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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 9 — May. 5, 2014
  • pp: 10119–10138
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Multi-resolution imaging with an optimized number and distribution of sampling points

Amedeo Capozzoli, Claudio Curcio, and Angelo Liseno  »View Author Affiliations


Optics Express, Vol. 22, Issue 9, pp. 10119-10138 (2014)
http://dx.doi.org/10.1364/OE.22.010119


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Abstract

We propose an approach of interest in Imaging and Synthetic Aperture Radar (SAR) tomography, for the optimal determination of the scanning region dimension, of the number of sampling points therein, and their spatial distribution, in the case of single frequency monostatic multi-view and multi-static single-view target reflectivity reconstruction. The method recasts the reconstruction of the target reflectivity from the field data collected on the scanning region in terms of a finite dimensional algebraic linear inverse problem. The dimension of the scanning region, the number and the positions of the sampling points are optimally determined by optimizing the singular value behavior of the matrix defining the linear operator. Single resolution, multi-resolution and dynamic multi-resolution can be afforded by the method, allowing a flexibility not available in previous approaches. The performance has been evaluated via a numerical and experimental analysis.

© 2014 Optical Society of America

1. Introduction

Imaging, Synthetic Aperture Radar (SAR) Tomography [1

1. A. Reigber and A. Moreira, “First demonstration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38(5), 2142–2152 (2000). [CrossRef]

3

3. A. Capozzoli, C. Curcio, and A. Liseno, “SAR tomography with optimized track distribution and controlled resolution, ” in Proceedings of the XIX Riunione Nazionale di Elettromagnetismo, (Roma, Italy, Sept. 10–14, 2012), pp. 174–177.

], and more generally Inverse Scattering problems [4

4. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, 1998).

] require the determination of qualitative or quantitative characteristics about a scenario under investigation from field data collected on a spatial domain, the scanning region. In such applications the system is typically active, since the target is properly and intentionally illuminated. However, in some imaging applications, also passive configurations are exploited, wherein the scenario scatters the environmental field naturally impinging on it [5

5. M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser Jr, and A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008). [CrossRef] [PubMed]

].

As long as active systems are considered, depending on the case, monostatic or multistatic, single-view or multi-view as well as single-frequency or multi-frequency acquisitions can be considered. In particular, in the monostatic multi-view case, the same antenna (also two very close antennas, one transmitting and one receiving, rigidly joined can be used) is exploited at the same time to illuminate the target and to collect the scattered signal, while in the multi-static single-view, the source illuminating the target is assumed fixed and the probe is moved just to measure the scattered field.

The use of such systems working in the mmW/THz frequency bandwidth is wide spreading for imaging in security and contraband detection applications, thanks to its capability of penetrating clothing barriers, to image items concealed by clothes [5

5. M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser Jr, and A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008). [CrossRef] [PubMed]

9

9. F. Qi, I. Ocket, D. Schreurs, and B. Nauwelaers, “A system-level simulator for indoor mmW SAR imaging and its applications,” Opt. Express 20(21), 23811–23820 (2012). [CrossRef] [PubMed]

]. Such systems belong to two classes. In the first one an array of detectors is located in the focal plane of a focusing system (lens/reflector). In the second one, the case of interest here, an Array (physical or synthetic) of transmitting/receiving Elements (AE) is considered, and the image is obtained by processing the collected data.

One of the key aspects when designing such systems is to determine the dimension of the scanning region, the number of AEs and their spatial distribution ensuring the information-gathering with the desired degree of accuracy [10

10. A. Capozzoli, C. Curcio, A. Liseno, and P. Vinetti, “Field sampling and field reconstruction: a new perspective,” Radio Sci. 45, RS6004 (2010). [CrossRef]

]. Indeed, the number and the distribution influence significantly the performance of the reconstruction, from one side, and from the other side the complexity of the hardware, in the case of physical arrays, or of the scanning procedure, in the case of synthetic arrays. Obviously, once both the optimal number and spatial distribution of the AEs are provided, also the optimal dimension of the region to be scanned is automatically obtained.

Accordingly, a criterion to determine the optimized number and the locations of the AEs for the desired resolution becomes mandatory, particularly when the expected number of AEs is large, due to the characteristics of the target and the dimensions of the scanning region [3

3. A. Capozzoli, C. Curcio, and A. Liseno, “SAR tomography with optimized track distribution and controlled resolution, ” in Proceedings of the XIX Riunione Nazionale di Elettromagnetismo, (Roma, Italy, Sept. 10–14, 2012), pp. 174–177.

]. In fact, lower resolutions should require a lesser number of properly spaced AEs, while higher resolutions should increase such a number. Accordingly, having at disposal an approach determining the optimal number of AEs and their locations allows to modulate the complexity of the system depending on the requested degree of resolution, see Fig. 1(a).
Fig. 1 a) Variable resolution; b) Multi-resolution; c) Dynamic multi-resolution.

Furthermore, two more features are attractive:

  • - multi-resolution, when we are interested in determining the number and the locations of AEs to reconstruct different portions of the target with different resolutions, see Fig. 1(b);
  • - dynamic multi-resolution, when we are interested in determining the number of AEs to be added, and their locations, to improve the resolution of an image already obtained only in a portion of the target where the attention is attracted, see Fig. 1(c).

The aim of the paper is to present an approach solving the problem above in the single frequency monostatic multi-view case or in the single frequency multi-static single-view one. The inspiring idea can be extended also to the multi-view multi-static case, as well as to multi-frequency acquisitions.

The approach is suited for both synthetic and physical arrays, and allows to find the optimal number of AEs and their optimal locations as a function of the desired resolution in the cases of single-resolution, multi-resolution and dynamic multi-resolution.

The paper is organized as follows. In Section 2 the geometry of the problem together with the considered electromagnetic modeling are presented. The representation of the target reflectivity is discussed in Section 3, while Section 4 concerns the solution algorithm. In Section 5 the resolution control is discussed. Finally, in Section 6 numerical and experimental results, proving the effectiveness of the approach, are shown. Conclusions are drawn in Section 7.

2. The geometry of the problem and the modeling

Let us refer to the single frequency case. We note that the approach could be extended also to the case of multi-frequency acquisitions [11

11. A. Capozzoli, C. Curcio, and A. Liseno, “Multi-frequency planar near-field scanning by means of singular-value decomposition (SVD) optimization,” IEEE Antennas Propag. Mag. 53, 212–221 (2011). [CrossRef]

].

For the sake of simplicity we will refer to a scalar problem, typical of imaging applications, even if its extension to a vector problem has no conceptual difficulties.

We assume that a planar object is under investigation, the target, located in the plane z = 0 of a reference system (Oxyz), which is contained in the rectangular region O = [-H'x, H'x]x[-H'y, H'y] and illuminated by a point-like source located at P0 = (x0,y0,z0), see Fig. 2.
Fig. 2 Geometry of the problem.
The field data are collected on a scanning region S = [-Hx, Hx]x[-Hy, Hy] of the plane z = d in the positions P = (x,y,d), see Fig. 2. In the following, the effects of the antenna response are neglected, since it is assumed point-like.

Obviously, P0 = P in the case of monostatic multi-view acquisitions, so that the antenna is exploited to illuminate the target and to collect the scattered field data at the same time; in the multi-static single-view case P0 is kept fixed and only P is varied.

We denote with ρ(x’,y’) the reflectivity of the target, assumed zero outside O, with u(x’,y’) the field scattered by the object on the plane z = 0, with s(x,y) the measured signal, and with k the propagation constant in the medium (k = 2π/λ, λ being the wavelength).

The solution of the problem of interest requires the determination of ρ, given s on S.

2.1. Accurate modelling

Let us set R = |R|, where
R¯(x,y,z;x,y,z')=(xx)x^+(yy)y^+(zz')z^,
(1)
and
G(x,y,z;x,y,z')=exp(jkR(x,y,z;x,y,z'))R(x,y,z;x,y,z').
(2)
Thanks to the Rayleigh-Sommerfeld theory [12

12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

], the signal s can be calculated as:
s(x,y)=12πOdxdyu(x,y)[zG(x,y,z;x,y,z')]z=d,z'=0,
(3)
for (x,y,d) S, where the scalar field u, apart from inessential constants, can be written as:
u(x,y)=ρ(x,y)exp(jkR(x0,y0,z0;x,y,0))R(x0,y0,z0;x,y,0).
(4)
As long as we set N as the number of the AEs located at Pn = (xn,yn,d), n = 1,...,N, on S, we have at disposal N measured data, s1 = s(x1,y1),…,sN = s(xN,yN) say, to retrieve the unknown function ρ.

By setting kz=k2kx2ky2, from the Weyl identity we have [13

13. T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields (IEEE, 1999).

]:
s(x,y)=14π2++dkxdkyexp[j(kxx+kyy)]exp[jkzd]Odxdyu(x,y)exp[j(kxx+kyy)],
(5)
which defines, thanks to Eq. (4), the operator mapping ρ onto s, say A1, which can be written in the abstract form as:

A1|ρA1(ρ)=s.
(6)

2.2. Approximate model

Simplified expressions for the operator A1 can be obtained as long as approximations frequently feasible in practice are introduced.

In particular, under the Fresnel approximation [14

14. G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13(9), 1816–1826 (1996). [CrossRef]

], with R>>λ, d>>Hx + H’x and d>>Hy + H’y, we have:
s˜(x,y)1jλd2Odxdyρ˜(x,y)exp[j2πλd((x+x0)x+(y+y0)y)],
(7)
where:
s˜(x,y)=exp(j2kd)exp[jk2d(x2+x02+y2+y02)]s(x,y),
(8)
and

ρ˜(x,y)=ρ(x,y)exp[jkd(x2+y2)],
(9)

As in the case of Eq. (5), the Eq. (7) can be written by introducing the operator A2 mapping ρ˜ onto s˜ in the abstract form
A2|ρ˜A2(ρ˜)=s˜,
(10)
making the discussion in Section 4 independent of the particular model and acquisition strategy at hand.

Obviously, in the case of multi-static single-view, (x0,y0) is kept fixed, while in the case of monostatic multi-view (x0,y0) is varied, being (x0,y0) = (x,y), and Eq. (7) assumes the well known Fourier transform based expression:
s˜(x,y)=Odxdyρ˜(x,y)exp[j4πλd(xx+yy)],
(11)
typical of some applications, e.g. SAR tomography [1

1. A. Reigber and A. Moreira, “First demonstration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38(5), 2142–2152 (2000). [CrossRef]

3

3. A. Capozzoli, C. Curcio, and A. Liseno, “SAR tomography with optimized track distribution and controlled resolution, ” in Proceedings of the XIX Riunione Nazionale di Elettromagnetismo, (Roma, Italy, Sept. 10–14, 2012), pp. 174–177.

].

3. The representation of the unknown

Now, let us introduce a very convenient representation for the unknown ρ. Indeed, the function ρ cannot be retrieved with any desired degree of accuracy due to the filtering properties of the propagation mechanism, allowing only the visible portion of ρ retrievable from field data collected some wavelength apart from the target.

According to the behaviour of exp[-jkzd], for d larger than few wavelengths, Eq. (5) shows that the support of the Fourier Transform (spectrum) of s is essentially limited to the visible domain:
V={(kx,ky)|kx2+ky2k2},
(12)
as long as ρ, as usual, has not super-directive [15

15. P. C. Clemmow, The Plane-Wave Spectrum Representation of Electromagnetic Fields (IEEE, 1996).

] characteristics, and the cumbersome problem of super-resolution is not considered [16

16. O. M. Bucci, A. Capozzoli, and G. D’Elia, “Regularizing strategy for image restoration and wave-front sensing by phase diversity,” J. Opt. Soc. Am. A 16(7), 1759–1768 (1999). [CrossRef]

].

This aspect is here quite relevant, since s is determined only by a filtered version of ρ, say ρv, having support limited to O and spectrum with support essentially limited to V.

To properly define ρv, we introduce the subspace, say B, of functions with support limited to O and spectrum with support essentially limited to V. Such a subspace can be determined by means of Generalized Prolate Spheroidal Wave Functions (GPSWFs) [17

17. B. R. Frieden and E. Wolf, eds., “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics 9 (North-Holland, 1971), pp. 311–407.

19

19. A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Phaseless antenna characterization by effective aperture field and data representations,” IEEE Trans. Antennas Propag. 57(1), 215–230 (2009). [CrossRef]

], which extend the usual properties of Prolate Spheroidal Wave Functions (PSWF) [18

18. H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: The dimension of essentially time- and band-limited signals,” Bell Syst. Tech. J. 41(4), 1295–1336 (1962). [CrossRef]

] to the case of non rectangular domains. If O is not small with respect to λ, the visible GPSWFs span the finite dimensional subspace B.

Obviously, ρv can be considered as just the projection of ρ onto B. And so, retrieving the target reflectivity from the field measurements is equivalent to retrieve a finite set of complex coefficients defining the expansion of ρv as a linear combination of the visible GPSWFs.

However, in practice, the use of the GPSWFs is limited by the computational burden and by the efforts needed for their accurate evaluation [20

20. G. D. de Villiers, F. B. T. Marchaud, and E. R. Pike, “Generalized Gaussian quadrature applied to an inverse problem in antenna theory: II. The two-dimensional case with circular symmetry,” Inverse Probl. 19(3), 755–778 (2003). [CrossRef]

], especially for large targets. And so it is of interest to employ an alternative set of basis functions, very convenient from the computational point of view, the Rectangular PSWFs (RPSWF), which has been proved “quasi-optimal” [19

19. A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Phaseless antenna characterization by effective aperture field and data representations,” IEEE Trans. Antennas Propag. 57(1), 215–230 (2009). [CrossRef]

].

To define RPSWFs, let us introduce RV = [-k,k] × [-k,k] as the smallest rectangle containing V. RPSWFs are given by:
Ψm(qx,qy)[cx,cy,x,y]=Πqx[cx,x]Πqy[cy,y],
(13)
where Πi[cw,w] is the i-th, 1D, PSWF with “space-bandwidth product” cw [17

17. B. R. Frieden and E. Wolf, eds., “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics 9 (North-Holland, 1971), pp. 311–407.

,18

18. H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: The dimension of essentially time- and band-limited signals,” Bell Syst. Tech. J. 41(4), 1295–1336 (1962). [CrossRef]

], cx = kH’x, cy = kH’y, and m(qx,qy) is the mapping relating the couple of indexes (qx,qy) to the index m according to a prefixed ordering.

We denote with μqx and μqythe eigenvalues associated to Πqx and Πqy, respectively [18

18. H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: The dimension of essentially time- and band-limited signals,” Bell Syst. Tech. J. 41(4), 1295–1336 (1962). [CrossRef]

], and with {νm}m=1+ the ordered (decreasing) sequence of the eigenvalues associated to Ψm, given by νm(qx,qy)=μqxμqy. When the dimensions of O are not small with respect to λ, {νm}m=1+ exhibits a sharp step-like behaviour, making νM+1/ν1 very small as long as M = Qx Qy, with Qx = Int[2 kH’x /π], and Qy = Int[2 kH’y /π], Int[x] being the integer part of x.

In this case {Ψm}m=1M span essentially all the functions having bounded support equal to O and spectral support essentially equal to RV, see Fig. 3.
Fig. 3 Supports of the {Ψm}m=1M in the spatial domain (a) and the spectral domain (b), where RV = [-k,k] × [-k,k] and RΩ = [-Ω,Ω] × [-Ω,Ω], with Ω<k.

As a consequence, ρv can be represented as
ρv(x,y)=m=1MrmΨm[cx,cy,x,y],
(14)
where rm are complex expansion coefficients to be calculated when determining ρv.

4. Solution algorithm

As stressed in the Introduction, the aim of the paper is to present a general approach to the determination of the number N of measurement points and their locations, providing the optimal reflectivity reconstruction, according to the required resolution.

We will denote by A the operator mapping the unknown onto the data, which assumes the form A1 when the Fresnel approximation cannot be applied, and A2 otherwise. Solving the problem at hand amounts at solving the abstract functional equation:

A(ρ)=s.
(15)

Following Eq. (14) the unknown of the problem turns into the vector r = [r1,…,rM], which gives, as shown above, ρ with the maximum retrievable resolution. Accordingly, the problem of determining the reflectivity ρ from the measured signal s can be recast in the form of a finite dimensional linear inverse problem.

By introducing the matrix S whose elements are given by:
Snm=<A(Ψm),δ(xxn,yyn))>,
(16)
δ being the two dimensional Dirac distribution, Eq. (15) turns into:
s¯=S¯¯(N;p¯)r¯,
(17)
where s = [s1,…,sN] and p = [(x1,y1),…,(xN,yN)].

Finding the reflectivity of the target amounts to inverting the algebraic linear system in Eq. (17).

Obviously, the inversion of the matrix S is generally ill-conditioned, and a regularization strategy should be adopted, e.g. a Truncated Singular Value Decomposition (TSVD) approach [4

4. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, 1998).

], in order to retrieve only the information robustly contained in the data.

However, as highlighted by the notation used in Eq. (17), the matrix S is not univocally determined. It depends parametrically on the number of sample points exploited, and on their locations. In other words, we have at disposal a family of matrixes S¯¯(N;p¯) and, before inverting one of them, the more convenient (the best conditioned) should be picked-up, by properly selecting N and p.

To increase the amount of retrievable information, the functional Φ[(x1,y1),…,(xN,yN)] as a function of (x1,y1),…,(xN,yN) must be maximized, to obtain the optimum AEs locations.

At this point two further aspects remain open. The first one involves the choice of the number N of the AEs. The second one involves the effective and efficient optimization of Φ.

Concerning the first, an iterative approach, schematically depicted in Fig. 4, can be followed.
Fig. 4 Flow Chart of the algorithm to determine N.
Starting with N equal to M, N is progressively increased, and at each step Φ is optimized getting the value Φopt. Obviously, Φopt as a function of N is expected to be an essentially increasing function with a knee defining the maximum N to be used, say Nopt, i.e. defining the maximum number of AEs to be exploited to retrieve as much information as possible about ρ, see Fig. 5.
Fig. 5 Φopt as a function of N: the saturation knee is highlighted together with the corresponding value of Nopt.

Concerning the second point, the optimization technique must be efficient, since the computational burden can be significant, due to the large number of unknowns involved, and effective, since we are interested in the point of global maximum of Φ, which should be determined avoiding the trapping of the optimization algorithm into local optima [22

22. A. Capozzoli and G. D’Elia, “Global optimization and antennas synthesis and diagnosis, part one: concepts, tools, strategies and performances,” Prog. Electromagn. Res. 56, 195–232 (2006).

].

Both aspects can be circumvented as long as a multi-stage optimization, starting with few unknowns and enlarging progressively its number, is exploited [10

10. A. Capozzoli, C. Curcio, A. Liseno, and P. Vinetti, “Field sampling and field reconstruction: a new perspective,” Radio Sci. 45, RS6004 (2010). [CrossRef]

]. However, such an approach becomes feasible only if a proper representation of the unknown positions (x1,y1),…,(xN,yN) is available. To this end, an auxiliary plane (ξ,η) is introduced and mapped onto S, through mapping functions g and h, to obtain the x and y coordinates of the AEs locations, see Fig. 6.
Fig. 6 Mapping function - a) Qualitative effects of the distortion of the regular grid in the (ξ,η) plane, providing the irregular grid in the (x,y) plane; b) Example of a mapping function providing an irregular sampling from the regular one in the one dimensional case.
In other words, the sampling grid in the (x,y) plane is obtained from a uniform grid in (−1,1) × (−1,1) of the (ξ,η) plane, (ξm,ηm) say, via the relationships:
{xm=h(ξm,ηm)ym=g(ξm,ηm),
(19)
wherein h and g are unknown and should be determined to define the location of the AEs. To make possible the numerical determination of h and g, and to modulate the number of the unknowns, the mapping functions are represented by using a proper expansion on a set of basis functions Lj's (e.g., Legendre polynomials, as employed in this paper):
{h(ξm,ηm)=r=1Rt=1ThrtLr(ξm)Ls(ηm)g(ξm,ηm)=r=1Rt=1TgrtLr(ξm)Ls(ηm),
(20)
hrt and grt being proper expansion coefficients [10

10. A. Capozzoli, C. Curcio, A. Liseno, and P. Vinetti, “Field sampling and field reconstruction: a new perspective,” Radio Sci. 45, RS6004 (2010). [CrossRef]

]. Accordingly, the determination of the AEs locations (x1,y1),…,(xN,yN) turns into the determination of the expansion coefficients hrt and grt, whose number can be modulated according to the needs, depending on the desired degree of distortion for the uniform grid (ξm,ηm) [10

10. A. Capozzoli, C. Curcio, A. Liseno, and P. Vinetti, “Field sampling and field reconstruction: a new perspective,” Radio Sci. 45, RS6004 (2010). [CrossRef]

].

5. Resolution control

Let us now introduce the concept of resolution control. As stressed in the Introduction, depending on the need, the resolution required could be less than the maximum allowed, in some cases, in some others only a snapshot of the target can be necessary at the beginning of the investigation, which is refined later on in those regions wherein the first rough estimate of the target focalizes the attention. In some further cases, it is of interest to reconstruct different portions of the target with different resolutions.

The resolution control can be obtained by looking for a smoothed version of ρv, instead of ρv, being the smoothing variable along the scene in the case of multi-resolution retrieving.

More formally, by firstly referring to the case of a resolution control uniform throughout the scene, we will introduce the concept of mollifier or smoothing operator.

In particular, the mollifier operator MΩ here exploited will be defined as the projector onto the subspace spanned by PSWFs defined as in Section 3, where RV = [-k,k] × [-k,k] is substituted by RΩ = [-Ω,Ω] × [-Ω,Ω], with Ω<k, see Fig. 3:
MΩ|ρρ.
(21)
The parameter Ω, the operator MΩ depends on, controls the bandwidth of that reflectivity one would like to retrieve, and defines the degree of smoothing provided by the mollifier, allowing the tuning of the resolution loss determined by the the smoothing process.

The problem now turns into finding a ρ which is as close as possible to ρ from some measured data. Obviously it is expected that retrieving ρ instead of ρ should require a lesser amount of data, and consequently a reduced number of properly located AEs.

In the case of multi-resolution, different approaches can be followed, exploiting the same inspiring concept. Here we use a simple strategy based on a segmentation of O into disjoined sub-regions, and on the use of a different value of Ω in each sub-region, see Fig. 7(a).
Fig. 7 Segmentation of O - a) Multi-resolution: O is subdivided into sub-regions with different values for Ω; b) Dynamic multi-resolution: the resolution configuration of a preliminary reconstruction involving N' AEs is changed by adding further N” AEs.
Indeed, the linear relationship between ρ and s allows calculating s as the superposition of the contributions produced by each sub-region, and the reflectivity of each sub-region can be mollified with a different value of Ω in each sub-region.

In the case of dynamic multi-resolution, at the first step a first estimate of the target reflectivity, for a given segmentation of O and a given set of values of Ω, is retrieved by using N' AEs, properly distributed according to the approach above; then further N” AEs are added and their locations are optimized to fulfil a new multi-resolution configuration defined by a new segmentation and a new set of parameters Ω, see Fig. 7(b).

6. Results

In this Section, we show numerical results as well as the results of a measurement campaign aimed at assessing the performance of the described approach.

In particular, the numerical analysis shows the advantages of the proposed approach, in the basic case of full resolution retrieving and in terms of number of AEs and quality of the reconstructions, when compared to more standard solutions. The experimental analysis proves the consistency of the approach when applied to real world data, and its capabilities in terms of variable single resolution, multiresolution and dynamic multiresolution retrieving.

6.1 Numerical results

The considered scattering scenario is made of a single metallic scattering wire located in the origin of the investigation domain. The wire length has been chosen significantly larger than the working wavelength, and than d, the distance between the target and the scanning region, so to simulate a 2D scenario. This choice is justified by the need of well assessing the performance of the method in terms of resolution capabilities.

A monostatic multi-view configuration has been considered, for a given dimension of the investigation domain H'x and of the distance d: in detail, we deal with H'x = 10λ and d = 5λ. The AEs positions have been determined without any constraint, so that they are assumed free to reside on the most convenient observation domain to optimize Φ. In other words, Hx is determined by the optimization process, which, accordingly, provides not only the optimal number and the locations of the AEs, but also the optimal extension of the scanning domain. A full resolution retrieving has been considered. The data have been corrupted by a Gaussian noise with a signal to noise ratio equal to 20dB.

After the optimization process a value for Hx of about 20λ has been calculated, with 46 AEs distributed as shown in Fig. 8.
Fig. 8 AEs positions distribution determined in the case of a full resolution retrieving of a single wire target numerically simulated.
The retrieved reflectivity is presented in Fig. 9 as the solid blue line, together with the reference (red dashed line), i.e. the projection of the real reflectivity onto the subspace spanned by the 45 PSWF's exploited to represent the target.
Fig. 9 Full resolution reconstruction of a single wire target reflectivity, numerically simulated. Blue: proposed approach. Red: reference.

The results show the capabilities of the proposed method to retrieve a target reflectivity very close to the reference one.

To estimate the relevance of the approach, the results have been compared with those achievable with standard techniques. In particular, three other cases have been considered. In the first case, case A, the retrieved reflectivity is obtained by adopting an half-wavelength spacing over the scanning region determined after the optimization problem, and by representing the target reflectivity by means of the PSWF's. In the second case, case B, the retrieved reflectivity is obtained by adopting an half-wavelength spacing over the scanning region determined after the optimization problem, and by representing the target reflectivity by means of “Dirac Pulses”, as numerous as the adopted AEs, uniformly spaced on the investigation domain. Finally, in the case C, the retrieved reflectivity is obtained by taking 46 AEs, the same number exploited by the optimized approach presented in this paper, regularly distributed in scanning region determined after the optimization problem, and by representing the target reflectivity by means of the PSWF's. It's worth noting that in all the three examined cases the target reflectivity has been determined from the field data by means of a truncated SVD-based inversion, with a truncation threshold comparable with the noise level.

Figure 10 shows the retrieved reflectivity in case A (blue solid line), in case B (cyan solid line), in case C (black solid line) and the reference (red dashed line).
Fig. 10 Full resolution reconstruction of a single wire target reflectivity, numerically simulated. Blue: case A. Cyan: case B. Black: case C. Red: reference.

The result in case A shows that the use of about 80 AEs, half a wavelength spaced on a scanning region determined by the proposed approach is redundant since it retrieves a target reflectivity very close to the reference, providing essentially the same results achievable using 46 AEs properly spaced on the same scanning region, as shown in Fig. 8. Furthermore it shows also that the dimension of the scanning region determined by the proposed approach identifies correctly the region containing the complete information to retrieve the target reflectivity with the desired (full) resolution. Obviously with a standard approach the value of Hx is generally unknown, and is typically determined so to keep the energy of the undetected field under a prefixed threshold.

The result in case B shows only a small worsening of the retrieving, if compared to the result in case A and to the reference, due to the different representation of the target reflectivity. This outcome is expected: in case B the same amount of information is exploited as in case A, and the inversion process is regularized the same way.

6.2 Experimental results

We are going now to describe the measurement set-up adopted throughout the experimental validation.

Scattering measurements at a frequency of 15GHz have been acquired, while the scattering scenario has been realized by copper tapes, vertically glued on polystyrene blocks, transparent in the frequency band at hand, see Fig. 11.
Fig. 11 Pictorial view of parts of the experimental setup.
As in the numerical analysis, the copper tapes length has been chosen significantly larger than the working wavelength, and than d, the distance between the target and the scanning region, so to simulate a 2D scenario. Accordingly the tapes has been taken 1.6m long and 2.5cm or 4cm wide, depending on the considered test case, to reproduce 2D scatterers with size comparable to λ.

The scattering scenario is illuminated, in a multi-static/single-view configuration, by a horn antenna located sufficiently far so to make the impinging field a canonical waveform, a plane wave on the investigated region. The scattered field is measured in the forward direction by a open-ended waveguide moved, by a linear positioner, in the direction orthogonal to the tapes and at a quota approximately equal to their middle.

Four different test-cases have been considered. In the first one, for fixed H'x and d, the AEs positions have been constrained to fully occupy the observation domain while optimizing the functional Φ. In other words, Hx is also fixed. In the second one, for fixed H'x and d, the AEs positions have no such a constraint and they are free to reside on the most convenient observation domain to optimize Φ. In other words, Hx is determined by the optimization process. The third test-case illustrates the multi-resolution capabilities of the approach, namely, how it is possible to deal with different resolutions for different parts of the investigation domain, see Fig. 1. Finally, in the fourth case, the dynamic resolution capabilities are highlighted, i.e., the possibility of appending further AEs whose positions are optimized to improve the resolution of the “scene”, see Fig. 1.

The parameter Ω will be expressed in terms of resolution Θ as Ω = π/Θ.

6.2.1 Constrained AEs positions

In this Subsection, we deal with H'x = 10λ and d = 28λ, while the AEs positions are constrained to belong to an observation domain with Hx = 20λ. Four different scattering configurations have been considered, one with no resolution limitation, and the others with resolution limited to Θ = 3λ, Θ = 5λ and Θ = 7λ.

Figure 12 shows the behavior of Φopt(N) for the different configurations from which deducing Nopt reported in Table 1.
Fig. 12 Constrained AEs positions. Behavior of Φopt(N) for the different scattering configurations considered in the constrained AEs positions case. Green: no resolution limitation. Red: Θ = 3λ. Blue: Θ = 5λ. Black: Θ = 7λ.

Table 1. Selected Number N of AEs for the Different Limitations on Resolution Considered for the Cases of Constrained and Unconstrained AEs Positions

table-icon
View This Table
As expected, the number of required AEs decreases with increasing resolution limitations. For the sake of brevity we report just the distribution of the AEs locations for the case of Θ = 3λ in Fig. 13.
Fig. 13 AEs positions distribution for Θ = 3λ. Red circles: constrained AEs positions. Blue stars: unconstrained AEs positions.

In order to check whether the actually achievable resolution matches the limitation value, reconstructions with an individual copper tape located at x' = 0λ and with three copper tapes located at x' = −3λ, x' = 0λ and x' = 3λ have been performed, see Figs. 14and 15, respectively.
Fig. 14 Constrained AEs positions. Reconstruction of an individual copper tape located at x' = 0λ. Green: no resolution limitation. Red: Θ = 3λ. Blue: Θ = 5λ. Black: Θ = 7λ.
Fig. 15 Constrained AEs positions. Reconstruction of three copper tapes located at x' = −3λ, x' = 0λ and x' = 3λ. Green: no resolution limitation. Red: Θ = 3λ. Blue: Θ = 5λ. Black: Θ = 7λ.
As it can be seen from Fig. 14, the widths of all the main lobes correctly match the expected resolutions. This is further confirmed by Fig. 11 highlighting that the three different copper tapes are not resolved in the limited resolution cases. Finally, it is worth underlining that the reconstructions in Fig. 14 compares satisfactory to analogous reconstructions obtained on numerical data simulating the scattering by a wire target located at x' = 0λ, see Fig. 16.
Fig. 16 Constrained AEs positions. Reconstruction of a simulated wire target located at x' = 0λ. Green: no resolution limitation. Red: Θ = 3λ. Blue: Θ = 5λ. Black: Θ = 7λ.

6.2.2 Unconstrained AEs positions

As for the foregoing Subsection, we deal with H'x = 10λ and d = 28λ, while the AEs locations are free to line up on an observation domain with unfixed extent Hx. The same four scattering configurations as before have been considered. In this way the approach is able to provide also the dimensions of the scanning region from which all the essential information of interest, for the desired degree of resolution, should be extracted.

Figure 17 shows the behavior of Φopt(N) for the different configurations from which deducing Nopt, again reported in Table 1.
Fig. 17 Unconstrained AEs positions. Behavior of Φopt(N) for the different scattering configurations considered in the unconstrained AEs positions case. Green: no resolution limitation. Red: Θ = 3λ. Blue: Θ = 5λ. Black: Θ = 7λ.
As expected, the number of required AEs decreases with increasing resolution limitations as before. We also observe that this number is smaller than the corresponding number for the constrained case and that, generally speaking, the values of the extent Hx of observation domain have been smaller than the value of 20λ previously considered, as also testified by Fig. 13.

Again to check whether the actually achievable resolution matches the limitation value, reconstructions with three copper tapes located at x' = −3λ, x' = 0λ and x' = 3λ have been performed, see Fig. 18.
Fig. 18 Unconstrained AEs positions. Reconstruction of three copper tapes located at x' = −3λ, x' = 0λ and x' = 3λ. Green: no resolution limitation. Red: Θ = 3λ. Blue: Θ = 5λ. Black: Θ = 7λ.
As it can be seen, the results meet our expectations.

6.2.3 Multi-resolution

The results depicted in this Subsection are aimed at showing the multi-resolution capability of the approach.

To this end, we have considered, as a first experimental test-case, a disconnected investigation domain composed by two segments, each 14λ wide and with center-to-center distance of 20λ. The distance between the investigation and the observation domains is again d = 28λ. On one of the two subdomains, the required resolution is Θ = 5λ, while that required for the other is Θ = 2λ. Figures 19 and 20 depict the distribution of the AEs positions and the inter-sample spacings for the considered test case, respectively.
Fig. 19 Distribution of the AEs positions for the experimental multi-resolution case.
Fig. 20 Inter-element spacing for the experimental multi-resolution case.
As it can be seen, the achieved spacings are significantly different from the usually considered λ/2 one. Moreover, four metallic tapes have been considered for this case and located at x' = −10λ, x' = −6λ, x' = 6λ and x' = 10λ, as in Fig. 11. In this way, the first two tapes fall in the first subdomain with 5λ resolution, while the second two tapes fall in the second subdomain with Θ = 2λ resolution. Due to the limitations of the linear scanner involved in the experimental setup, to avoid obtaining an exceedingly large observation domain, the AEs positions have been constrained to belong to a measurement region with Hx = 30λ. As it can be seen from Fig. 21 and as expected, the first two tapes are not resolved, while the second copper tapes do.
Fig. 21 Multi-resolution. Reconstruction of the four copper tapes illustrated in Fig. 11 and located at x' = −10λ, x' = −6λ, x' = 6λ, and x' = 10λ. Red curve: reconstruction of the first two copper tapes on the first subdomain. Blue curve: reconstruction of the second two copper tapes on the second subdomain.
Once again, the result in Fig. 21 satisfactorily matches a simulated reference, see Fig. 22.
Fig. 22 Multi-resolution. Reconstruction of the four simulated point targets located at x' = −10λ, x' = −6λ, x' = 6λ, and x' = 10λ. Red curve: reconstruction of the first two point targets on the first subdomain. Blue curve: reconstruction of the second two point targets on the second subdomain.

The second test-case considers a disconnected investigation domain composed by two segments, each 14λ wide and with center-to-center distance of 20λ. Monostatic multi-view simulated data have been considered, typical of interest in SAR tomography for remote sensing applications, with a distance between the investigation and observation domains d = 103λ. On one of the two subdomains, the required resolution is Θ = 8λ, while that required for the other is Θ = 2λ. In Fig. 23, the distribution of the AEs locations is depicted and, as it can be seen, the sampling step is smaller in correspondence to the region where the better resolution is required.
Fig. 23 Distribution of the AEs positions for the numerical multi-resolution case.
Finally, in Fig. 24, the successful reconstructions of step-like profiles is shown.
Fig. 24 Multi-resolution. Profile reconstruction for Θ = 2λ and Θ = 8λ. Blue: reference profile. Black: reconstructed profile.

6.4 Dynamic multi-resolution

The last test case of our interest presented in this Subsection concerns showing the dynamic multi-resolution capability of the approach.

To this end, we have considered a (connected) investigation domain with H'x = 10λ and d = 28λ and unconstrained AEs locations. We have then performed a first acquisition, say a “low-resolution” acquisition, by limiting the resolution to Θ = 7λ, on the AEs positions illustrated in Fig. 25.
Fig. 25 Dynamic multi-resolution. Blue crosses: Distribution of the AEs positions for Θ = 7λ. Red circles: added AEs positions to achieve a Θ = 2λ.
Figure 26 (red curve) shows the reconstruction of an individual copper tape located at x' = 0λ.
Fig. 26 Dynamic multi-resolution. Reconstruction of an individual copper tape located at x' = 0λ. Blue curve: “low-resolution” limited to 7λ. Red curve: Θ = 2λ with further optimized AEs positions. Green curve: reconstruction obtained by dismissing the original AEs positions of the “low-resolution” case.
Subsequently, by fixing the “low-resolution” AEs locations, we have optimized further 24 AEs locations by the proposed approach, again illustrated in Fig. 25 (red circles), by limiting the resolution to Θ = 2λ. The new reconstruction of the individual tape occurs with the desired resolution. Relevantly, from Fig. 26, it can be seen how, by dismissing the original AEs corresponding to the “low-resolution” case, the reconstruction significantly worsens.

7. Conclusions

An approach for the optimal determination of the scanning region dimension, of the number of sampling points therein, and their spatial distribution, for imaging applications has been proposed. The method recasts the reconstruction of the target reflectivity from the field data on the scanning region in terms of a finite dimensional algebraic linear inverse problem. The number and the positions are optimally determined by optimizing the singular value behavior of the matrix defining the linear operator. The method can be extended to more complex modeling of the reflecting behavior of the target, of the probe response and of the illuminating configuration as long as the linear relationship between the unknowns and the data is preserved. Furthermore it can be extended to the case of non planar targets and/or scanning domains.

The performance of the method has been evaluated via numerical and experimental analysis.

Extensions to the case of partially coherent or fully incoherent radiation, of interest in the case of passive systems, can be pursued according to the guidelines in [23

23. A. Capozzoli, C. Curcio, and A. Liseno, “Experimental field reconstruction of incoherent sources,” Prog. Electromagn. Res. B 47, 219–239 (2013).

].

GPU computing can be used when very large investigation domains are considered [11

11. A. Capozzoli, C. Curcio, and A. Liseno, “Multi-frequency planar near-field scanning by means of singular-value decomposition (SVD) optimization,” IEEE Antennas Propag. Mag. 53, 212–221 (2011). [CrossRef]

].

References and links

1.

A. Reigber and A. Moreira, “First demonstration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38(5), 2142–2152 (2000). [CrossRef]

2.

A. Capozzoli, G. D’Elia, A. Liseno, P. Vinetti, M. Nannini, A. Reigber, R. Scheiber, and V. Severino, “SAR tomography with optimized constellation and its application to forested scenes,” Atti Fondazione Giorgio Ronchi LXV, 367–375 (2010).

3.

A. Capozzoli, C. Curcio, and A. Liseno, “SAR tomography with optimized track distribution and controlled resolution, ” in Proceedings of the XIX Riunione Nazionale di Elettromagnetismo, (Roma, Italy, Sept. 10–14, 2012), pp. 174–177.

4.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, 1998).

5.

M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser Jr, and A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008). [CrossRef] [PubMed]

6.

L. Zhang, Y. Hao, C. G. Parini, and J. Dupuy, “An investigation of antenna element spacing on the quality of the millimetre wave imaging,” in Proceedings of the IEEE Antennas Prop.Int. Symp. (San Diego, CA, Jul. 5–11, 2008), pp. 1–4.

7.

S. Yeom, D.-S. Lee, J.-Y. Son, M.-K. Jung, Y. S. Jang, S.-W. Jung, and S.-J. Lee, “Real-time outdoor concealed-object detection with passive millimeter wave imaging,” Opt. Express 19(3), 2530–2536 (2011). [CrossRef] [PubMed]

8.

B. Recur, A. Younus, S. Salort, P. Mounaix, B. Chassagne, P. Desbarats, J.-P. Caumes, and E. Abraham, “Investigation on reconstruction methods applied to 3D terahertz computed tomography,” Opt. Express 19(6), 5105–5117 (2011). [CrossRef] [PubMed]

9.

F. Qi, I. Ocket, D. Schreurs, and B. Nauwelaers, “A system-level simulator for indoor mmW SAR imaging and its applications,” Opt. Express 20(21), 23811–23820 (2012). [CrossRef] [PubMed]

10.

A. Capozzoli, C. Curcio, A. Liseno, and P. Vinetti, “Field sampling and field reconstruction: a new perspective,” Radio Sci. 45, RS6004 (2010). [CrossRef]

11.

A. Capozzoli, C. Curcio, and A. Liseno, “Multi-frequency planar near-field scanning by means of singular-value decomposition (SVD) optimization,” IEEE Antennas Propag. Mag. 53, 212–221 (2011). [CrossRef]

12.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

13.

T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields (IEEE, 1999).

14.

G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13(9), 1816–1826 (1996). [CrossRef]

15.

P. C. Clemmow, The Plane-Wave Spectrum Representation of Electromagnetic Fields (IEEE, 1996).

16.

O. M. Bucci, A. Capozzoli, and G. D’Elia, “Regularizing strategy for image restoration and wave-front sensing by phase diversity,” J. Opt. Soc. Am. A 16(7), 1759–1768 (1999). [CrossRef]

17.

B. R. Frieden and E. Wolf, eds., “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics 9 (North-Holland, 1971), pp. 311–407.

18.

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: The dimension of essentially time- and band-limited signals,” Bell Syst. Tech. J. 41(4), 1295–1336 (1962). [CrossRef]

19.

A. Capozzoli, C. Curcio, G. D’Elia, and A. Liseno, “Phaseless antenna characterization by effective aperture field and data representations,” IEEE Trans. Antennas Propag. 57(1), 215–230 (2009). [CrossRef]

20.

G. D. de Villiers, F. B. T. Marchaud, and E. R. Pike, “Generalized Gaussian quadrature applied to an inverse problem in antenna theory: II. The two-dimensional case with circular symmetry,” Inverse Probl. 19(3), 755–778 (2003). [CrossRef]

21.

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7(2), 163–165 (1973). [CrossRef]

22.

A. Capozzoli and G. D’Elia, “Global optimization and antennas synthesis and diagnosis, part one: concepts, tools, strategies and performances,” Prog. Electromagn. Res. 56, 195–232 (2006).

23.

A. Capozzoli, C. Curcio, and A. Liseno, “Experimental field reconstruction of incoherent sources,” Prog. Electromagn. Res. B 47, 219–239 (2013).

OCIS Codes
(110.0110) Imaging systems : Imaging systems
(280.0280) Remote sensing and sensors : Remote sensing and sensors

ToC Category:
Imaging Systems

History
Original Manuscript: November 13, 2013
Revised Manuscript: December 23, 2013
Manuscript Accepted: January 7, 2014
Published: April 21, 2014

Citation
Amedeo Capozzoli, Claudio Curcio, and Angelo Liseno, "Multi-resolution imaging with an optimized number and distribution of sampling points," Opt. Express 22, 10119-10138 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10119


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References

  1. A. Reigber, A. Moreira, “First demonstration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38(5), 2142–2152 (2000). [CrossRef]
  2. A. Capozzoli, G. D’Elia, A. Liseno, P. Vinetti, M. Nannini, A. Reigber, R. Scheiber, V. Severino, “SAR tomography with optimized constellation and its application to forested scenes,” Atti Fondazione Giorgio Ronchi LXV, 367–375 (2010).
  3. A. Capozzoli, C. Curcio, A. Liseno, “SAR tomography with optimized track distribution and controlled resolution, ” in Proceedings of the XIX Riunione Nazionale di Elettromagnetismo, (Roma, Italy, Sept. 10–14, 2012), pp. 174–177.
  4. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, 1998).
  5. M. R. Fetterman, J. Grata, G. Jubic, W. L. Kiser, A. Visnansky, “Simulation, acquisition and analysis of passive millimeter-wave images in remote sensing applications,” Opt. Express 16(25), 20503–20515 (2008). [CrossRef] [PubMed]
  6. L. Zhang, Y. Hao, C. G. Parini, and J. Dupuy, “An investigation of antenna element spacing on the quality of the millimetre wave imaging,” in Proceedings of the IEEE Antennas Prop.Int. Symp. (San Diego, CA, Jul. 5–11, 2008), pp. 1–4.
  7. S. Yeom, D.-S. Lee, J.-Y. Son, M.-K. Jung, Y. S. Jang, S.-W. Jung, S.-J. Lee, “Real-time outdoor concealed-object detection with passive millimeter wave imaging,” Opt. Express 19(3), 2530–2536 (2011). [CrossRef] [PubMed]
  8. B. Recur, A. Younus, S. Salort, P. Mounaix, B. Chassagne, P. Desbarats, J.-P. Caumes, E. Abraham, “Investigation on reconstruction methods applied to 3D terahertz computed tomography,” Opt. Express 19(6), 5105–5117 (2011). [CrossRef] [PubMed]
  9. F. Qi, I. Ocket, D. Schreurs, B. Nauwelaers, “A system-level simulator for indoor mmW SAR imaging and its applications,” Opt. Express 20(21), 23811–23820 (2012). [CrossRef] [PubMed]
  10. A. Capozzoli, C. Curcio, A. Liseno, P. Vinetti, “Field sampling and field reconstruction: a new perspective,” Radio Sci. 45, RS6004 (2010). [CrossRef]
  11. A. Capozzoli, C. Curcio, A. Liseno, “Multi-frequency planar near-field scanning by means of singular-value decomposition (SVD) optimization,” IEEE Antennas Propag. Mag. 53, 212–221 (2011). [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  13. T. B. Hansen and A. D. Yaghjian, Plane-Wave Theory of Time-Domain Fields (IEEE, 1999).
  14. G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13(9), 1816–1826 (1996). [CrossRef]
  15. P. C. Clemmow, The Plane-Wave Spectrum Representation of Electromagnetic Fields (IEEE, 1996).
  16. O. M. Bucci, A. Capozzoli, G. D’Elia, “Regularizing strategy for image restoration and wave-front sensing by phase diversity,” J. Opt. Soc. Am. A 16(7), 1759–1768 (1999). [CrossRef]
  17. B. R. Frieden and E. Wolf, eds., “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics 9 (North-Holland, 1971), pp. 311–407.
  18. H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III: The dimension of essentially time- and band-limited signals,” Bell Syst. Tech. J. 41(4), 1295–1336 (1962). [CrossRef]
  19. A. Capozzoli, C. Curcio, G. D’Elia, A. Liseno, “Phaseless antenna characterization by effective aperture field and data representations,” IEEE Trans. Antennas Propag. 57(1), 215–230 (2009). [CrossRef]
  20. G. D. de Villiers, F. B. T. Marchaud, E. R. Pike, “Generalized Gaussian quadrature applied to an inverse problem in antenna theory: II. The two-dimensional case with circular symmetry,” Inverse Probl. 19(3), 755–778 (2003). [CrossRef]
  21. F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7(2), 163–165 (1973). [CrossRef]
  22. A. Capozzoli, G. D’Elia, “Global optimization and antennas synthesis and diagnosis, part one: concepts, tools, strategies and performances,” Prog. Electromagn. Res. 56, 195–232 (2006).
  23. A. Capozzoli, C. Curcio, A. Liseno, “Experimental field reconstruction of incoherent sources,” Prog. Electromagn. Res. B 47, 219–239 (2013).

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