Abstract

The inverse problem in optics, which is closely related to the classical question of the resolving power, is reconsidered as a communication channel problem. The main result is the evaluation of the maximum number Mε of ε-distinguishable messages (ε being a bound on the noise of the image) that can be conveyed back from the image to reconstruct the object. We study the case of coherent illumination. By using the concept of Kolmogorov’s ε-capacity, we obtain Mε2Slog(1ε) for ε0, where S is the Shannon number. Moreover, we show that the ε-capacity in inverse optical imaging is nearly equal to the amount of information on the object that is contained in the image. We thus compare the results obtained through the classical information theory—which is based on probability theory—with those derived from a form of topological information theory, based on the Kolmogorov ε-entropy and ε-capacity, which are concepts related to the evaluation of the massiveness of compact sets.

© 2009 Optical Society of America

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References

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  1. H. Wolter, “On basic analogies and principal differences between optical and electronic information,” in Progress in Optics, Vol. I, E.Wolf, ed. (North Holland, 1961), pp. 155-210.
    [CrossRef]
  2. G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799-804 (1969), and references cited therein.
    [CrossRef] [PubMed]
  3. J. Hadamard, Lectures on the Cauchy Problem in Linear Partial Differential Equations (Yale U. Press, 1923).
  4. G. A. Viano, “On the extrapolation of optical image data,” J. Math. Phys. 17, 1160-1165 (1976).
    [CrossRef]
  5. M. Bertero, C. De Mol, and G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H.P.Baltes, ed., Topics in Current Physics, Vol. 20 (Springer, 1980), pp. 161-214.
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon, 1959), p. 166.
  7. A. Stern and B. Javidi, “Shannon number and information capacity of three-dimensional integral imaging,” J. Opt. Soc. Am. A 21, 1602-1612 (2004).
    [CrossRef]
  8. A. Tikhonov and V. Arsenine, Méthodes de Résolution de Problèmes Mal Poses (Mir, 1976).
  9. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind (Pitman, 1984).
  10. J. M. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl. 31, 682-716 (1970).
    [CrossRef]
  11. E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855-877 (1998).
    [CrossRef]
  12. A. N. Kolmogorov and V. M. Tihomirov, “ε-entropy and ε-capacity of sets in fuctional spaces,” Am. Math. Soc. Transl. 17, 277-364 (1961).
  13. D. Gabor, “Light and information,” in Progress in Optics, Vol. I, E.Wolf, ed. (North Holland, 1961), pp. 111-153.
  14. D. Slepian and H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty--I,” Bell Syst. Tech. J. 40, 43-64 (1961).
  15. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty--IV: Extension to many dimensions; Generalized prolate spheroidal wave functions,” Bell Syst. Tech. J. 43, 3009-3057 (1964).
  16. D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745-1759 (1965).
  17. B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, Vol. IX, E.Wolf, ed. (North Holland, 1972), pp. 311-407.
  18. N. Magnoli and G. A. Viano, “The source identification problem in electromagnetic theory,” J. Math. Phys. 38, 2366-2388 (1997).
    [CrossRef]
  19. A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, 1976), Chap. 6.
  20. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960), Chap. 6.
  21. E. De Micheli and G. A. Viano, “Probabilistic regularization in inverse optical imaging,” J. Opt. Soc. Am. A 17, 1942-1951 (2000).
    [CrossRef]
  22. G. G. Lorentz, Approximation of Functions (Holt, Rinehart and Winston, 1966), Chap. 10.
  23. E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283-309 (2002).
    [CrossRef]
  24. R. T. Prosser, “The ε-entropy and ε-capacity of certain time-varying channels,” J. Math. Anal. Appl. 16, 553-573 (1966).
    [CrossRef]
  25. I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions IV, Applications of Harmonic Analysis (Academic, 1964).
  26. I. M. Gelfand and A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. 12, 199-246 (1959).
  27. A. J. Jerri, “The Shannon sampling theorem--its various extensions and applications: A tutorial review,” Proc. IEEE 65, 1565-1596 (1977).
    [CrossRef]
  28. T. Kato, Perturbation Theory for Linear Operators (Springer, 1966), p. 522.
  29. F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163-165 (1973).
    [CrossRef]
  30. M. Bertero and C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, Vol. XXXVI, E.Wolf, ed. (North-Holland, 1996), pp. 129-178.
    [CrossRef]

2004 (1)

2002 (1)

E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283-309 (2002).
[CrossRef]

2000 (1)

1998 (1)

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855-877 (1998).
[CrossRef]

1997 (1)

N. Magnoli and G. A. Viano, “The source identification problem in electromagnetic theory,” J. Math. Phys. 38, 2366-2388 (1997).
[CrossRef]

1977 (1)

A. J. Jerri, “The Shannon sampling theorem--its various extensions and applications: A tutorial review,” Proc. IEEE 65, 1565-1596 (1977).
[CrossRef]

1976 (1)

G. A. Viano, “On the extrapolation of optical image data,” J. Math. Phys. 17, 1160-1165 (1976).
[CrossRef]

1973 (1)

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163-165 (1973).
[CrossRef]

1970 (1)

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

1969 (1)

1966 (1)

R. T. Prosser, “The ε-entropy and ε-capacity of certain time-varying channels,” J. Math. Anal. Appl. 16, 553-573 (1966).
[CrossRef]

1965 (1)

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745-1759 (1965).

1964 (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty--IV: Extension to many dimensions; Generalized prolate spheroidal wave functions,” Bell Syst. Tech. J. 43, 3009-3057 (1964).

1961 (2)

A. N. Kolmogorov and V. M. Tihomirov, “ε-entropy and ε-capacity of sets in fuctional spaces,” Am. Math. Soc. Transl. 17, 277-364 (1961).

D. Slepian and H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty--I,” Bell Syst. Tech. J. 40, 43-64 (1961).

1959 (1)

I. M. Gelfand and A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. 12, 199-246 (1959).

Arsenine, V.

A. Tikhonov and V. Arsenine, Méthodes de Résolution de Problèmes Mal Poses (Mir, 1976).

Balakrishnan, A. V.

A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, 1976), Chap. 6.

Bertero, M.

M. Bertero and C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, Vol. XXXVI, E.Wolf, ed. (North-Holland, 1996), pp. 129-178.
[CrossRef]

M. Bertero, C. De Mol, and G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H.P.Baltes, ed., Topics in Current Physics, Vol. 20 (Springer, 1980), pp. 161-214.
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1959), p. 166.

De Micheli, E.

E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283-309 (2002).
[CrossRef]

E. De Micheli and G. A. Viano, “Probabilistic regularization in inverse optical imaging,” J. Opt. Soc. Am. A 17, 1942-1951 (2000).
[CrossRef]

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855-877 (1998).
[CrossRef]

De Mol, C.

M. Bertero, C. De Mol, and G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H.P.Baltes, ed., Topics in Current Physics, Vol. 20 (Springer, 1980), pp. 161-214.
[CrossRef]

M. Bertero and C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, Vol. XXXVI, E.Wolf, ed. (North-Holland, 1996), pp. 129-178.
[CrossRef]

Franklin, J. M.

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

Frieden, B. R.

B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, Vol. IX, E.Wolf, ed. (North Holland, 1972), pp. 311-407.

Gabor, D.

D. Gabor, “Light and information,” in Progress in Optics, Vol. I, E.Wolf, ed. (North Holland, 1961), pp. 111-153.

Gelfand, I. M.

I. M. Gelfand and A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. 12, 199-246 (1959).

I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions IV, Applications of Harmonic Analysis (Academic, 1964).

Gori, F.

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163-165 (1973).
[CrossRef]

Groetsch, C. W.

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind (Pitman, 1984).

Guattari, G.

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163-165 (1973).
[CrossRef]

Hadamard, J.

J. Hadamard, Lectures on the Cauchy Problem in Linear Partial Differential Equations (Yale U. Press, 1923).

Javidi, B.

Jerri, A. J.

A. J. Jerri, “The Shannon sampling theorem--its various extensions and applications: A tutorial review,” Proc. IEEE 65, 1565-1596 (1977).
[CrossRef]

Kato, T.

T. Kato, Perturbation Theory for Linear Operators (Springer, 1966), p. 522.

Kolmogorov, A. N.

A. N. Kolmogorov and V. M. Tihomirov, “ε-entropy and ε-capacity of sets in fuctional spaces,” Am. Math. Soc. Transl. 17, 277-364 (1961).

Lorentz, G. G.

G. G. Lorentz, Approximation of Functions (Holt, Rinehart and Winston, 1966), Chap. 10.

Magnoli, N.

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855-877 (1998).
[CrossRef]

N. Magnoli and G. A. Viano, “The source identification problem in electromagnetic theory,” J. Math. Phys. 38, 2366-2388 (1997).
[CrossRef]

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960), Chap. 6.

Pollack, H. O.

D. Slepian and H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty--I,” Bell Syst. Tech. J. 40, 43-64 (1961).

Prosser, R. T.

R. T. Prosser, “The ε-entropy and ε-capacity of certain time-varying channels,” J. Math. Anal. Appl. 16, 553-573 (1966).
[CrossRef]

Slepian, D.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745-1759 (1965).

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty--IV: Extension to many dimensions; Generalized prolate spheroidal wave functions,” Bell Syst. Tech. J. 43, 3009-3057 (1964).

D. Slepian and H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty--I,” Bell Syst. Tech. J. 40, 43-64 (1961).

Sonnenblick, E.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745-1759 (1965).

Stern, A.

Tihomirov, V. M.

A. N. Kolmogorov and V. M. Tihomirov, “ε-entropy and ε-capacity of sets in fuctional spaces,” Am. Math. Soc. Transl. 17, 277-364 (1961).

Tikhonov, A.

A. Tikhonov and V. Arsenine, Méthodes de Résolution de Problèmes Mal Poses (Mir, 1976).

Toraldo di Francia, G.

Viano, G. A.

E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283-309 (2002).
[CrossRef]

E. De Micheli and G. A. Viano, “Probabilistic regularization in inverse optical imaging,” J. Opt. Soc. Am. A 17, 1942-1951 (2000).
[CrossRef]

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855-877 (1998).
[CrossRef]

N. Magnoli and G. A. Viano, “The source identification problem in electromagnetic theory,” J. Math. Phys. 38, 2366-2388 (1997).
[CrossRef]

G. A. Viano, “On the extrapolation of optical image data,” J. Math. Phys. 17, 1160-1165 (1976).
[CrossRef]

M. Bertero, C. De Mol, and G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H.P.Baltes, ed., Topics in Current Physics, Vol. 20 (Springer, 1980), pp. 161-214.
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1959), p. 166.

Wolter, H.

H. Wolter, “On basic analogies and principal differences between optical and electronic information,” in Progress in Optics, Vol. I, E.Wolf, ed. (North Holland, 1961), pp. 155-210.
[CrossRef]

Ya. Vilenkin, N.

I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions IV, Applications of Harmonic Analysis (Academic, 1964).

Yaglom, A. M.

I. M. Gelfand and A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. 12, 199-246 (1959).

Am. Math. Soc. Transl. (2)

A. N. Kolmogorov and V. M. Tihomirov, “ε-entropy and ε-capacity of sets in fuctional spaces,” Am. Math. Soc. Transl. 17, 277-364 (1961).

I. M. Gelfand and A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Transl. 12, 199-246 (1959).

Bell Syst. Tech. J. (3)

D. Slepian and H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty--I,” Bell Syst. Tech. J. 40, 43-64 (1961).

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty--IV: Extension to many dimensions; Generalized prolate spheroidal wave functions,” Bell Syst. Tech. J. 43, 3009-3057 (1964).

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of zero order,” Bell Syst. Tech. J. 44, 1745-1759 (1965).

J. Integral Equ. Appl. (1)

E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283-309 (2002).
[CrossRef]

J. Math. Anal. Appl. (2)

R. T. Prosser, “The ε-entropy and ε-capacity of certain time-varying channels,” J. Math. Anal. Appl. 16, 553-573 (1966).
[CrossRef]

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

J. Math. Phys. (2)

N. Magnoli and G. A. Viano, “The source identification problem in electromagnetic theory,” J. Math. Phys. 38, 2366-2388 (1997).
[CrossRef]

G. A. Viano, “On the extrapolation of optical image data,” J. Math. Phys. 17, 1160-1165 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163-165 (1973).
[CrossRef]

Proc. IEEE (1)

A. J. Jerri, “The Shannon sampling theorem--its various extensions and applications: A tutorial review,” Proc. IEEE 65, 1565-1596 (1977).
[CrossRef]

SIAM J. Math. Anal. (1)

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855-877 (1998).
[CrossRef]

Other (14)

A. Tikhonov and V. Arsenine, Méthodes de Résolution de Problèmes Mal Poses (Mir, 1976).

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind (Pitman, 1984).

B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, Vol. IX, E.Wolf, ed. (North Holland, 1972), pp. 311-407.

A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, 1976), Chap. 6.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960), Chap. 6.

G. G. Lorentz, Approximation of Functions (Holt, Rinehart and Winston, 1966), Chap. 10.

I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions IV, Applications of Harmonic Analysis (Academic, 1964).

D. Gabor, “Light and information,” in Progress in Optics, Vol. I, E.Wolf, ed. (North Holland, 1961), pp. 111-153.

T. Kato, Perturbation Theory for Linear Operators (Springer, 1966), p. 522.

M. Bertero and C. De Mol, “Super-resolution by data inversion,” in Progress in Optics, Vol. XXXVI, E.Wolf, ed. (North-Holland, 1996), pp. 129-178.
[CrossRef]

J. Hadamard, Lectures on the Cauchy Problem in Linear Partial Differential Equations (Yale U. Press, 1923).

M. Bertero, C. De Mol, and G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H.P.Baltes, ed., Topics in Current Physics, Vol. 20 (Springer, 1980), pp. 161-214.
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, 1959), p. 166.

H. Wolter, “On basic analogies and principal differences between optical and electronic information,” in Progress in Optics, Vol. I, E.Wolf, ed. (North Holland, 1961), pp. 155-210.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Schematic of coherent light image formation in a one-dimensional diffraction-limited optical system (see also [2]).

Fig. 2
Fig. 2

Eigenvalues λ k (filled dots) of the kernel in Eq. (3) with Shannon number S = 12.7 .

Equations (66)

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

F ( ω ) = 1 2 π + f ( x ) e i ω x d x ,
g ( y ) = 1 2 π Ω Ω F ( ω ) e i ω y d ω .
g ( y ) = 1 2 π Ω Ω e i ω y d ω X 0 2 X 0 2 f ( x ) e i ω x d x = X 0 2 X 0 2 sin [ Ω ( x y ) ] π ( x y ) f ( x ) d x .
( A f ) ( y ) = X 0 2 X 0 2 sin [ Ω ( x y ) ] π ( x y ) f ( x ) d x = g ( y ) , ( X 0 2 y X 0 2 ) .
M ε 2 S log ( 1 ε ) ,
A f + n = g , ( g = g + n ) ,
A f g Y ε , ( ε = constant ) ,
B f Z E , ( E = constant ) ,
Φ ( f ) = A f g Y 2 + μ 2 B f Z 2 , ( μ = ε E ) ,
B f Z = ( k = 0 β k 2 f k 2 ) 1 2 < E , ( E = constant ) ,
f = A * g A * A + ( ε E ) 2 B * B ,
f = k = 0 λ k g k λ k 2 + ( ε E ) 2 β k 2 ψ k .
lim ε 0 f f X = 0 , ( E = fixed ) .
f ( 1 ) k = 0 k β g k λ k ψ k ,
λ k β k ε E .
lim ε 0 f f ( 1 ) = 0 , ( E = fixed ) .
B f Z f L 2 ( X 0 2 , X 0 2 ) = ( k = 0 f k 2 ) 1 2 E ,
( E = constant ) .
f ( 2 ) = k = 0 λ k g k λ k 2 + ( ε E ) 2 ψ k ,
f ( 3 ) k = 0 k I g k λ k ψ k ,
λ k ε E .
lim ε 0 ( [ f f ( 2 ) ] , v ) = 0 , [ v L 2 ( X 0 2 , X 0 2 ) ; E = fixed ] .
lim ε 0 ( [ f f ( 3 ) ] , v ) = 0 , [ v L 2 ( X 0 2 , X 0 2 ) ; E = fixed ] .
A ξ + ζ = η ,
R η η = A R ξ ξ A * + R ζ ζ ,
R ξ η = R ξ ξ A * .
R ζ ζ = ε 2 H ,
λ k ξ k + ζ k = η k , ( k = 0 , 1 , 2 , ) ,
p ξ k ( x ) = 1 2 π ρ k exp ( x 2 2 ρ k 2 ) , ( k = 0 , 1 , 2 , ) ,
p ζ k ( x ) = 1 2 π ε ν k exp ( x 2 2 ε 2 ν k 2 ) , ( k = 0 , 1 , 2 , ) .
p η k ( y x ) = 1 2 π ε ν k exp [ ( y λ k x ) 2 2 ε 2 ν k 2 ] = 1 2 π ε ν k exp [ λ k 2 2 ε 2 ν k 2 ( x y λ k ) 2 ] .
p ξ k ( x y ) = p ξ k ( x ) p η k ( y x ) p η k ( y ) ,
p ξ k ( x g ) = A k exp ( x 2 2 ρ k 2 ) exp [ λ k 2 2 ε 2 ν k 2 ( x g k λ k ) 2 ] .
J = { k N : λ k ρ k ε ν k } ,
N = { k N : λ k ρ k < ε ν k } .
ξ k = { g k λ k , if k J , 0 , if k N . }
T η = k J g k λ k ψ k .
E { ξ T η 2 } = k N ρ k 2 + k J ε 2 ν k 2 λ k 2 < .
lim ε 0 E { ξ T η 2 } = 0 ,
H ε ( E ) C ε ( E ) H ε 2 ( E ) .
k = 0 k I λ k ε N ε ( E ) ;
k = 0 k I log λ k ε log N ε ( E ) = H ε ( E ) .
H ε 2 ( E ) k I ( ε 4 ) { log ( 1 ε ) + log 6 + 1 2 log k I ( ε 4 ) } ,
k = 0 k I log λ k ε = k = 0 S 1 log λ k ε + k = S k I log λ k ε ,
H ε ( E ) S log ( 1 ε ) .
M ε ( E ) 2 S log ( 1 ε ) ε 0 .
H ε 2 ( E ) ε 0 k I ( ε 4 ) log ( 1 ε ) 1 2 log 2 ( 1 ε ) .
S log ( 1 ε ) C ε ( E ) 1 2 log 2 ( 1 ε ) .
J ( ξ k , η k ) = 1 2 ln ( 1 r k 2 ) , ( k = 0 , 1 , 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 + ( ε ν k ) 2 , ( k = 0 , 1 , 2 , ) ,
J ( ξ k , η k ) = 1 2 ln ( 1 + λ k 2 ρ k 2 ε 2 ν k 2 ) , ( k = 0 , 1 , 2 , ) .
J ( ξ k , η k ) < 1 2 ln 2 , ( k N ) .
J k I J ( ξ k , η k ) = k J ln 1 + λ k 2 ρ k 2 ε 2 ν k 2 .
J = k J ln 1 + λ k 2 ρ k 2 ε 2 ν k 2 k J ln λ k ρ k ε ν k ,
J = k J J ( ξ k , η k ) k J ln λ k ε .
k J J ( ξ k , η k ) k J ln λ k ε = k = 0 k I ln λ k ε ,
J = k J J ( ξ k , η k ) S ln ( 1 ε ) .
M ε ( E ) = 2 C ε ( E ) 2 S log ( 1 ε ) 2 J = 2 { k J J ( ξ k , η k ) } ,
J k J ln λ k ρ k ε ν k k J ln ( 1 ε ) = k I ( ε ) ln ( 1 ε ) .
J k I ( ε ) ln ( 1 ε ) 1 2 ln 2 ( 1 ε ) ,
C ε ( E ) S log ( 1 ε ) ,
M ε ( E ) 2 S log ( 1 ε ) .
C ε ( E ) 1 2 log 2 ( 1 ε ) ε 0 .
J = k J J ( ξ k , η k ) S ln ( 1 ε ) .
J 1 2 ln 2 ( 1 ε ) .
M ε ( E ) = 2 C ε ( E ) 2 { k J J ( ξ k , η k ) } ε 0 .

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