Abstract

We develop a probabilistic approach to the problem of signal recovery from noisy data. In particular, in constructing the approximation, we introduce a truncation method that is related to an order–disorder transition point of the Fourier coefficients of the data. Through a statistical method we show how to determine this truncation point without using any prior information.

© 1993 Optical Society of America

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References

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  1. A. Tikhonov, V. Arsenine, Méthodes de Résolution de Problèmes Mal Posés (Mir, Moscow, 1976).
  2. G. Wahba, S. Wold, “A completely automatic French curve: fitting spline functions by cross-validation,” Commun. Stat. 4, 1–17 (1975).
  3. G. Wahba, S. Wold, “Periodic splines for spectral density estimation: the use of cross-validation for determining the degree of smoothing,” Commun. Stat. 4, 125–141 (1975).
  4. G. Wahba, “Practical approximation solutions to linear operator equations when the data are noisy,” SIAM J. Numer. Anal. 14, 651–667 (1977).
    [CrossRef]
  5. A. V. Balakrishnan, Applied Functional Analysis (Springer, New York, 1976), Chap. 6.
  6. G. A. Viano, “Fredholm integral equations of first kind and the method of correlogram,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, San Diego, Calif., 1987), pp. 151–164. See also the references cited therein.
  7. J. N. Franklin, “Well-posed stochastic extensions of ill-posed problems,”J. Math. Anal. Appl. 31, 682–716 (1970).
    [CrossRef]
  8. I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. Ser. II 12, 199–246 (1959).
  9. G. Toraldo di Francia, “Degrees of freedom of an image,”J. Opt. Soc. Am. 59, 799–804 (1969).
    [CrossRef] [PubMed]
  10. G. Toraldo di Francia, “Some recent progress in classical optics,” Riv. Nuovo Cimento 1, 460–484 (1969).
  11. E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
    [CrossRef]
  12. M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin, London, 1966), Vol. 3, Chap. 45.

1977 (1)

G. Wahba, “Practical approximation solutions to linear operator equations when the data are noisy,” SIAM J. Numer. Anal. 14, 651–667 (1977).
[CrossRef]

1975 (2)

G. Wahba, S. Wold, “A completely automatic French curve: fitting spline functions by cross-validation,” Commun. Stat. 4, 1–17 (1975).

G. Wahba, S. Wold, “Periodic splines for spectral density estimation: the use of cross-validation for determining the degree of smoothing,” Commun. Stat. 4, 125–141 (1975).

1970 (1)

J. N. Franklin, “Well-posed stochastic extensions of ill-posed problems,”J. Math. Anal. Appl. 31, 682–716 (1970).
[CrossRef]

1969 (2)

G. Toraldo di Francia, “Degrees of freedom of an image,”J. Opt. Soc. Am. 59, 799–804 (1969).
[CrossRef] [PubMed]

G. Toraldo di Francia, “Some recent progress in classical optics,” Riv. Nuovo Cimento 1, 460–484 (1969).

1959 (1)

I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. Ser. II 12, 199–246 (1959).

1931 (1)

E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Arsenine, V.

A. Tikhonov, V. Arsenine, Méthodes de Résolution de Problèmes Mal Posés (Mir, Moscow, 1976).

Balakrishnan, A. V.

A. V. Balakrishnan, Applied Functional Analysis (Springer, New York, 1976), Chap. 6.

Franklin, J. N.

J. N. Franklin, “Well-posed stochastic extensions of ill-posed problems,”J. Math. Anal. Appl. 31, 682–716 (1970).
[CrossRef]

Gel’fand, I. M.

I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. Ser. II 12, 199–246 (1959).

Hille, E.

E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Kendall, M. G.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin, London, 1966), Vol. 3, Chap. 45.

Stuart, A.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin, London, 1966), Vol. 3, Chap. 45.

Tamarkin, J. D.

E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Tikhonov, A.

A. Tikhonov, V. Arsenine, Méthodes de Résolution de Problèmes Mal Posés (Mir, Moscow, 1976).

Toraldo di Francia, G.

G. Toraldo di Francia, “Degrees of freedom of an image,”J. Opt. Soc. Am. 59, 799–804 (1969).
[CrossRef] [PubMed]

G. Toraldo di Francia, “Some recent progress in classical optics,” Riv. Nuovo Cimento 1, 460–484 (1969).

Viano, G. A.

G. A. Viano, “Fredholm integral equations of first kind and the method of correlogram,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, San Diego, Calif., 1987), pp. 151–164. See also the references cited therein.

Wahba, G.

G. Wahba, “Practical approximation solutions to linear operator equations when the data are noisy,” SIAM J. Numer. Anal. 14, 651–667 (1977).
[CrossRef]

G. Wahba, S. Wold, “A completely automatic French curve: fitting spline functions by cross-validation,” Commun. Stat. 4, 1–17 (1975).

G. Wahba, S. Wold, “Periodic splines for spectral density estimation: the use of cross-validation for determining the degree of smoothing,” Commun. Stat. 4, 125–141 (1975).

Wold, S.

G. Wahba, S. Wold, “Periodic splines for spectral density estimation: the use of cross-validation for determining the degree of smoothing,” Commun. Stat. 4, 125–141 (1975).

G. Wahba, S. Wold, “A completely automatic French curve: fitting spline functions by cross-validation,” Commun. Stat. 4, 1–17 (1975).

Yaglom, A. M.

I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. Ser. II 12, 199–246 (1959).

Acta Math. (1)

E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Am. Math. Soc. Transl. Ser. II (1)

I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. Ser. II 12, 199–246 (1959).

Commun. Stat. (2)

G. Wahba, S. Wold, “A completely automatic French curve: fitting spline functions by cross-validation,” Commun. Stat. 4, 1–17 (1975).

G. Wahba, S. Wold, “Periodic splines for spectral density estimation: the use of cross-validation for determining the degree of smoothing,” Commun. Stat. 4, 125–141 (1975).

J. Math. Anal. Appl. (1)

J. N. Franklin, “Well-posed stochastic extensions of ill-posed problems,”J. Math. Anal. Appl. 31, 682–716 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

Riv. Nuovo Cimento (1)

G. Toraldo di Francia, “Some recent progress in classical optics,” Riv. Nuovo Cimento 1, 460–484 (1969).

SIAM J. Numer. Anal. (1)

G. Wahba, “Practical approximation solutions to linear operator equations when the data are noisy,” SIAM J. Numer. Anal. 14, 651–667 (1977).
[CrossRef]

Other (4)

A. V. Balakrishnan, Applied Functional Analysis (Springer, New York, 1976), Chap. 6.

G. A. Viano, “Fredholm integral equations of first kind and the method of correlogram,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, San Diego, Calif., 1987), pp. 151–164. See also the references cited therein.

A. Tikhonov, V. Arsenine, Méthodes de Résolution de Problèmes Mal Posés (Mir, Moscow, 1976).

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin, London, 1966), Vol. 3, Chap. 45.

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Figures (5)

Fig. 1
Fig. 1

Two correlograms for the data sets { g ¯ k } 1 200 corresponding to the function f(y) = |y log y|. The filled circles correspond to = 10−2. The open circles correspond to = 10−4.

Fig. 2
Fig. 2

Reconstructions of the function f(y) = |y log y| with the use of the truncations indicated by the correlograms of Fig. 1. The solid curve represents the true function. The filled squares represent the reconstruction that we obtained by using two eigenfunctions ( = 10−2). The open squares represent the reconstruction obtained by using seven eigenfunctions ( = 10−4).

Fig. 3
Fig. 3

Correlogram for the data set { g ¯ k } 1 200 corresponding to the function f(y) = 6y exp(−6y) + 6(1 − y)exp[−6(1 − y)] and with = 10−4. It shows that the components g ¯ k, with k even, are nearly zero (see text for explanation).

Fig. 4
Fig. 4

Correlogram for the data set { g ¯ 2 k + 1 } 1 99 corresponding to the function f(y) = 6y exp(−6y) + 6(1 − y)exp[−6(1 − y)] and with = 10−2, N = 100.

Fig. 5
Fig. 5

Reconstruction of the function f(y) = 6y exp(−6y) + 6(1 − y)exp[−6(1 − y)] with = 10−2. The solid curve represents the true function. The filled squares represent the reconstruction that we obtained by using two eigenfunctions of odd index as indicated by the correlogram of Fig. 4. The hollow squares show the reconstruction obtained by using three eigenfunctions of odd index (one beyond those permitted by the correlogram method).

Equations (40)

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( A f ) ( x ) = a b K ( x , y ) f ( y ) d y = g ( x )             ( a x b ) ,             ( A f = g ) ,
K ( x , y ) = K ( y , x ) ¯ , a b a b K ( x , y ) 2 d x d y < .
lim k λ k = 0.
- a a sin Ω ( x - y ) π ( x - y ) f ( y ) d y = g ( x ) ,
A f + n = g ¯             ( g ¯ = g + n ) ,
A ξ + ζ = η .
R η η = A R ξ ξ A + R ζ ζ ,
R ξ η = E [ ξ , η * ] = E [ ξ , ξ * A * + ζ * ] = R ξ ξ A * = R ξ ξ A .
R ζ ζ = 2 N ,
Tr R ξ ξ = k = 1 ( R ξ ξ ψ k , ψ k ) = k = 1 ρ k 2 = Γ < .
E { ξ - F η 2 } = T r ( R ξ ξ - R ξ ξ A F * - F A R ξ ξ + F R η η F * ) ,
F ^ g ¯ = k I g ¯ k λ k ψ k             g ¯ k = ( g ¯ , ψ k ) ,
I α = { k k α ( ) } ,
k α ( ) = max { n N : k = 1 n ( ρ k 2 + 2 ν k 2 λ k 2 ) Γ } .
E { ξ - F ^ α η 2 } = k = k α ( ) + 1 ρ k 2 + k = 1 k α ( ) ( ν k λ k ) 2 ,
lim 0 k α ( ) = + .
Γ < k = 1 k α 1 ( ) ( ρ k 2 + i 2 ν k 2 λ k 2 ) k = 1 M ( ρ k 2 + i 2 ν k 2 λ k 2 ) .
Γ < k = 1 M ρ k 2 k = 1 ρ k 2 = Γ ,
lim 0 { k = k α ( ) + 1 ρ k 2 + k = 1 k α ( ) ( ν k λ k ) 2 } = 0.
lim 0 k = k α ( ) + 1 ρ k 2 = 0.
k = 1 k α ( ) ( ν k λ k ) 2 + k = 1 k α ( ) ρ k 2 Γ = k = 1 ρ k 2 ,
k = 1 k α ( ) ( ν k λ k ) 2 k = k α ( ) + 1 ρ k 2 .
k k S λ k ρ k ,
k > k S λ k ρ k < .
k k S λ k ρ k ν k ,
k > k S λ k ρ k < ν k .
F ^ S g ¯ = k I S g ¯ k λ k ψ k
E { ξ - F ^ S η 2 } = k = k S ( ) + 1 ρ k 2 + k = 1 k S ( ) ( ν k λ k ) 2 .
lim 0 { k = k S ( ) + 1 ρ k 2 + k = 1 k S ( ) ( ν k λ k ) 2 } = 0.
lim 0 k = k S ( ) + 1 ρ k 2 = 0.
k = k α ( ) + 1 k S ( ) ( ν k λ k ) 2 < k = k α ( ) + 1 k S ( ) ρ k 2 < k = k α ( ) + 1 ρ k 2 ,
J k ( ξ k , η k ) = - ½ log ( 1 - r k 2 ) ,
r k 2 = E [ ξ k , η k ] 2 E [ ξ k 2 ] E [ η k 2 ] = ( λ k ρ k ) 2 ( λ k ρ k ) 2 + ( ν k ) 2 ,
J k ( ξ k , η k ) = ½ log { 1 + ( λ k ρ k ) 2 ( ν k ) 2 } .
δ n = k = 1 N - n ( g ¯ k - g ¯ ¯ k ) ( g ¯ k + n - g ¯ ¯ k + n ) { k = 1 N - n ( g ¯ k - g ¯ ¯ k ) 2 k = 1 N - n ( g ¯ k + n - g ¯ ¯ k + n ) 2 } 1 / 2 ( n = 0 , 1 , 2 , , N - k ) ,
g ¯ ¯ k = 1 N - n k = 1 N - n g ¯ k ;             g ¯ ¯ k + n = 1 N - n k = 1 N - n g ¯ k + n .
( A f ) ( x ) = 0 1 K ( x , y ) f ( y ) d y = g ( x )             ( 0 x 1 ) ,
{ K ( x , y ) = ( 1 - x ) y ( 0 y x 1 ) K ( x , y ) = x ( 1 - y ) ( 0 x y 1 ) .
{ ψ k = 2 sin ( k π x ) λ k = 1 / ( k 2 π 2 ) .
f k S ( ) ( y ) = k = 1 k S ( ) g ¯ k λ k ψ k ( y )

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