Abstract

This paper investigates the signal-subspace method approach to solve the electromagnetic inverse scattering problem using intensity-only (phase-free) data. Due to the polarization of electromagnetic fields, the relationship between the rank of the multistatic matrix and the number of small scatterers is different from that associated with the scalar wave. Multiple scattering between scatterers is considered, and the inverse scattering problem of determining the polarization tensors is nonlinear, which, however, is solved by the proposed analytical approach where no associated forward problem is iteratively evaluated.

© 2008 Optical Society of America

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  1. E. A. Marengo, R. D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc. Am. A 24, 3619-3635 (2007).
    [CrossRef]
  2. H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, “MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions,” SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
    [CrossRef]
  3. M. Cheney, “The linear sampling method and the MUSIC algorithm,” Inverse Probl. 17, 591-595 (2001).
    [CrossRef]
  4. Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiply scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542-3549 (2007).
    [CrossRef]
  5. E. A. Marengo and F. K. Gruber, “Subspace-based localization and inverse scattering of multiply scattering point targets,” EURASIP J. Appl. Signal Process. 2007, 17342 (2007).
    [CrossRef]
  6. D. H. Chambers and J. G. Berryman, “Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field,” IEEE Trans. Antennas Propag. 52, 1729-1738 (2004).
    [CrossRef]
  7. D. H. Chambers and J. G. Berryman, “Target characterization using decomposition of the time-reversal operator: Electromagnetic scattering from small ellipsoids,” Inverse Probl. 22, 2145-2163 (2006).
    [CrossRef]
  8. D. H. Chambers and A. K. Gautesen, “Time-reversal operator for a single spherical scatterer,” J. Acoust. Soc. Am. 109, 2616-2624 (2001).
    [CrossRef] [PubMed]
  9. 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]
  10. E. A. Marengo, “Further theoretical considerations for time-reversal MUSIC imaging of extended scatterers,” in IEEE/SP 14th Workshop on Statistical Signal Processing (IEEE, 2007), pp. 304-306.
    [CrossRef]
  11. 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]
  12. E. A. Marengo and F. K. Gruber, “Noniterative analytical formula for inverse scattering of multiply scattering point targets,” J. Acoust. Soc. Am. 120, 3782-3788 (2006).
    [CrossRef]
  13. X. Chen and Y. Zhong, “A robust noniterative method for obtaining scattering strengths of multiply scattering point targets,” J. Acoust. Soc. Am. 122, 1325-1327 (2007).
    [CrossRef] [PubMed]
  14. C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1998).
    [CrossRef]
  15. H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements (Springer-Verlag, 2004).
  16. X. Chen, “MUSIC imaging applied to total internal reflection tomography,” J. Opt. Soc. Am. A 25, 357-364 (2008).
    [CrossRef]
  17. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1986).
  18. A. Kirsch, “The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media,” Inverse Probl. 18, 1025-1040 (2002).
    [CrossRef]

2008 (1)

2007 (7)

X. Chen and Y. Zhong, “A robust noniterative method for obtaining scattering strengths of multiply scattering point targets,” J. Acoust. Soc. Am. 122, 1325-1327 (2007).
[CrossRef] [PubMed]

E. A. Marengo, R. D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc. Am. A 24, 3619-3635 (2007).
[CrossRef]

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, “MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions,” SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
[CrossRef]

Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiply scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542-3549 (2007).
[CrossRef]

E. A. Marengo and F. K. Gruber, “Subspace-based localization and inverse scattering of multiply scattering point targets,” EURASIP J. Appl. Signal Process. 2007, 17342 (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, “Further theoretical considerations for time-reversal MUSIC imaging of extended scatterers,” in IEEE/SP 14th Workshop on Statistical Signal Processing (IEEE, 2007), pp. 304-306.
[CrossRef]

2006 (2)

D. H. Chambers and J. G. Berryman, “Target characterization using decomposition of the time-reversal operator: Electromagnetic scattering from small ellipsoids,” Inverse Probl. 22, 2145-2163 (2006).
[CrossRef]

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

2005 (1)

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]

2004 (2)

D. H. Chambers and J. G. Berryman, “Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field,” IEEE Trans. Antennas Propag. 52, 1729-1738 (2004).
[CrossRef]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements (Springer-Verlag, 2004).

2002 (1)

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

2001 (2)

M. Cheney, “The linear sampling method and the MUSIC algorithm,” Inverse Probl. 17, 591-595 (2001).
[CrossRef]

D. H. Chambers and A. K. Gautesen, “Time-reversal operator for a single spherical scatterer,” J. Acoust. Soc. Am. 109, 2616-2624 (2001).
[CrossRef] [PubMed]

1998 (1)

C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1998).
[CrossRef]

1986 (1)

A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1986).

Ammari, H.

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, “MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions,” SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
[CrossRef]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements (Springer-Verlag, 2004).

Berryman, J. G.

D. H. Chambers and J. G. Berryman, “Target characterization using decomposition of the time-reversal operator: Electromagnetic scattering from small ellipsoids,” Inverse Probl. 22, 2145-2163 (2006).
[CrossRef]

D. H. Chambers and J. G. Berryman, “Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field,” IEEE Trans. Antennas Propag. 52, 1729-1738 (2004).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Chambers, D. H.

D. H. Chambers and J. G. Berryman, “Target characterization using decomposition of the time-reversal operator: Electromagnetic scattering from small ellipsoids,” Inverse Probl. 22, 2145-2163 (2006).
[CrossRef]

D. H. Chambers and J. G. Berryman, “Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field,” IEEE Trans. Antennas Propag. 52, 1729-1738 (2004).
[CrossRef]

D. H. Chambers and A. K. Gautesen, “Time-reversal operator for a single spherical scatterer,” J. Acoust. Soc. Am. 109, 2616-2624 (2001).
[CrossRef] [PubMed]

Chen, X.

X. Chen, “MUSIC imaging applied to total internal reflection tomography,” J. Opt. Soc. Am. A 25, 357-364 (2008).
[CrossRef]

Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiply scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542-3549 (2007).
[CrossRef]

X. Chen and Y. Zhong, “A robust noniterative method for obtaining scattering strengths of multiply scattering point targets,” J. Acoust. Soc. Am. 122, 1325-1327 (2007).
[CrossRef] [PubMed]

Cheney, M.

M. Cheney, “The linear sampling method and the MUSIC algorithm,” Inverse Probl. 17, 591-595 (2001).
[CrossRef]

Devaney, A. J.

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]

Gautesen, A. K.

D. H. Chambers and A. K. Gautesen, “Time-reversal operator for a single spherical scatterer,” J. Acoust. Soc. Am. 109, 2616-2624 (2001).
[CrossRef] [PubMed]

Gruber, F. K.

E. A. Marengo and F. K. Gruber, “Subspace-based localization and inverse scattering of multiply scattering point targets,” EURASIP J. Appl. Signal Process. 2007, 17342 (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, “Noniterative analytical formula for inverse scattering of multiply scattering point targets,” J. Acoust. Soc. Am. 120, 3782-3788 (2006).
[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]

Hernandez, R. D.

Horn, A.

A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1986).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Iakovleva, E.

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, “MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions,” SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
[CrossRef]

Johnson, C. R.

A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1986).

Kang, H.

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements (Springer-Verlag, 2004).

Kirsch, A.

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

Lesselier, D.

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, “MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions,” SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
[CrossRef]

Lev-Ari, H.

Marengo, E. A.

E. A. Marengo, R. D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc. Am. A 24, 3619-3635 (2007).
[CrossRef]

E. A. Marengo and F. K. Gruber, “Subspace-based localization and inverse scattering of multiply scattering point targets,” EURASIP J. Appl. Signal Process. 2007, 17342 (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, “Further theoretical considerations for time-reversal MUSIC imaging of extended scatterers,” in IEEE/SP 14th Workshop on Statistical Signal Processing (IEEE, 2007), pp. 304-306.
[CrossRef]

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

Perruson, G.

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, “MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions,” SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
[CrossRef]

Simonetti, F.

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]

Zhong, Y.

X. Chen and Y. Zhong, “A robust noniterative method for obtaining scattering strengths of multiply scattering point targets,” J. Acoust. Soc. Am. 122, 1325-1327 (2007).
[CrossRef] [PubMed]

Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiply scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542-3549 (2007).
[CrossRef]

EURASIP J. Appl. Signal Process. (1)

E. A. Marengo and F. K. Gruber, “Subspace-based localization and inverse scattering of multiply scattering point targets,” EURASIP J. Appl. Signal Process. 2007, 17342 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

D. H. Chambers and J. G. Berryman, “Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field,” IEEE Trans. Antennas Propag. 52, 1729-1738 (2004).
[CrossRef]

Y. Zhong and X. Chen, “MUSIC imaging and electromagnetic inverse scattering of multiply scattering small anisotropic spheres,” IEEE Trans. Antennas Propag. 55, 3542-3549 (2007).
[CrossRef]

IEEE Trans. Image Process. (1)

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]

Inverse Probl. (3)

M. Cheney, “The linear sampling method and the MUSIC algorithm,” Inverse Probl. 17, 591-595 (2001).
[CrossRef]

D. H. Chambers and J. G. Berryman, “Target characterization using decomposition of the time-reversal operator: Electromagnetic scattering from small ellipsoids,” Inverse Probl. 22, 2145-2163 (2006).
[CrossRef]

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

J. Acoust. Soc. Am. (4)

D. H. Chambers and A. K. Gautesen, “Time-reversal operator for a single spherical scatterer,” J. Acoust. Soc. Am. 109, 2616-2624 (2001).
[CrossRef] [PubMed]

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]

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

X. Chen and Y. Zhong, “A robust noniterative method for obtaining scattering strengths of multiply scattering point targets,” J. Acoust. Soc. Am. 122, 1325-1327 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

SIAM J. Sci. Comput. (USA) (1)

H. Ammari, E. Iakovleva, D. Lesselier, and G. Perruson, “MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions,” SIAM J. Sci. Comput. (USA) 29, 674-709 (2007).
[CrossRef]

Other (4)

A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1986).

E. A. Marengo, “Further theoretical considerations for time-reversal MUSIC imaging of extended scatterers,” in IEEE/SP 14th Workshop on Statistical Signal Processing (IEEE, 2007), pp. 304-306.
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scatttering of Light by Small Particles (Wiley, 1998).
[CrossRef]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements (Springer-Verlag, 2004).

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Figures (3)

Fig. 1
Fig. 1

Singular values of the MSR matrix in the first example.

Fig. 2
Fig. 2

Singular values of the MSR matrix in the second example.

Fig. 3
Fig. 3

Comparison between retrieval errors using intensity-only and full-information electric field data.

Equations (34)

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G ̿ 0 ( r , r ) = [ ( G ̿ 0 ( r , r ) ) x , ( G ̿ 0 ( r , r ) ) y , ( G ̿ 0 ( r , r ) ) z ] ,
χ ̿ ( r , r ) = [ ( χ ̿ ( r , r ) ) x , ( χ ̿ ( r , r ) ) y , ( χ ̿ ( r , r ) ) z ] ,
G ¯ ( m , L , l ) ( r ) = { i k ( G ̿ 0 ( r , r m ) ) l if L = E ( χ ̿ ( r , r m ) ) l if L = M } ,
Q ( m , L , l ) = { η 0 I l l ( r m ) if L = E K l l ( r m ) if L = M } ,
E ¯ sca ( r ) = s = 1 6 M G ¯ s ( r ) Q ( s ) .
I ¯ = s 1 = 1 6 M s 2 s 1 Q ( s 1 , s 2 , 1 ) U ¯ ( s 1 , s 2 ) + s 1 = 1 6 M s 2 > s 1 Q ( s 1 , s 2 , 2 ) V ¯ ( s 1 , s 2 ) ,
U ¯ ( s 1 , s 2 ) = [ G ¯ ( m 1 , L 1 , l 1 ) * G ¯ ( m 2 , L 2 , l 2 ) ] R ,
V ¯ ( s 1 , s 2 ) = [ G ¯ ( m 1 , L 1 , l 1 ) * G ¯ ( m 2 , L 2 , l 2 ) ] I ,
Q ( s 1 , s 2 , j ) = { 2 Q R ( s 1 , s 2 ) ( 1 1 2 δ s 1 , s 2 ) j = 1 2 Q I ( s 1 , s 2 ) j = 2 } ,
Q ( s 1 , s 2 ) = Q * ( s 1 ) Q ( s 2 ) .
( χ ̿ ) x * ( χ ̿ ) y = 1 R d g d R 2 [ 0 0 ( x x ) ( y y ) ] ,
( G ̿ 0 ) x * ( χ ̿ ) x = i k ϑ * ( R ) 1 R d g d R [ 0 ( x x ) ( y y ) ( z z ) ( x x ) ( y y ) ( z z ) ] ,
( G ̿ 0 ) x * ( χ ̿ ) x + ( G ̿ 0 ) y * ( χ ̿ ) y + ( G ̿ 0 ) z * ( χ ̿ ) z = [ 0 0 0 ] .
r K = α 2 β 3 γ 5 τ ,
Λ ̿ = diag [ P ̿ 1 , P ̿ 2 , , P ̿ M , P ̿ M + 1 , P ̿ M + 2 , , P ̿ 2 M ] ,
Λ ̿ = L ̿ S ̿ Λ ̿ S ̿ T L ̿ T ,
I ¯ t = [ U ̿ , V ̿ ] [ Q ¯ t ( 1 ) Q ¯ t ( 2 ) ] ,
[ Q ¯ t ( 1 ) Q ¯ t ( 2 ) ] = [ U ̿ , V ̿ ] I ¯ t ,
Q ¯ t = Λ ̿ S ̿ T L ̿ T ( T ¯ t + Φ ̿ L ̿ S ̿ Q ¯ t ) ,
q ¯ t = Λ ̿ ( C ¯ t f t + D ¯ t ) .
[ q ¯ t ] i = [ Λ ̿ ] i ( [ C ¯ t ] i f t + [ D ¯ t ] i ) , i = 1 , 2 , , I tot .
[ q ̿ 1 : 3 ] i = [ Λ ̿ ] i ( [ C ̿ 1 : 3 ] i f ̿ 1 : 3 + [ D ̿ 1 : 3 ] i ) .
[ Λ ̿ ] i 1 = ( [ C ̿ 1 : 3 ] i f ̿ 1 : 3 + [ D ̿ 1 : 3 ] i ) [ q ̿ 1 : 3 ] i 1 .
[ Λ ̿ ] i 1 = ( [ C ̿ 2 : 4 ] i f ̿ 2 : 4 + [ D ̿ 2 : 4 ] i ) [ q ̿ 2 : 4 ] i 1 .
f ̿ 1 : 3 + Y ̿ i f ̿ 2 : 4 Z ̿ i = W ̿ i ,
X ̿ i f ¯ 1 : 4 = W ¯ i ,
X ̿ i f ¯ 1 : 3 = W ¯ i ,
f 1 + Y i f 2 Z i = W i ,
[ Λ ̿ ] i = [ q ̿ ] i ( [ C ̿ ] i f ̿ + [ D ̿ ] i ) , i = 1 , 2 , , I tot ,
Δ = i = 1 I tot Λ ̿ i Λ ̃ i 2 i = 1 I tot Λ ̿ i 2 × 100 % ,
G ̿ 0 ( r , r ) = ( I ̿ + 1 k 2 [ 2 x 2 2 x y 2 x z 2 y x 2 y 2 2 y z 2 z x 2 z y 2 z 2 ] ) g ( r , r ) ,
χ ̿ ( r , r ) = [ 0 ( z z ) ( y y ) ( z z ) 0 ( x x ) ( y y ) ( x x ) 0 ] 1 R d g d R .
[ 1 Y ̿ i ( 1 , 1 ) Z ̿ i ( 1 , 1 ) Y ̿ i ( 1 , 2 ) Z ̿ i ( 2 , 1 ) Y ̿ i ( 1 , 3 ) Z ̿ i ( 3 , 1 ) 0 Y ̿ i ( 2 , 1 ) Z ̿ i ( 1 , 1 ) Y ̿ i ( 2 , 2 ) Z ̿ i ( 2 , 1 ) Y ̿ i ( 2 , 3 ) Z ̿ i ( 3 , 1 ) 0 Y ̿ i ( 3 , 1 ) Z ̿ i ( 1 , 1 ) Y ̿ i ( 3 , 2 ) Z ̿ i ( 2 , 1 ) Y ̿ i ( 3 , 3 ) Z ̿ i ( 3 , 1 ) ] .
[ 1 Y ̿ i ( 1 , 1 ) Z ̿ i ( 1 , 1 ) Y ̿ i ( 1 , 2 ) Z ̿ i ( 2 , 1 ) 0 Y ̿ i ( 2 , 1 ) Z ̿ i ( 1 , 1 ) Y ̿ i ( 2 , 2 ) Z ̿ i ( 2 , 1 ) ] .

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