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

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References

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  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.
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    [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.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [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).
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    [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).

2013

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

2012

2011

2010

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).

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

2009

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]

2008

2006

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).

2003

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]

2000

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]

1999

1996

1973

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

1962

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]

Abraham, E.

Bucci, O. M.

Capozzoli, A.

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

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]

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

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).

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]

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).

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]

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.

Caumes, J.-P.

Chassagne, B.

Curcio, C.

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

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]

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

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]

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.

D’Elia, G.

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).

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]

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).

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]

de Villiers, G. D.

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]

Desbarats, P.

Fetterman, M. R.

Forbes, G. W.

Gori, F.

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

Grata, J.

Guattari, G.

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

Jang, Y. S.

Jubic, G.

Jung, M.-K.

Jung, S.-W.

Kiser, W. L.

Landau, H. J.

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]

Lee, D.-S.

Lee, S.-J.

Liseno, A.

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

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]

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

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).

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]

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.

Marchaud, F. B. T.

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]

Moreira, A.

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]

Mounaix, P.

Nannini, M.

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).

Nauwelaers, B.

Ocket, I.

Pike, E. R.

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]

Pollak, H. O.

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]

Qi, F.

Recur, B.

Reigber, A.

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).

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]

Salort, S.

Scheiber, R.

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).

Schreurs, D.

Severino, V.

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).

Son, J.-Y.

Vinetti, P.

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).

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

Visnansky, A.

Yeom, S.

Younus, A.

Atti Fondazione Giorgio Ronchi

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).

Bell Syst. Tech. J.

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]

IEEE Antennas Propag. Mag.

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]

IEEE Trans. Antennas Propag.

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]

IEEE Trans. Geosci. Remote Sens.

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]

Inverse Probl.

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]

J. Opt. Soc. Am. A

Opt. Commun.

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

Opt. Express

Prog. Electromagn. Res.

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).

Prog. Electromagn. Res. B

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

Radio Sci.

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

Other

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.

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.

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

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

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

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

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.

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Figures (26)

Fig. 1
Fig. 1

a) Variable resolution; b) Multi-resolution; c) Dynamic multi-resolution.

Fig. 2
Fig. 2

Geometry of the problem.

Fig. 3
Fig. 3

Supports of the { Ψ m } m = 1 M 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.

Fig. 4
Fig. 4

Flow Chart of the algorithm to determine N.

Fig. 5
Fig. 5

Φopt as a function of N: the saturation knee is highlighted together with the corresponding value of Nopt.

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.

Fig. 7
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.

Fig. 8
Fig. 8

AEs positions distribution determined in the case of a full resolution retrieving of a single wire target numerically simulated.

Fig. 9
Fig. 9

Full resolution reconstruction of a single wire target reflectivity, numerically simulated. Blue: proposed approach. Red: reference.

Fig. 10
Fig. 10

Full resolution reconstruction of a single wire target reflectivity, numerically simulated. Blue: case A. Cyan: case B. Black: case C. Red: reference.

Fig. 11
Fig. 11

Pictorial view of parts of the experimental setup.

Fig. 12
Fig. 12

Constrained AEs positions. Behavior of Φ o p t ( 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.

Fig. 13
Fig. 13

AEs positions distribution for Θ = 3λ. Red circles: constrained AEs positions. Blue stars: unconstrained AEs positions.

Fig. 14
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
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.

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.

Fig. 17
Fig. 17

Unconstrained AEs positions. Behavior of Φ o p t ( 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.

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.

Fig. 19
Fig. 19

Distribution of the AEs positions for the experimental multi-resolution case.

Fig. 20
Fig. 20

Inter-element spacing for the experimental multi-resolution case.

Fig. 21
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.

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.

Fig. 23
Fig. 23

Distribution of the AEs positions for the numerical multi-resolution case.

Fig. 24
Fig. 24

Multi-resolution. Profile reconstruction for Θ = 2λ and Θ = 8λ. Blue: reference profile. Black: reconstructed profile.

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.

Fig. 26
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.

Tables (1)

Tables Icon

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

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

R ¯ (x,y,z; x , y ,z')=(x x ) x ^ +(y y ) y ^ +(zz') z ^ ,
G(x,y,z; x , y ,z')= exp(jkR(x,y,z; x , y ,z')) R(x,y,z; x , y ,z') .
s(x,y)= 1 2π O d x d y u( x , y )[ z G(x,y,z; x , y ,z') ] z=d,z'=0 ,
u( x , y )=ρ( x , y ) exp(jkR( x 0 , y 0 , z 0 ; x , y ,0)) R( x 0 , y 0 , z 0 ; x , y ,0) .
s(x,y)= 1 4 π 2 + + d k x d k y exp[j( k x x+ k y y)]exp[j k z d] O d x d y u( x , y )exp[j( k x x + k y y )],
A 1 |ρ A 1 (ρ)=s.
s ˜ (x,y) 1 jλ d 2 O d x d y ρ ˜ ( x , y )exp[ j 2π λd ((x+ x 0 ) x +(y+ y 0 ) y ) ] ,
s ˜ (x,y)=exp(j2kd)exp[ j k 2d ( x 2 + x 0 2 + y 2 + y 0 2 ) ]s(x,y),
ρ ˜ ( x , y )=ρ( x , y )exp[ j k d ( x 2 + y 2 ) ],
A 2 | ρ ˜ A 2 ( ρ ˜ )= s ˜ ,
s ˜ (x,y)= O d x d y ρ ˜ ( x , y )exp[ j 4π λd (x x +y y ) ] ,
V={ ( k x , k y )| k x 2 + k y 2 k 2 },
Ψ m( q x , q y ) [ c x , c y ,x,y]= Π q x [ c x ,x] Π q y [ c y ,y],
ρ v ( x , y )= m=1 M r m Ψ m [ c x , c y , x , y ] ,
A(ρ)=s.
S nm =<A( Ψ m ),δ(x x n ,y y n ))>,
s ¯ = S ¯ ¯ (N; p ¯ ) r ¯ ,
Φ= s=1 S σ s σ 1 ,
{ x m = h ( ξ m , η m ) y m = g ( ξ m , η m ) ,
{ h ( ξ m , η m ) = r = 1 R t = 1 T h r t L r ( ξ m ) L s ( η m ) g ( ξ m , η m ) = r = 1 R t = 1 T g r t L r ( ξ m ) L s ( η m ) ,
M Ω |ρ ρ .

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