Abstract

The inherent stability or instability in reconstructing an object field, in the presence of observation noise, for a class of ill-posed problems is investigated for situations in which constraints are imposed on the object fields. The class of ill-posed problems includes inversion of truncated Fourier transforms. Two kinds of constraint are considered. It is shown that if the object field is restricted to a subset of L2 space over Rn that is bounded, closed, convex, and has nonempty interior, then a (nonlinear) least-squares estimate always exists but is unstable. It is also shown that if one is primarily concerned with the situation in which the object field belongs to a compact parallelepiped in L2, aligned in a natural way, there is a satisfactory, stable linear estimate that is optimal according to a min–max criterion. This also leads to a nonlinear modification for the case in which the object field is actually restricted to the parallelepiped. A summary of some relevant mathematical background is included.

© 1987 Optical Society of America

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References

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  1. A proper definition of white noise in Hilbert space cannot be made with a (conventional) countably additive probability measure. One approach is to use finitely additive measures. See Ref. 5, Appendix B and further references given there. All this does not really matter here, because when one projects on any finite-dimensional subspace, the induced white-noise vector is well defined in the usual way.
  2. N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, translated by M. Nestell (Ungar, New York, 1961), Vol. 1.
  3. F. Riesz, B. Sz. Nagy, Functional Analysis, translated by F. Boron (Ungar, New York, 1951).
  4. J. L. C. Sanz, T. S. Huang, “Unified Hilbert space approach to iterative least-squares signal restoration,”J. Opt. Soc. Am. 73, 1455–1465 (1983).
    [CrossRef]
  5. L. S. Joyce, W. L. Root, “Precision bounds in superresolution processing,” J. Opt. Soc. Am. A 1, 149–168 (1984).
    [CrossRef]
  6. F. J. Beutler, W. L. Root, “The operator pseudoinverse in control and systems identification,” in Generalized Inverses and Applications, Z. Nashed, ed. (Academic, New York, 1976), pp. 397–494.
  7. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–701 (1978).
    [CrossRef]
  8. J. B. Abbiss, M. Defrise, C. DeMol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration in the presence of noise: an error analysis,”J. Opt. Soc. Am. 73, 1470–1475 (1983).
    [CrossRef]
  9. C. K. Rushforth, R. W. Harris, “Restoration, resolution and noise,”J. Opt. Soc. Am. 58, 539–545 (1968).
    [CrossRef]
  10. J. B. Abbiss, C. DeMol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration from experimental data,” Opt. Acta 30, 107–24 (1983).
    [CrossRef]
  11. A. Papoulis, “A new algorithm in spectral analysis and band-limited extraction,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  12. A. Albert, Regression and the Moore–Penrose Pseudoinverse (Academic, New York, 1972).
  13. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,”IEEE Trans Med. Imaging MI-1, 81–84 (1982).
    [CrossRef]
  14. W. L. Root, “Estimation in identification theory,” in Proceedings of the Ninth Allerton Conference on Circuit and System Theory (U. Illinois Press, Urbana, Ill., 1971), pp. 1–10.
  15. P. H. Fiske, W. L. Root, “Identifiability of slowly varying systems,” Inf. Control 32, 201–230 (1976).
    [CrossRef]

1984 (1)

1983 (3)

1982 (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,”IEEE Trans Med. Imaging MI-1, 81–84 (1982).
[CrossRef]

1978 (1)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–701 (1978).
[CrossRef]

1976 (1)

P. H. Fiske, W. L. Root, “Identifiability of slowly varying systems,” Inf. Control 32, 201–230 (1976).
[CrossRef]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extraction,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1968 (1)

Abbiss, J. B.

J. B. Abbiss, M. Defrise, C. DeMol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration in the presence of noise: an error analysis,”J. Opt. Soc. Am. 73, 1470–1475 (1983).
[CrossRef]

J. B. Abbiss, C. DeMol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration from experimental data,” Opt. Acta 30, 107–24 (1983).
[CrossRef]

Akhiezer, N. I.

N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, translated by M. Nestell (Ungar, New York, 1961), Vol. 1.

Albert, A.

A. Albert, Regression and the Moore–Penrose Pseudoinverse (Academic, New York, 1972).

Beutler, F. J.

F. J. Beutler, W. L. Root, “The operator pseudoinverse in control and systems identification,” in Generalized Inverses and Applications, Z. Nashed, ed. (Academic, New York, 1976), pp. 397–494.

Defrise, M.

DeMol, C.

J. B. Abbiss, M. Defrise, C. DeMol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration in the presence of noise: an error analysis,”J. Opt. Soc. Am. 73, 1470–1475 (1983).
[CrossRef]

J. B. Abbiss, C. DeMol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration from experimental data,” Opt. Acta 30, 107–24 (1983).
[CrossRef]

Dhadwal, H. S.

J. B. Abbiss, C. DeMol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration from experimental data,” Opt. Acta 30, 107–24 (1983).
[CrossRef]

J. B. Abbiss, M. Defrise, C. DeMol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration in the presence of noise: an error analysis,”J. Opt. Soc. Am. 73, 1470–1475 (1983).
[CrossRef]

Fiske, P. H.

P. H. Fiske, W. L. Root, “Identifiability of slowly varying systems,” Inf. Control 32, 201–230 (1976).
[CrossRef]

Glazman, I. M.

N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, translated by M. Nestell (Ungar, New York, 1961), Vol. 1.

Harris, R. W.

Huang, T. S.

Joyce, L. S.

Nagy, B. Sz.

F. Riesz, B. Sz. Nagy, Functional Analysis, translated by F. Boron (Ungar, New York, 1951).

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extraction,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Riesz, F.

F. Riesz, B. Sz. Nagy, Functional Analysis, translated by F. Boron (Ungar, New York, 1951).

Root, W. L.

L. S. Joyce, W. L. Root, “Precision bounds in superresolution processing,” J. Opt. Soc. Am. A 1, 149–168 (1984).
[CrossRef]

P. H. Fiske, W. L. Root, “Identifiability of slowly varying systems,” Inf. Control 32, 201–230 (1976).
[CrossRef]

F. J. Beutler, W. L. Root, “The operator pseudoinverse in control and systems identification,” in Generalized Inverses and Applications, Z. Nashed, ed. (Academic, New York, 1976), pp. 397–494.

W. L. Root, “Estimation in identification theory,” in Proceedings of the Ninth Allerton Conference on Circuit and System Theory (U. Illinois Press, Urbana, Ill., 1971), pp. 1–10.

Rushforth, C. K.

Sanz, J. L. C.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,”IEEE Trans Med. Imaging MI-1, 81–84 (1982).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,”IEEE Trans Med. Imaging MI-1, 81–84 (1982).
[CrossRef]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–701 (1978).
[CrossRef]

IEEE Trans Med. Imaging (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,”IEEE Trans Med. Imaging MI-1, 81–84 (1982).
[CrossRef]

IEEE Trans. Circuits Syst. (2)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,”IEEE Trans. Circuits Syst. CAS-25, 694–701 (1978).
[CrossRef]

A. Papoulis, “A new algorithm in spectral analysis and band-limited extraction,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Inf. Control (1)

P. H. Fiske, W. L. Root, “Identifiability of slowly varying systems,” Inf. Control 32, 201–230 (1976).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Acta (1)

J. B. Abbiss, C. DeMol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration from experimental data,” Opt. Acta 30, 107–24 (1983).
[CrossRef]

Other (6)

A. Albert, Regression and the Moore–Penrose Pseudoinverse (Academic, New York, 1972).

A proper definition of white noise in Hilbert space cannot be made with a (conventional) countably additive probability measure. One approach is to use finitely additive measures. See Ref. 5, Appendix B and further references given there. All this does not really matter here, because when one projects on any finite-dimensional subspace, the induced white-noise vector is well defined in the usual way.

N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, translated by M. Nestell (Ungar, New York, 1961), Vol. 1.

F. Riesz, B. Sz. Nagy, Functional Analysis, translated by F. Boron (Ungar, New York, 1951).

F. J. Beutler, W. L. Root, “The operator pseudoinverse in control and systems identification,” in Generalized Inverses and Applications, Z. Nashed, ed. (Academic, New York, 1976), pp. 397–494.

W. L. Root, “Estimation in identification theory,” in Proceedings of the Ninth Allerton Conference on Circuit and System Theory (U. Illinois Press, Urbana, Ill., 1971), pp. 1–10.

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Equations (72)

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s ( ξ ) = E b ( ξ , x ) σ ( x ) d x ,             ξ F ,
s = B σ .
s = B σ + w
F F b ( ξ , x ) 2 d ξ d x < .
s ( ξ ) = E exp [ - j 2 π ( x , ξ ) ] σ ( x ) d x ,             ξ F ,
B * B ϕ n = λ n ϕ n ;
ψ n λ n - 1 / 2 B ϕ n ,             n = 1 , 2 , ,
B B * ψ n = λ n ψ n             ( same λ n ) ,
ϕ n = λ n - 1 / 2 B * ψ n .
B y = n = 1 λ n 1 / 2 y , ϕ n ψ n
B * B y = n = 1 λ n y , ϕ n ϕ n .
B l - 1 ( z ) = n = 1 λ n - 1 / 2 z , ψ n ϕ n ,             z R ( B ) .
[ B * B σ ] ( x ) = E g ( x , x ) σ ( x ) d x ,
g ( x , x ) = g ( x , x ) ¯ = F b ( ξ , x ) ¯ b ( ξ , x ) d ξ .
n = 1 λ n < .
[ B * B σ ] ( x ) = [ Q T * P T Q σ ] ( x ) = E ρ ( x - x ) σ ( x ) d x ,
ρ ( x ) F exp [ j 2 π ( x , ξ ) ] d ξ .
δ s = inf σ Σ s - B σ .
s - B σ s = δ s
σ s < σ
σ = C s = ( B r ) - 1 P R s ,             s D ( C ) ,
B σ n - s δ s             as n .
σ n             as n .
C = B + = ( B * B ) - 1 B * .
E σ - σ ^ 2 = Tr [ C Γ C * ] ,
C = ( B * Γ - 1 B ) - 1 B * Γ - 1 .
C α = [ α I + B * B ] - 1 B * ,             α > 0.
s = B Q k σ + w ,
s = B Q k σ k + w .
P k s = B k σ k + P k w ,
Π = { y H : y , ϕ n b n } .
σ ^ N = ( B N * B N ) - 1 B N * P N s
σ ^ N = σ N + ζ N ,
ζ N = ( B N * B N ) - 1 B N * ( P N w )
( B * B ) - 1 ϕ j = μ j 2 ϕ j ,             j = 1 , 2 , ,
μ j 2 λ j - 1 .
( B N * B N ) - 1 ϕ j = μ j 2 ϕ j ,             j = 1 , 2 , , N .
E ζ , ϕ n ζ , ϕ m ¯ = c 2 μ n m δ n m ,
y = i , j = 1 N α i j σ ^ N , ϕ j ϕ i .
y - σ N = i N [ ( a i i - 1 ) σ i + j i a i j σ j ] ϕ i + i N [ j N a i j ζ , ϕ j ] ϕ i ,
E y - σ N 2 = i N | ( a i i - 1 ) σ i + j i N a i j σ j | 2 + c 2 i , j N a i j 2 μ j 2 .
G N ( a ) i N [ a i i - 1 b i + j i N a i j b j ] 2 + c 2 i , j N a i j 2 μ j 2 .
a i i = b i 2 b i 2 + c 2 μ i 2 ,             i = 1 , , N .
σ ^ ^ N = i = 1 N b i 2 b i 2 + c 2 μ i 2 σ ^ N , ϕ i ϕ i
= i = 1 N b i 2 μ i b i 2 + c 2 μ i 2 s , ψ i ϕ i
G N = i N | b i 2 b i 2 + c 2 μ i 2 - 1 | 2 b i 2 + i N | b i 2 b i 2 + c 2 μ i 2 | 2 c 2 μ i 2 = i N ( c 2 μ i 2 b i b i 2 + c 2 μ i 2 ) 2 + c 2 i N ( b i 2 μ i b i 2 + c 2 μ i 2 ) 2 = c 2 i N μ i 2 b i 2 b i 2 + c 2 μ i 2 ,
G c 2 i = 1 μ i 2 b i 2 b i 2 + c 2 μ i 2
C s = i = 1 b i 2 μ i b i 2 + c 2 μ i 2 s , ψ i ϕ i .
σ ^ ^ = C s .
C s i = 1 μ i s , ψ i ϕ i = B + s
σ ^ ^ Π = C Π s .
z m = 1 m 1 m σ n ,
B z m - s 1 m n = 1 m B σ n - s .
δ s B z m - s δ s + 1 m n = 1 m 2 - n δ s + m - 1 .
δ s B σ 0 - s B σ 0 - B z m + B z m - s δ s
B σ 0 - s = δ s ,
B [ α σ 0 + ( 1 - α ) σ 0 ] - s α B σ 0 - s + ( 1 - α ) B σ 0 - s = δ s ,
δ s B σ - s B σ - B σ 0 n + B σ 0 n - s B σ - σ 0 n + δ s δ s
s = ( 1 - α ) s ˜ + α s = B [ ( 1 - α ) σ ˜ + α σ ] .
s - s = ( 1 - α ) s ˜ - ( 1 - α ) s = ( 1 - α ) s ˜ - s = ( 1 - α ) B σ ˜ - s < B σ ˜ - s = δ s .
σ - σ ˜ = - α σ ˜ + α σ = α σ - σ ˜ .
s n = B σ n = B σ ˜ + K B ϕ n = s ˜ + λ n 1 / 2 K ψ n ,             n = 1 , 2 , .
σ N = i = 1 N a i i σ ^ ^ N , ϕ i ϕ i
σ ^ ^ N = i = 1 N b i 2 b i 2 + c 2 μ i 2 σ ^ N , ϕ i ϕ i .
σ N ( a ) = i , j = 1 N a i j σ ^ N , ϕ i ϕ i .
sup σ E σ N ( a ) - σ N 2 sup σ E σ N ( a ) - σ N 2 ,
inf a sup σ E σ N ( a ) - σ N 2 = inf a sup σ E σ N ( a ) - σ N 2 .
E σ N ( a ) - σ N 2 = i N | ( a i i - 1 ) σ i + j i N a i j σ j | 2 + c 2 i , j N a i j 2 μ j 2 .
E σ N ( a ) - σ N 2 = A + B + C + D ,
A = i N α i i 2 σ i 2 , B = j N i j N α i j 2 σ j 2 , C = j N k j N σ j σ ¯ k ( i N α i j α ¯ i k ) , D = c 2 i , j N a i j 2 μ j 2 .
E σ N ( a ) - σ N 2 = A + D ,
E σ N ( a ) - σ N 2 | σ max { θ 1 , , θ N } E σ N ( a ) - σ N 2 | σ

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