Abstract

A signal-subspace method is derived for the localization and imaging of unknown scatterers using intensity-only wave field data (lacking field phase information). The method is an extension of the time-reversal multiple-signal-classification imaging approach to intensity-only data. Of importance, the derived methodology works within exact scattering theory including multiple scattering.

© 2007 Optical Society of America

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References

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  1. G. Gbur and E. Wolf, "Diffraction tomography without phase information," Opt. Lett. 27, 1890-1892 (2002).
    [CrossRef]
  2. G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. M. H. Maleki, A. J. Devaney, and A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 10, 1356-1363 (1992).
    [CrossRef]
  8. M. H. Maleki and A. J. Devaney, "Phase retrieval and intensity-only reconstruction algorithms for optical diffraction tomography," J. Opt. Soc. Am. A 10, 1086-1092 (1993).
    [CrossRef]
  9. O. M. Bucci, L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on open lines," J. Opt. Soc. Am. A 23, 2566-2577 (2006).
    [CrossRef]
  10. L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on a closed curve," J. Opt. Soc. Am. A 21, 622-631 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the memetic algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2753 (2003).
    [CrossRef]
  14. H. Lev-Ari and A. J. Devaney, "The time-reversal technique re-interpreted: Subspace-based signal processing for multi-static target location," Proc. of the 2000 IEEE Sensor Array and Multichannel Signal Processing Workshop (IEEE, 2000)509-513.
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    [CrossRef] [PubMed]
  16. S. K. Lehman and A. J. Devaney, "Transmission mode time-reversal super-resolution imaging," J. Acoust. Soc. Am. 113, 2742-2753 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  18. A. J. Devaney, E. A. Marengo, and F. K. Gruber, "Time-reversal-based imaging and inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 118, 3129-3138 (2005).
    [CrossRef]
  19. A. Kirsch, "The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media," Inverse Probl. 18, 1025-1040 (2002).
    [CrossRef]
  20. S. Hou, K. Solna, and H. Zhao, "Imaging of location and geometry for extended targets using the response matrix," J. Comput. Phys. 199, 317-338 (2004).
    [CrossRef]
  21. S. Hou, K. Solna, and H. Zhao, "A direct imaging algorithm for extended targets," Inverse Probl. 22, 1151-1178 (2006).
    [CrossRef]
  22. S. Hou, K. Solna, and H. Zhao, "A direct imaging method using far field data," Inverse Probl. 23, 1533-1546 (2007).
    [CrossRef]
  23. E. A. Marengo, F. K. Gruber, and F. Simonetti, "Time-reversal MUSIC imaging of extended targets," IEEE Trans. Image Process. 16, 1967-1984 (2007).
    [CrossRef] [PubMed]
  24. F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory (Springer, 2006).
  25. R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory (Chapman & Hall/CRC, 2001).
    [CrossRef]
  26. R. O. Schmidt, "Multiple emitter location and signal parameter estimation," IEEE Trans. Antennas Propag. 34, 276-280 (1986).
    [CrossRef]
  27. C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice Hall, 1992).
  28. E. A. Marengo and F. K. Gruber, "Subspace-based localization and inverse scattering of multiply scattering point targets," EURASIP J. Advances in Signal Processing 2007, 17342, 16 pp. (2007).
  29. J. H. Taylor, Scattering Theory (Wiley, 1972).
  30. R. Pierri and A. Tamburrino, "On the local minima problem in conductivity imaging via a quadratic approach," Inverse Probl. 13, 1547-1568 (1997).
    [CrossRef]
  31. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).
  32. A. J. Devaney, "Super-resolution processing of multi-static data using time-reversal and MUSIC," Northeastern University, 2000-year report available at http://www.ece.neu.edu/faculty/devaney/ajd/preprints.htm.
  33. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
    [CrossRef]
  34. R. Pierri and F. Soldovieri, "On the information content of the radiated fields in the near zone over bounded domains," Inverse Probl. 14, 321-337 (1998).
    [CrossRef]
  35. D. A. B. Miller, "Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths," Appl. Opt. 39, 1681-1699 (2000).
    [CrossRef]
  36. P. C. Hansen, Rank Deficient and Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
    [CrossRef]
  37. M. Tanter, J.-L. Thomas, and M. Fink, "Time reversal and the inverse filter," J. Acoust. Soc. Am. 108, 223-234 (2000).
    [CrossRef]
  38. E. A. Marengo, "Further theoretical considerations for time-reversal MUSIC imaging of extended scatterers," invited paper, in IEEE Statistical Signal Processing Workshop 2007, Madison, Wisconsin, USA (IEEE, 2007), pp. 304-306. ISBN: 978-1-4244-1198-6.
    [CrossRef]
  39. L. L. Foldy, "The multiple scattering of waves: I. General theory of isotropic scattering by randomly distributed scatterers," Phys. Rev. 67, 107-119 (1945).
    [CrossRef]
  40. M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287-310 (1951).
    [CrossRef]
  41. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).
  42. L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
    [CrossRef]
  43. R. K. Snieder and J. A. Scales, "Time-reversed imaging as a diagnostic of wave and particle chaos," Phys. Rev. E 58, 5668-5675 (1998).
    [CrossRef]
  44. E. A. Marengo and F. K. Gruber, "Non-iterative analytical formula for inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 120, 3782-3788 (2006).
    [CrossRef]

2007 (3)

S. Hou, K. Solna, and H. Zhao, "A direct imaging method using far field data," Inverse Probl. 23, 1533-1546 (2007).
[CrossRef]

E. A. Marengo, F. K. Gruber, and F. Simonetti, "Time-reversal MUSIC imaging of extended targets," IEEE Trans. Image Process. 16, 1967-1984 (2007).
[CrossRef] [PubMed]

E. A. Marengo and F. K. Gruber, "Subspace-based localization and inverse scattering of multiply scattering point targets," EURASIP J. Advances in Signal Processing 2007, 17342, 16 pp. (2007).

2006 (4)

S. Hou, K. Solna, and H. Zhao, "A direct imaging algorithm for extended targets," Inverse Probl. 22, 1151-1178 (2006).
[CrossRef]

E. A. Marengo and F. K. Gruber, "Non-iterative analytical formula for inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 120, 3782-3788 (2006).
[CrossRef]

O. M. Bucci, L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on open lines," J. Opt. Soc. Am. A 23, 2566-2577 (2006).
[CrossRef]

A. Litman and K. Belkebir, "Two-dimensional inverse profiling problem using phaseless data," J. Opt. Soc. Am. A 23, 2737-2746 (2006).
[CrossRef]

2005 (3)

2004 (4)

S. Hou, K. Solna, and H. Zhao, "Imaging of location and geometry for extended targets using the response matrix," J. Comput. Phys. 199, 317-338 (2004).
[CrossRef]

F. K. Gruber, E. A. Marengo, and A. J. Devaney, "Time-reversal imaging with multiple signal classification considering multiple scattering between the targets," J. Acoust. Soc. Am. 115, 3042-3047 (2004).
[CrossRef]

G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).
[CrossRef]

L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on a closed curve," J. Opt. Soc. Am. A 21, 622-631 (2004).
[CrossRef]

2003 (3)

S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the memetic algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2753 (2003).
[CrossRef]

C. Prada and J. L. Thomas, "Experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix," J. Acoust. Soc. Am. 114, 235-243 (2003).
[CrossRef] [PubMed]

S. K. Lehman and A. J. Devaney, "Transmission mode time-reversal super-resolution imaging," J. Acoust. Soc. Am. 113, 2742-2753 (2003).
[CrossRef] [PubMed]

2002 (3)

2000 (2)

1998 (2)

R. K. Snieder and J. A. Scales, "Time-reversed imaging as a diagnostic of wave and particle chaos," Phys. Rev. E 58, 5668-5675 (1998).
[CrossRef]

R. Pierri and F. Soldovieri, "On the information content of the radiated fields in the near zone over bounded domains," Inverse Probl. 14, 321-337 (1998).
[CrossRef]

1997 (2)

R. Pierri and A. Tamburrino, "On the local minima problem in conductivity imaging via a quadratic approach," Inverse Probl. 13, 1547-1568 (1997).
[CrossRef]

T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

1993 (1)

1992 (2)

A. J. Devaney, "Diffraction tomographic reconstruction from intensity data," IEEE Trans. Image Process. 1, 221-228 (1992).
[CrossRef] [PubMed]

M. H. Maleki, A. J. Devaney, and A. Schatzberg, "Tomographic reconstruction from optical scattered intensities," J. Opt. Soc. Am. A 10, 1356-1363 (1992).
[CrossRef]

1986 (1)

R. O. Schmidt, "Multiple emitter location and signal parameter estimation," IEEE Trans. Antennas Propag. 34, 276-280 (1986).
[CrossRef]

1951 (1)

M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287-310 (1951).
[CrossRef]

1945 (1)

L. L. Foldy, "The multiple scattering of waves: I. General theory of isotropic scattering by randomly distributed scatterers," Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Appl. Opt. (1)

EURASIP J. Advances in Signal Processing (1)

E. A. Marengo and F. K. Gruber, "Subspace-based localization and inverse scattering of multiply scattering point targets," EURASIP J. Advances in Signal Processing 2007, 17342, 16 pp. (2007).

IEEE Trans. Antennas Propag. (1)

R. O. Schmidt, "Multiple emitter location and signal parameter estimation," IEEE Trans. Antennas Propag. 34, 276-280 (1986).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo, "Electromagnetic detection of dielectric scatterers using phaseless synthetic and real data and the memetic algorithm," IEEE Trans. Geosci. Remote Sens. 41, 2745-2753 (2003).
[CrossRef]

IEEE Trans. Image Process. (2)

A. J. Devaney, "Diffraction tomographic reconstruction from intensity data," IEEE Trans. Image Process. 1, 221-228 (1992).
[CrossRef] [PubMed]

E. A. Marengo, F. K. Gruber, and F. Simonetti, "Time-reversal MUSIC imaging of extended targets," IEEE Trans. Image Process. 16, 1967-1984 (2007).
[CrossRef] [PubMed]

Inf. Sci. (N.Y.) (1)

G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).
[CrossRef]

Inverse Probl. (5)

A. Kirsch, "The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media," Inverse Probl. 18, 1025-1040 (2002).
[CrossRef]

S. Hou, K. Solna, and H. Zhao, "A direct imaging algorithm for extended targets," Inverse Probl. 22, 1151-1178 (2006).
[CrossRef]

S. Hou, K. Solna, and H. Zhao, "A direct imaging method using far field data," Inverse Probl. 23, 1533-1546 (2007).
[CrossRef]

R. Pierri and F. Soldovieri, "On the information content of the radiated fields in the near zone over bounded domains," Inverse Probl. 14, 321-337 (1998).
[CrossRef]

R. Pierri and A. Tamburrino, "On the local minima problem in conductivity imaging via a quadratic approach," Inverse Probl. 13, 1547-1568 (1997).
[CrossRef]

J. Acoust. Soc. Am. (6)

C. Prada and J. L. Thomas, "Experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix," J. Acoust. Soc. Am. 114, 235-243 (2003).
[CrossRef] [PubMed]

S. K. Lehman and A. J. Devaney, "Transmission mode time-reversal super-resolution imaging," J. Acoust. Soc. Am. 113, 2742-2753 (2003).
[CrossRef] [PubMed]

F. K. Gruber, E. A. Marengo, and A. J. Devaney, "Time-reversal imaging with multiple signal classification considering multiple scattering between the targets," J. Acoust. Soc. Am. 115, 3042-3047 (2004).
[CrossRef]

A. J. Devaney, E. A. Marengo, and F. K. Gruber, "Time-reversal-based imaging and inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 118, 3129-3138 (2005).
[CrossRef]

M. Tanter, J.-L. Thomas, and M. Fink, "Time reversal and the inverse filter," J. Acoust. Soc. Am. 108, 223-234 (2000).
[CrossRef]

E. A. Marengo and F. K. Gruber, "Non-iterative analytical formula for inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am. 120, 3782-3788 (2006).
[CrossRef]

J. Comput. Phys. (1)

S. Hou, K. Solna, and H. Zhao, "Imaging of location and geometry for extended targets using the response matrix," J. Comput. Phys. 199, 317-338 (2004).
[CrossRef]

J. Opt. Soc. Am. A (8)

Microwave Opt. Technol. Lett. (1)

T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

L. L. Foldy, "The multiple scattering of waves: I. General theory of isotropic scattering by randomly distributed scatterers," Phys. Rev. 67, 107-119 (1945).
[CrossRef]

Phys. Rev. E (1)

R. K. Snieder and J. A. Scales, "Time-reversed imaging as a diagnostic of wave and particle chaos," Phys. Rev. E 58, 5668-5675 (1998).
[CrossRef]

Rev. Mod. Phys. (1)

M. Lax, "Multiple scattering of waves," Rev. Mod. Phys. 23, 287-310 (1951).
[CrossRef]

Other (12)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).
[CrossRef]

E. A. Marengo, "Further theoretical considerations for time-reversal MUSIC imaging of extended scatterers," invited paper, in IEEE Statistical Signal Processing Workshop 2007, Madison, Wisconsin, USA (IEEE, 2007), pp. 304-306. ISBN: 978-1-4244-1198-6.
[CrossRef]

P. C. Hansen, Rank Deficient and Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998).
[CrossRef]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory (Wiley, 1983).

A. J. Devaney, "Super-resolution processing of multi-static data using time-reversal and MUSIC," Northeastern University, 2000-year report available at http://www.ece.neu.edu/faculty/devaney/ajd/preprints.htm.

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

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory (Springer, 2006).

R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory (Chapman & Hall/CRC, 2001).
[CrossRef]

C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice Hall, 1992).

J. H. Taylor, Scattering Theory (Wiley, 1972).

H. Lev-Ari and A. J. Devaney, "The time-reversal technique re-interpreted: Subspace-based signal processing for multi-static target location," Proc. of the 2000 IEEE Sensor Array and Multichannel Signal Processing Workshop (IEEE, 2000)509-513.

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Figures (19)

Fig. 1
Fig. 1

Schematic of the remote sensing system for intensity-only localization and imaging of pointlike or extended scatterers.

Fig. 2
Fig. 2

Conceptual depiction of the support regions D and D D in R 3 and R 6 spaces, respectively.

Fig. 3
Fig. 3

Indicated diagonal line dividing the support region D D in R 6 corresponds to the subset of that support that is associated with the points ( R , R ) R 6 and their propagators U ( R , R ) used in the noncomputationally intensive version of the intensity-only signal-subspace-based imaging algorithm of this work.

Fig. 4
Fig. 4

Geometry for the first set of pointlike scatterer simulations.

Fig. 5
Fig. 5

Intensity-only pseudospectrum image of the two point targets under perfect data conditions (121 experiments).

Fig. 6
Fig. 6

Singular values of the data matrix for the case of two pointlike scatterers under (a) noiseless conditions and (b) 20 dB SNR.

Fig. 7
Fig. 7

Intensity-only pseudospectrum image of the two point targets under 20 dB SNR (121 experiments).

Fig. 8
Fig. 8

Slices of the pseudospectrum A [ R = ( X , Y , Z ) , R ] under 20 dB SNR corresponding to different values of Z (different constant-Z planes).

Fig. 9
Fig. 9

Minimum target separability (resolution) versus SNR of intensity-and-phase versus intensity-only data conditions, where the latter involves the two variants (intensive and non-intensive). The upper curve corresponds to the case of intensity-only imaging via the noncomputationally intensive signal-subspace-based method of this paper. The middle curve corresponds to the case of intensity-only imaging via the computationally intensive signal-subspace-based method of this paper. As expected, the computationally intensive version of the method outperforms its nonintensive counterpart. Finally, the lower curve corresponds to the case of time-reversal MUSIC imaging using both amplitude and phase data. As expected, it clearly performs better than the approaches based on intensity-only data.

Fig. 10
Fig. 10

Slices of the pseudospectrum A [ R = ( X , Y , Z ) , R ] for a second example under 24 dB SNR corresponding to different values of Z (different constant-Z planes).

Fig. 11
Fig. 11

Pseudospectrum A p [ R = ( X , Y , Z = 0 ) ] with far-field data gathered by placing the aperture in the same configuration as in Fig. 4 but with the aperture plane at Z = 200 instead of at Z = 15 (121 experiments). The peaks in the image correspond to the correct target positions, as desired.

Fig. 12
Fig. 12

Slices of the pseudospectrum A [ R = ( X , Y , Z ) , R ] for a four-pointlike-scatterers case where the scatterers’ positions have been divided between two different planes: one target is located at plane Z = 1.0 and three additional targets at plane Z = 0 under 50 dB SNR. The slices correspond to different values of Z (different constant-Z planes).

Fig. 13
Fig. 13

Singular values of the data matrix for the case of four point-like scatterers under (a) noiseless conditions and (b) 50 dB SNR.

Fig. 14
Fig. 14

Slices of the pseudospectrum A [ R = ( X , Y , Z ) , R ] for a six-pointlike scatterers case where the scatterers’ positions have been divided between two different planes ( Z = 1.0 , Z = 0 ) containing an equal number of scatterers under 50 dB SNR. The slices correspond to different values of Z (different constant-Z planes).

Fig. 15
Fig. 15

Singular values of the data matrix for the case of six pointlike scatterers under (a) noiseless conditions and (b) 50 dB SNR.

Fig. 16
Fig. 16

Imaging array formed by 25 × 25 = 625 elements and a square-loop scatterer whose shape we wish to deduce from scattered field intensity measurements at the array.

Fig. 17
Fig. 17

Intensity-only pseudospectrum image for the square-loop scatterer under perfect data conditions. The sharpest image corresponds to the correct target plane of Z = 0 , as desired.

Fig. 18
Fig. 18

Intensity-only pseudospectrum image for the square-loop scatterer under 50 dB SNR. The sharpest image corresponds to the correct target plane of Z = 0 , as desired.

Fig. 19
Fig. 19

Singular values of the data matrix for the case of the square-loop scatterer under (a) noiseless conditions and (b) 50 dB SNR.

Equations (34)

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( 2 + k 2 ) ψ t inc ( r ) = ρ t ( r ) ,
ρ t ( r ) = n = 1 N T α n , t δ ( r Y n ) ,
ψ t inc ( r ) = n = 1 N T α n , t G ( r , Y n ) ,
G ( r , r ) = exp ( i k r r ) 4 π r r .
( 2 + k 2 ) ψ t ( r ) = V ( r ) ψ t ( r ) ρ t ( r ) ,
ψ t s ( r ) = d r G ( r , r ) I D ( r ) Q t ( r ) ,
Q t ( r ) = V ( r ) [ ( I G V ˜ ) 1 ψ t inc ] ( r ) ,
( G V ˜ ψ t ) ( r ) = d r G ( r , r ) V ( r ) ψ t ( r ) .
I t ( r ) = ψ t s ( r ) 2 = d r d r H ( r ; r , r ) Q t ( r , r ) ,
H ( r ; r , r ) = G * ( r , r ) G ( r , r ) I D ( r ) I D ( r )
Q t ( r , r ) = Q t * ( r ) Q t ( r ) .
I t ( r ) = d r H ( r ; r , r ) Q t ( r , r ) + 2 R [ d r { x < x , y < y , z < z } d r H ( r ; r , r ) Q t ( r , r ) ] = d r H ( r ; r , r ) Q t ( r , r ) + 2 d r { x < x , y < y , z < z } d r R [ H ( r ; r , r ) ] R [ Q t ( r , r ) ] 2 d r { x < x , y < y , z < z } d r I [ H ( r ; r , r ) ] I [ Q t ( r , r ) ] ,
H ¯ ( r ; r , r , j ) = { R [ H ( r ; r , r ) ] j = 1 I [ H ( r ; r , r ) ] j = 2
Q ¯ t ( r , r , j ) = { 2 R [ Q t ( r , r ) ] [ 1 1 2 δ ( r r ) ] j = 1 2 I [ Q t ( r , r ) ] j = 2
I t ( r ) = d r { x x , y y , z z } d r H ¯ ( r ; r , r , 1 ) Q ¯ t ( r , r , 1 ) + d r { x < x , y < y , z < z } d r H ¯ ( r ; r , r , 2 ) Q ¯ t ( r , r , 2 ) ,
I t = [ I t ( Z 1 ) , I t ( Z 2 ) , , I t ( Z N R ) ] R N R .
Q ¯ t Q ¯ t X = j = 1 2 d r d r w ( r , r , j ) Q ¯ t ( r , r , j ) Q ¯ t ( r , r , j ) ,
w ( r , r , j ) = { 1 if ( r , r , j ) D [ D { x x , y y , z z } ] { 1 } 1 if ( r , r , j ) D [ D { x < x , y < y , z < z } ] { 2 } , 0 otherwise
I t I t Y = n = 1 N R I t ( n ) I t ( n ) .
I t ( n ) = ( P Q ¯ t ) ( n ) = j = 1 2 d r d r w ( r , r , j ) H ¯ ( Z n ; r , r , j ) Q ¯ t ( r , r , j ) n = 1 , 2 , , N R .
I t ( n ) = ( P Q ¯ t ) ( n ) = m = 1 M m m H ¯ ( Z n ; X m , X m , 1 ) Q ¯ t ( X m , X m , 1 ) + m = 1 M m < m H ¯ ( Z n ; X m , X m , 2 ) Q ¯ t ( X m , X m , 2 )
U ( R , R ) ( n ) = R [ G * ( Z n , R ) G ( Z n , R ) ] ,
V ( R , R ) ( n ) = I [ G * ( Z n , R ) G ( Z n , R ) ] n = 1 , 2 , , N R .
I t = P Q ¯ t = m = 1 M m m Q ¯ t ( X m , X m , 1 ) U ( X m , X m ) + m = 1 M m m Q ¯ t ( X m , X m , 2 ) V ( X m , X m ) ,
S y = Span [ U ( X m , X m ) , V ( X m , X m ) ,
m = 1 , 2 , , M ; m m ] ,
S y = Span [ U ( X m , X m ) , V ( X m , X m ) ,
m = 1 , 2 , , M ; m m ] ,
A ( R , R ) = [ p > M 2 W p U ( R , R ) Y 2 + p > M 2 W p V ( R , R ) Y 2 ] 1
A p ( R ) = max R [ A ( R , R ) ] ,
( P Q ¯ t ) ( n ) = p = 1 N R σ p W p ( n ) < R p Q ¯ t > X ,
I t = P Q ¯ t p = 1 L σ p W p R p Q ¯ t X
A ( R , R ) = [ p > L W p U ( R , R ) Y 2 + p > L W p V ( R , R ) Y 2 ] 1
A p ( R ) = max R [ A ( R , R ) ]

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