Abstract

Fisher information can be used as a surrogate for task-based measures of image quality based on ideal observer performance. A new and improved derivation of the Fisher information approximation for ideal-observer detectability is provided. This approximation depends only on the presence of a weak signal and does not depend on Gaussian statistical assumptions. This is also not an asymptotic result and therefore applies to imaging, where there is typically only one dataset, albeit a large one. Applications to statistical mixture models for image data are presented. For Gaussian and Poisson mixture models the results are used to connect reconstruction error with ideal-observer detection performance. When the task is the estimation of signal parameters of a weak signal, the ensemble mean squared error of the posterior mean estimator can also be expanded in powers of the signal amplitude. There is no linear term in this expansion, and it is shown that the quadratic term involves a Fisher information kernel that generalizes the standard Fisher information. Applications to imaging mixture models reveal a close connection between ideal performance on these estimation tasks and detection tasks for the same signals. Finally, for tasks that combine detection and estimation, we may also define a detectability that measures performance on this combined task and an ideal observer that maximizes this detectability. This detectability may also be expanded in powers of the signal amplitude, and the quadratic term again involves the Fisher information kernel. Applications of this approximation to imaging mixture models show a relation with the pure detection and pure estimation tasks for the same signals.

© 2010 Optical Society of America

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References

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  1. J. Shao, Mathematical Statistics (Springer, New York, 1999).
  2. F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images” 23, 2406–2414 (2006).
  3. H. H. Barrett, C. K. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers and likelihood generating functions,” 15, 1520–1535 (1998).
  4. E. Clarkson, “Estimation receiver operating characteristic curve and ideal observers for combined detection/estimation tasks,” 24, 91–98 (2007).
  5. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).
  6. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  7. S. M. Kay, Fundamentals of Statistical Signal Processing II: Detection Theory (Prentice Hall, 2008).

2008 (1)

S. M. Kay, Fundamentals of Statistical Signal Processing II: Detection Theory (Prentice Hall, 2008).

2007 (2)

E. Clarkson, “Estimation receiver operating characteristic curve and ideal observers for combined detection/estimation tasks,” 24, 91–98 (2007).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

2006 (1)

F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images” 23, 2406–2414 (2006).

2004 (1)

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

1999 (1)

J. Shao, Mathematical Statistics (Springer, New York, 1999).

1998 (1)

H. H. Barrett, C. K. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers and likelihood generating functions,” 15, 1520–1535 (1998).

Abbey, C. K.

H. H. Barrett, C. K. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers and likelihood generating functions,” 15, 1520–1535 (1998).

Barrett, H. H.

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

H. H. Barrett, C. K. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers and likelihood generating functions,” 15, 1520–1535 (1998).

Clarkson, E.

E. Clarkson, “Estimation receiver operating characteristic curve and ideal observers for combined detection/estimation tasks,” 24, 91–98 (2007).

F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images” 23, 2406–2414 (2006).

H. H. Barrett, C. K. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers and likelihood generating functions,” 15, 1520–1535 (1998).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing II: Detection Theory (Prentice Hall, 2008).

Myers, K. J.

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

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

Shao, J.

J. Shao, Mathematical Statistics (Springer, New York, 1999).

Shen, F.

F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images” 23, 2406–2414 (2006).

Other (7)

J. Shao, Mathematical Statistics (Springer, New York, 1999).

F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images” 23, 2406–2414 (2006).

H. H. Barrett, C. K. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers and likelihood generating functions,” 15, 1520–1535 (1998).

E. Clarkson, “Estimation receiver operating characteristic curve and ideal observers for combined detection/estimation tasks,” 24, 91–98 (2007).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

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

S. M. Kay, Fundamentals of Statistical Signal Processing II: Detection Theory (Prentice Hall, 2008).

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Equations (114)

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Λ ( g ) = pr ( g | H 1 ) pr ( g | H 0 ) ,
Λ a ( g ) = pr ( g | a ) pr ( g | a 0 ) .
F ( a ) = s c ( g | a ) s c ( g | a ) g | a .
pr ( x | H i ) = ( 2 π σ 2 ) 1 2 exp [ ( x i ) 2 2 σ 2 ] .
Λ a , σ ( g , x ) = pr ( g | a ) pr ( x | H 1 ) pr ( g | a 0 ) pr ( x | H 0 ) = Λ a ( g ) Λ σ ( x ) .
AUC Λ = 1 1 4 π | Λ 1 2 + i α ( g ) g | H 0 | 2 d α α 2 + 1 4 .
AUC ( a , σ ) = 1 1 4 π | Λ a 1 2 + i α ( g ) g | a 0 | 2 | Λ σ 1 2 + i α ( x ) x | H 0 | 2 × d α α 2 + 1 4 .
AUC ( a , σ ) = 1 1 4 π | Λ a 1 2 + i α ( g ) g | a 0 | 2 × exp ( α 2 + 1 4 2 σ 2 ) d α α 2 + 1 4 ,
AUC ( a ) = 1 1 4 π | Λ a 1 2 + i α ( g ) g | a 0 | 2 d α α 2 + 1 4 .
AUC ( a , σ ) = 1 2 + 1 2 erf [ 1 2 d ( a , σ ) ] .
AUC ( a , σ ) = 1 1 4 π exp [ ( α 2 + 1 4 ) d 2 ( a , σ ) ] d α α 2 + 1 4 .
γ ( a ) = lim σ γ ( a , σ ) = lim σ d 2 ( a , σ ) = d 2 ( a ) .
C exp [ | z | 2 γ ( a , σ ) | z | 2 σ 2 ] d z | z | 2 = C | Λ a z ( g ) g | a | 2 exp ( | z | 2 σ 2 ) d z | z | 2 .
γ ( a 0 , σ ) = 0 ,
a k γ ( a 0 , σ ) = 0 ,
2 a k a l γ ( a 0 , σ ) = 2 F k l ( a 0 ) ,
3 a k a l a m γ ( a 0 , σ ) = a 0 k F l m ( a 0 ) + a 0 l F m k ( a 0 ) + a 0 m F k l ( a 0 ) .
4 a 4 γ ( a 0 , σ ) = L ( a 0 ) K ( a 0 ) σ 2 ,
d 2 ( a ) ( a a 0 ) F ( a 0 ) ( a a 0 ) .
pr ( g | a ) = B pr ( g | b + a s ) pr ( b ) d b .
pr ( g ) = pr ( g | 0 ) = B pr ( g | b ) pr ( b ) d b .
pr ( b | g ) = pr ( g | b ) pr ( b ) pr ( g ) .
s c ¯ ( g ) = B s c ( g | b ) pr ( b | g ) d b .
s c ¯ ( g ) 0 = G B s c ( g | b ) pr ( b | g ) pr ( g ) d b d g = B G s c ( g | b ) pr ( g | b ) pr ( b ) d g d b = B [ G b pr ( g | b ) d g ] pr ( b ) d b = 0 .
F SKE ( 0 ) = tr [ s s s c ¯ ( g ) s c ¯ ( g ) 0 ] = | s s c ¯ ( g ) | 2 0 .
pr ( g | a ) = S B pr ( g | b + a s ) pr ( b ) pr ( s ) d b d s .
F SKS ( 0 ) = tr [ s ¯ s ¯ s c ¯ ( g ) s c ¯ ( g ) 0 ] = | s ¯ s c ¯ ( g ) | 2 0 .
F RSKE ( 0 ) = tr [ s s s c ¯ ( g ) s c ¯ ( g ) 0 ] = | s s c ¯ ( g ) | 2 0 s ,
b ̂ ( g ) = B b pr ( b | g ) d b ,
F SKE ( 0 ) = | s K 1 n ̂ ( g ) | 2 0 ,
F SKS ( 0 ) = | s ¯ K 1 n ̂ ( g ) | 2 0 ,
F RSKE ( 0 ) = | s K 1 n ̂ ( g ) | 2 0 s .
| s K 1 n ̂ ( g ) | 2 0 = s K 1 s | s K 1 [ b b ̂ ( g ) ] | 2 g | b b ,
s K 1 s | s K 1 [ b b ̃ ( g ) ] | 2 0 | s K 1 n ̂ ( g ) | 2 0 s K 1 s ,
s ( K + K b ) 1 s | s K 1 n ̂ ( g ) | 2 0 s K 1 s ,
| s K 1 n ̂ ( g ) | 2 0 s K 1 s { 1 [ b b ̃ ( g ) ] K 1 [ b b ̃ ( g ) ] 0 } .
| s K 1 n ̂ ( g ) | 2 0 s K 1 s { 1 K 1 2 H 2 f b f g 2 0 } .
b ̂ m ( g ) = [ B 1 b m pr ( b | g ) d b ] 1 .
[ s s c ¯ ( g ) ] 2 0 s 2 m = 1 M { 1 b m 0 g m 2 [ 1 b ̂ m ( g m ) 1 b m ] 2 0 } .
s s c ( g | b ) = 1 2 tr [ K b 1 ( D s K b ) ] + s n ( g , b ) + 1 2 n ( g , b ) × ( D s K b ) n ( g , b ) ,
s s c ¯ ( g ) = B s s c ( g | b ) pr ( b | g ) d b .
U ( a ) = V { G u [ v ̂ ( g | a ) , v ] pr ( g | v , a ) d g } pr ( v ) d v .
pr ( g | a ) = V pr ( g | v , a ) pr ( v ) d v ,
pr ( v | g , a ) = pr ( g | v , a ) pr ( v ) pr ( g | a ) .
U ( a ) = G { V u [ v ̂ ( g | a ) , v ] pr ( v | g , a ) d v } pr ( g | a ) d g .
v ̂ ( g | a ) = V v pr ( v | g , a ) d v .
v ̂ ( g | a ) = V v pr ( v ) pr ( g | v , a ) d v pr ( g | a ) .
v ̂ ( g | a 0 ) = v ¯ = V v pr ( v ) d v ,
H U ( a 0 ) = ( v v ¯ ) s c ( g | v , a 0 ) v ( v v ¯ ) s c ( g | v , a 0 ) v g | a 0 .
F ( a 0 | v , v ) = s c ( g | v , a 0 ) s c ( g | v , a 0 ) g | a 0 .
H U ( a 0 ) = F ( a 0 | v , v ) ( v v ¯ ) ( v v ¯ ) v , v .
U ( a ) = 1 1 2 tr ( K v ) + 1 2 ( a a 0 ) H U ( a 0 ) ( a a 0 ) + ,
pr ( g | v , a ) = B pr ( g | b + a s ( v ) ) pr ( b ) d b ,
sc ( g | v , 0 ) = s ( v ) B s c ( g | b ) pr ( b | g ) d b = s ( v ) s c ¯ ( g ) .
H U ( 0 ) = d 2 d a 2 U ( 0 ) = tr [ K v s K v s s c ¯ ( g ) s c ¯ ( g ) 0 ] = K v s s c ¯ ( g ) 2 0 .
AEROC ( a ) u ¯ 0 { 1 2 + 1 2 erf [ 1 2 d ( a ) ] } .
u ¯ 0 = V u ( v 0 , v ) pr ( v ) d v ,
v 0 = arg max v [ V u ( v , v ) pr ( v ) d v ] .
H ( a 0 ) = 1 u ¯ 0 2 u ( v 0 , v ) F ( a 0 | v , v ) u ( v 0 , v ) v , v .
H ( a 0 ) = H ( 0 ) = tr [ s ̃ s ̃ s c ¯ ( g ) s c ¯ ( g ) 0 ] = | s ̃ s c ¯ ( g ) | 2 0 .
D 0 = exp [ | z | 2 γ ( a , σ ) ] = exp ( | z | 2 γ ) ,
D 1 = | z | 2 γ 1 D 0 ,
D 2 = ( | z | 2 γ 2 + | z | 4 γ 1 2 ) D 0 ,
D 3 = ( | z | 2 γ 3 + 3 | z | 4 γ 1 γ 2 | z | 6 γ 1 3 ) D 0 ,
D 4 = ( | z | 2 γ 4 + 4 | z | 4 γ 1 γ 3 + 3 | z | 4 γ 2 2 6 | z | 6 γ 1 2 γ 2 + | z | 8 γ 1 4 ) D 0 .
D ̃ 0 = | Λ a z ( g ) g | a | 2 = | Λ z | 2
D ̃ 1 = z Λ z 1 Λ 1 Λ z ¯ + c.c. ,
D ̃ 2 = | z | 2 Λ z 2 Λ 1 2 Λ z ¯ + z Λ z 1 Λ 2 Λ z ¯ + | z | 2 Λ z 1 Λ 1 × Λ z ¯ 1 Λ 1 + c.c. ,
D ̃ 3 = | z | 2 ( z 2 ) Λ z 3 Λ 1 3 Λ z ¯ 3 | z | 2 Λ z 2 Λ 1 Λ 2 Λ z ¯ 2 z ¯ | z | 2 Λ z 2 Λ 1 2 Λ z ¯ 1 Λ 1 + z Λ z 1 Λ 3 Λ z ¯ + 2 | z | 2 Λ z 1 Λ 2 Λ z ¯ 1 Λ 1 z | z | 2 Λ z 1 Λ 1 Λ z ¯ 2 Λ 1 2 + | z | 2 Λ z 1 Λ 1 Λ z ¯ 1 Λ 2 + c.c. ,
D ̃ 4 = | z | 2 ( z 2 ) ( z 3 ) Λ z 4 Λ 1 4 Λ z ¯ 6 | z | 2 ( z 2 ) × Λ z 3 Λ 1 2 Λ 2 Λ z ¯ 3 z ¯ | z | 2 ( z 2 ) Λ z 3 Λ 1 3 Λ z ¯ 1 Λ 1 3 | z | 2 Λ z 2 Λ 2 2 Λ z ¯ 4 | z | 2 Λ z 2 Λ 1 Λ 3 Λ z ¯ 9 z ¯ | z | 2 Λ z 2 Λ 1 Λ 2 Λ z ¯ 1 Λ 1 + 3 | z | 4 Λ z 2 Λ 1 2 Λ z ¯ 2 Λ 1 2 3 z ¯ | z | 2 Λ z 2 Λ 1 2 Λ z ¯ 1 Λ 2 + z Λ z 1 Λ 4 Λ z ¯ + 3 | z | 2 Λ z 1 Λ 3 Λ z ¯ Λ 1 3 z | z | 2 Λ z 1 Λ 2 Λ z ¯ 2 Λ 1 2 + 3 | z | 2 Λ z 1 Λ 2 Λ z ¯ 1 Λ 2 z | z | 2 ( z ¯ 2 ) Λ z 1 Λ 1 × Λ z ¯ 3 Λ 1 3 3 z | z | 2 Λ z 1 Λ 1 Λ z ¯ 2 Λ 1 Λ 2 + | z | 2 Λ z 1 Λ 1 × Λ z ¯ 1 Λ 3 + c.c.
D ̃ 0 = 1 ,
D ̃ 1 = 0 ,
D ̃ 2 = 2 | z | 2 Λ 1 2 ,
D ̃ 3 = | z | 2 ( z 2 ) Λ 1 3 3 | z | 2 Λ 1 Λ 2 + c.c. ,
D ̃ 4 = | z | 2 ( z 2 ) ( z 3 ) Λ 1 4 6 | z | 2 ( z 2 ) Λ 1 2 Λ 2 3 | z | 2 Λ 2 2 4 | z | 2 Λ 1 Λ 3 + 3 | z | 4 Λ 1 2 Λ 1 2 + c.c. ,
C exp ( | z | 2 σ 2 ) d z = i π σ exp ( 1 4 σ 2 ) = I ( σ )
γ 1 = a γ ( a 0 , σ ) = 0 .
γ 2 = 2 a 2 γ ( a 0 , σ ) = 2 Λ 1 2 = 2 F ( a 0 ) .
C z exp ( | z | 2 σ 2 ) d z = C z ¯ exp ( | z | 2 σ 2 ) d z = 1 2 I ( σ ) .
γ 3 = 3 a 3 γ ( a 0 , σ ) = 3 Λ 1 3 + 6 Λ 1 Λ 2 .
d d a 0 F ( a 0 ) = d d a 0 R M [ pr ( g | a 0 ) ] 2 pr ( g | a 0 ) d M g .
γ 3 = 3 a 3 γ ( a 0 , σ ) = 3 d d a 0 F ( a 0 ) .
C z 2 exp ( | z | 2 σ 2 ) d z = C z ¯ 2 exp ( | z | 2 σ 2 ) d z = I ( σ ) ( 1 4 σ 2 2 )
C | z | 2 exp ( | z | 2 σ 2 ) d z = I ( σ ) ( 1 4 + σ 2 2 ) .
I ( σ ) [ γ 4 + 3 ( 1 + 2 σ 2 ) Λ 1 2 2 ] ,
I ( σ ) [ ( σ 2 15 2 ) Λ 1 4 + 18 Λ 1 2 Λ 2 6 Λ 2 2 8 Λ 1 Λ 3 + ( 6 + 3 σ 2 ) Λ 1 2 2 ] .
c ̂ ( g ) = C c [ B pr ( c | b ) pr ( b | g ) d b ] d c .
c ̂ ( g ) = B L b pr ( b | g ) d b = L b ̂ ( g ) .
| s K 1 n ̂ ( g ) | 2 0 = s K 1 [ g b ̂ ( g ) ] [ g b ̂ ( g ) ] 0 K 1 s .
[ g b ] [ b b ̂ ( g ) ] 0 = g [ b b ̂ ( g ) ] 0 b [ b b ̂ ( g ) ] 0
g [ b b ̂ ( g ) ] 0 = g [ b b ̂ ( g ) ] b | g g = 0 .
b [ b b ̂ ( g ) ] = [ b b ̂ ( g ) ] [ b b ̂ ( g ) ] .
| s K 1 n ̂ ( g ) | 2 0 = s K 1 s | s K 1 [ b b ̂ ( g ) ] | 2 g | b b .
| s K 1 [ b b ̂ ( g ) ] | 2 0 = | c c ̂ ( g ) | 2 0 .
| s K 1 [ b b ̂ ( g ) ] | 2 0 | s K 1 [ b b ̃ ( g ) ] | 2 0 .
s K 1 s | s K 1 [ b b ̃ ( g ) ] | 2 0 | s K 1 n ̂ ( g ) | 2 0 s K 1 s .
s ( K + K b ) 1 s | s K 1 n ̂ ( g ) | 2 0 s K 1 s .
[ s s c ¯ ( g ) ] 2 0 = [ m = 1 M s m g m b ̂ m ( g m ) s m ] 2 0 s 2 m = 1 M [ g m b ̂ m ( g m ) 1 ] 2 0 .
g m 2 [ 1 b ̂ m ( g m ) 1 b m ] 2 0 .
[ g m b m 1 ] 2 0 = 1 b m 0 .
2 g m [ 1 b ̂ m ( g m ) 1 b m ] [ g m b m 1 ] 0 .
[ 1 b ̂ m ( g m ) 1 b m ] b | g = 0 .
g m [ 1 b ̂ m ( g m ) 1 b m ] b | g = g m 2 b ̂ m ( g m ) [ 1 b ̂ m ( g m ) 1 b m ] b | g = 0 .
2 g m [ 1 b ̂ m ( g m ) 1 b m ] [ g m b m 1 ] 0 = 2 g m 2 [ 1 b ̂ m ( g m ) 1 b m ] 2 0 .
[ s s c ¯ ( g ) ] 2 0 s 2 m = 1 M { 1 b m 0 g m 2 [ 1 b ̂ m ( g m ) 1 b m ] 2 0 } .
a v ̂ ( g | a ) = V s c ( g | v , a ) [ v v ̂ ( g | a ) ] pr ( v ) pr ( g | v , a ) d v V pr ( v ) pr ( g | v , a ) d v ,
a v ̂ ( g | a ) = V s c ( g | v , a ) [ v v ̂ ( g | a ) ] pr ( v | g , a ) d v .
a v ̂ ( g | a 0 ) = V s c ( g | v , a 0 ) ( v v ¯ ) pr ( v ) d v ,
U 1 ( a ) = G a v ̂ ( g | a ) × [ | v V u ( v , v ) pr ( v | g , a ) d v | v ̂ ( g | a ) ] pr ( g | a ) d g
U 2 ( a ) = V { G u [ v ̂ ( g | a ) , v ] a pr ( g | v , a ) d g } pr ( v ) d v .
U 2 ( a 0 ) = V u ( v ¯ , v ) [ G | a pr ( g | v , a ) | a 0 d g ] pr ( v ) d v ,
U 4 = V u ( v ¯ , v ) [ G | a a pr ( g | v , a ) | a 0 d g ] pr ( v ) d v
U 3 = V [ G a v ̂ ( g | a 0 ) × | v u ( v , v ) | v ¯ | a pr ( g | v , a ) | a 0 d g ] pr ( v ) d v .
U 3 = G [ V a v ̂ ( g | a 0 ) ( v v ¯ ) s c ( g | v , a 0 ) pr ( v ) d v ] × pr ( g | a 0 ) d g .

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