.As it can be seen, the reconstruction achieved by shift compensation is more accurate and the normalized mean square error (NMSE) attains lower values (see details in the figure's caption).
The second example is a noncircular scatterer, i.e., the kite with a leading dimension of 1λ .Also in this case, the shift compensation of the pivot points allows to achieve a better reconstruction, without holes in the central part of the scatterer and a lower reconstruction error, see Fig. 6.In this case, it is worth to notice that some pivot points are not taken exactly within the scatterer's support because the LSM map does not exactly resemble the actual target's shape due to concavity of the kite [27].Anyway, the reconstruction is quantitative although slightly underestimates the actual dimension of the target.
As a final comment, it is worth to underline that the developed analysis is only a proof of concept as the exact shift of the pivot points is not known in any actual instances, depending on turn of the local value of the contrast.However, in the light of future developments, the value of the contrast can be estimated by other auxiliary techniques or in distorted approaches, wherein the shift compensation can be refined at each step of the iterative procedure.

VII. DISCUSSION AND CONCLUSION
In this article, an analytical expression for the VE internal field has been derived, able to describe the virtual transmitted field designed via LSM.The developed analysis reveals several interesting features which allow a deeper understanding of the design equation used for the VE paradigm.
First of all, outcomes of ( 12)- (13) show that, if the expansion coefficients can be assumed (approximately) constant up to the truncation index, the internal field resembles the cylindrical symmetry of the Bessel function of zero order, but for a shift of the focusing point which can be assumed negligible only for low contrast and/or small targets. 1 This result is quite well in agreement with the physical interpretation of the LSM in terms of focusing strategy given in [16].On the other hand, the focusing interpretation is no longer valid when the scatterer shows larger and larger contrast, since the expansion coefficients in (12) cannot be assumed constant with m.Interestingly, such circumstance is also numerically shown in the framework of a linearized VE based reconstruction approach proposed in [15], wherein the indicator 2 proposed to appraise the validity of the method amounts to 3.85, which is the value reported in Table II for a radius of 0.5λ and dielectric constant ϵ s = 1.5.The numerical analysis suggested by the analytical findings fully agrees with the one reported in [15].
It is also interesting to discuss the implication of the analytical findings in the light of the results provided by [17], wherein it is shown that the L 2 -norm of ( 8) diverges for all sampling points different from the sampling point at the center of the cylinder.The present analysis also identifies such a peculiar point because it is the only one which ensures exact circular symmetry of the internal field, whatever the amplitude and size of the scatterer.Instead, in all the other pivot points, the internal field cannot exactly match a cylindrical focused pattern and needs to increase its energy by involving nonradiating currents [29].
Finally, the obtained results are also in agreement with the discussion in [18], wherein it is shown that a circular symmetric pattern can be also radiated by any arbitrary shaped source, not necessarily a circular or mainly focused one, owing to the occurrence of nonradiating currents [29].Indeed, also not focused internal fields are compatible with the FFE solution as shown in Fig. 2(f).
However, it is important to note that when the regularization scheme is considered, the nonradiating currents are minimized and the radiating ones are predominant and are expected to be circularly symmetric as discussed in [16].This circumstance allows to interpret the LSM as nearly optimal focusing strategy in the neighborhood of the pivot point [16], [30].
On the other hand, both the shift of the focusing point and the deformation of the cylindrical wave pattern explain, in a straightforward manner, the reason why the performance of LSM based VE recovery strategies worsen in imaging strong scatterers with leading dimension larger than λ 0 .For instance, the shift of the focusing point can prevent the accuracy of the underlying source point field approximation in the linearized inversion method [8], as shown by means of the examples in Section VI.Furthermore, in [9], the penalty term enforcing circular symmetry of the contrast sources with respect to the pivot points could not act properly in case of significant deformation and shift of the cylindrical field pattern.Finally, in [11], the algebraic approach could not work properly since the polynomial expansion of the Bessel function around the pivot point would be not properly exploited.
It is also worth noticing that (13) can suggest some counteracting strategies to avoid performance degradation of the VE.For instance, a proper compensation procedure could be adopted to correct the actual focusing point as shown in Section VI.Last but not least, results in case of strong targets show that total field distributions other than circular ones could be sought, so they suggest the use of different design equations to enlarge the class of retrievable scatterers by means of VE inversion strategies [31].
A last comment must be spent to discuss the extension to the more realistic case concerned with the 3-D problem.In this respect, it is worth to recall that the VE framework up to now has been introduced and discussed only for 2-D scalar problem.In order to extend and tailor the VE framework to the more challenging 3-D case, two possibilities can be of interest.The first one is concerned with scalar fields, such as acoustic waves, wherein the problem still holds scalar and the rationale underlying VE and the above analytical developments can be extended and tailored.Conversely, for vector fields, the VE design entails the finding of the optimal polarization as additional problem's degree of freedom [32].Then, one could split the problem as three scalar separate focusing problems by considering three different independent polarizations and recasting the problem accordingly.Also in this case, the extension of the suggested analysis are expected to help in gaining more insights into the VE settings.

APPENDIX
In this Appendix, the analytical expression of the coefficients ratio b m in expansion ( 12) is derived.By recalling the expressions (4)-( 5 Further manipulations can be performed by means of the expression (6.521) in [33] to obtain b m = 2ȷ π(k 0 a) 2 χ 1 0 ρ J m (k 0 aρ)J m (k s aρ)dρ wherein ρ is the radial variable spanning the unitary disk.The above expression of b m has a more straightforward readability than the previous ones.

Fig. 2 .
Fig.2.Numerical analysis in case of r p = (0.125λ 0 , 0.216λ 0 ).(a)-(c) LSM spatial indicators with superimposed the contour of the target support.Subplots (d)-(f) show amplitudes of the normalized VE total fields induced within the circular targets; subplots (g)-(i) show the pertaining Bessel functions J 0 focused in the considered pivot point.This latter is superimposed as red asterisk, while the actual focusing point, shifted according to(13), is shown with a black cross.

Fig. 3 .
Fig. 3. Numerical analysis for ring target.(a), (e), (i) LSM spatial indicators with superimposed the contour of the target support.The subplots (b)-(d), (f)-(h), and (j)-(l) show the amplitudes of the normalized VE total fields induced within the circular target for different pivot points.The considered pivot points are superimposed as red asterisks, while the actual focusing points, shifted according to (13), are shown with black crosses.

Fig. 4 .
Fig. 4. Numerical analysis for kyte target.(a), (e), (i) LSM spatial indicators with superimposed the contour of the target support.The subplots (b)-(d), (f)-(h), and (j)-(l) show the amplitudes of the normalized VE total fields induced within the circular target for different pivot points.The considered pivot points are superimposed as red asterisks, while the actual focusing points, shifted according to (13), are shown with black crosses.
) and substituting in b m , a compact expression can be obtainedb m = J m (k 0 a)H (2) ′ m (k 0 a) − J ′ m (k 0 a)H (2) m (k 0 a) √ ϵ s J m (k 0 a)J ′ m (k s a) − J ′ m (k 0 a)J m (k s a) .(A.1)Then, substituting the expression for the Hankel function and its first derivative (H(2)   m = J m − jY m and H (2) ′ m = J ′ m − jY ′ m ), recalling the Wronskian of the Bessel functions (J m (x)Y m (x) ′ − Y m (x)J ′ m (x) = 2/π x)and the recursive formula of the derivative of Bessel functions (J ′ (αx) = α J m−1 (αx) − (m/α)J m (αx)), the following ultimate expressions for b m can be obtained b m = −2ȷ π k 0 a √ ϵ s J m (k 0 a)J m−1 (k s a) − J m−1 (k 0 a)J m (k s a) .(A.2)

TABLE II PERFORMANCE
INDICATOR TO EVALUATE THE BEHAVIOR OF THE EXPANSION COEFFICIENTS (A.3)