Abstract

We describe a pattern-recognition processor that is optimal for detection and location of a target with white Gaussian random gray levels on a white random spatially disjoint background. We show that this algorithm consists of correlations of the silhouette of the reference object with preprocessed versions of the scene image. This result can provide a theoretical basis for pattern-recognition techniques that use nonlinear preprocessing of images before correlation.

© 1996 Optical Society of America

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References

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1995

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1992

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1964

A. VanderLugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[CrossRef]

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Gianino, P. D.

Goudail, F.

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Horner, J. L.

Javidi, B.

Laude, V.

Réfrégier, Ph.

VanderLugt, A.

A. VanderLugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[CrossRef]

Wang, J.

Willet, P.

Zhang, G.

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Figures (2)

Fig. 1
Fig. 1

(a) Shape of the target used in the test. (b) Scene used in the test, with probability-density function parameters ma = mb = 1.0, σa = 0.1, and σb = 0.3. (c) Result of the processing of (b) with the optimal processor defined in Eq. (4). (d) Result of the processing of (b) with an optimal trade-off filter optimized for peak sharpness and robustness and additive noise. The same kinds of results are obtained with other linear filters. (c), (d) Plots of the maximum of each line of the correlation plane.

Fig. 2
Fig. 2

(a) Scene used in the test. (b) Result of the processing of (a) with the optimal processor for white-noise statistics defined in Eq. (6). Linear filtering techniques such as matched, phase-only, and optimal trade-off filters fail to locate the target in this case, whereas the processor of Eq. (6) succeeds, although it is not optimal for that case.

Equations (10)

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s i = a i w i - j + b i [ 1 - w i - j ] .
P [ H j s ] = 1 2 π σ a N a σ b N b exp { - 1 σ a 2 i = 1 N [ s i - m a ] 2 w i - j - 1 σ b 2 i = 1 N [ s i - m b ] 2 [ 1 - w i - j ] } ,
( 1 σ b 2 - 1 σ a 2 ) i = 1 N s i 2 w i - j + 2 ( m a σ a 2 - m b σ b 2 ) i = 1 N s i w i - j + { ( m b 2 σ b 2 - m 2 σ a 2 ) i = 1 N w i - j - 1 σ b 2 i = 1 N [ s i - m b ] 2 } .
F j ( 1 ) = [ { ( σ a 2 - σ b 2 ) s 2 + 2 ( σ b 2 m a - σ a 2 m b ) s } w ] j .
F j ( 1 ) = ( Δ v / v ) [ ( s - m u ) 2 w ] j + 2 Δ m [ s w ] j ,
F j ( 2 ) = - N a log ( σ j 2 ) - N b log ( σ ¯ j 2 ) ,
σ j 2 = 1 N a { i w i - j = 1 } [ s i - 1 N a { i w i - j = 1 } s i ] 2 ,
σ ¯ j 2 = 1 N b { i w i - j = 0 } [ s i - 1 N b { i w i - j = 0 } s i ] 2 .
σ j 2 = 1 N a [ s 2 w ] j - 1 N a 2 [ s w ] j 2 ,
σ ¯ j 2 = 1 N b ( i = 1 N s i 2 - [ s 2 w ] j ) - 1 N b 2 ( i = 1 N s i - [ s w ] j ) 2 .

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