1. Introduction

Inverse scattering problems amount to recovering scattering objects (in terms of their positions, shape and possibly physical parameters) by scattered field measurements. Such problems are relevant in all those applicative contexts where it is convenient or one is forced to inspect without physically penetrating inside the object under test. A few of examples are remote sensing, non-destructive diagnostic and microwave tomography.

In this framework many inversion methods have been developed ranging from linearized approaches [1], to non-linear inversion schemes which perform the reconstruction by minimizing cost functionals accounting for the mismatch between the data and the model [2], and more recently, to the so-called qualitative methods, which still in the framework of non-approximate models for the scattering, aim at reconstructing the scatterers’ support by means of certain indicator functions [3] ¹.

As is well known, inverse scattering problems suffer from ill-posedness and non-linearity. Moreover, in some situations, one also has to tackle the further difficulty that only phaseless data are available. This occurs when frequency grows up and the field phase can be overwhelmed by noise or could require too expensive measurement setup.

Several contributions can be found in the literature which address inverse scattering problems from phaseless data. Some of the pioneering contributions in the case of weak scattering objects are reported, for example, in [4] and [5]. In particular, in [4], under the Born model, the scattering amplitude is approximated in terms of the total field amplitude and the reconstruction is obtained by backpropagating such an approximate scattering amplitude. In this case, it was shown that the reconstruction so obtained are affected by an error term which becomes negligible, within the object spatial domain, if the scattering amplitude is collected in far zone. A similar procedure is also presented in [5] where the Rytov approximation is employed. Moreover, in [6] a hybrid approach is presented to deal with the breakdown of the Rytov approximation in far-field zone.

A more general approach, in the sense that it can deal with scatterers beyond the linear scattering regimes, is presented in [7] where the solution is searched for by a single step inversion procedure consisting of the minimization of a suitable cost functional accounting for the mismatch between the intensity-only data and the the ones predicted by the model acting on an estimated scattering object.

Alternatively, in [8], a two step inversion procedure is proposed. First, the scattered field is retrieved (both in amplitude and in phase) from the square amplitude of the total field (scattered plus incident fields). Then, starting from the estimated scattered field a further optimization stage leads to the reconstruction. This approach has the advantage that splitting the reconstruction procedure in two steps allows one to better control the overall non-linearity of the problem as compared to the single step procedure. In particular, the first step amounts to a phase retrieval problem which takes advantage from the presence of an interfering term (between the scattered and the incident fields) which increases the data dimensionality and thus increase the reliability of solution against local minima [9]. Finally, it is worth mentioning the very recent approach presented in [10] where a signal sub-space based method is presented.

When the available data concerns the amplitude of the scattered field, the phase retrieval problem becomes more challenging with respect to the case of total field amplitude data [11, 12]. In fact, in this case, measurements under different scattering conditions are required to increase the ratio between the number of data and the number of unknowns in order to obtain reliable reconstructions. In particular, in [11] and [12] a two step reconstruction procedure, as discussed in connection to the work in [8], is adopted where the phase retrieval stage is tackled by collecting the scattering amplitude over two different planar bounded measurement domain in the near zone of the scatterers.

In this paper, we are concerned with the inverse scattering problem of reconstructing the contour of metallic planar objects from the scattered field amplitude measurements. In particular, as previously discussed, we adopt a two step procedure where first the phase of the scattered field is retrieved. However, at variance of all the above mentioned contributions where a singlefrequency measurement configuration is employed, here, instead, we adopt frequency diversity. More in detail, in order to accommodate the need of independent data for avoiding local minima problem in the phase retrieval stage we collect the scattered field amplitude over a single planar bounded measurement domain at two different working frequencies.

As is shown afterwards, the problem can be cast as done in [11] so it is expected the reconstruction algorithm shares the same features in terms of local minima and achievable performances of the method reported in [11]. However, as now the measurements are taken over a single domain the proposed technique has the advantage of requiring a more practical and less expensive measurement set up.

It is worth mentioning that phase retrieval using multiple illumination wavelengths has recently presented in [13]. In such a paper, a successive alternating projections technique is employed and the wavelength is changed after each couple of projections by converting the phase of the field obtained at a given wavelength at the object plane for the next employedwavelength. The algorithm was tested for 20 different wavelengths.

The method we propose herein, instead, is developed, and is shown to work, for only two different frequencies (even though it can be easily generalized for more frequencies) which are exploited simultaneously for phase retrieval. More in detail, the phase retrieval problem is cast as the inversion of a quadratic non-linear operator by minimizing a cost functional accounting for the square amplitude of the scattered field as data of the problem. This formulation has allowed a thorough analysis of the effect of the parameters of the problem on the local minima problem [9, 14]. In particular, it has been pointed out how the increase in the ratio between the number of independent data and the number of unknowns allows us to achieve a favorable effect on the local minima problem.

The paper is organized as follows. Section 2 is devoted to present the formulation of the inverse scattering problem and to the description of the inversion algorithm. In particular, as we are interested in metallic planar scatterers, the Physical Optics approximation will be adopted in setting the mathematical relationship linking the scatterers to their scattered field. In Section 3, details concerning the implementation of the inversion procedure emphasizing the required discretization related on the representation for the unknown and the data sets are given. Finally, a numerical analysis which outlines the feasibility and reliability of the proposed algorithm is provided in Section 4. Conclusions follow.

2. Formulation of the problem

We are concerned with the problem of determining the contour of planar metallic scatterers from the knowledge of only the amplitude of their scattered field collected over a bounded planar domain in near-field zone.

More in detail, we consider the following scattering configuration. The planar metallic objects reside within the so-called object-aperture 𝒪=[-X_O,X_O]×[-Y_O,Y_O] which is a subset of the x-y plane located at z=0. Whereas the amplitude of the scattered field is collected over a single planar measurement-domain 𝓓=[-X_M,X_M]×[-Y_M,Y_M] in the x-y plane at z=z ₁. As incident field we consider a normally impinging plane wave (i.e., along -z direction) at two different frequencies, say f ₁ and f ₂, with f ₂>f ₁. The background medium is assumed to be the free-space whose dielectric permittivity and magnetic permeability are denoted by ε ₀ and μ ₀, respectively.

Accordingly, say Γ⊆𝒪 the scatterers’ supports (not necessarily connected in the case of more than one scatterers) and U _Γ the corresponding characteristic function, that is

then the problem can be rigorously stated as follows: retrieve U _Γ from the amplitude of the scattered field M ₁ and M ₂ collected over the measurement aperture 𝓓 at two different frequencies f ₁ and f ₂.

In setting up the approach, first we need the mathematical relationship between U _Γ and M_i, with i∊(1,2). Hence, we start by considering the expression of the scattered field which arises from the interaction between the incident plane wave and the metallic planar scatterers.

As aforementioned, the incident field is a plane wave of the type E̲_inc=E_incexp(jk_iz)î_x, with $k_i=2\pi f_i\sqrt{{\epsilon }_0{\mu }_0}$ being the wavenumber of the free-space at the frequency f_i. Note that the time dependence exp(j2π f_it) has been assumed and suppressed.

As the scatterers we are considering are metallic and planar their scattered field can be well expressed by adopting the Physical Optics (PO) approximation [15]. By doing so, then, the scattered field can be written as

where r̲ is the field point within the observation aperture 𝓓, G̳(r̲-r̲′,f_i) is the dyadic Green’s function and J̲_PO(r̲′,f_i) is the current induced over the scattering objects under the PO approximation. In particular,

where $\zeta =\sqrt{\frac{{\mu }_0}{{\epsilon }_0}}$ is the free-space characteristic impedance, r̲′ ranges within 𝒪 and î_n is the unitary vector normal to the scatterers’ surface, which for the case at hand coincides with î_z.

Hereinafter, we choose to exploit only the x-component of the scattered field. Hence, the problem turns to be scalar. For such a case, by exploiting Eq. (3), Eq. (2) simplifies as

where for simplicity of notation we have denoted the x-component of the scattered field still as E_S but discarding the vectorial notation.

By resorting to the spectral representation of the Green’s function G_xx(r̲-r̲′,f_i) [16] the scattered field over the measurement aperture 𝓓 at z=z ₁ can be rewritten as

where u, ν and w_i are the spectral variables conjugated of those spatial x, y and z, respectively, with $w_i=\sqrt{k_i^2-u^2-v^2}$ , k_i=2π/λ_i and λ_i are the free-space wavenumber and the corresponding wavelength, respectively, at the frequency f_i, $F_i=\frac{1}{4{\pi }^2{k}_i}$ and Û _Γ is the Fourier transform of the induced current support

with 𝓕 being the two dimensional Fourier transform operator.

Now, starting from the amplitude of the scattered field over 𝓓 at the two frequencies f ₁ and f ₂ we aim at reconstructing the function U _Γ. To this end, a reconstruction procedure in two steps is adopted. In the first step we retrieve Û _Γ by the data M ₁ and M ₂. Then, U _Γ is retrieved from Û _Γ.

It is clear that the first step of the reconstruction procedure consists in solving a phase retrieval problem. Such a problem has been extensively studied and many approaches are present in the literature. Here, we adopt an approach similar to the one already exploited for the phase retrieval problem in antenna diagnostics and, very recently, in inverse scattering problem [11].

Accordingly, in the first step the problem is cast as the inversion for Û _Γ of the following non-linear operator

achieved by searching for the global minimum of the cost functional

where ∥·∥ is the usual quadratic norm in the data space.

Note that in eqs. (7) and (8) the constant E_inc has been assumed equal to one and the square amplitude of the scattered field M ² ₁ and M ² ₂ have been considered as data instead of M ₁ and M ₂.

Such a choice offers the advantage to deal with the inversion of a non-linear quadratic operator for which some conclusions have been derived to overcome the problems of false solutions corresponding to the local minima of the non-quadratic functional (8) within which the minimizing procedure can be trapped. More in detail, based on the geometrical features of the mapping (7), it was shown in a number of papers (see [9] fore example) that the problem of local minima can be avoided if an adequate ratio between independent data and unknowns is available. This circumstance can be meet by assuming the knowledge of the scattered field amplitude in more than one measurement condition. In the paper by Soldovieri and Pierri [11] this was achieved by collecting the scattered field amplitude data at a single frequency over two different measurement apertures still located in the near-field zone of the scatterers.

Here, the role of the two set of data collected over two different planar domains is played by the two set of data acquired on a single measurement domain but at two different frequencies.

This is made it possible as the the unknown of the problem is the support function U _Γ which does not depend on the frequency whereas the way the induced surface current depends on frequency is a priori known due to the adopted PO approximation. In particular, for the configuration considered in this paper, the induced surface current does not depend on the frequency (as is evident from Eq. (3)). Accordingly, observations (of the scattered field intensity) at different frequencies can be exploited for conveying an increased amount of information on the unknown, and simultaneously adopted in the reconstruction procedure. From a mathematical point of view, this leads to the the inversion of an operator similar to the one in [11] but now the diversity between the two set of intensity data is achieved by exploiting the frequency diversity.

In other words, for the case considered herein, diversity between the two sets of intensity data stems from the differences in the kernels of ℒ ₁ and ℒ ₂, that is from the exponential terms exp[-jz ₁(w ₂-w ₁)], whereas in the approach presented in [11] this same role was played by the exponential term exp[-jw(z ₂-z ₁)], with w defined as above but at the single employed frequency, and z ₂ and z ₁ were the quotas of the two planar measurement domains.

Therefore, since exp[-jz ₁(w ₂-w ₁)] has unit modulus (in the visible domain), the possibility to achieve diversity between the two sets of near field data is related to the variability of its phase. By rewriting such a term as exp $\left[-j2\pi z_1\left(\frac{1}{{\lambda }_2}-\frac{1}{{\lambda }_1}\right)\left(w_2-w_1\right)\right]$ , with λ ₁ and λ ₂ being the wavelengths at f ₁ and f ₂ respectively, it is evident the positive role of the increase in the spacing between f ₁ and f ₂. Indeed, as f ₁ and f ₂ are more far apart, the phase term is increasingly varying in the spectral plane and more different the two sets of the intensity data are.

It is worth recalling that, the possibility of exploiting intensity data collected over two different apertures, and equivalently for two different frequencies in this case, is also helpful in resolving the intrinsic ambiguities of the phase retrieval problem [17], aside from very particular (for instance symmetric) situations. This, indeed, results from the further constraint provided by the knowledge of a second set of intensity data (over a different aperture or at a different frequency) to which the solution has to obey [18].

Therefore, the proposed two frequencies approach shares the main advantages of the method adopted in [11] as to the local minima problem and phase ambiguities. In addition it also allows one to adopt a more practical and convenient configuration as data over a single aperture, even though at two different frequencies, are required.

Once Û _Γ has been retrieved, the current support is obtained by passing Û _Γ through the adjoint of the Fourier operator 𝓕, that is

Ũ _Γ being the reconstructed version of U _Γ.

The implementation of the reconstruction algorithm and the details related to the discretization of Eq. (8) are discussed in next section.

3. The solution algorithm

In this section we aim at describing the discretizing procedure upon which the the reconstruction scheme rely on.

Accordingly, the phase retrieval stage is achieved by minimizing a suitably discretized version of the functional (8), that is

with $\underset{\_}{Û}$ _Γ, ${\widetilde{\underset{¯}{M}}}_1^2$ and ${\widetilde{\underset{¯}{M}}}_2^2$ being finite dimensional (discretized) version of the corresponding entities and $\underset{\_}{\underset{\_}{𝓛}}$ ₁ and $\underset{\_}{\underset{\_}{𝓛}}$ ₂ are the propagator operators at the two different frequencies as in (5) discretized according to the representation adopted for the data and the unknown. Note, that ${\widetilde{\underset{¯}{M}}}_1^2$ and ${\widetilde{\underset{¯}{M}}}_2^2$ are corrupted (by measurement errors and noise) version of M ² ₁ and M ² ₂.

As is known, while numerically implementing an algorithm for minimizing Eq. (10), data and unknown representations play a crucial role in ensuring the reliability of the solution. Indeed, the ratio between the number of data and the number of unknowns is strongly related to the occurrence of false solutions. Moreover, due to the ill-posedness of the problem a judicious choice of such representations helps in restoring the stability of the solution.

To this end, mathematical features of the involved entities have to adopted. However, optimal nonredundant representation schemes [19], that exploit precisely the mathematical features and the available a priori information, not necessarily are simple to be implemented. Therefore, here we choose a driscretizing scheme which exploit the a priori information as guide line making safe the possibility to implement the reconstruction algorithm easily.

In order to numerically implement the solution algorithm the first choice concerns the representation of the unknown function Û _Γ. In this case the a priori knowledge we have says us that Û _Γ, being the Fourier transform of a function compactly supported over 𝒪, is an entire band-limited function [20]. Accordingly, Û _Γ can be expanded in a sampling series as

where Û _Γnm are the samples of Û _Γ taken at the Nyquist rate, the truncation indexes N and M are chosen as explained below and sinc(x), as usual, is defined as

In particular, as the measurement aperture 𝓓 is chosen some wavelengths apart from the aperture 𝒪 where the scatterers reside, it can be argued that the integration appearing within the propagator operators ℒ ₁ and ℒ ₂ can be limited to only the visible part of the plane wave spectrum as evanescent waves give rise to an exponentially decaying (with z) contribution. Therefore, the truncation indexes are chosen so that

Choice in Eq. (15) entails that the spectrum samples searched for as actual unknowns are the ones belonging to the visible domain attained at the maximum adopted frequency. This, on one side allows us to implicitly regularize the problem. On the other side this choice fixes the achievable resolution while, from Û _Γ, U _Γ is reconstructed by means of (9).

It can be also argued that, fixed N and M, the best way (in terms of the norm of the representation error) to represent a band-limited function projected over a compact support (in our case the circle of radius k ₂ in the u-v plane) is by means of a finite series of the singular functions of the operator P_VC𝓕P _𝒪, where 𝓕 is defined as above and P _𝒪 and P_VC are projector operators over the investigation domain 𝒪 and the circular visible domain, respectively. However, the representation adopted in Eq. (11) is useful to achieve a very efficient implementation of the solution algorithm as such a choice enables most of the operations to be performed thanks to the FFT technique.

As we have assumed that evanescent waves are negligible then the scattered field, as function of x and y, can be considered an almost (related to the actual negligibility of the evanescent part) bandlimited function whose band is enclosed in the square [-k_i,k_i]×[-k_i,k_i] (for each adopted frequency). Accordingly, the scattered field square amplitude has a bandwidth which is enclosed in the square [-2k_i,2k_i]×[-2k_i,2k_i] and this suggest to acquire data at the Nyquist rate π/(2k_i)=λ_i/4 along x and y.

This representation may results redundant. It has been shown that exploiting the a priori information concerning the scatterer dimension allows to reduce the number of measurements but generally require a nonuniform sampling step in x and y [19, 21]. Therefore, once again, our choice simplifies the implementation operation as it allows for an heavy use of the FFT algorithm.

As far as the minimization of the functional in (10) is concerned, an iterative procedure based on the Polak-Ribiere method [22] is applied which requires the computation of the gradient of the cost function (10) with respect to the real and imaginary parts of the Fourier coefficients Û _Γnm to evaluate the updating direction in the minimizing procedure.

As mentioned, the increase of the ratio between the number of independent data and the dimension of the unknown space has a favorable effect on the local minima problem. Thus, for the test cases presented below, we adopt a minimization strategy based on L minimization steps where the accuracy of the solution is improved by gradually increasing the number of unknowns [23]. In particular, at the l-th step we consider N=2l+1 and M=2l+1. The obtained solution is then used as starting point in the next step by considering N=2(l+1)+1 and M=2(l+1)+1. This procedure continues until all the samples falling within the visible domain at the higher frequency, as dictated by Eq. (15), are considered. Accordingly, a resolution of λ ₂/2, along both x and y, is expected while reconstructing U _Γ.

Finally, the result can be further improved by adopting the weighted formulation [14]

where c=a./b means a vector whose components are given as c_n=a_n/b_n and η is factor avoiding division by zero.

After that the phase retrieval problem has been solved and an estimate of the characteristic function is obtained, the shape of the metallic plate is determined as the region where the reconstruction is significantly different from zero. To this end, we introduce a threshold under which the reconstruction is discarded. In particular, the threshold is chosen at -13dB below the maximum of the reconstruction which corresponds to the sidelobe level of the expected point spread function.

4. Numerical analysis

This section is devoted to present the feasibility of the solution procedure by considering two test cases concerned with different shapes of the metallic planar scatterers.

For both the test cases, we consider two frequencies such that f ₂=2f ₁, a measurement aperture 𝓓=[-16λ ₂,16λ ₂]×[-16λ ₂,16λ ₂] located at the quota z ₁=5λ ₂ and an object-aperture 𝒪=[-15λ ₂,15λ ₂]×[-15λ ₂,15λ ₂]. Accordingly, the scattered field data at the lower frequency f ₁ consist of 64×64 measurements taken λ ₂/2 apart each other both along x and y, while for the frequency f ₂ we have 128×128 measurements spaced by λ ₂/4.

 

Fig. 1. a: actual scatterer. b: reconstruction obtained by considering up to (2N ₁+1)(2M ₁+ 1) unknown coefficients for the Û _Γ representation in the minimization of the cost functional (10). c: reconstruction obtained by enlarging the number of unknown coefficients up to (2N+1)(2M+1). d: reconstruction obtained by simultaneously searching for all the (2N+1)(2M+1) unknowns by first minimizing the cost functional (10) and then adopting the results as starting point for minimizing the weighted functional (14). Results are shown in dB scale.

Download Full Size | PDF

In all the cases the starting point of the minimization procedure is assumed in a random way.

The first test case refers to a square plate scatterer whose support is given by Γ=[-5λ ₂,5λ ₂]×[-5λ ₂,5λ ₂] (see panel a of Fig. 1). The corresponding reconstructions are reported in panels b, c and d of the same figure whereas the behavior of the cost functional Φ (as given by Eq. (10)) normalized to the square norm data, that is Φ($\underset{\_}{Û}$ _Γ)/∥(M̃ ² ₁, M̃ ² ₂)∥², is shown as a blue line in Fig. 2 as the minimization procedure evolves. Markers * denote the transition between two successive minimization steps, namely when the minimization employing N=2l+1 and M=2l+1 arrests and the one with an enlarged number of unknowns N=2(l+1)+1 and M=2(l+1)+1 starts. In particular, panel b of Fig. 1 refers to the case in which the minimization procedure considers up to the unknown coefficients Û _Γnm laying within the visible domain related to the frequency f ₁, that is

Instead, in panel c of the same figure the reconstruction have been obtained by enlarging the number of unknowns so as to consider all the visible part dictated by f ₂. As can be seen, even though the reconstruction error quantified as ∥U _Γ-Û _Γ∥²/∥U _Γ∥² is not particularly low (for example in the case of panel c it is equal to 0.12) the reconstruction results allow us to correctly determine the location and the geometry of the metallic plate.

 

Fig. 2. Behavior of the error in the data space defined as Φ(U̲ _Γ)/∥(M̃ ² ₁, M̃ ² ₂)∥². Blue line refers to the case reported in Fig. 1 whereas red line refers to the case reported in Fig. 3. Markers * denotes the transition by two different minimization steps accordingly to the progressive increasing of the searched for number of unknown coefficients.

Download Full Size | PDF

The advantage offered by the adopted minimization procedure against false solutions is made it evident by comparing the reconstructions reported in panels c and d of Fig. 1. In fact, the reconstruction shown in panel d have been obtained by a single minimization of (10) by simultaneously searching for all the 2N×2M unknowns. In particular, in order to improve the reconstruction, the procedure have been followed by a weighted minimization according to the functional in Eq. (14) (instead for the case of panel c such further step did not give a relevant improvement and hence is not reported here). In this case, the error in the data space evaluated as for previous case, when the procedure arrests returns -34.7 dB which is much higher than the one obtained for the case reported in panel c (see blue line of Fig. 2). Indeed, due to the reduced ration between the number of data and unknowns the procedurewas not able to achieve the true solution.

Let us turn to consider the second example. It refers to a metallic object consisting of a plate as in the previous case but, now, there is a hole at the center of side 4λ ₂ (see Fig. 3, panel a).

The same analysis as in Fig. 1 has been performed and the reconstructions are shown in Fig. 3, where the various panels are the analogues of the ones in Fig. 1. Also in this more critical case (note, by looking at the red line of Fig. 2, that when the minimization procedure arrests the residual error in the data space is higher than the one of the previous case) the reconstruction procedure works very well in retrieving the scatterer’s contour and we can appreciate the favorable effect of the strategy based on the progressive enlargement of the unknowns to be searched for on the reliability and accuracy of the retrieved solution.

 

Fig. 3. a: actual scatterer. b: reconstruction obtained by considering up to (2N ₁+1)(2M ₁+1) unknown coefficients for the Û _Γ representation in the minimization of the cost functional (10). c: reconstruction obtained by enlarging the number of unknown coefficients up to (2N+1)(2M+1). d: reconstruction obtained by simultaneously searching for all the (2N+1)(2M+1) unknowns by first minimizing the cost functional (10) and then adopting the results as starting point for minimizing the weighted functional (14). Results are shown in dB scale.

Download Full Size | PDF

Finally, we verify the stability of the reconstruction procedure against noisy data. In Fig. 4 the reconstruction obtained by intensity data corrupted by an additive 10% uniformly distributed noise is shown for the case reported in Fig. 1 (panel c). As expected in this case the reconstruction error increases (now ∥U _Γ-Û _Γ∥²/∥U _Γ∥² returns 0.24 whereas in the previous example it was 0.12). Nevertheless, the scatterer’s contour is retrieved and the reconstruction is very similar to corresponding reconstruction reported in Fig. 1.

As concluding remarks we outline that the same performances in terms of reconstruction error can be obtained by the single-frequency two plane configuration [11]. In fact, the reconstructions reported in this paper and those shown in [11] are very similar. However, we once again stress the practical advantages offered by the two-frequency single plane configuration addressed herein.

5. Conclusion

The problem of reconstructing the support U _Γ of metallic planar scatterers which are known to reside within the object aperture from the amplitude of their scattered field has been addressed. A two step inversion procedure has been presented. First, a phase retrieval problem has solved to retrieve (both in amplitude and phase) the Fourier transform Û _Γ of the support function U _Γ. Then, U _Γ has been determined through 𝓕^† Û _Γ. The phase retrieval stage has taken advantage from considering as data the square amplitude of the scattered field which has allowed to cast the problem as the inversion of quadratic operator whose features concerning the occurrence of local minima has been well understood [9]. In particular, it is known that to avoid false solutions an adequate ratio of the number of data over the number of the unknowns is required. To this end, previous methods relying on a single frequency measurement set up [14, 12] need the data to be collected over multiple measurement domains. Here, instead, we have considered the amplitude data acquired over a single measurement domain and two different frequencies have been adopted to increase the number of independent data. By doing so we have obtained the advantage to deal with a more practical and less expensive measurement set up.

 

Fig. 4. Reconstruction for the same scatterer and for the same situation as in Fig. 1 panel c but data have been corrupted by an additive 10% uniformly distributed noise. Result is shown in dB scale.

Download Full Size | PDF

Numerical examples which have been reported testified the good performance achievable by the proposed reconstruction method also against the case of data corrupted by noise.

Finally, it worth stressing that we have considered a canonical situation where measurements are taken over a plane parallel to the target plane. This condition can be difficult to maintain experimentally, specially in the optical regime. Therefore, as a future developments it could be interesting to investigate tilting effect of the measurement plane on the reconstructed results.