Abstract

We present a statistical-analysis technique for a nonlinear joint transform correlator (JTC) based on two assumptions: the noise and the signal spectra are identical, and the signal energy is small relative to the noise energy. The first assumption, while admittedly convenient, is also defensible in that it is a worst case and in that image and scene noise can be similar in texture. The second is also reasonable, given that even a clearly visible signal may have small energy compared with the scene noise if it is of limited extent; in any case, the results appear moderately faithful even for the case that signal and noise energies are equal. We discover that the optimal Fourier-plane transformation is spatially variant and tends to remove the Fourier amplitudes of the input image, and indeed functions in a way very similar to the spatially variant binary JTC. We also see that the classic (or spatially invariant linear) JTC is a very inferior technique for signallike noise, that the best spatially variant binary JTC uses a threshold proportional to the noise power spectrum, and that, if a spatially invariant binary-thresholded JTC is desired, then the median Fourier-plane value is an excellent choice of threshold. The performance predictions are verified by simulation and appear to be reasonable even for the highly nonlinear binary schemes.

© 1995 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [PubMed]
  12. S. Kassam, Signal Detection in Non-Gaussian Noise (Springer-Verlag, New York, 1987).
  13. H. Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1987).

1994

1992

W. Hahn, D. Flannery, “Basic design elements of the binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

1991

1990

K. Fielding, J. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

1989

1984

1970

1966

Fielding, K.

K. Fielding, J. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Flannery, D.

W. Hahn, D. Flannery, “Basic design elements of the binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Gianino, P.

Goodman, J.

Hahn, W.

W. Hahn, D. Flannery, “Basic design elements of the binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Horner, J.

K. Fielding, J. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

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

Javidi, B.

Kabrisky, M.

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Kassam, S.

S. Kassam, Signal Detection in Non-Gaussian Noise (Springer-Verlag, New York, 1987).

Kline, J.

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Kozma, A.

Laude, V.

P. Réfrégier, V. Laude, B. Javidi, “Nonlinear joint transform correlation, an optimum solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).
[PubMed]

P. Réfrégier, V. Laude, “Critical analysis of filtering techniques for optical pattern recognition,” in Proceedings of the Euro-American Workshop on Optical Pattern Recognition, P. Réfrégier, B. Javidi, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 58–83.

Mills, J.

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Poor, H.

H. Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1987).

Réfrégier, P.

P. Réfrégier, V. Laude, B. Javidi, “Nonlinear joint transform correlation, an optimum solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).
[PubMed]

P. Réfrégier, V. Laude, “Critical analysis of filtering techniques for optical pattern recognition,” in Proceedings of the Euro-American Workshop on Optical Pattern Recognition, P. Réfrégier, B. Javidi, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 58–83.

Rogers, S.

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Rotz, F.

Tang, Q.

VanderLugt, A.

Wang, J.

Weaver, C.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

W. Hahn, D. Flannery, “Basic design elements of the binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

K. Fielding, J. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Opt. Lett.

Other

P. Réfrégier, V. Laude, “Critical analysis of filtering techniques for optical pattern recognition,” in Proceedings of the Euro-American Workshop on Optical Pattern Recognition, P. Réfrégier, B. Javidi, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), pp. 58–83.

S. Kassam, Signal Detection in Non-Gaussian Noise (Springer-Verlag, New York, 1987).

H. Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, New York, 1987).

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Figures (8)

Fig. 1
Fig. 1

Nonlinear JTC.

Fig. 2
Fig. 2

Analytical and simulated performance of optimal (linear spatially variant, or LSV) JTC (bandwidth parameter W = 6; the solid curve is from theory; the circles refer to simulation results).

Fig. 3
Fig. 3

Analytical and simulated performance of the classical (linear spatially-invariant, or LSI) JTC (bandwidth parameter W = 6; the solid curve is from theory; the circles refer to simulation results).

Fig. 4
Fig. 4

Analytical and simulated performance of the binary spatially variant (BSV) JTC (bandwidth parameter W = 6; the solid curve is from theory; the circles refer to simulation results).

Fig. 5
Fig. 5

Performance of the binary spatially invariant (BSI) JTC versus bandwidth W for various thresholds [t is the threshold; k is the maximum value of ψ(α, β)]; T is the spatial extent of the Fourier plane.

Fig. 6
Fig. 6

Performance of the BSI JTC versus the threshold for various spectral shapes [t/k is the ratio of the threshold to the maximum value of ψ(α, β); the circles refer to performance with the median as the threshold, from theory].

Fig. 7
Fig. 7

Performance of the BSI JTC with the median as the threshold (the solid line is from theory; the circles refer to simulation results).

Fig. 8
Fig. 8

Analytical and simulated performance of various JTC’s versus bandwidth W (signal strength parameter θ = 0.5; the solid curves are from theory; other marks refer to simulation results). LSV, linear spatially variant (i.e., optimal); BSV, binary spatially variant; BSI, binary spatially invariant; LSI, linear spatially invariant (i.e., the classical JTC).

Equations (73)

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a ( x , y ) = s ( x , y ) + r ( x - x 0 , y ) ,
s ( x , y ) = r ( x + x 0 , y ) + ν ( x , y ) .
A ( α , β ) = - S S - S S a ( x , y ) exp [ - j ( α x + β y ) ] d x d y .
h ( x ^ , y ^ ) = | - T T - T T g α , β [ B ( α , β ) ] exp [ j ( x ^ α + y ^ β ) ] d α d β | 2 ,
a ( x , y ) = ν ( x , y ) + r ( x + x 0 , y ) + r ( x - x 0 , y ) ,
A ( α , β ) = N ( α , β ) + R ( α , β ) exp ( j α x 0 ) + R ( α , β ) exp ( - j α x 0 ) ,
R ( α , β ) = R ( α , β ) exp [ j Φ r ( α , β ) ] , N ( α , β ) = N ( α , β ) exp [ j Φ n ( α , β ) ]
ψ n ( α , β ) = - - ɛ { ν ( x , y ) ν ( x + u , y + v ) } × exp [ - j ( u α + v β ) ] d u d v
ɛ { N ( α , β ) 2 } = ɛ { - S 0 - S S - S 0 - S S ν ( x , y ) ν ( u , v ) × exp { - j [ ( x - u ) α + ( y - v ) β ] } × d x d y d u d v }
ɛ { N ( α , β ) 2 } - S 0 - S S ψ n ( α , β ) d x d y = 2 S 2 ψ n ( α , β ) ψ ( α , β ) .
N ( α , β ) = z ( α , β ) [ ψ ( α , β ) ] 1 / 2 ,
R ( α , β ) = θ [ ψ ( α , β ) ] 1 / 2 exp [ j Φ r ( α , β ) ] .
a θ ( x , y ) = ν ( x , y ) + r ( x + x 0 , y ) + r ( x - x 0 , y ) .
A θ ( α , β ) = N ( α , β ) + R ( α , β ) exp ( j α x 0 ) + R ( α , β ) exp ( - j α x 0 ) ,
B θ ( α , β ) = N ( α , β ) 2 + 2 R ( α , β ) 2 + 2 Re { R ( α , β ) N * ( α , β ) exp ( - j α x 0 ) } + 2 Re { R ( α , β ) N * ( α , β ) exp ( j α x 0 ) } + 2 R ( α , β ) 2 cos ( 2 α x 0 ) ,
B θ ( α , β ) = z 2 ( α , β ) ψ ( α , β ) + 2 θ 2 ψ ( α , β ) + 4 θ z ( α , β ) ψ ( α , β ) cos [ Φ r ( α , β ) - Φ n ( α , β ) ] cos ( α x 0 ) + 2 θ 2 ψ ( α , β ) cos ( 2 α x 0 ) ,
Re { R ( α , β ) N * ( α , β ) [ exp ( j α x 0 ) + exp ( - j α x 0 ) ] } = 2 Re { R ( α , β ) N * ( α , β ) cos ( α x 0 ) } = 2 Re { R ( α , β ) exp [ j Φ r ( α , β ) ] N ( α , β ) × exp [ - j Φ n ( α , β ) ] cos ( α x 0 ) } = 2 R ( α , β ) N ( α , β ) × cos [ Φ r ( α , β ) - Φ n ( α , β ) ] cos ( α x 0 ) .
B 0 ( α , β ) = z 2 ( α , β ) ψ ( α , β ) ,
B ˙ 0 ( α , β ) = 4 z ( α , β ) ψ ( α , β ) cos [ Φ r ( α , β ) - Φ n ( α , β ) ] × cos ( α x 0 ) ,
B ¨ 0 ( α , β ) = 4 ψ ( α , β ) [ 1 + cos ( 2 α x 0 ) ] .
h ( x ^ , y ^ ) = IFT { g α , β [ B θ ( α , β ) ] } 2 ,
ɛ { h ( x ^ , y ^ ) } = - T T - T T - T T - T T ɛ { g α , β [ B θ ( α , β ) ] g γ , δ [ B θ ( γ , δ ) ] } × exp { j [ ( α - γ ) x ^ + ( β - ) y ^ ] } d α d β d γ d .
ɛ { g α , β [ B θ ( α , β ) ] g γ , δ [ B θ ( γ , δ ) ] } = [ ɛ { g α , β 2 [ B θ ( α , β ) ] } - ( ɛ { g α , β [ B θ ( α , β ) ] } ) 2 ] × δ ( α - γ ) δ ( β - δ ) + ɛ { g α , β [ B θ ( α , β ) ] } × ɛ { g γ , δ [ B θ ( γ , δ ) ] } ,
ɛ { h ( x ^ , y ^ ) } = - T T - T T [ ɛ { g α , β 2 [ B θ ( α , β ) ] } - ( ɛ { g α , β [ B θ ( α , β ) ] } ) 2 ] d α d β + | - T T - T T ɛ { g α , β [ B θ ( α , β ) ] } × exp [ j ( α x ^ + β y ^ ) ] d α d β | 2 .
g α , β [ B θ ( α , β ) ] g α , β [ B 0 ( α , β ) ] + θ B ˙ 0 ( α , β ) g ˙ α , β [ B 0 ( α , β ) ] + ½ θ 2 B ¨ 0 ( α , β ) g ˙ α , β [ B 0 ( α , β ) ] + ½ θ 2 [ B ˙ 0 ( α , β ) ] 2 g ¨ α , β [ B 0 ( α , β ) ]
ɛ { g α , β [ B 0 ( α , β ) ] } = ɛ { g α , β [ z 2 ( α , β ) ψ ( α , β ) ] } .
ɛ { g α , β [ B 0 ( α , β ) ] } = ɛ { g α , β [ z 2 ψ ( α , β ) ] } = 0 g α , β [ z 2 ψ ( α , β ) ] f z ( z ) d z .
ɛ { B ˙ 0 ( α , β ) g ˙ α , β [ B 0 ( α , β ) ] } = 4 ψ ( α , β ) cos ( α x 0 ) ɛ { z g ˙ α , β [ z 2 ψ ( α , β ) ] } × ɛ { cos [ Φ r ( α , β ) - Φ n ( α , β ) ] } = 0 ,
ɛ { B ¨ 0 ( α , β ) g ˙ α , β [ B 0 ( α , β ) ] } = 4 [ 1 + cos ( 2 α x 0 ) ] × ψ ( α , β ) ɛ { g ˙ α , β [ z 2 ψ ( α , β ) ] } .
ɛ { [ B ˙ 0 ( α , β ) ] 2 g ¨ α , β [ B 0 ( α , β ) ] } = 16 ψ 2 ( α , β ) cos 2 ( α x 0 ) ɛ { z 2 g ¨ α , β [ z 2 ψ ( α , β ) ] } × ɛ { cos 2 [ Φ r ( α , β ) - Φ n ( α , β ) ] }
= 8 ψ 2 ( α , β ) cos 2 ( α x 0 ) ɛ { z 2 g ¨ α , β [ z 2 ψ ( α , β ) ] } ,
ɛ { g α , β [ B θ ( α , β ) ] } ɛ { g α , β [ z 2 ψ ( α , β ) ] } + 2 θ 2 [ 1 + cos ( 2 α x 0 ) ] ψ ( α , β ) × ɛ { g ˙ α , β [ z 2 ψ ( α , β ) ] } + 4 θ 2 ψ 2 ( α , β ) cos 2 ( α x 0 ) × ɛ { z 2 g ¨ α , β [ z 2 ψ ( α , β ) ] } .
ɛ { g α , β [ B θ ( α , β ) ] } ɛ { g } + 2 θ 2 [ 1 + cos ( 2 α x 0 ) ] × ( ψ ɛ { g ˙ } + ψ 2 ɛ { z 2 g ¨ } ) ,
ɛ { g α , β 2 [ B θ ( α , β ) ] } ɛ { g 2 } + 4 θ 2 [ 1 + cos ( 2 α x 0 ) ] × [ ψ ɛ { g g ˙ } + ψ 2 ɛ { z 2 g g ¨ } + ψ 2 ɛ { ( z g ˙ ) 2 } ] + 4 θ 4 ψ 2 [ 1 + cos ( α x 0 ) ] 2 ɛ { ( g ˙ ) 2 } + 16 θ 4 ψ 3 [ 1 + cos ( 2 α x 0 ) ] 2 ɛ { z 2 g ˙ g ¨ } + 6 θ 4 ψ 4 [ 1 + cos ( α x 0 ) ] 2 ɛ { ( z g ˙ ) 4 } .
ɛ { h n ( x ^ , y ^ ) } = - T T - T T [ ɛ { g α , β 2 [ B θ ( α , β ) ] } - ( ɛ { g α , β [ B θ ( α , β ) ] } ) 2 ] d α d β ,
ɛ { h n ( x ^ , y ^ ) } - T T - T T [ ɛ { g 2 } + 4 θ 2 × ( ψ ɛ { g g ˙ } + ψ 2 ɛ { z 2 g g ¨ } + ψ 2 ɛ { z 2 g ˙ 2 } ) - ( ɛ { g } ) 2 - 4 θ 2 ɛ { g } ( ψ ɛ { g ˙ } + ψ 2 ɛ { z 2 g ¨ } ) ] d α d β ,
ɛ { h s ( x ^ , y ^ ) } = | - T T - T T ɛ { g α , β [ B θ ( α , β ) ] } × exp [ j ( α x ^ + β y ^ ) ] d α d β | 2 .
ɛ { h s ( x ^ , y ^ ) } | - T T - T T ( ɛ { g } + 2 θ 2 [ 1 + cos ( 2 α x 0 ) ] × ( ψ ɛ { g ˙ } + ψ 2 ɛ { z 2 g ¨ } ) ) exp [ j ( α x ^ + β y ^ ) ] d α d β | 2 ,
ɛ { h s ( 2 x 0 , 0 ) } θ 4 [ - T T - T T ( ψ ɛ { g ˙ } + ψ 2 ɛ { z 2 g ¨ } ) d α d β ] 2 .
f μ ( μ ) = 1 μ f z ( μ ) ,
ψ ɛ { g ˙ } = - g ( 0 ) f μ ( 0 ) - 0 g ( μ ψ ) f ˙ μ ( μ ) d μ ,
ψ 2 ɛ { z 2 g ¨ } = g ( 0 ) f μ ( 0 ) + 0 g ( μ ψ ) [ 2 f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ ,
ψ ɛ { g g ˙ } = - 1 2 g 2 ( 0 ) f μ ( 0 ) - 1 2 0 g 2 ( μ ψ ) f ˙ μ ( μ ) d μ ,
ψ 2 ɛ { z 2 g g ¨ + z 2 g ˙ 2 } = 1 2 g 2 ( 0 ) f μ ( 0 ) + 1 2 0 g 2 ( μ ψ ) × [ 2 f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ ,
SNR = ɛ { h s ( x ^ , y ^ ) } [ ɛ { h n ( x ^ , y ^ ) } ] - 1 ( - T T - T T { 0 g ( μ ψ ) [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ } d α d β ) 2 × { - T T - T T [ 0 g 2 ( μ ψ ) { f μ ( μ ) + 2 θ 2 [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] } d μ - [ 0 g ( μ ψ ) f μ ( μ ) d μ ] × ( 0 g ( μ ψ ) { f μ ( μ ) + 4 θ 2 [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] } d μ ) ] × d α d β } - 1 ,
θ 2 = energy in r ( x , y ) expected energy in ν ( x , y ) .
SNR θ 4 ( - T T - T T { 0 g ( μ ψ ) [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ } d α d β ) 2 - T T - T T { 0 g 2 ( μ ψ ) f μ ( μ ) d μ - [ 0 g ( μ ψ ) f μ ( μ ) d μ ] 2 } d α d β .
g ¯ ( α , β ) 0 g α , β [ μ ψ ( α , β ) ] f μ ( μ ) d μ
{ - T T - T T 0 g α , β [ μ ψ ( α , β ) ] [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ d α d β } 2 = ( - T T - T T 0 { g α , β [ μ ψ ( α , β ) ] - g ¯ ( α , β ) } × [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) d μ d α d β ] ) 2 ( - T T - T T 0 { g α , β [ μ ψ ( α , β ) ] - g ¯ ( α , β ) } 2 f μ ( μ ) d μ d α d β ) × { - T T - T T 0 [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] 2 f μ ( μ ) d μ d α d β }
SNR θ 4 { - T T - T T 0 [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] 2 f μ ( μ ) d μ d α d β } ,
g α , β [ μ ψ ( α , β ) ] - g ¯ ( α , β ) = κ [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] f μ ( μ )
f μ ( μ ) = exp ( - μ ) u ( μ ) ,
g α , β [ μ ψ ( α , β ) ] = κ ( μ - 1 ) + g ¯ ( α , β )
g α , β [ μ ψ ( α , β ) ] = μ
g α , β ( μ ) = μ ψ ( α , β ) ,
SNR opt = 4 θ 4 T 2 1 + 4 θ 2 .
SNR LSI = θ 4 [ - T T - T T ψ ( α , β ) d α d β ] 2 ( 1 + 4 θ 2 ) - T T - T T ψ 2 ( α , β ) d α d β ,
SNR LSI 4 T 2 θ 4 1 + 4 θ 2
ψ ( α , β ) = κ exp [ - ( α 2 + β 2 ) / 2 W 2 ]
SNR LSI θ 4 4 π W 2 1 + 4 θ 2 .
ψ ( α , β ) = κ 1 + [ ( α 2 + β 2 ) / W 2 ] 2 ,
SNR LSI θ 4 π 2 W 2 1 + 4 θ 2 .
[ - - ψ ( α W , β W ) d α d β ] 2 - - ψ 2 ( α W , β W ) d α d β ,
W 2 [ - - ψ ( α ^ , β ^ ) d α ^ d β ^ ] 2 - - ψ 2 ( α ^ , β ^ ) d α ^ d β ^ .
g α , β ( x ) = u [ x - τ ψ ( α , β ) ] ,
SNR BJTC - SV = 4 θ 4 T 2 τ 2 exp ( - τ ) 1 - exp ( - τ ) + 2 θ 2 τ [ 1 - 2 exp ( - τ ) ] .
g α , β ( x ) = u ( x - τ ) ,
0 g α , β [ μ ψ ( α , β ) ] [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ = 0 u [ μ ψ ( α , β ) - τ ] ( μ - 1 ) exp ( - μ ) d μ = τ ψ ( α , β ) exp [ - τ / ψ ( α , β ) ] ,
0 g α , β 2 [ μ ψ ( α , β ) ] f μ ( μ ) d μ = 0 u [ μ ψ ( α , β ) - τ ] exp ( - μ ) d μ = exp [ - τ / ψ ( α , β ) ] = 0 g α , β [ μ ψ ( α , β ) ] f μ ( μ ) d μ .
SNR BJTC - SI = θ 4 [ - T T - T T τ ψ exp ( - τ / ψ ) d α d β ] 2 - T T - T T [ exp ( - τ / ψ ) ( 1 + 2 θ 2 τ ψ ) - exp ( - 2 τ / ψ ) ( 1 + 4 θ 2 τ ψ ) ] d α d β .
SNR BJTC - SI = θ 4 [ A ( τ / κ ) ] 2 B ( τ / κ ) - B ( 2 τ / κ ) + 2 θ 2 [ A ( τ / κ ) - A ( 2 τ / κ ) ] ,
A ( x ) = 2 π x exp ( - x ) [ ( 1 + 2 x ) π W 2 erf ( x T 2 / W 2 ) 8 x 3 / 2 - T 2 4 x exp ( - x T 4 / W 4 ) ] , B ( x ) = 2 π exp ( - x ) π W 2 erf ( x T 2 / W 2 ) 4 x ,
τ median = κ [ 1 + 1 4 ( T W ) 4 ] ,

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