Abstract

Accuracy of target position estimation, defined as the variance of location errors, is evaluated when a noisy target is embedded on a nonoverlapping background. It is shown, with some assumptions, that the generalized matched filter minimizes this variance. We also investigate the performance of various correlation filters in terms of location accuracy. Computer simulations are made to compare the results obtained with the generalized matched filter with those of other filters.

© 1996 Optical Society of America

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References

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  1. L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics XXXII, E. Wolf, ed. (Elsevier, Amsterdam, 1993), pp. 145–201.
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1994

1993

1992

1991

1990

U. Mahlab, M. Fleisher, J. Shamir, “Error probability in optical pattern recognition,” Opt. Commun. 77, 415–422 (1990).
[CrossRef]

1984

1964

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

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

DeLaurentis, J. M.

Dickey, F. M.

Fleisher, M.

M. Fleisher, U. Mahlab, J. Shamir, “Target location measurement by optical correlators: performance criterion,” Appl. Opt. 31, 230–235 (1992).
[CrossRef] [PubMed]

U. Mahlab, M. Fleisher, J. Shamir, “Error probability in optical pattern recognition,” Opt. Commun. 77, 415–422 (1990).
[CrossRef]

Gianino, P. D.

Horner, J. L.

Jacobs, I. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965).

Javidi, B.

Mahlab, U.

M. Fleisher, U. Mahlab, J. Shamir, “Target location measurement by optical correlators: performance criterion,” Appl. Opt. 31, 230–235 (1992).
[CrossRef] [PubMed]

U. Mahlab, M. Fleisher, J. Shamir, “Error probability in optical pattern recognition,” Opt. Commun. 77, 415–422 (1990).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

Parchekani, F.

Refregier, P.

Romero, L. A.

Shamir, J.

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.

Wozencraft, J. M.

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965).

Yaroslavsky, L. P.

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics XXXII, E. Wolf, ed. (Elsevier, Amsterdam, 1993), pp. 145–201.
[CrossRef]

Zalman, G.

Zhang, G.

Appl. Opt.

IEEE Trans. Inf. Theory

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

J. Opt. Soc. Am. A

Opt. Commun.

U. Mahlab, M. Fleisher, J. Shamir, “Error probability in optical pattern recognition,” Opt. Commun. 77, 415–422 (1990).
[CrossRef]

Opt. Lett.

Other

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

J. M. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1986).

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics XXXII, E. Wolf, ed. (Elsevier, Amsterdam, 1993), pp. 145–201.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Test reference object t(x), inverse support function w(x), and first-order derivative w(x) of inverse support function.

Fig. 2
Fig. 2

Standard deviation (St.D) of location errors (in percent) as a function of nonoverlapping noise mean for the set of correlation filters (CMF, POF, GMF, MMSE, GMF_OPT). Standard deviations of white overlapping and nonoverlapping noises are equal to unity: (a) nonoverlapping noise is white, (b) nonoverlapping noise is colored with correlation coefficient of 0.95.

Fig. 3
Fig. 3

Standard deviation of location errors (in percent) as a function of standard deviation of additive white noise for the set of correlation filters (CMF, POF, GMF, MMSE, GMF_OPT). The nonoverlapping background is a constant bias mb with values (a) mb = 0, (b) mb = 0.1, (c) mb = 0.3, and (d) mb = 0.5.

Fig. 4
Fig. 4

Ratio between standard deviation of location errors and standard deviation of nonoverlapping white noise (in percent) versus ratio between standard deviations of additive white noise and nonoverlapping white noise for the set of correlation filters (CMF, POF, GMF, MMSE, GMF_OPT). The mean of the nonoverlapping background, mb, is (a) mb = 0, (b) mb = 0.1, (c) mb = 0.3, and (d) mb = 0.5.

Fig. 5
Fig. 5

Ratio between standard deviation of location errors and standard deviation of nonoverlapping colored noise (in percent) versus ratio between standard deviations of additive white noise and nonoverlapping colored noise for the set of correlation filters (CMF, POF, GMF, MMSE, GMF_OPT). The mean mb and the correlation coefficient ρ of the nonoverlapping background are (a) mb = 0, ρ = 0.5; (b) mb = 0, ρ = 0.95; (c) mb = 0.1, ρ = 0.5; (d) mb = 0.1, ρ = 0.95; (e) mb = 0.3, ρ = 0.5; (f) mb = 0.3, ρ = 0.95; (g) mb = 0.5, ρ = 0.5; and (h) mb = 0.5, ρ = 0.95.

Tables (1)

Tables Icon

Table 1 Critical Values of mb and Corresponding Standard Deviations of Location Errors for Correlation Filters with Different Correlation Coefficients While the Variances of Overlapping and Nonoverlapping Noises are Fixed

Equations (65)

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T ( f ) = t ( x ) exp ( j 2 π f x ) d x .
s ( x , x 0 ) = t ( x x 0 ) + b ( x , x 0 ) + n ( x ) .
b ( x , x 0 ) = b ( x ) w ( x x 0 ) .
c ( x , x 0 ) = c t ( x x 0 ) + c b ( x , x 0 ) + c n ( x ) ,
c t ( x ) = t ( ξ ) h ( x ξ ) d ξ
c b ( x , x 0 ) = b ( ξ , x 0 ) h ( x ξ ) d ξ ,
c n ( x ) = n ( ξ ) h ( x ξ ) d ξ
c ( x , x 0 ) = E [ c ( x , x 0 ) ] + c b 0 ( x , x 0 ) + c n ( x ) ,
E [ c ( Δ + x 0 , x 0 ) ] = E [ c ( x 0 , x 0 ) ] + E [ c ( x , x 0 ) ] | x = x 0 Δ + E [ c ( x , x 0 ) ] | x = x 0 Δ 2 2 + o ( Δ 2 ) ,
d 2 E [ c ( x , x 0 ) ] | x = x 0 = 4 π 2 f 2 H ( f ) × [ T ( f ) + m b W ( f ) ] d f .
d 1 E [ c ( x , x 0 ) ] | x = x 0 = 2 π j f H ( f ) × [ T ( f ) + m b W ( f ) ] d f .
c ( x , x 0 ) x | x = Δ 0 + x 0 = 0.
Δ 0 = d 1 + r b + r n d 2 ,
r b c b 0 ( x , x 0 ) x | x = Δ 0 + x 0 , r n c n ( x ) x | x = Δ 0 + x 0
E ( Δ 0 ) = d 1 d 2 = j f H ( f ) [ T ( f ) + m b W ( f ) ] d f 2 π f 2 H ( f ) [ T ( f ) + m b W ( f ) ] d f .
Var ( Δ 0 ) = Var ( r b ) + Var ( r n ) d 2 2 ,
Var ( r n ) = 4 π 2 f 2 N ( f ) | H ( f ) | 2 d f .
Var ( r b ) = 4 π 2 f 1 f 2 H ( f 1 ) H * ( f 2 ) B 0 ( f ) × W ( f 1 f ) W * ( f 2 f ) exp [ j 2 π Δ 0 × ( f 1 f 2 ) d f d f 1 d f 2 ,
Var ( r b ) 4 π 2 f 1 f 2 H ( f 1 ) H * ( f 2 ) B 0 ( f ) × W ( f 1 f ) W * ( f 2 f ) d f d f 1 d f 2 .
Var ( r b ) 4 π 2 f 2 B 0 ( f ) | H ( f ) | 2 d f .
Var ( Δ 0 ) [ f 1 f 2 H ( f 1 ) H * ( f 2 ) B 0 ( f ) W ( f 1 f ) W * ( f 2 f ) ] d f d f 1 d f 2 + f 2 N ( f ) | H ( f ) | 2 d f 4 π 2 | f 2 H ( f ) [ T ( f ) + m b W ( f ) ] d f | 2
E [ exp ( j 2 π f Δ 0 ) ] α δ ( f ) ,
Var ( r b ) 4 π 2 α f 2 | H ( f ) | 2 [ B 0 ( f ) ° | W ( f ) | 2 ] d f ,
Var ( Δ 0 ) f 2 | H ( f ) | 2 { [ α B 0 ( f ) | W ( f ) | 2 ] + N ( f ) } d f 4 π 2 | f 2 H ( f ) [ T ( f ) + m b W ( f ) d f ] | 2 ,
H opt ( f ) = T * ( f ) + m b W * ( f ) [ α B 0 ( f ) | W ( f ) | 2 ] + N ( f ) .
Var opt ( Δ 0 ) = { 4 π 2 [ f 2 | T ( f ) + m b W ( f ) | 2 [ α B 0 ( f ) | W ( f ) | 2 ] + N ( f ) d f ] } 1 .
H opt ( f ) T * ( f ) B 0 ( f ) + N ( f ) ,
Var opt ( Δ 0 ) { 4 π 2 [ f 2 | T ( f ) | 2 B 0 ( f ) + N ( f ) d f ] } 1 .
Var opt ( Δ 0 ) = ( 4 π 2 { ( f x cos ϕ + f y sin ϕ ) 2 | T ( f x , f y ) + m b W ( f x , f y ) | 2 [ α B 0 ( f x , f y ) | W ( f x , f y ) | 2 ] + N ( f x , f y ) d f x d f y } ) 1 ,
Var ( r b ) 4 π 2 F f 1 f 2 H ( f 1 ) H * ( f 2 ) B 0 ( f ) W ( f 1 f ) × W * ( f 2 f ) d f d f 1 d f 2 = 4 π 2 F B 0 ( f ) | F f 1 H ( f 1 ) W ( f 1 f ) d f 1 | 2 d f .
Var ( r b ) 4 π 2 F B 0 ( f ) [ F f 1 2 | H ( f 1 ) | 2 × | W ( f 1 f ) | 2 d f 1 F d f 1 ] d f = 4 π 2 F F f 2 | H ( f ) | 2 [ B 0 ( f ) | W ( f ) | 2 ] d f .
T = + g ( x ) d x g ( 0 ) .
F = + G ( f ) d f G ( 0 ) .
δ ( x ) g ( x ) = lim Δ 0 g ( x + Δ ) + g ( x Δ ) 2 = g ( x + 0 ) + g ( x 0 ) 2 ,
g ( x ) = lim Δ 0 g ( x + Δ / 2 ) g ( x Δ / 2 ) Δ .
Var ( Δ 0 ) Var b | f 1 H ( f 1 ) W ( f 1 f ) d f 1 | 2 d f + Var n f 2 | H ( f ) 2 d f 4 π 2 | f 2 H ( f ) [ T ( f ) + m b W ( f ) ] d f | 2 .
H CMF ( f ) = T * ( f ) / ( Var b + Var n ) .
m b = f 2 | T ( f ) | 2 d f / f 2 T * ( f ) W ( f ) d f ,
w ( x 1 ) = lim Δ 0 w ( x 1 + Δ / 2 ) w ( x 1 Δ / 2 ) Δ = δ ( x 1 ) , w ( x 2 ) = lim Δ 0 w ( x 2 + Δ / 2 ) w ( x 2 Δ / 2 ) Δ = δ ( x 2 ) .
d 1 = 2 π j [ f | T ( f ) | 2 d f + m b f T * ( f ) W ( f ) d f ] .
H POF ( f ) = T * ( f ) / | T ( f ) | .
m b = f 2 | T ( f ) | d f / f 2 T * ( f ) / | T ( f ) | d f .
d 2 = 4 π 2 f 2 | T ( f ) + m b W ( f ) | 2 d f .
d 2 = | t ( x ) + m b w ( x ) | 2 d x .
t ( x ) + m b w ( x ) = 0
H MMSE ( f ) = T * ( f ) + m b W * ( f ) | T ( f ) + m b W ( f ) | 2 + [ α B 0 ( f ) | W ( f ) | 2 ] + N ( f ) .
R ( τ ) = Var b exp ( λ | τ | ) ,
B 0 ( f ) = Var b 2 λ λ 2 + ( 2 π f ) 2 .
Var ( Δ 0 ) Var b 2 λ [ λ 2 + ( 2 π f ) 2 ] 1 | f 1 H ( f 1 ) W ( f 1 f ) d f 1 | 2 d f + Var n f 2 | H ( f ) | 2 d f 4 π 2 | f 2 H ( f ) [ T ( f ) + m b W ( f ) d f ] | 2 .
m b = f 2 | T ( f ) | 2 [ λ 2 + ( 2 π f ) 2 ] / { Var b 2 λ + Var n [ λ 2 + ( 2 π f ) 2 ] } d f f 2 T * ( f ) W ( f ) [ λ 2 + ( 2 π f ) 2 ] / { Var b 2 λ + Var n [ λ 2 + ( 2 π f ) 2 ] } d f .
Var ( r b ) = E ( r b r b * ) = E { b 0 ( Δ 0 + x 0 ξ ) × w ( Δ 0 ξ ) h ( ξ ) [ b 0 ( Δ 0 + x 0 β ) ] × w ( Δ 0 β ) h ( β ) ] * d ξ d β } ,
Var ( r b ) = R b ( β ξ ) w ( Δ 0 ξ ) h ( ξ ) × [ w ( Δ 0 β ) h ( β ) ] * d ξ d β = { R b ( γ ) [ h ( γ ) w ( Δ 0 γ ) ] [ h ( γ ) × w ( Δ 0 + γ ) ] * } γ = 0 .
Var ( r b ) = 4 π 2 B 0 ( f ) [ f H ( f ) W ( f ) exp ( j 2 π Δ 0 f ) ] × [ f H ( f ) W ( f ) exp ( j 2 π Δ 0 f ) ] * d f ,
Var ( r b ) = 4 π 2 f 1 f 2 H ( f 1 ) H * ( f 2 ) B 0 ( f ) × W ( f 1 f ) w * ( f 2 f ) × exp [ j 2 π Δ 0 ( f 1 f 2 ) ] d f d f 1 d f 2 .
s ( x , x 0 ) = t ( x x 0 ) + w ( x x 0 ) b ( x ) + n ( x ) = [ t ( x x 0 ) + w ( x x 0 ) m b ] + [ w ( x x 0 ) b 0 ( x ) + n ( x ) ] .
R w b ( x 1 , x 2 ) = R b 0 ( x 1 x 2 ) + + W ( f 1 ) W ( f 2 ) × exp [ i 2 π ( f 1 x 1 + f 2 x 2 ) ] × CF x 0 ( f 1 + f 2 ) d f 1 d f 2 ,
CF x 0 ( f 1 + f 2 ) α δ ( f 1 + f 2 ) ,
R w b ( x 1 , x 2 ) α R b 0 ( x 1 x 2 ) + | W ( f ) | 2 × exp [ i 2 π f ( x 1 x 2 ) ] d f ,
H w b ( f ) = α B 0 ( f ) | W ( f ) | 2 .
E { c [ ( Δ + ρ 0 ) u , x 0 , y 0 ] } = E [ c ( ρ 0 u ; x 0 , y 0 ) ] + ( u ) E [ c ( ρ u ; x 0 , y 0 ) ] | ρ = ρ 0 Δ + ( u ) 2 E [ c ( ρ u ; x 0 , y 0 ) ] | ρ = ρ 0 Δ 2 2 + o ( Δ 2 ) ,
d 2 = 4 π 2 ( f x cos ϕ + f y sin ϕ ) 2 H ( f x , f y ) [ T ( f x , f y ) + m b W ( f x , f y ) ] d f x d f y ,
N n ( f x , f y ) = 4 π 2 ( f x cos ϕ + f y sin ϕ ) 2 N ( f x , f y ) ,
Var ( r n ) = 4 π 2 ( f x cos ϕ + f y sin ϕ ) 2 × | H ( f x , f y ) | 2 N ( f x , f y ) d f x d f y .
Var ( r b ) 4 π 2 α ( f x cos ϕ + f y sin ϕ ) 2 × | H ( f x , f y ) | 2 [ B 0 ( f x , f y ) | W ( f x , f y ) | 2 ] d f x d f y ,
Var ( Δ 0 ) ( f x cos ϕ + f y sin ϕ ) 2 | H ( f x , f y ) | 2 [ α B 0 ( f x , f y ) | W ( f x , f y ) | 2 + N ( f x , f y ) ] d f x d f y 4 π 2 ( f x cos ϕ + f y sin ϕ ) 2 H ( f x , f y ) [ T ( f x , f y ) + m b W ( f x , f y ) ] d f x d f y .

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