Abstract

The canonical problem of detecting and localizing missing scatterers (faults) inside a known grid of small cross section perfect electric conducting cylinders is dealt with. The case of a TM scalar two-dimensional geometry is considered. The scattering by a fault is modeled as the radiation of a proper magnetic current, by exploiting the Green’s function of the complete grid. An approximated linear model of the scattering is proposed and discussed in terms of the achievable probability of detection, also in the case of two faults, and checked against model error and noisy synthetic data.

© 2011 Optical Society of America

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References

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  1. K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals, Optical Engineering (CRC, 2005).
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  2. C. Elachi, “Waves in active and passive periodic structures: a review,” IEEE Trans. Microwave Theory Tech. 24, 1666–1698(1976).
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  5. A. Ludwig and Y. Leviatan, “Analysis of arbitrary defects in photonic crystals by use of the source-model technique,” J. Opt. Soc. Am. A 21, 1334–1343 (2004).
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  6. T. Isernia, V. Pascazio, and R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (July 2001).
    [CrossRef]
  7. J. P. Groby and D. Lesselier, “Localization and characterization of simple defects in finite-sized photonic crystals,” J. Opt. Soc. Am. A 25, 146–152 (2008).
    [CrossRef]
  8. R. F. Harrington, Time Harmonic Electromagnetic Fields(Wiley, 2001).
    [CrossRef]
  9. R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029(2005).
    [CrossRef]
  10. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  11. A. Brancaccio, G. Leone, and R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
    [CrossRef]
  12. R. Zhou, H. Zhang, and H. Xin, “Metallic wire antennas as low-effective index of refraction medium for directive antenna application,” IEEE Trans. Antennas Propag. 58, 79–87(2010).
    [CrossRef]

2010 (1)

R. Zhou, H. Zhang, and H. Xin, “Metallic wire antennas as low-effective index of refraction medium for directive antenna application,” IEEE Trans. Antennas Propag. 58, 79–87(2010).
[CrossRef]

2008 (1)

2005 (1)

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029(2005).
[CrossRef]

2004 (1)

2003 (1)

1998 (1)

1997 (1)

1976 (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” IEEE Trans. Microwave Theory Tech. 24, 1666–1698(1976).

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Brancaccio, A.

Elachi, C.

C. Elachi, “Waves in active and passive periodic structures: a review,” IEEE Trans. Microwave Theory Tech. 24, 1666–1698(1976).

Groby, J. P.

Harrington, R. F.

R. F. Harrington, Time Harmonic Electromagnetic Fields(Wiley, 2001).
[CrossRef]

Isernia, T.

T. Isernia, V. Pascazio, and R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (July 2001).
[CrossRef]

Leone, G.

Lesselier, D.

Leviatan, Y.

Liseno, A.

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029(2005).
[CrossRef]

Ludwig, A.

Maystre, D.

Pascazio, V.

T. Isernia, V. Pascazio, and R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (July 2001).
[CrossRef]

Pierri, R.

T. Isernia, V. Pascazio, and R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (July 2001).
[CrossRef]

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029(2005).
[CrossRef]

A. Brancaccio, G. Leone, and R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

Romano, J.

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029(2005).
[CrossRef]

Solimene, R.

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029(2005).
[CrossRef]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Tayeb, G.

Xin, H.

R. Zhou, H. Zhang, and H. Xin, “Metallic wire antennas as low-effective index of refraction medium for directive antenna application,” IEEE Trans. Antennas Propag. 58, 79–87(2010).
[CrossRef]

Yasumoto, K.

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals, Optical Engineering (CRC, 2005).
[CrossRef]

Zhang, H.

R. Zhou, H. Zhang, and H. Xin, “Metallic wire antennas as low-effective index of refraction medium for directive antenna application,” IEEE Trans. Antennas Propag. 58, 79–87(2010).
[CrossRef]

Zhou, R.

R. Zhou, H. Zhang, and H. Xin, “Metallic wire antennas as low-effective index of refraction medium for directive antenna application,” IEEE Trans. Antennas Propag. 58, 79–87(2010).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. Pierri, R. Solimene, A. Liseno, and J. Romano, “Linear distribution imaging of thin metallic cylinders under mutual scattering,” IEEE Trans. Antennas Propag. 53, 3019–3029(2005).
[CrossRef]

R. Zhou, H. Zhang, and H. Xin, “Metallic wire antennas as low-effective index of refraction medium for directive antenna application,” IEEE Trans. Antennas Propag. 58, 79–87(2010).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

T. Isernia, V. Pascazio, and R. Pierri, “On the local minima in a tomographic imaging technique,” IEEE Trans. Geosci. Remote Sens. 39, 1596–1607 (July 2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

C. Elachi, “Waves in active and passive periodic structures: a review,” IEEE Trans. Microwave Theory Tech. 24, 1666–1698(1976).

J. Opt. Soc. Am. A (5)

Other (3)

R. F. Harrington, Time Harmonic Electromagnetic Fields(Wiley, 2001).
[CrossRef]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).

K. Yasumoto, Electromagnetic Theory and Applications for Photonic Crystals, Optical Engineering (CRC, 2005).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Grid of small cross section cylindrical scatterers inside an x, y reference system. The elements of the grid are numbered starting from the top left side.

Fig. 2
Fig. 2

Illustrating the scattering by a PEC object (first row) and a missing PEC object (second row).

Fig. 3
Fig. 3

Integral path relative to Eq. (13).

Fig. 4
Fig. 4

Grid of N = 121 scatterers. Different fault positions at m = 10 , 67, 70 are indicated in red. The observation points are taken along the green circle. Blue dots denotes two source positions at θ s = 0 , π / 4 .

Fig. 5
Fig. 5

Approximated (black line) and exact (red line) scattered field amplitude as functions of the observation angle for two different views θ s = 0 (left column) and θ s = π / 4 (right column). As expected the amplitude ratio is constant for the two different views, equal to 4 dB for m = 10 , 62 (top and center) and to 5 dB for m = 72 (bottom).

Fig. 6
Fig. 6

Grid of N = 121 scatterers. The observation points are taken along the green circle. Blue dots denotes three source positions at θ s = 0 , 2 π / 3 , 2 π / 3 .

Fig. 7
Fig. 7

Singular values behavior: (top) one view θ s = 0 ; (bottom) three views θ s = 0 , 2 π / 3 , 2 π / 3 . Configuration: N scatterers: N = 36 (dashed); N = 121 (solid); N = 441 (dotted).

Fig. 8
Fig. 8

Spatial content for the three view configuration and N = 441 scatterers.

Fig. 9
Fig. 9

Probability of detection for different threshold levels, N = 121 .

Fig. 10
Fig. 10

Two “separated” faults at m = 56 and m = 63 : reconstructed fault positions are indicated in white (left panel); amplitude of γ ̲ ˜ and threshold level (right panel). Data are noisy with SNR = 20 dB . The actual faults are well detected.

Fig. 11
Fig. 11

Two close faults at m = 62 and m = 63 : reconstructed fault positions are indicated in white (left panel); amplitude of γ ̲ ˜ and threshold level (right panel). Data are noisy with SNR = 20 dB . The actual faults are well detected.

Fig. 12
Fig. 12

Two faults at m = 27 and m = 75 : reconstructed fault positions are indicated in white (left panel); amplitude of γ ̲ ˜ and threshold level (right panel). Data are noisy with SNR = 20 dB . Only the fault at m = 75 is detected.

Fig. 13
Fig. 13

Q m for the case of Fig. 10: noiseless data (left panel); noisy data SNR = 20 dB (right panel).

Fig. 14
Fig. 14

Q m for the case of Fig 11: noiseless data (left panel); noisy data SNR = 20 dB (right panel).

Equations (49)

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J ms = i ^ n × E N 1 .
E s ( r ) = C G e N ( r , r ) × J ms ( r ) d r ,
E N ( r o , r s ) = E N ( r o , r s ) i ^ z = β ζ I e 4 [ H 0 ( 2 ) ( β | r o r s | ) + n = 1 N a n ( r s ) H 0 ( 2 ) ( β | r o r n | ) ] i ^ z ,
A ̲ ̲ · a ̲ = b ̲ .
A n m = { 1 if     n = m J 0 ( β a ) H 0 ( 2 ) ( β a ) H 0 ( 2 ) ( β d n m ) if     n m ,
b n ( r s ) = J 0 ( β a ) H 0 ( 2 ) ( β a ) H 0 ( 2 ) ( β | r n r s | ) , n = 1 , , N ,
G e N ( r o , r s ) = j 4 [ H 0 ( 2 ) ( β | r o r s | ) + n = 1 N a n ( r s ) H 0 ( 2 ) ( β | r o r n | ) ] .
G e N ( r o , r s ) = j 4 [ H 0 ( 2 ) ( β | r s r o | ) + n = 1 N a n ( r o ) H 0 ( 2 ) ( β | r s r n | ) ] .
E s ( r o , r s ) = E N 1 ( r o , r s ) E N ( r o , r s ) ,
E s ( r o , r s ) = C J ms ( r , r s ) × G e N ( r o , r ) · , i z d r ,
J ms × G e N · i ^ z = ( i ^ n × E N 1 i ^ z ) × G e N · i ^ z = G e N · i ^ n E N 1
E s ( r o , r s ) = E N 1 ( r m , r s ) C G e N ( r o , r ) · i ^ ρ , d r ,
G e N ( r o , r ) · i ^ ρ = G e N ( r o , ρ + r m ) ρ = j β 4 H 1 ( 2 ) ( β | r o r m ρ | ) ( r o r m ρ ) · i ^ ρ | r o r m ρ | + j β 4 n = 1 N a n ( r o ) H 1 ( 2 ) ( β | r n r m ρ | ) ( r n r m ρ ) · i ^ ρ | r n r m ρ | ,
I ( R ) = C H 1 ( 2 ) ( β | R ρ | ) ( R ρ ) · i ^ ρ | R ρ | d C ,
I ( 0 ) = C H 1 ( 2 ) ( β a ) d C = 2 π a H 1 ( 2 ) ( β a ) ,
I ( R ) = 0 2 π H 1 ( 2 ) ( β | R ρ | ) cos ( χ + φ ) a d φ
H 1 ( 2 ) ( β | R ρ | ) cos ( χ + φ ) = p H p + 1 ( 2 ) ( β R ) J p ( β a ) cos [ ( p + 1 ) φ ] .
I ( R ) = 2 π a H 0 ( 2 ) ( β R ) J 1 ( β a ) .
E s ( r o , r s ) = E N 1 ( r m , r s ) j π β a 2 J 1 ( β a ) × [ H 0 ( 2 ) ( β | r o r m | ) + n m a n ( r o ) H 0 ( 2 ) ( β d n m ) + a m ( r o ) H 1 ( 2 ) ( β a ) J 1 ( β a ) ] .
H 0 ( 2 ) ( β | r o r m | ) + n m a n ( r o ) H 0 ( 2 ) ( β d n m ) = a m ( r o ) H 0 ( 2 ) ( β a ) J 0 ( β a ) .
E s ( r o , r s ) = a m ( r o ) E N 1 ( r m , r s ) J 0 ( β a ) .
E N 1 ( r m , r s ) = β ζ I e 4 [ H 0 ( 2 ) ( β | r m r s | ) + n m a ¯ n ( r s ) H 0 ( 2 ) ( β d n m ) ] ,
a ¯ n = { a n if     n m 0 if     n = m .
E s ( r o , r s ) = β ζ I e 4 J 0 ( β a ) a m ( r o ) [ H 0 ( 2 ) ( β | r m r s | ) + n m a n ( r s ) H 0 ( 2 ) ( β d n m ) ] .
E s ( r o , r s ) = a m ( r o ) a m ( r s ) ,
E s ( r o h , r s l ) = n = 1 N a n ( r o h ) a n ( r s l ) γ n ,
E ̲ s = L ̲ ̲ · γ ̲ ,
L ̲ ̲ γ ̲ C N E ̲ s C N o N s ,
γ ̲ ˜ = l = 1 N T E ̲ s · v ̲ l * σ l u ̲ l + l = 1 N T N ̲ · v ̲ l * σ l u ̲ l ,
S C ( n ) = l = 1 N T | u l n | 2 ,
R N ̲ ˜ ( n , n ) = E [ N ˜ n , N ˜ n ] = N 0 2 l = 1 N T u l n u l n * σ l 2 ,
v a r N ̲ ˜ ( n ) = N 0 2 l = 1 N T | u l n | 2 σ l 2
p | N ˜ n | ( α ) = α v a r N ̲ ˜ ( n ) exp [ α 2 / ( 2 v a r N ̲ ˜ ( n ) ) ] ,
p | γ ˜ n | ( α ) = α v a r N ̲ ˜ ( n ) exp [ ( α 2 + S C 2 ( n ) ) / ( 2 v a r N ̲ ˜ ( n ) ) ] I 0 ( α S C ( n ) / v a r N ̲ ˜ ( n ) ) )
P FA ( n ) = exp [ A th 2 / ( 2 v a r N ̲ ˜ ( n ) ) ] ,
P D ( n ) = Q ( S C ( n ) v a r N ̲ ˜ ( n ) , A th v a r N ̲ ˜ ( n ) ) ,
Δ E s ( r o , r s ) = β ζ I e 4 J 0 ( β a ) a m ( r o ) T ̲ T · Δ a ̲ ( r s ) ,
T ̲ T = [ H 0 ( 2 ) ( β d 1 m ) , H 0 ( 2 ) ( β d 2 m ) , , H 0 ( 2 ) ( β d m 1 m ) , 0 , H 0 ( 2 ) ( β d m + 1 m ) , , H 0 ( 2 ) ( β d N m ) . ]
A ¯ ̲ ̲ = A ̲ ̲ Δ A ̲ ̲ ,
Δ A i j = { 0 if     i m and j m A i j if     i = m or j = m 0 if     i = j = m .
a ˜ ̲ ( r s ) = ( A ¯ ̲ ̲ ) 1 · b ̲ ,
a ˜ n = { a ¯ n if    n m b n if    n = m .
Δ a ˜ ̲ ( r s ) = ( A ̲ ̲ Δ A ̲ ̲ ) 1 · Δ A ̲ ̲ · a ̲ ( r s ) .
Δ A ̲ ̲ · a ̲ ( r s ) = J 0 ( β a ) H 0 ( 2 ) ( β a ) T ˜ ̲ a m ( r s ) + T ̲ ,
T ˜ n = { T n if    n m 1 if     n = m ,
T n = { 0 if     n m J 0 ( β a ) H 0 ( 2 ) ( β a ) H 0 ( 2 ) ( β | r s r m | ) if     n = m .
Δ E s ( r o , r s ) = β ζ I e 4 H 0 ( 2 ) ( β a ) a m ( r o ) a m ( r s ) T ̲ T · ( A ̲ ̲ Δ A ̲ ̲ ) 1 T ̲ .
E s ACT ( r o , r s ) = E s ( r o , r s ) [ 1 ( J 0 ( β a ) H 0 ( 2 ) ( β a ) ) 2 T ̲ T ( A ̲ ̲ Δ A ̲ ̲ ) 1 T ̲ ] ,
P D ( n ) = Q ( S C ( n ) | A f n | v a r N ̲ ˜ ( n ) , A th v a r N ̲ ˜ ( n ) ) ,

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