Abstract

We investigate a channelized-ideal observer (CIO) with Laguerre–Gauss (LG) channels to approximate ideal-observer performance in detection tasks involving non-Gaussian distributed lumpy backgrounds and a Gaussian signal. A Markov-chain Monte Carlo approach is employed to determine the performance of both the ideal observer and the CIO using a large number of LG channels. Our results indicate that the CIO with LG channels can approximate ideal-observer performance within error bars, depending on the imaging system, object, and channel parameters. The CIO also outperforms a channelized-Hotelling observer using the same channels. In addition, an alternative approach for estimating the CIO is investigated. This approach makes use of the characteristic functions of channelized data and employs an approximation method to the area under the receiver operating characteristic curve. The alternative approach provides good estimates of the performance of the CIO with five LG channels. However, for large channel cases, more efficient computational methods need to be developed for the CIO to become useful in practice.

© 2007 Optical Society of America

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References

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  1. H. H. Barrett, "Objective assessment of image quality: Effects of quantum noise and object variability," J. Opt. Soc. Am. A 7, 1266-1278 (1990).
    [CrossRef] [PubMed]
  2. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Part 1) (Academic, 1968).
  3. H. H. Barrett, C. K. Abbey, and E. Clarkson, "Objective assessment of image quality III: ROC metrics, ideal observers, and likelihood-generating functions," J. Opt. Soc. Am. A 15, 1520-1535 (1998).
    [CrossRef]
  4. E. Clarkson and H. H. Barrett, "Approximation to ideal-observer performance on signal-detection tasks," Appl. Opt. 39, 1783-1794 (2000).
    [CrossRef]
  5. M. A. Kupinski, J. W. Hoppin, E. Clarkson, and H. H. Barrett, "Ideal observer computation using Markov-chain Monte Carlo," J. Opt. Soc. Am. A 20, 430-438 (2003).
    [CrossRef]
  6. S. Park, M. A. Kupinski, E. Clarkson, and H. H. Barrett, "Ideal-observer performance under signal and background uncertainty," 5A in Information Processing in Medical Imaging, Vol. 2732 in Lecture Notes in Computer Science, C.J.Taylor and J.A.Noble, eds. (Springer-Verlag, 2003), pp. 342-353.
    [CrossRef]
  7. K. J. Myers and H. H. Barrett, "Addition of a channel mechanism to the ideal observer model," J. Opt. Soc. Am. A 4, 2447-2457 (1987).
    [CrossRef] [PubMed]
  8. B. D. Gallas and H. H. Barrett, "Validating the use of channels to estimate the ideal linear observer," J. Opt. Soc. Am. A 20, 1725-1738 (2003).
    [CrossRef]
  9. H. H. Barrett, C. K. Abbey, B. D. Gallas, and M. Eckstein, "Stabilized estimates of Hotelling-observer detection performance in patient-structured noise," Proc. SPIE 3340, 27-43 (1998).
    [CrossRef]
  10. S. Park, M. A. Kupinski, E. Clarkson, and H. H. Barrett, "Efficient channels for the ideal observer," Proc. SPIE 5372, 12-21 (2004).
    [CrossRef]
  11. S. Park, "Signal detection with random backgrounds and random signals," Ph.D. dissertation (The University of Arizona, 2004).
  12. S. Park, E. Clarkson, H. H. Barrett, M. A. Kupinski, and K. J. Myers, "Performance of a channelized-ideal observer using Laguerre-Gauss channels for detecting a Gaussian signal at a known location in different lumpy backgrounds," Proc. SPIE 6146, 61460P (2006).
    [CrossRef]
  13. J. P. Rolland and H. H. Barrett, "Effect of random background inhomogeneity on observer detection performance," J. Opt. Soc. Am. A 9, 649-658 (1992).
    [CrossRef] [PubMed]
  14. E. Clarkson, M. A. Kupinski, and J. W. Hoppin, "Assessing the accuracy of estimates of the likelihood ratio," Proc. SPIE 5034, 135-143 (2003).
    [CrossRef]
  15. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  16. W. Kaplan, Advanced Calculus, 4th ed. (Addison-Wesley, 1993).
  17. E. Clarkson, M. A. Kupinski, and H. H. Barrett, "Transformation of characteristic functionals through imaging systems," Opt. Express 10, 536-539 (2002).
    [PubMed]
  18. E. Clarkson, M. A. Kupinski, and H. H. Barrett, "A probabilistic development of the MRMC method," Acad. Radiol. 13, 1410-1421 (2006).
    [CrossRef] [PubMed]
  19. B. D. Gallas, "One-shot estimate of MRMC variance: AUC," Acad. Radiol. 13, 353-362 (2006).
    [CrossRef]

2006 (3)

S. Park, E. Clarkson, H. H. Barrett, M. A. Kupinski, and K. J. Myers, "Performance of a channelized-ideal observer using Laguerre-Gauss channels for detecting a Gaussian signal at a known location in different lumpy backgrounds," Proc. SPIE 6146, 61460P (2006).
[CrossRef]

E. Clarkson, M. A. Kupinski, and H. H. Barrett, "A probabilistic development of the MRMC method," Acad. Radiol. 13, 1410-1421 (2006).
[CrossRef] [PubMed]

B. D. Gallas, "One-shot estimate of MRMC variance: AUC," Acad. Radiol. 13, 353-362 (2006).
[CrossRef]

2004 (1)

S. Park, M. A. Kupinski, E. Clarkson, and H. H. Barrett, "Efficient channels for the ideal observer," Proc. SPIE 5372, 12-21 (2004).
[CrossRef]

2003 (3)

2002 (1)

2000 (1)

1998 (2)

H. H. Barrett, C. K. Abbey, B. D. Gallas, and M. Eckstein, "Stabilized estimates of Hotelling-observer detection performance in patient-structured noise," Proc. SPIE 3340, 27-43 (1998).
[CrossRef]

H. H. Barrett, C. K. Abbey, and E. Clarkson, "Objective assessment of image quality III: ROC metrics, ideal observers, and likelihood-generating functions," J. Opt. Soc. Am. A 15, 1520-1535 (1998).
[CrossRef]

1992 (1)

1990 (1)

1987 (1)

Acad. Radiol. (2)

E. Clarkson, M. A. Kupinski, and H. H. Barrett, "A probabilistic development of the MRMC method," Acad. Radiol. 13, 1410-1421 (2006).
[CrossRef] [PubMed]

B. D. Gallas, "One-shot estimate of MRMC variance: AUC," Acad. Radiol. 13, 353-362 (2006).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (6)

Opt. Express (1)

Proc. SPIE (4)

E. Clarkson, M. A. Kupinski, and J. W. Hoppin, "Assessing the accuracy of estimates of the likelihood ratio," Proc. SPIE 5034, 135-143 (2003).
[CrossRef]

S. Park, E. Clarkson, H. H. Barrett, M. A. Kupinski, and K. J. Myers, "Performance of a channelized-ideal observer using Laguerre-Gauss channels for detecting a Gaussian signal at a known location in different lumpy backgrounds," Proc. SPIE 6146, 61460P (2006).
[CrossRef]

H. H. Barrett, C. K. Abbey, B. D. Gallas, and M. Eckstein, "Stabilized estimates of Hotelling-observer detection performance in patient-structured noise," Proc. SPIE 3340, 27-43 (1998).
[CrossRef]

S. Park, M. A. Kupinski, E. Clarkson, and H. H. Barrett, "Efficient channels for the ideal observer," Proc. SPIE 5372, 12-21 (2004).
[CrossRef]

Other (5)

S. Park, "Signal detection with random backgrounds and random signals," Ph.D. dissertation (The University of Arizona, 2004).

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Part 1) (Academic, 1968).

S. Park, M. A. Kupinski, E. Clarkson, and H. H. Barrett, "Ideal-observer performance under signal and background uncertainty," 5A in Information Processing in Medical Imaging, Vol. 2732 in Lecture Notes in Computer Science, C.J.Taylor and J.A.Noble, eds. (Springer-Verlag, 2003), pp. 342-353.
[CrossRef]

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

W. Kaplan, Advanced Calculus, 4th ed. (Addison-Wesley, 1993).

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Figures (13)

Fig. 1
Fig. 1

The left picture shows a 64 × 64 noiseless Gaussian-signal image. The middle and right pictures show 64 × 64 LB images with N ¯ = 5 and 12 , respectively.

Fig. 2
Fig. 2

The four rows show 64 × 64 LG channels with a u = 5 , 10 , 16 , and 23 from top to bottom. The three columns show 64 × 64 LG channels with p = 0 (left), p = 4 (middle), and p = 9 (right).

Fig. 3
Fig. 3

Plots (a)–(e) show profiles of the real part of an estimate of the CF under the signal-absent hypothesis, ψ v H 0 ( ω ) , calculated at surfaces where only ω k 0 , k = 1 , , 5 . For these plots, N ¯ = 12 , a u = 5 , N c = 5 , and L ω = 0.025 .

Fig. 4
Fig. 4

As Fig. 3, for the imaginary part.

Fig. 5
Fig. 5

As Fig. 3, but for ψ v H 1 ( ω ) .

Fig. 6
Fig. 6

As Fig. 4, but for ψ v H 1 ( ω ) .

Fig. 7
Fig. 7

Plots (a)–(e) show estimates of the marginal PDFs of v k , k = 1 , , 5 , obtained by integrating estimates of the PDF functions, pr ( v H j ) , j = 0 , 1 , respectively, over { v n n k , n = 1 , , 5 } . For these plots, N ¯ = 12 , a = 5 , and N c = 5 .

Fig. 8
Fig. 8

Plots (a)–(e) show empirical histograms of v k , k = 1 , , 5 , obtained from 5000 channelized signal-absent lumpy-background images. For these plots, N ¯ = 12 , a = 5 , and N c = 5 .

Fig. 9
Fig. 9

Performance of the ideal observer and the CIO using 5 to 40 LG channels with the MCMC algorithm. The lumpy images with N ¯ = 5 were used. The channel width parameters are 5 and 10 for (a) and 16 and 23 for (b). The dotted curve represents ± 2 standard errors for ideal-observer performance. All error bars are ± 2 standard errors by the one-shot method.

Fig. 10
Fig. 10

As Fig. 9, but for N ¯ = 12 .

Fig. 11
Fig. 11

As Fig. 9, but for the ideal observer and the CHO.

Fig. 12
Fig. 12

As Fig. 11, but for N ¯ = 12 .

Fig. 13
Fig. 13

Comparison of the MCMC algorithm and the approach with the method of G ( 0 ) approximation to the AUC and the IFFT algorithm for estimating the performance of the CIO using five LG channels. The mean number N ¯ of lumps is 5 and 12, respectively, for (a) and (b). Values for the width parameter a u of LG channels are 5 , 10 , 16 , and 23. Error bars for MCMC estimates are ± 2 standard errors by the one-shot method. For the approach with the G ( 0 ) approximation, errors are approximation errors.

Tables (4)

Tables Icon

Table 1 Object and System Parameters

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Table 2 Performance of the Ideal Observer, the CIO, and the CHO a

Tables Icon

Table 3 Comparison of the Markov-Chain Monte Carlo Algorithm and the Approach Using the Method of G ( 0 ) Approximation to the AUC and the IFFT Algorithm to Determine the Performance of the CIO Using Five LG Channels a

Tables Icon

Table 4 Efficiencies of the CIO and the CHO Relative to the Ideal Observer a

Equations (58)

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g = H f + n ,
h m ( r ) = h 2 π w 2 exp [ ( r p m ) t ( r p m ) 2 w 2 ] ,
H 0 : g = H f b + n ,
H 1 : g = H ( f b + f s ) + n ,
p r ( n H j ) = [ ( 2 π ) M det ( K n ) ] 1 2 exp [ 1 2 n t K n 1 n ] = [ 2 π σ n 2 ] M 2 exp [ n t n 2 σ n 2 ] ,
g m = S d r h m ( r ) f ( r ) + n m ,
f b f b ( r ) = n = 1 N l b ( r c n a b , σ b ) ,
l b ( r c n a b , σ b ) = a b exp ( ( r c n ) t ( r c n ) 2 σ b 2 ) .
f s = l s ( r c s a s , σ s ) = a s exp ( ( r c s ) t ( r c s ) 2 σ s 2 ) ,
Λ ( g ) = pr ( g H 1 ) pr ( g H 0 ) ,
Λ ( g ) = d θ Λ BKE ( g b ( θ ) ) pr ( θ g , H 0 ) ,
pr ( θ g , H 0 ) = pr ( g b ( θ ) , H 0 ) pr ( θ ) d θ pr ( g b ( θ ) , H 0 ) pr ( θ ) .
Λ BKE ( g b ( θ ) ) = C 1 exp ( { g b ( θ ) } t K n 1 s ) ,
Λ ̂ ( g ) = 1 J J 0 i = J 0 + 1 J Λ BKE [ g b ( θ ( i ) ) ] ,
v = Tg = T ( H f + n ) .
u p ( r a u ) = 2 a u exp ( π r 2 a u 2 ) L p ( 2 π r 2 a u 2 ) ,
L p ( x ) = k = 0 p ( 1 ) k ( p k ) x k k ! .
Λ ( v ) = pr ( v H 1 ) pr ( v H 0 ) ,
Λ ( v ) = d θ Λ BKE ( v b ( θ ) ) pr ( θ v , H 0 ) ,
Λ BKE ( v b ( θ ) ) = pr ( v b ( θ ) , H 1 ) pr ( v b ( θ ) , H 0 )
pr ( θ v , H 0 ) = pr ( v b ( θ ) , H 0 ) pr ( θ ) d θ pr ( v b ( θ ) , H 0 ) pr ( θ ) .
Λ ̂ ( v ) = 1 J J 0 i = J 0 + 1 J Λ BKE ( v b ( θ ( i ) ) ) ,
pr ( v b ( θ ) , H j ) = exp ( ( 2 σ n 2 ) 1 [ T ( g b ( θ ) j s ) ] t [ TT t ] 1 [ T ( g b ( θ ) j s ) ] ) ( 2 π ) N c det ( TK n T t ) .
Λ BKE ( v b ( θ ) ) = exp ( 1 σ n 2 [ T ( g b ( θ ) s 2 ) ] t [ TT t ] 1 [ Ts ] ) .
α ( θ ̃ , θ ( i ) ) = min [ 1 , pr ( θ ̃ g , H 0 ) q ( θ ̃ θ ( i ) ) pr ( θ ( i ) g , H 0 ) q ( θ ( i ) θ ̃ ) ] ,
t v = ( K v 1 v s ) t v ,
K v = TK b T t + σ n 2 TT t ,
ψ v ( ω ) = exp ( 2 π i ω v ) v ,
Ψ f ( ξ ) = exp ( 2 π i ξ f ) f ,
Ψ f b ( ξ ) = exp ( N ¯ [ 1 Ψ l b ( ξ ) ] ) ,
Ψ l b ( ξ ) = exp ( 2 π i d r ξ ( r ) l b ( r c a b , σ b ) ) c .
Ψ f s ( ξ ) = exp ( 2 π i d r ξ ( r ) l s ( r c s a s , σ s ) ) .
ψ v ( ω ) = ψ g ( T t ω ) .
ψ v H j ( ω ) = Ψ f H j [ H T t ω ] ψ n ( T t ω ) , j = 0 , 1 ,
Ψ f H 0 [ H T t ω ] = Ψ f b [ H T t ω ]
Ψ f H 1 [ H T t ω ] = Ψ f b [ H T t ω ] Ψ f s [ H T t ω ] .
Ψ f b [ H T t ω ] = exp [ N ¯ { 1 Ψ l b [ H T t ω ] } ] ,
Ψ l b [ H T t ω ] = exp ( 2 π i d r H T t ω ( r ) l b ( r c a b , σ b ) ) c
Ψ f s [ H T t ω ] = exp ( 2 π i d r H T t ω ( r ) l s ( r c s a s , σ s ) ) .
pr ( v H j ) = d N c ω ψ v H j ( ω ) exp ( 2 π i ω v ) , j = 0 , 1 ,
AUC Λ 1 2 + 1 2 erf ( 1 2 SNR G ( 0 ) ) ,
G ( 0 ) = 4 log ( Λ ( v ) H 0 ) .
G ( 0 ) = 4 log ( d N c v pr ( v H 1 ) pr ( v H 0 ) pr ( v H 0 ) ) = 4 log ( d N c v pr ( v H 1 ) pr ( v H 0 ) ) ,
Ψ l b [ H T t ω ] 1 N m c k = 1 N m c exp ( 2 π i d r H T t ω ( r ) l b ( r c k a b , σ b ) )
= 1 N m c k = 1 N m c exp ( 2 π i m = 1 M { T t ω } m d r h m ( r ) l b ( r c k a b , σ b ) )
= 1 N m c k = 1 N m c exp ( 2 π i ω t Ty k ) ,
y k = { d r h m ( r ) l b ( r c k a b , σ b ) } m , m = 1 , , M .
Ψ f s [ H T t ω ] = exp ( 2 π i ω t Tx ) ,
x = { d r h m ( r ) l s ( r c s a s , σ s ) } m , m = 1 , , M .
G ( 0 ) 4 log ( p ( N v ) N d pr ( v p H 1 ) pr ( v p H 0 ) ) .
Ψ f s ( ξ ) = exp ( 2 π i d r ξ ( r ) l s ( r c s a s , σ s ) ) c s .
g ¯ = g n ,
ψ g ¯ ( ρ ) = exp ( 2 π i ρ { H f } ) g ¯ = exp ( 2 π i { H ρ } f ) f = Ψ f ( H ρ ) ,
ψ g ( ρ ) = exp ( 2 π i ρ { H f + n } ) g
= exp ( 2 π i ρ g ¯ ) g ¯ exp ( 2 π i ρ n ) n
= ψ g ¯ ( ρ ) ψ n ( ρ ) ,
ψ g ¯ ( ρ ) = Ψ f ( H ρ ) = Ψ f b + f s ( H ρ ) = Ψ f b ( H ρ ) Ψ f s ( H ρ ) ,
ψ g ( ρ ) = Ψ f b ( H ρ ) Ψ f s ( H ρ ) ψ n ( ρ ) .

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