Abstract

The relation between eigenvalues of the kernel covariance matrix in inversion problems and the number of pieces of information that can be extracted is discussed. It is shown that an example given by Wang and Goulard, which was claimed to show that the eigenvalue analysis underestimated the information content, was incorrectly calculated.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Twomey, J. Franklin Inst. 279 (2), 95 (1965).
    [CrossRef]
  2. J. Y. Wang, R. Goulard, Purdue University Research Report, Project 5543 (1972).
  3. Fractional powers do not apply in general in matrix algebra, but with diagonal matrices such as Λ their significance is obvious (and unambiguous).
  4. S. Twomey, H. B. Howell, Appl. Opt. 6 (12), 2125 (1967.
    [CrossRef] [PubMed]

1967 (1)

1965 (1)

S. Twomey, J. Franklin Inst. 279 (2), 95 (1965).
[CrossRef]

Goulard, R.

J. Y. Wang, R. Goulard, Purdue University Research Report, Project 5543 (1972).

Howell, H. B.

Twomey, S.

Wang, J. Y.

J. Y. Wang, R. Goulard, Purdue University Research Report, Project 5543 (1972).

Appl. Opt. (1)

J. Franklin Inst. (1)

S. Twomey, J. Franklin Inst. 279 (2), 95 (1965).
[CrossRef]

Other (2)

J. Y. Wang, R. Goulard, Purdue University Research Report, Project 5543 (1972).

Fractional powers do not apply in general in matrix algebra, but with diagonal matrices such as Λ their significance is obvious (and unambiguous).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Tables (1)

Tables Icon

Table I f(x), δ(x), g and and Their Magnitudes

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

g i = a b k i ( x ) f ( x ) d x + i             ( i = 1 , 2 , N )
C = ( k 1 2 ( x ) d x k 1 ( x ) k 2 ( x ) d x k 1 ( x ) k 2 ( x ) d x k 2 2 ( x ) d x ) .
f ( x ) = ξ ˜ * ϕ ˜ ( x ) = ϕ ˜ * ( x ) ξ ˜ ( x )
g ˜ = k ˜ ( x ) f ( x ) d x = k ˜ ( x ) ϕ ˜ * ( x ) ξ d x = [ k ˜ ( x ) k ˜ * ( x ) d x ] W ξ ˜ ,
g ˜ = C W ξ ˜ ,
ϕ ˜ ( x ) ϕ ˜ * ( x ) d x = Λ - 1 / 2 U * C U Λ - 1 / 2 = Λ - 1 / 2 Λ Λ - 1 / 2 = I .
g ˜ = C U Λ 1 / 2 ξ ˜ ,
ξ ˜ = Λ - 1 / 2 U * g ˜
f ( x ) = k ˜ * ( x ) U Λ - 1 U * g ˜ .
δ ( x ) 2 = δ 2 ( x ) d x = ( ˜ ) * Λ - 1 U * k ˜ ( x ) k ˜ * ( x ) d x U Λ - 1 ˜ = ( ˜ ) * Λ - 1 U * C U Λ - 1 ˜ = ( ˜ ) * Λ - 1 U * U Λ Λ - 1 ˜ = ( ˜ ) * Λ - 1 ˜ 1 .
δ ( x ) f ( x ) - 1 = ( λ 1 - 1 1 2 + λ 2 - 1 2 2 + + λ N - 1 N 2 ) 1 / 2 ,
N - 1 / 2 ( λ 1 - 1 + λ 2 - 1 + + λ N - 1 ) 1 / 2 .
( 1 2 + 2 2 + + N 2 ) 1 / 2 ( λ 1 ξ 1 2 + λ 2 ξ 2 2 + + λ N ξ N 2 ) - 1 / 2 ,
N 1 / 2 ( λ 1 + λ 2 + + λ N ) - 1 / 2 .
[ δ ( x ) 2 f ( x ) - 2 ] 1 / 2 ( 2 g - 2 ) - 1 / 2 = N - 1 [ j λ j - 1 j λ j ] 1 / 2 .
a b k i ( x ) k j ( x ) d x
exp [ - ½ ( x - d j ) 2 ] ,
0 10 exp [ - ½ ( x - d j ) 2 - ½ ( x - d k ) 2 ] d x
C = ( 1.7724 1.5861 1.1365 0.6520 0.2996 0.1102 0.0325 1.5861 1.7725 1.5861 1.1365 0.6520 0.2996 0.1102 1.1365 1.5861 1.7725 1.5861 1.1365 0.6520 0.2996 0.6520 1.1365 1.5861 1.7725 1.5861 1.1365 0.6520 0.2996 0.6520 1.1365 1.5861 1.7725 1.5861 1.1365 0.1102 0.2996 0.6520 1.1365 1.5861 1.7725 1.5861 0.0325 0.1102 0.2996 0.6520 1.1365 1.5861 1.7724 )
λ j = 7.40 , 3.61 , 1.13 , 0.234 , 0.0323 , 0.00281 , 0.000121 ,
( 0.2751 0.3664 0.4317 0.4554 0.4317 0.3664 0.2751 ) ( 0.4747 0.4470 0.2737 - 0.0000 - 0.2737 - 0.4470 - 0.4747 ) ( 0.5475 0.1753 - 0.2590 - 0.4525 - 0.2590 0.1753 0.5475 ) ( - 0.4895 0.2461 0.4470 0.0000 - 0.4470 - 0.2461 - 0.4895 ) ( - 0.3462 0.5172 0.0309 - 0.4725 0.0309 0.5172 - 0.3462 ) ( - 0.1872 0.4895 - 0.4747 0.0000 0.4746 - 0.4895 0.1872 ) ( 0.0681 - 0.2599 0.4956 - 0.6038 0.4956 - 0.2599 0.0681 ) .
2.26 × 10 11 , 2.23 × 10 7 , 1.54 × 10 4 , 29.7 , × 8.67 × 10 - 2 , 1.84 × 10 - 4 , 9.63 × 10 - 8 .
0 10 k i ( x ) k j ( x ) d x
0 10 exp [ - ( x - d j ) 2 d x = - d j 10 - d j exp ( - ξ 2 ) d ξ π = 1.772
[ j λ j - 1 j λ j ] 1 / 2
[ λ N - 1 λ 1 ] 1 / 2

Metrics