Abstract

The design of an optimum receiver for pattern recognition is based on multiple-alternative hypothesis testing with unknown parameters for detecting and locating a noisy target or a noise-free target in scene noise that is spatially nonoverlapping with this target. The optimum receiver designed for a noise-free target has the interesting property of detecting, without error, a noise-free target that has unknown illumination by using operations that are independent of the scene-noise statistics. We investigate the performance of the optimum receiver designed for nonoverlapping target and scene noise in terms of rotation and scale sensitivity of the input targets and discrimination against similar objects. Because it is not possible in practical systems to have a completely noise-free target, we examine how the performance of the optimum receiver designed for a noise-free target is affected when there is some overlapping noise on the target. The application of the optimum receiver to binary character recognition is described. Computer simulation results are provided.

© 1995 Optical Society of America

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References

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  1. B. Javidi, Ph. Refregier, P. Willett, “Optimum receiver design for pattern recognition with nonoverlapping target and scene noise,” Opt. Lett. 18, 1660–1662 (1993).
    [CrossRef] [PubMed]
  2. B. Javidi, J. Wang, “Limitation of the classical definition of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise,” Appl. Opt. 31, 6826–6829 (1992).
    [CrossRef] [PubMed]
  3. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  4. D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
    [CrossRef]
  5. B. E. Saleh, C. T. Malvin, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 17.
    [CrossRef]
  6. D. Casasent, D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  7. D. Casasent, “Unified synthetic discriminant function computational formula,” Appl. Opt. 22, 1620–1627 (1984).
    [CrossRef]
  8. F. M. Dickey, L. A. Romero, “Normalized correlation for pattern recognition,” Opt. Lett. 16, 1186–1188 (1991).
    [CrossRef] [PubMed]
  9. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  10. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).
  11. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]

1993 (1)

1992 (1)

1991 (1)

1989 (2)

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

1984 (2)

D. Casasent, “Unified synthetic discriminant function computational formula,” Appl. Opt. 22, 1620–1627 (1984).
[CrossRef]

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

1976 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Casasent, D.

D. Casasent, “Unified synthetic discriminant function computational formula,” Appl. Opt. 22, 1620–1627 (1984).
[CrossRef]

D. Casasent, D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef] [PubMed]

Dickey, F. M.

Flannery, D. L.

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Gianino, P. D.

Horner, J. L.

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

Javidi, B.

Malvin, C. T.

B. E. Saleh, C. T. Malvin, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 17.
[CrossRef]

Psaltis, D.

Refregier, Ph.

Romero, L. A.

Saleh, B. E.

B. E. Saleh, C. T. Malvin, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 17.
[CrossRef]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Wang, J.

Willett, P.

Appl. Opt. (5)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

D. L. Flannery, J. L. Horner, “Fourier optical signal processors,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Other (2)

B. E. Saleh, C. T. Malvin, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 17.
[CrossRef]

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

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Figures (15)

Fig. 1
Fig. 1

(a) Three targets (helicopters) with different illuminations a in white background noise. a for targets 1, 2, and 3 are 0.5, 1, and 2, respectively. The noise mean value is 0.5 and its standard deviation is 0.3. The maximum value of the target with a = 1 is unity. The size of the targets is 25 × 39 pixels. (b) Noise-free helicopter image with illumination of unity (45 × 160 pixels).

Fig. 2
Fig. 2

Optimum receiver’s output for the input image of Fig. 1(a).

Fig. 3
Fig. 3

(a) Reference image (tank). (b) Three targets (tanks) with different illuminations a in a real scene that contains another vehicle at its center. a for targets 1, 2, and 3 are 0.5, 1, and 2, respectively. (c) Plot of the output of the optimum receiver when Fig. 3(b) is the input image. Only the output pixels with values above −100 are shown in the plot. The three peaks that correspond to the three targets are 0. (d) Plot of the output of the phase-only filter for the input image of Fig. 3(b). (e) Plot of the output of the matched filter for the input image of Fig. 3(b).

Fig. 4
Fig. 4

Images used in discrimination and rotation sensitivity tests. Pr is the reference image.

Fig. 5
Fig. 5

Variations in the peak of (a) the optimum receiver, (b) the phase-only filter (dashed-dotted curve) and matched filter (solid curve) versus the angle of rotation of the target Pr shown in Fig. (4). The filters’ output peaks are normalized to a maximum value of unity.

Fig. 6
Fig. 6

Variations in the peak of (a) the optimum receiver, (b) the phase-only filter (dashed-dotted curve) and matched filter (solid curve) versus the angle of rotation of the target helicopter shown in Fig. 1(b). The filters’ output peaks are normalized to a maximum value of unity. The failure of the system to detect the target is represented by f.

Fig. 7
Fig. 7

Variations in the peak of (a) the optimum receiver, (b) the phase-only filter (dashed-dotted curve) and matched filter (solid curve) versus the angle of rotation of the target tank [shown in Fig. 3(a) with a size of 55 × 150]. The filters’ output peaks are normalized to a maximum of unity. The failure of the system to detect the target is represented by f.

Fig. 8
Fig. 8

Three rotated tanks in a real scene. Tanks 1, 2, and 3 are rotated 3°, 5°, and 7° with respect to the reference, respectively. The size of the tanks is 55 × 150 pixels. The size of the real scene is 256 × 256 pixels.

Fig. 9
Fig. 9

Rotation sensitivity test of the output of the optimum receiver for the input image of Fig. 8. Only the output pixels with values above −600 are shown in the plot.

Fig. 10
Fig. 10

Scale sensitivity test for (a) the optimum receiver, (b) the phase-only filter (dashed–dotted curve) and matched filter (solid curve) with the image of the helicopter [shown in Fig. 1(b)] as target. The filters’ output peaks are normalized to a maximum of unity. The failure of the system to detect the target is represented by f.

Fig. 11
Fig. 11

Scale sensitivity test for (a) the optimum receiver, (b) the phase-only filter (dashed-dotted curve) and matched filter (solid curve) with the image of the tank [shown in Fig. 3(a) with a size of 55 × 150] as target. The filters’ output peaks are normalized to a maximum of unity. The failure of the system to detect the target is represented by f.

Fig. 12
Fig. 12

Helicopter with its three scaled versions in a scene. Helicopters 1, 2, 3, and 4 have 0, +15%, 15%, and +10% of scale change, respectively. The size of the top helicopter image is 32 × 110 pixels. The size of the scene is 256 × 256 pixels.

Fig. 13
Fig. 13

Scale sensitivity test of the output of the optimum receiver for the image of Fig. 12. Only the output pixels with values above −50 are shown in the plot.

Fig. 14
Fig. 14

Input scene of Fig. 3(b) with added zero-mean white Gaussian noise with a standard deviation of σ = 0.1. The maximum of the pixel values in the input scene of Fig. 3(b) is unity.

Fig. 15
Fig. 15

Plot of the output of the optimum receiver when Fig. 14 is the input image. Only the output pixels with values above −100 are shown in the plot. The reference image is the tank shown in Fig. 3(a).

Tables (2)

Tables Icon

Table 1 Output of the Optimum Receiver, the Matched Filter and the Phase-Only Filter for the Input Images of Fig. (4)

Tables Icon

Table 2 Data Obtained from Monte Carlo Simulations and the Mathematical Expressions of Eqs. (13) and (14) for the Image of Fig. 1(B)

Equations (48)

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H j : r ( t ) = [ a s ( t t j ) + n d ( t ) ] w ( t t j ) + n j ( t ) ,
n j ( t ) = [ n B ( t ) + n d ( t ) ] [ 1 w ( t t j ) ] .
log [ p ( r / H j ; a ˆ ) ] = log [ p n j ( r ¯ j ) ] [ 1 / 2 σ d 2 ] C N ( t j ) ,
C N ( t j ) = i = 1 m r 2 ( t i ) w ( t i t j ) [ i = 1 m r ( t i ) s ( t i t j ) ] 2 i = 1 m s 2 ( t i t j ) ,
log [ p ( r / H j ; a ˆ ) ] = { 1 / [ 2 ( σ n 2 + σ d 2 ) ] } × i = 1 m m s { [ 1 w ( t i t j ) ] r ( t i ) m n } 2 [ 1 / ( 2 σ d 2 ) C N ( t j ) ] .
i = 1 m r ( t i ) s ( t i t j ) 0 . 5 i = 1 m r 2 ( t i ) w ( t i t j )
s ( t ) = w ( t ) ,
0 . 5 i = 1 m r ( t i ) w ( t i t j ) .
C N ( t j ) = i = 1 m r 2 ( t i ) w ( t i t j ) [ i = 1 m r ( t i ) s ( t i t j ) ] i = 1 m s 2 ( t i t j ) 2 ,
r ( t i ) = r ( t i ) + n ( t i ) ,
E { C N ( t j ) } = σ 2 ( W 1 ) + C N ( t j ) ,
VAR { C N ( t j ) } = σ 4 [ 2 ( W 1 ) ] + 4 σ 2 C N ( t j ) ,
E { C N ( t 0 ) } = σ 2 ( W 1 ) ,
VAR { C N ( t 0 ) } = 2 σ 4 ( W 1 ) ,
C N ( t j ) = i = 1 m r 2 ( t i ) w ( t i t j ) [ i = 1 m r ( t i ) s ( t i t j ) ] i = 1 m s 2 ( t i t j ) 2 ,
r ( t i ) = r ( t i ) + n ( t i ) .
C N ( t j ) = i = 1 m r 2 ( t i ) w ( t i t j ) + i = 1 m n 2 ( t i ) w ( t i t j ) + i = 1 m 2 r ( t i ) n ( t i ) w ( t i t j ) ( 1 / s ) × [ i = 1 m r ( t i ) s ( t i t j ) + i = 1 m n ( t i ) s ( t i t j ) ] 2 .
E { C N ( t j ) } = i = 1 m r 2 ( t i ) w ( t i t j ) + σ 2 i = 1 m w ( t i t j ) ( 1 / s ) [ i = 1 m r ( t i ) s ( t i t j ) ] 2 ( 1 / s ) i = 1 m k = 1 m E { n ( t i ) n ( t k ) } × s ( t i t j ) s ( t k t j ) .
E { n ( t i ) n ( t k ) } = σ 2 δ ( i , k ) = { σ 2 ; i = k 0 ; i k .
E { C N ( t j ) } = i = 1 m r 2 ( t i ) w ( t i t j ) ( 1 / s ) × [ i = 1 m r ( t i ) s ( t i t j ) ] 2 + σ 2 ( W 1 ) = C N ( t j ) + σ 2 ( W 1 ) ,
W { C N ( t 0 ) } = σ 2 ( W 1 ) .
VAR [ C N ( t j ) ] = E { [ C N ( t j ) E { C N ( t j ) } ] 2 }
= E { ( i = 1 m n 2 ( t i ) w ( t i t j ) + 2 i = 1 m r ( t i ) n ( t i ) w ( t i t j ) ( 1 / s ) [ i = 1 m n ( t i ) s ( t i t j ) ] 2 ( 2 / s ) i = 1 m r ( t i ) s ( t i t j ) i = 1 m n ( t i ) × s ( t i t j ) σ 2 ( W 1 ) ) 2 } = A + B + C + D + E + F G H I J K L + M + N + O ,
A = E { i = 1 m k = 1 m n 2 ( t i ) n 2 ( t k ) w ( t i t j ) w ( t k t j ) } ,
B = 4 E { i = 1 m k = 1 m n ( t i ) n ( t k ) r ( t i ) w ( t i t j ) w ( t k t j ) } ,
C = E { ( 1 / s 2 ) [ i = 1 m n ( t i ) s ( t i t j ) ] 4 } ,
D = E { ( 4 / s 2 ) [ i = 1 m r ( t i ) s ( t i t j ) ] 2 i = 1 m k = 1 m n ( t i ) × n ( t k ) s ( t i t j ) s ( t k t j ) } ,
E = σ 4 ( W 1 ) 2 ,
F = E { 4 i = 1 m k = 1 m n 2 ( t i ) n ( t k ) r ( t k ) w ( t i t j ) w ( t k t j ) } ,
G = E { ( 2 / s ) i = 1 m n 2 ( t i ) w ( t i t j ) [ i = 1 m n ( t k ) s ( t k t j ) ] 2 } ,
H = E { ( 4 / s ) i = 1 m n 2 ( t i ) w ( t i t j ) l = 1 m r ( t l ) s ( t l t j ) } × k = 1 m n ( t k ) s ( t k t j ) } ,
I = E { 2 σ 2 ( W 1 ) i = 1 m n 2 ( t i ) w ( t i t j ) } ,
J = E { ( 4 / s ) [ i = 1 m n ( t i ) r ( t i ) w ( t i t j ) ] × [ i = 1 m n ( t i ) s ( t i t j ) ] 2 } ,
K = E { ( 8 / s ) i = 1 m r ( t i ) n ( t i ) w ( t i t j ) × l = 1 m r ( t i ) s ( t i t j ) k = 1 m n ( t k ) s ( t k t j ) } ,
L = E { 4 σ 2 ( W 1 ) i = 1 m r ( t i ) n ( t i ) w ( t i t j ) } ,
M = E { ( 4 / s 2 ) [ i = 1 m n ( t i ) s ( t i t j ) ] 2 × l = 1 m r ( t i ) s ( t i t j ) k = 1 m n ( t k ) s ( t k t j ) } ,
N = E { 2 σ 2 ( W 1 ) s [ i = 1 m n ( t i ) s ( t i t j ) ] 2 } ,
O = E { 4 σ 2 ( W 1 ) s i = 1 m r ( t i ) s ( t i t j ) i = 1 m n ( t i ) s ( t l t j ) } .
A = i = 1 m k = 1 m E { n 2 ( t i ) n 2 ( t k ) } w ( t i t j ) w ( t k t j ) ,
E { n 2 ( t i ) n 2 ( t k ) } = { σ 4 ; t i t k 3 σ 4 ; t i t k
= σ 4 [ 1 δ ( t i t k ) ] + 3 σ 4 δ ( t i t k ) = σ 4 + 2 σ 4 δ ( t i t k ) ,
A = σ 4 i = 1 m k = 1 m w ( t i t j ) w ( t k t j ) + 2 σ 4 i = 1 m k = 1 m w ( t i t j ) w ( t k t j ) δ = σ 4 W 2 + 2 σ 4 W . ( t i t k )
C = ( 1 / s 2 ) i = 1 m k = 1 m l = 1 m p = 1 m E { n ( t i ) n ( t k ) n ( t l ) n ( t p ) } × s ( t i t j ) s ( t k t j ) s ( t l t j ) s ( t p t j ) ,
E { n ( t i ) n ( t k ) n ( t l ) n ( t p ) } = { 3 σ 4 if t i = t k = t l = t p σ 4 if ( t i = t k ) ( t l = t p ) , ( t i = t l ) ( t k = t p ) , or ( t i = t p ) ( t l = t k ) , ( three combinations ) 0 otherwise
C = ( 1 / s 2 ) { 3 σ 4 i = 1 m s 4 ( t i t j ) + 3 σ 4 [ i = 1 m s 2 ( t i t j ) l = 1 l i m s 2 ( t i t j ) ] } = 3 σ 4
VAR [ C N ( t j ) ] = ( σ 4 W 2 + 2 σ 4 W ) + [ 4 σ 2 k = 1 m r 2 ( t k ) w ( t k t j ) ] + 3 σ 4 + { 4 σ 2 s [ i = 1 m r ( t i ) s ( t i t j ) ] 2 } + [ σ 4 ( W 1 ) 2 ] + 0 2 σ 4 ( W + 2 ) + 0 [ 2 σ 4 ( W 1 ) W ] + 0 { 8 σ 2 s [ i = 1 m r ( t i ) s ( t i t j ) ] 2 } + 0 + 0 + 2 σ 4 ( W 1 ) + 0 ,
VAR [ C N ( t j ) ] = σ 4 [ 2 ( W 1 ) ] + 4 σ 2 C N ( t j ) .
VAR { C N ( t 0 ) } = σ 4 [ 2 ( W 1 ) ] .

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