Abstract

As band-limited interpolation is possible to some extent, whereas band-limited extrapolation is strictly forbidden, it is natural to state the deconvolution problem in terms of weighted interpolation (in the frequency domain). The corresponding regularization principle, which is to be compared with that of Tikhonov, proves to be intimately related to the notion of resolution: the aperture to be synthesized explicitly appears in the definition of the stabilizing component of the objective function. The choice of this synthetic aperture of course depends on the error analysis. The study developed in this paper shows, in particular, how the amount and the nature of the interpolation to be performed play a central part in the error propagation. The corresponding analytic parameters are exhibited. This survey, which is illustrated with the aid of a simulated example, also indicates how to estimate the degree of confidence of any other deconvolution technique (clean, filtered singular-value decomposition, maximum-entropy, etc.).

© 1987 Optical Society of America

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References

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  1. J. L. C. Sanz, T. S. Huang, “Unified Hilbert space approach to iterative least-squares linear signal restoration,”J. Opt. Soc. Am. 73, 1455–1465 (1983).
    [CrossRef]
  2. J. B. Abiss, N. Defrise, C. De Mole, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration in the presence of noise and error analysis,”J. Opt. Soc. Am. 73, 1470–1475 (1983).
    [CrossRef]
  3. W. L. Root, “Ill-posedness and precision in object field reconstruction problems,” J. Opt. Soc. Am. A 4, 171–179 (1987).
    [CrossRef]
  4. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).
  5. V. Vapnik, Estimation of Dependences Based on Empirical Data (Springer-Verlag, New York, 1982).
  6. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  7. A. Lannes, “Abstract holography,”J. Math. Anal. Appl. 74, 530–559 (1980).
    [CrossRef]
  8. A. Lannes, J.-Ph. Pérez, Optique de Fourier en Microscopie Electronique (Masson, Paris, 1983).
  9. Given some complex vector space E, there exists a real vector space E0underlying E, the elements of which coincide with those of E. For example, if E is the one-dimensional complex space generated by some given vector e, we have E={λe:λ∈C}={αe+β(ie):α,β∈R}=E0. In this case E0is a two-dimensional real vector space. More generally, when E is finite dimensional, the dimension of E0is twice as large as that of E. In practice, E0can be regarded as the working space of the computer.
  10. M. Hestenes, E. Stiefel, “Method of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
    [CrossRef]
  11. J. W. Daniel, “The conjugate gradient method for linear and nonlinear operator equations,” SIAM J. Numer. Anal. 4, 10–26 (1967).
    [CrossRef]
  12. E. Durand, Solutions Numériques des Équations Algébriques (Masson, Paris, 1960), Vol. I.
  13. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty, IV: Extension to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
  14. C. J. Bowkamp, “On spheroidal wave functions of order zero,”J. Math. Phys. 26, 79–92 (1957).
  15. A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing,” Opt. Acta (to be published).
  16. U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (method clean),” Astron. Astrophys. 65, 345–356 (1978).
  17. R. Kikuchi, B. H. Soffer, “Maximum entropy image restoration. I. The entropy expression,”J. Opt. Soc. Am. 67, 1656–1665 (1977).
    [CrossRef]

1987 (1)

1983 (2)

1980 (1)

A. Lannes, “Abstract holography,”J. Math. Anal. Appl. 74, 530–559 (1980).
[CrossRef]

1978 (2)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (method clean),” Astron. Astrophys. 65, 345–356 (1978).

1977 (1)

1967 (1)

J. W. Daniel, “The conjugate gradient method for linear and nonlinear operator equations,” SIAM J. Numer. Anal. 4, 10–26 (1967).
[CrossRef]

1964 (1)

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

1957 (1)

C. J. Bowkamp, “On spheroidal wave functions of order zero,”J. Math. Phys. 26, 79–92 (1957).

1952 (1)

M. Hestenes, E. Stiefel, “Method of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[CrossRef]

Abiss, J. B.

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Bowkamp, C. J.

C. J. Bowkamp, “On spheroidal wave functions of order zero,”J. Math. Phys. 26, 79–92 (1957).

Casanove, M. J.

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing,” Opt. Acta (to be published).

Daniel, J. W.

J. W. Daniel, “The conjugate gradient method for linear and nonlinear operator equations,” SIAM J. Numer. Anal. 4, 10–26 (1967).
[CrossRef]

De Mole, C.

Defrise, N.

Dhadwal, H. S.

Durand, E.

E. Durand, Solutions Numériques des Équations Algébriques (Masson, Paris, 1960), Vol. I.

Hestenes, M.

M. Hestenes, E. Stiefel, “Method of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[CrossRef]

Huang, T. S.

Kikuchi, R.

Lannes, A.

A. Lannes, “Abstract holography,”J. Math. Anal. Appl. 74, 530–559 (1980).
[CrossRef]

A. Lannes, J.-Ph. Pérez, Optique de Fourier en Microscopie Electronique (Masson, Paris, 1983).

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing,” Opt. Acta (to be published).

Pérez, J.-Ph.

A. Lannes, J.-Ph. Pérez, Optique de Fourier en Microscopie Electronique (Masson, Paris, 1983).

Root, W. L.

Roques, S.

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing,” Opt. Acta (to be published).

Sanz, J. L. C.

Schwarz, U. J.

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (method clean),” Astron. Astrophys. 65, 345–356 (1978).

Slepian, D.

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

Soffer, B. H.

Stiefel, E.

M. Hestenes, E. Stiefel, “Method of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Vapnik, V.

V. Vapnik, Estimation of Dependences Based on Empirical Data (Springer-Verlag, New York, 1982).

Youla, D. C.

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Astron. Astrophys. (1)

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (method clean),” Astron. Astrophys. 65, 345–356 (1978).

Bell Syst. Tech. J. (1)

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

IEEE Trans. Circuits Syst. (1)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

J. Math. Anal. Appl. (1)

A. Lannes, “Abstract holography,”J. Math. Anal. Appl. 74, 530–559 (1980).
[CrossRef]

J. Math. Phys. (1)

C. J. Bowkamp, “On spheroidal wave functions of order zero,”J. Math. Phys. 26, 79–92 (1957).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

J. Res. Nat. Bur. Stand. (1)

M. Hestenes, E. Stiefel, “Method of conjugate gradients for solving linear systems,”J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[CrossRef]

SIAM J. Numer. Anal. (1)

J. W. Daniel, “The conjugate gradient method for linear and nonlinear operator equations,” SIAM J. Numer. Anal. 4, 10–26 (1967).
[CrossRef]

Other (6)

E. Durand, Solutions Numériques des Équations Algébriques (Masson, Paris, 1960), Vol. I.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

V. Vapnik, Estimation of Dependences Based on Empirical Data (Springer-Verlag, New York, 1982).

A. Lannes, J.-Ph. Pérez, Optique de Fourier en Microscopie Electronique (Masson, Paris, 1983).

Given some complex vector space E, there exists a real vector space E0underlying E, the elements of which coincide with those of E. For example, if E is the one-dimensional complex space generated by some given vector e, we have E={λe:λ∈C}={αe+β(ie):α,β∈R}=E0. In this case E0is a two-dimensional real vector space. More generally, when E is finite dimensional, the dimension of E0is twice as large as that of E. In practice, E0can be regarded as the working space of the computer.

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing,” Opt. Acta (to be published).

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Figures (9)

Fig. 1
Fig. 1

General diagram for stable reconstruction.

Fig. 2
Fig. 2

Illustration of the regularization principle.

Fig. 3
Fig. 3

The original object and its experimental image.

Fig. 4
Fig. 4

Resolution parameters.

Fig. 5
Fig. 5

Weighted frequency gaps.

Fig. 6
Fig. 6

Estimation of λ and μ for η = ηe.

Fig. 7
Fig. 7

Choice of the starting function vΦt.

Fig. 8
Fig. 8

Localization of the eigenvalues.

Fig. 9
Fig. 9

Stable reconstruction with and without nonnegativity constraint.

Equations (97)

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Φ ^ i ( u ) = def R p Φ i ( x ) exp ( - 2 i π x · u ) d x ,
Φ ^ i ( u ) - h ( u ) Φ ^ o ( u ) σ i ( u ) .
h ( u ) = psf ^ ( u ) ,             psf ( x ) = h ^ ( - x ) .
ϕ = def [ R p ϕ ( x ) 2 d x ] 1 / 2 .
ϕ 1 ϕ 2 = def R p ϕ 1 * ( x ) ϕ 2 ( x ) d x .
SNR ( u ) = def Φ ^ i ( u ) σ i ( u ) .
A * A Φ = A * Ψ i .
Φ r - Φ s / Φ s .
Φ ^ s ( u ) = s ( u ) Φ ^ o ( u ) ,
h r s 2 s 2 H r s ( u ) 2 d u R p s ( u ) d u = χ 2 .
Δ ν = def 2 sup u H r u .
V Φ s ( x ) d x = V o Φ o ( x ) d x ,
Φ ^ t = def s t Φ ^ i ,
s t = def { s / h if SNR α t 0 otherwise ,
q ( ϕ ) = def g · ( Φ ^ t - ϕ ^ ) 2 ,
( on H r )     g = { 1 if SNR α t ( SNR - α t ) / ( α t - α t ) if α t SNR < α t 0 if SNR < α t .
q ( ϕ ) = Ψ i - A ϕ 2 ,
Ψ i = U * g U Φ t = U * g s t U Φ i ,             A ϕ = def U * g U ϕ .
Ψ ^ s = g s t h Φ ^ o + k r s Φ ^ o ,
k r = def { 1 outside H r if SNR < α t 0 otherwise .
Ψ ^ i - Ψ ^ s = g s t · ( Φ ^ i - h Φ ^ o ) - k r s Φ ^ o .
Φ ^ o ( u ) σ o ( u ) .
Ψ i - Ψ s 2 = Ψ ^ i - Ψ ^ s 2 g s t σ i 2 + k r s σ o 2 .
Ψ i - Ψ s , = def i 2 + o 2 ,
i = def g s t σ i , o = def k r s σ o .
F i = { ψ i E o = def L 2 ( R p ) : ψ ^ i ( u ) = 0 if u H r and SNR < α t } .
( A ϕ ψ i ) = ( U * g U ϕ ψ i ) = ( ϕ U * g * U ψ i ) = ( ϕ v U * g U ψ i ) .
A * ψ i = v A ψ i = v U * g U ψ i ,             A * A ϕ = v A 2 ϕ = v U * g 2 U ϕ .
w = def 1 - g 2 ,
B B ( v , w ) :             B ( v , w ) ϕ = def v U * w U ϕ             ( ϕ = v ϕ E ) .
λ = def λ 1 > λ 2 > > λ k > λ k + 1 > > 0.
μ = def μ 1 < μ 2 < < μ k < μ k + 1 < < 1.
f j = def 1 μ j + A e j
Φ = j 1 β j μ j + e j ,             β j = def ( f j Ψ i ) = ( f j Ψ ) .
q i ( ϕ ) = def ( 1 - k r ) · g ( Φ ^ t - Φ ^ ) 2 ,             q r ( ϕ ) = def k r ϕ ^ 2 .
lim n Φ - ϕ n + 1 Φ - ϕ n = 0.
R 0 = D 0 = 1 ;             R n + 1 = R n - ω n X D n ,             D n + 1 = R n + 1 + γ n D n .
S 0 ( X ) , , S m - 1 ( X ) , S m ( X ) , S m + 1 ( X ) , , S n ( X ) ,
N C n ( 1 - δ μ ) .
δ Φ = def Φ - Φ s ,             Ψ = def Ψ - Ψ s .
Φ = j 1 δ β j μ j + e j ,             δ β j = def ( f j δ Ψ ) ,
δ Φ 2 = δ Ψ 2 j 1 κ j 2 μ j + ,             κ j 2 = def δ β j 2 δ Ψ 2 .
Ψ i - Ψ s 2 = Ψ i - Ψ 2 + Ψ - Ψ s 2 = Ψ i - Ψ 2 + δ Ψ 2 ;
δ Ψ ρ ,             ρ = def 2 - δ 2 ,             δ = def Ψ i - Ψ .
ϕ n - Φ s ϕ n Θ n ,             Θ n = def 1 μ ρ n ϕ n ,
1 μ = def κ 2 1 - δ μ + j = 1 N κ j 2 μ j + ,             κ 2 + j = 1 N κ j 2 = 1 ,
ρ n = def 2 - δ n 2 ,             δ n 2 = def g ( Φ ^ t - ϕ ^ n 2 .
( B ϕ ) ( x ) = v ( x ) V w ^ ( x - x ) ϕ ( x ) d x .
v ( x ) = def v ( τ x + x 0 ) , τ = def [ v ( x ) d x ] 1 / p , x 0 = def 1 τ p x v ( x ) d x .
w ( u ) = def w ( v u + u 0 ) ,             ν = def [ v ( u ) d u ] 1 / p u 0 = def 1 ν p u w ( u ) d u .
w η ( u ) = def w ( u ) / η ) ,
η = def τ ν .
( M ϕ ) ( x ) = def ϕ ( τ x + x 0 ) exp ( - 2 i π τ x · u 0 ) .
B = M - 1 BM ,             B = def B ( v , w η ) .
( B Φ ) ( x ) = η p v ( x ) v w ^ { η ( x - x ) } Φ ( x ) d x .
( Φ B Φ ) = ( Φ U * w η Φ ^ ) = ( Φ ^ w η Φ ^ ) = ( w η 1 / 2 Φ ^ w η 1 / 2 Φ ^ ) ,
λ = λ v , w ( η ) = sup Φ * = 1 ( Φ B Φ ) Φ 2 = sup Φ * = 1 w η 1 / 2 Φ ^ 2 .
( B η Φ ) ( x ) = def v ( x ) v w ^ { η ( x - x ) } Φ ( x ) d x .
λ v , w ( η ) = η p Λ η ,             Λ η = Λ - η .
λ v , w ( η ) = η p m = 0 ( - 1 ) m l m η 2 m ,             l 0 = 1.
l 1 = 2 J 2 ,
l 2 = 2 J 4 - J 2 2 + a 2 2 ( u ) w ( u ) d u + b 2 2 ( x ) v ( x ) d x ,
J m = def ( 2 π ) m m ! ( x · u ) m v ( x ) w ( u ) d x d u ,
a m ( u ) = def ( 2 π ) m m ! ( x · u ) m v ( x ) d x
b m ( x ) = def ( 2 π ) m m ! ( x · u ) m w ( u ) d u .
J ( v , w ) = def J = def J 2 = 2 π 2 ( x · u ) 2 v ( x ) w ( u ) d x d u
ν v , w ( η ) = η m = 0 ( - 1 ) m l m η 2 m
l 1 = 1 6 c 1 , l 2 = 1 15 c 2 + 1 180 c 1 2 , l 3 = 1 28 c 3 + 1 126 c 1 c 2 - 1 1080 c 1 3 , l 4 = 1 45 c 4 + 1 350 c 2 2 + 1 180 c 1 c 3 - 19 12600 c 1 2 c 2 + 1 8100 c 1 4 , l 5 = 1 66 c 5 + 2 495 c 2 c 3 + 2 495 c 1 c 4 - 1 1350 c 1 c 2 2 - 1 1080 c 1 2 c 3 + 11 56700 c 1 3 c 2 - 1 97200 c 1 5 .
c m = def ( 2 π ) 2 m ( 2 m ) ! u 2 m w ( u ) d u .
[ λ ] 5 = def η m = 0 5 ( - 1 ) m l m η 2 m
h r s 2 = ( h r s h r s ) = ( s h r s ) = ( U s ^ h r U s ^ ) = ( s ^ d r U * h r U s ^ ) ,
( s ^ B ( d r , h r ) s ^ ) s 2 = χ 2 .
( B Φ ) ( x ) = g r ( x ) U * [ h r ( y / η ) Φ ^ ( y ) ] = η p g r ( x ) G r h ^ r { η ( x - x ) } Φ ( x ) d x ,
η = [ G r d x ] 1 / p [ H r d u ] 1 / p .
E ( G r ) = def { Φ L 2 ( G r ) :             Φ ( - x ) = Φ ( x ) , Φ ( x ) = Φ * ( x ) }
Λ ( η ) = η p [ 1 - 2 J ( g r , h r ) η 2 + O ( η 4 ) ] ,
D ( β ; H r ) = def { x : b ( x ; H r ) β } ,
b ( x ; H r ) = def 2 π 2 H r ( x · u ) 2 d u .
b ( x ; H r ) = i = 1 p b i i x i 2 + 2 ( i , j ) b i j x i x j ,             b i j = def 2 π 2 H r u i u j d u .
τ r = def [ D r d x ] 1 / p = η ( χ ) / ν r ,             ν r = def [ H r d u ] 1 / p .
s ( u ) = s ( τ r u ; H r , χ ) ,             s ^ ( x ) = 1 τ r p s ^ ( x τ r ; H r , χ ) ) ,
s ^ ( x ; H r , χ ) = def Φ r ( x ) Φ ^ r ( 0 ) .
h ^ i ( x ) = { 1 if x 1 / 2 0 otherwise ;             h i ( u ) = sinc u = def sin π u π u .
θ = def Φ - Φ s Φ Θ ,             Θ = def 1 μ ρ Φ .
Θ e = def 1 μ e Φ t ,             Φ t = Φ t .
Θ e = def [ 1 2 ( 1 μ e + 1 ) ] 1 / 2 Φ ^ t .
ϕ n + 1 = P c { ϕ n + ω r n } = v n + 1 + ( ϕ n + ω r n ) ,
r n = v U * g 2 ( Φ ^ t - ϕ ^ n ) , v n + 1 + ( x ) = def { 1 if ϕ n ( x ) + ω r n ( x ) > 0. 0 otherwise
Φ c - Φ s Φ c - Φ + Φ - Φ s .
Φ 0 E ( G r ) ,             Φ n + 1 = B ( Φ n Φ n ) = g r ( x ) U * [ h r ( u / η ) Φ ^ n 1 ( u ) ] ,
1 Λ lim n Φ n = P Φ 0 P Φ 0 .
D ( β ) = def D ( β ; W ) = def { x : b ( x ; W ) β } .
σ ( β ) = D ( β ) V d x ,             σ 0 ( β ) = D ( β ) D d x .
J ( V , W ) = 0 β d σ ( β ) .
J ( V , W ) = 0 β d σ 0 ( β ) - 0 β d ( β ) ,             ( β ) = def σ 0 ( β ) - σ ( β ) .
J ( V , W ) = J ( D , W ) + 0 ( β ) d β J ( D , W ) .
E={λe:λC}={αe+β(ie):α,βR}=E0.

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