Abstract

Object reconstruction in weighted Hilbert space is related to the Miller regularization theory [ K. Miller, SIAM J. Math. Anal. 1, 52 ( 1970)]. Experimental results illustrating the power of the method are presented.

© 1983 Optical Society of America

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References

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  1. Optical Society of America Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints, Incline Village, Nevada, January 12–14, 1983.
  2. M. Bertero, C. De Mol, and G. A. Viano, "The stability of inverse problems," in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.
    [CrossRef]
  3. A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  4. D. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  5. J. B. Abbiss, C. De Mol, and H. S. Dhadwal, "Regularised iterative and non-iterative procedures for object restoration from experimental data," Opt. Acta 30, 107–124 (1983).
    [CrossRef]
  6. K. Miller, "Least-squares methods for ill-posed problems with a prescribed bound," SIAM J. Math. Anal. 1, 52–74 (1970).
    [CrossRef]
  7. W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973).

1983 (1)

J. B. Abbiss, C. De Mol, and H. S. Dhadwal, "Regularised iterative and non-iterative procedures for object restoration from experimental data," Opt. Acta 30, 107–124 (1983).
[CrossRef]

1978 (1)

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

1975 (1)

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

1970 (1)

K. Miller, "Least-squares methods for ill-posed problems with a prescribed bound," SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Abbiss, J. B.

J. B. Abbiss, C. De Mol, and H. S. Dhadwal, "Regularised iterative and non-iterative procedures for object restoration from experimental data," Opt. Acta 30, 107–124 (1983).
[CrossRef]

Bertero, M.

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

De Mol, C.

J. B. Abbiss, C. De Mol, and H. S. Dhadwal, "Regularised iterative and non-iterative procedures for object restoration from experimental data," Opt. Acta 30, 107–124 (1983).
[CrossRef]

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

Dhadwal, H. S.

J. B. Abbiss, C. De Mol, and H. S. Dhadwal, "Regularised iterative and non-iterative procedures for object restoration from experimental data," Opt. Acta 30, 107–124 (1983).
[CrossRef]

Miller, K.

K. Miller, "Least-squares methods for ill-posed problems with a prescribed bound," SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Papoulis, A.

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

Rudin, W.

W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973).

Viano, G. A.

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

Youla, D.

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

IEEE Trans. Circuits Syst. (2)

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

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

Opt. Acta (1)

J. B. Abbiss, C. De Mol, and H. S. Dhadwal, "Regularised iterative and non-iterative procedures for object restoration from experimental data," Opt. Acta 30, 107–124 (1983).
[CrossRef]

SIAM J. Math. Anal. (1)

K. Miller, "Least-squares methods for ill-posed problems with a prescribed bound," SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Other (3)

W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973).

Optical Society of America Topical Meeting on Signal Recovery and Synthesis with Incomplete Information and Partial Constraints, Incline Village, Nevada, January 12–14, 1983.

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

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Figures (3)

Fig. 1
Fig. 1

Examples of (a) original object, f(t), (32 × 32), (b) the passband in x space (11 × 11), (c) the corresponding low-pass filtered image, (d) the chosen weighting function p(t), and (e) the corresponding estimate for f(t), f ˜ (t) obtained after inverting a 121 × 121 matrix [Eq. (9)]

Fig. 2
Fig. 2

(a) Passband in x space giving a bow-tie data set, (b) low-pass-filtered image corresponding to the object shown in Fig. 1(a), and (c) estimate using the p(t) shown in Fig. 1(d).

Fig. 3
Fig. 3

(a) Estimate for f(t) corresponding to Fig. 1(e) but with a low level of white noise added to the data (signal-to-noise ratio, 200:1), and (b) the corresponding estimate when the p(t) of Fig. 1(d) has 5 × 10−3 added to it, making it nonzero outside the known object support.

Equations (23)

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f ^ ( x n ) = Ω f ( t ) exp ( i t x n ) d t .
c n = ( f , ϕ n ) F .
f = n = 1 N a n ϕ n + f 2
n = 1 N ( ϕ n , ϕ m ) F a n = c m .
f ˜ = n = 1 N a n ϕ n .
f - n = 1 N a n ϕ n F 2
( f , ϕ ) F = Ω p - 1 ( t ) f ( t ) ϕ * ( t ) d t ,
f ˜ = n = 1 N a n ϕ n
c n = f ^ ( x n ) = Ω f ( t ) exp ( i t x n ) d t = ( f , ϕ n ) F ,
c n = m = 1 N a m ( ϕ m , ϕ n ) F = m = 1 N a m p ^ ( x n - x m ) ,
A f + n = f ^ ,
A f - f ^ F ^ ɛ ,             B f E
Φ ( f ) = A f - f ^ F 2 + ( / E ) 2 B f 2 2 .
f ˜ = [ A * A + ( / E ) 2 B * B ] - 1 A * f ^ ,
A f = n = 1 N ( f , ϕ n ) F e n = n = 1 N f ^ ( x n ) e n = f ^ ,
( f ^ , g ^ ) F ^ = m = 1 N q - 1 ( x m ) f ^ ( x m ) g ^ * ( x m )
( n = 1 N ( f , ϕ n ) F e n , f ^ ) F ^ = n = 1 N ( f , ϕ n ) F ( e n , f ^ ) F ^ = ( f , n = 1 N ( f ^ , e n ) F ^ ϕ n ) F = ( f , A * f ^ ) F ,
A * f ^ = n = 1 N ( f ^ , e n ) F ^ ϕ n .
A * A f = A * [ n = 1 N ( f , ϕ n ) F e n ] = m = 1 N ( n = 1 N ( f , ϕ n ) F e n , e m ) F ^ ϕ m = m = 1 , n = 1 N ( f , ϕ n ) F ( e n , e m ) F ^ ϕ m = n = 1 N ( f , ϕ n ) F ϕ n ,
n = 1 N ( f ˜ , ϕ n ) F ϕ n + ( / E ) 2 f ˜ = n = 1 N ( f ^ , e n ) F ^ ϕ n .
f ˜ = n = 1 N a n ϕ n
n = 1 N ( ( m = 1 N a m ϕ m ) , ϕ n ) F ϕ n + ( / E ) 2 n = 1 N a n ϕ n = n = 1 N ( f ^ , e n ) F ^ ϕ n .
m = 1 N [ ( ϕ m , ϕ n ) F + ( / E ) 2 δ n m ] a m = f ^ ( x n ) ,