Abstract

An optimal distortion-invariant filter for detecting a distorted target in input noise is designed. The input noise consists of two kinds of noise, overlapping additive noise and nonoverlapping background noise. We obtain the filter function by statistically maximizing the peak-to-output-energy ratio criterion, which is defined as the ratio of the square of the expected value of the output signal at the target location to the expected value of the average output energy. This results in a filter output with a well-defined output peak at the target location and a low output-noise floor. This filter is designed to take into account the effects of both overlapping additive noise and nonoverlapping background noise, the finite size of the input data, and the target distortion. The special cases of detecting a distorted target in nonoverlapping background noise and detecting a distorted target in overlapping additive noise are discussed. Computer simulation results are provided to show the performance of the filter.

© 1995 Optical Society of America

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    [CrossRef]

1994

B. Javidi, J. Wang, “Design of filters to detect a noisy target in non-overlapping background noise,” J. Opt. Soc. Am. 11, 2604–2612 (1994).
[CrossRef]

1993

B. Javidi, P. Réfrégier, P. K. Willet, “Design of an optimum receiver for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1160–1162 (1993).
[CrossRef]

P. Réfrégier, B. Javidi, G. Zhang, “Minimum mean square error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1455 (1993).
[CrossRef] [PubMed]

1992

1991

F. M. Dickey, L. A. Romero, “Normalized correlation for pattern recognition,” Opt. Lett. 16, 1186–1188 (1991).
[CrossRef] [PubMed]

P. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the wiener approach,” Opt. comput. process. 1, 245–265 (1991).

1990

1989

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

1987

1984

1982

1976

1969

1964

A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

1960

J. L. Turin, “An introduction to matched filters.” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

Arsenault, H. H.

Casasent, D.

Caulfield, H. J.

Dickey, F. M.

Figue, J.

P. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the wiener approach,” Opt. comput. process. 1, 245–265 (1991).

Flannery, D. L.

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

Hassebrook, L.

Horner, J. L.

J. L. Horner, “Metrics for assessing pattern recognition performance,” Appl. Opt.165–166 (1992).
[CrossRef]

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

B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, Boston, Mass., 1994).

Hsu, Y.-N.

Javidi, B.

B. Javidi, J. Wang, “Design of filters to detect a noisy target in non-overlapping background noise,” J. Opt. Soc. Am. 11, 2604–2612 (1994).
[CrossRef]

B. Javidi, P. Réfrégier, P. K. Willet, “Design of an optimum receiver for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1160–1162 (1993).
[CrossRef]

P. Réfrégier, B. Javidi, G. Zhang, “Minimum mean square error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1455 (1993).
[CrossRef] [PubMed]

B. Javidi, J. Wang, “Limitation of the classic 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]

B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, Boston, Mass., 1994).

Mahalanobis, A.

Maloney, W. T.

Psaltis, D.

Réfrégier, P.

B. Javidi, P. Réfrégier, P. K. Willet, “Design of an optimum receiver for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1160–1162 (1993).
[CrossRef]

P. Réfrégier, B. Javidi, G. Zhang, “Minimum mean square error filter for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1453–1455 (1993).
[CrossRef] [PubMed]

P. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the wiener approach,” Opt. comput. process. 1, 245–265 (1991).

P. Réfrégier, “Filter design for optimal pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
[CrossRef] [PubMed]

Romero, L. A.

Turin, J. L.

J. L. Turin, “An introduction to matched filters.” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Vijaya Kumar, B. V. K.

Wang, J.

Willet, P. K.

B. Javidi, P. Réfrégier, P. K. Willet, “Design of an optimum receiver for pattern recognition with spatially disjoint signal and scene noise,” Opt. Lett. 18, 1160–1162 (1993).
[CrossRef]

Zhang, G.

Appl. Opt.

IEEE Trans. Inf. Theory

A. VanderLugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

IRE Trans. Inf. Theory

J. L. Turin, “An introduction to matched filters.” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

J. Opt. Soc. Am.

B. Javidi, J. Wang, “Design of filters to detect a noisy target in non-overlapping background noise,” J. Opt. Soc. Am. 11, 2604–2612 (1994).
[CrossRef]

Opt. comput. process.

P. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with the wiener approach,” Opt. comput. process. 1, 245–265 (1991).

Opt. Lett.

Proc. IEEE

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

Other

B. Javidi, J. L. Horner, Real-Time Optical Information Processing (Academic, Boston, Mass., 1994).

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Figures (10)

Fig. 1
Fig. 1

True-class target at different rotation angles and the false-class objects used in the tests: (a-1)–(a-9) the training set of the true-class target, the Lamborghini with out-of-plane rotation of 0–80 deg. The rotation increment is 10 deg. (b–1)–(b-5) The false-class objects, a Ferrari, an ambulance, a helicopter, and APC, and a jet. (c-1)–(c-9) The nontraining true-class targets, the Lamborghini with out-of-plane rotation of 31–39 deg. The rotation increment is 1 deg.

Fig. 2
Fig. 2

Measurements for the true-class targets and the false-class objects in nonoverlapping background white noise: (a) example of a target in nonoverlapping background white noise, (b) output POE ratio [Eq. (17)], (c) output peak intensity at the target location.

Fig. 3
Fig. 3

Measurements for the true-class targets and the false-class objects in nonoverlapping background colored noise: (a) example of a target in nonoverlapping background colored noise, (b) output POE ratio [Eq. (17)], (c) output peak intensity at the target location.

Fig. 4
Fig. 4

Output measurement for the true-class targets and the false-class objects in overlapping additive white noise: (a) example of a target in overlapping additive white noise, (b) output POE ratio [Eq. (15)], (c) output peak intensity at the target location.

Fig. 5
Fig. 5

Output measurement for the true-class targets and the false-class objects in overlapping additive colored noise: (a) example of a target in overlapping additive colored noise, (b) output POE ratio [Eq. (15)], (c) output peak intensity at the target location.

Fig. 6
Fig. 6

Multiple true-class targets and false-class objects used in the tests.

Fig. 7
Fig. 7

Computer simulation results for multiple true-class targets and false-class objects in nonoverlapping background white noise and overlapping additive white noise: (a) input scene, (b) 3D plot of the filter output.

Fig. 8
Fig. 8

Computer simulation results for the multiple true-class targets and false-class objects in nonoverlapping background colored noise and overlapping additive colored noise: (a) input scene, (b) 3D plot of the filter output.

Fig. 9
Fig. 9

Computer simulation for a realistic example. An input image contains the multiple true-class targets and false-class objects in a realistic nonoverlapping background noise. The input image is deteriorated by the overlapping additive white noise: (a) input scene, (b) input scene in (a) degraded by additive overlapping noise, (c) 3D plot of the filter output.

Fig. 10
Fig. 10

Computer simulation results for multiple true-class targets and false-class objects in strong nonoverlapping background white noise and overlapping additive white noise: (a) input scene, (b) 3D plot of the filter output.

Equations (41)

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s ( t ) = i = 1 N a i r i ( t τ ) + n b ( t ) [ w 0 ( t ) i 1 N a i w ri ( t τ ) ] + n a ( t ) w 0 ( t ) ,
p ( a i = 0 ) = p ( a j = 0 ) = p 0 i , j = 1 , . . . , N ,
p ( a i = 1 ) = p ( a j = 1 ) = p 1 i , j = 1 , . . . , N ,
p 0 + p 1 = 1 ,
p ( a i = 0 | a j = 1 ) = { 1 i j 0 i = j i , j = 1 , . . . , N ,
p ( a i = 1 | a j = 1 ) = { 0 i j 1 i = j i , j = 1 , . . . , N ,
p 1 = 1 N o r p 0 N 1 N .
POE = | E [ y ( τ , τ ) ] | 2 / E { [ y ( t , τ ) ] 2 ¯ } ,
H opt * ( ω ) = E [ S ( ω , τ ) exp ( j ω τ ) ] E [ | S ( ω , τ ) | 2 ] ,
H opt * ( ω ) = ( i 1 N { R i ( ω ) + m b W 1 i ( ω ) + m a [ | W 0 ( ω ) | 2 / d ] } ) / i 1 N { | R i ( ω ) + m b W 1 i ( ω ) + m a [ | W 0 ( ω ) | 2 / d ] | 2 + 1 2 π W 2 i ( ω ) * N b 0 ( ω ) + 1 2 π | W 0 ( ω ) | 2 * N a 0 ( ω ) + ( m b + m a ) 2 [ W 2 i ( ω ) | W 1 i ( ω ) | 2 ] } ,
W 1 i ( ω ) = [ | W 0 ( ω ) | 2 / d ] W ri ( ω ) ,
W 2 i ( ω ) = | W 0 ( ω ) | 2 + | W ri ( ω ) | 2 { 2 | W 0 ( ω ) | 2 Real [ W ri ( ω ) ] / d } ,
d = W 0 ( 0 ) = w 0 ( t ) d t .
s ( t ) = i 1 N a i r i ( t τ ) + n a ( t ) w 0 ( t ) .
H opt on * ( ω ) = i 1 N { R i ( ω ) + m a [ | W 0 ( ω ) | 2 / d ] } i 1 N { | R i ( ω ) + m a [ | W 0 ( ω ) | 2 / d ] | 2 + 1 2 π | W 0 ( ω ) | 2 * N a 0 ( ω ) + m a 2 [ W 2 i ( ω ) | W 1 i ( ω ) | 2 ] } .
s ( t ) = i 1 N a i r i ( t τ ) + n a ( t ) .
H opt on * ( ω ) = i 1 N [ R i ( ω ) + m a δ ( ω ) ] i 1 N { | R i ( ω ) + m a δ ( ω ) | 2 + 1 2 π N a 0 ( ω ) } .
s ( t ) = i 1 N a i r i ( t τ ) + n b ( t ) [ w 0 ( t ) i 1 N a i w ri ( t τ ) ] .
H opt non * ( ω ) = i 1 N [ R i ( ω ) + m b W 1 i ( ω ) ] i 1 N { | R i ( ω ) + m b W 1 i ( ω ) | 2 + 1 2 π W 2 i ( ω ) * N b 0 ( ω ) + m b 2 [ W 2 i ( ω ) | W 1 i ( ω ) | 2 ] } .
s ( t ) = i 1 N a i r i ( t τ ) + n b ( t ) [ 1 i 1 N a i w ri ( t τ ) ] .
H opt non * ( ω ) = i 1 N [ R i ( ω ) + m b W 1 i 0 ( ω ) ] i 1 N { | R i ( ω ) + m b W 1 i 0 ( ω ) | 2 + 1 2 π | W 1 i 0 ( ω ) | 2 * N b 0 ( ω ) } ,
W 1 i 0 ( ω ) = δ ( ω ) W ri ( ω ) .
p ( a 1 = 0 ) = p ( a 1 = 0 , a 2 = 1 , a 3 = 0 , . . . , a N = 0 ) + p ( a 1 = 0 , a 2 = 0 , a 3 = 1 , . . . , a N = 0 ) + . . . + p ( a 1 = 0 , a 2 = 0 , a 3 = 0 , . . . , a N = 1 ) = ( N 1 ) p ( a 1 = 0 , a 2 = 1 , a 3 = 0 , . . . , a N = 0 ) = ( N 1 ) p ( a 1 = 0 , a 3 = 0 , . . . , a N = 0 | a 2 = 1 ) × p ( a 2 = 1 ) .
p ( a 1 = 0 , a 3 = 0 , . . . , a N = 0 | a 2 = 1 ) = 1 .
p ( a 1 = 0 ) = ( N 1 ) p ( a 2 = 1 ) .
p 0 = ( N 1 ) p 1 = ( N 1 ) ( 1 p 0 ) .
y ( t , τ ) = 1 2 π H ( ω ) S ( ω , τ ) exp ( j ω t ) d ω ,
E [ y ( τ , τ ) ] = 1 2 π H ( ω ) E [ S ( ω , τ ) exp ( j ω τ ) ] d ω .
E [ [ y ( τ , τ ) ] 2 ] ¯ = 1 2 π L | H ( ω ) | 2 E [ | S ( ω , τ ) | 2 ] d ω ,
POE = | H ( ω ) E [ S ( ω , τ ) exp ( j ω τ ) ] d ω | 2 2 π L | H ( ω ) | 2 E [ | S ( ω , τ ) | 2 ] d ω .
POE L 2 π | E [ S ( ω , τ ) exp ( j ω τ ) ] { E [ | S ( ω , τ ) | 2 ] } 1 / 2 | 2 d ω .
H opt * ( ω ) = E [ S ( ω , τ ) exp ( j ω τ ) ] E [ | S ( ω , τ ) | 2 ] .
s i ( t ) = r i ( t τ ) + n b ( t ) [ w 0 ( t ) w ri ( t τ ) ] + n a ( t ) w 0 ( t ) .
s ( t ) = i = 1 N a i s i ( t ) i = 1 N ( a i 1 N ) [ n b ( t ) + n a ( t ) ] w 0 ( t ) .
S ( ω , τ ) = i = 1 N a i S i ( ω , τ ) ,
E a [ S ( ω , τ ) ] = 1 N i = 1 N S i ( ω , τ ) ,
E a [ | S ( ω , τ ) | 2 ] = E a [ | i = 1 N a i S i ( ω , τ ) | 2 ] = 1 N i = 1 N | S i ( ω , τ | 2 .
E [ S ( ω , τ ) exp ( j ω τ ) ] = E { E a [ S ( ω , τ ) exp ( j ω τ ) } = 1 N i = 1 N E [ S i ( ω , τ ) exp ( j ω τ ) ] ,
E [ | S ( ω , τ ) | 2 ] = E { E a [ | S ( ω , τ ) | 2 ] } = 1 N i = 1 N E [ | S i ( ω , τ ) | 2 ] .
E [ S i ( ω , τ ) exp ( j ω τ ) ] = R i ( ω ) + m b W 1 i ( ω ) + m a [ | W 0 ( ω ) | 2 / d ] ,
E [ | S i ( ω , τ ) | 2 ] = | R i ( ω ) + m b W 1 i ( ω ) + m a [ | W 0 ( ω ) | 2 / d ] | 2 + 1 2 π W 2 i ( ω ) * N b 0 ( ω ) + 1 2 π | W 0 ( ω ) | 2 * N a 0 ( ω ) + ( m b + m a ) 2 [ W 2 i ( ω ) | W 1 i ( ω ) | 2 ] .

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