Abstract

A new method for the restoration of two-dimensional (2-D) images obtained through a circularly band-limited system is given. Object and image are decomposed into circular harmonics, and it is observed that the imaging system acts separately on each harmonic. We show that superresolution is, in practice, attainable with a small number of one-dimensional iterations. The method presents several advantages on the conventional 2-D algorithms of the Gerchberg type. The computing effort in particular can be much reduced. Performances of our method on computer-generated images are presented.

© 1985 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, 1455–65 (1983).
    [CrossRef]
  2. A. M. Darling, T. J. Hall, M. A. Fiddy, “Stable, noniterative object reconstruction from incomplete data using a priori knowledge,” J. Opt. Soc. Am. 73, 1466–1469 (1983).
    [CrossRef]
  3. L. S. Joyce, W. L. Root, “Precision bounds in superresolution processing,” J. Opt. Soc. Am. A 1, 149–168 (1984).
    [CrossRef]
  4. M. Bertero, C. de Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging, a singular value analysis. IV. The case of uncertain localization or nonuniform illumination of the object,” Opt. Acta, 31, 923–945 (1984).
    [CrossRef]
  5. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  6. P. de Santis, F. Gori, “On an iterative method for super-resolution,” Opt. Acta 22, 691–695 (1975).
    [CrossRef]
  7. A. Papoulis, “A new algorithm in spectral analysis and band-limited signal extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  8. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  9. G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
    [CrossRef]
  10. M. Bertero, C. de Mol, 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]
  11. R. J. Marks, “Gerchberg’s extrapolation algorithm in two dimensions,” Appl. Opt. 20, 1815–1820 (1981).
    [CrossRef]
  12. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
    [CrossRef]
  13. M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
    [CrossRef]
  14. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. IV: Extensions to many dimensions; generalized prolate spheroidal wave functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964).
    [CrossRef]
  15. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  16. F. Riesz, B. S. Nagy, Functional Analysis (Ungar, New York, 1951).
  17. F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs. Singular value analysis for two cases of low-pass filtering,” Inverse Problems 1, 67–85 (1985).
    [CrossRef]

1985 (1)

F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs. Singular value analysis for two cases of low-pass filtering,” Inverse Problems 1, 67–85 (1985).
[CrossRef]

1984 (2)

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

M. Bertero, C. de Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging, a singular value analysis. IV. The case of uncertain localization or nonuniform illumination of the object,” Opt. Acta, 31, 923–945 (1984).
[CrossRef]

1983 (2)

1982 (2)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

1981 (1)

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]

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

1975 (2)

P. de Santis, F. Gori, “On an iterative method for super-resolution,” Opt. Acta 22, 691–695 (1975).
[CrossRef]

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

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1964 (1)

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

Bertero, M.

M. Bertero, C. de Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging, a singular value analysis. IV. The case of uncertain localization or nonuniform illumination of the object,” Opt. Acta, 31, 923–945 (1984).
[CrossRef]

M. Bertero, C. de Mol, 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]

Cesini, G.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Darling, A. M.

de Mol, C.

M. Bertero, C. de Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging, a singular value analysis. IV. The case of uncertain localization or nonuniform illumination of the object,” Opt. Acta, 31, 923–945 (1984).
[CrossRef]

M. Bertero, C. de Mol, 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 Santis, P.

P. de Santis, F. Gori, “On an iterative method for super-resolution,” Opt. Acta 22, 691–695 (1975).
[CrossRef]

Fiddy, M. A.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs. Singular value analysis for two cases of low-pass filtering,” Inverse Problems 1, 67–85 (1985).
[CrossRef]

P. de Santis, F. Gori, “On an iterative method for super-resolution,” Opt. Acta 22, 691–695 (1975).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs. Singular value analysis for two cases of low-pass filtering,” Inverse Problems 1, 67–85 (1985).
[CrossRef]

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Hall, T. J.

Huang, T. S.

Joyce, L. S.

Lucarini, G.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Marks, R. J.

Nagy, B. S.

F. Riesz, B. S. Nagy, Functional Analysis (Ungar, New York, 1951).

Palma, C.

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Papoulis, A.

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

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Pike, E. R.

M. Bertero, C. de Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging, a singular value analysis. IV. The case of uncertain localization or nonuniform illumination of the object,” Opt. Acta, 31, 923–945 (1984).
[CrossRef]

Riesz, F.

F. Riesz, B. S. Nagy, Functional Analysis (Ungar, New York, 1951).

Root, W. L.

Sanz, J. L. C.

Sezan, M. I.

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Slepian, D.

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

Stark, H.

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Viano, G. A.

M. Bertero, C. de Mol, 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]

Walker, J. G.

M. Bertero, C. de Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging, a singular value analysis. IV. The case of uncertain localization or nonuniform illumination of the object,” Opt. Acta, 31, 923–945 (1984).
[CrossRef]

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (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, 81–94 (1982).
[CrossRef]

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

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

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

IEEE Trans. Circuits Syst. (2)

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

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

IEEE Trans. Med. Imaging (2)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982).
[CrossRef]

Inverse Problems (1)

F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs. Singular value analysis for two cases of low-pass filtering,” Inverse Problems 1, 67–85 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Acta (4)

M. Bertero, C. de Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging, a singular value analysis. IV. The case of uncertain localization or nonuniform illumination of the object,” Opt. Acta, 31, 923–945 (1984).
[CrossRef]

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

P. de Santis, F. Gori, “On an iterative method for super-resolution,” Opt. Acta 22, 691–695 (1975).
[CrossRef]

G. Cesini, G. Guattari, G. Lucarini, C. Palma, “An iterative method for restoring noisy images,” Opt. Acta 25, 501–508 (1978).
[CrossRef]

Other (3)

M. Bertero, C. de Mol, 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]

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

F. Riesz, B. S. Nagy, Functional Analysis (Ungar, New York, 1951).

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Figures (9)

Fig. 1
Fig. 1

Plots of qN,n(l) = 1 − (1 − μN,n)l for different harmonics and different space–bandwidth products a. The number of iterations l is given close to each curve [for convenience, continuous curves join the values of qN,n(l) for the same l].

Fig. 2
Fig. 2

Object utilized in example 1.

Fig. 3
Fig. 3

Reconstruction of the image in Fig. 4 obtained by iterating the first seven harmonics according to Table 1.

Fig. 4
Fig. 4

Image of the object in Fig. 2 (a = 10).

Fig. 5
Fig. 5

Plots of the 1-D restorations for each harmonic component corresponding to the reconstruction in Fig. 3. Dotted–dashed line, object; dashed line, image; solid line, reconstruction.

Fig. 6
Fig. 6

Reconstruction of the image in Fig. 8 obtained by iterating the first seven harmonics according to Table 1.

Fig. 7
Fig. 7

Object utilized in example 2.

Fig. 8
Fig. 8

Image of the object in Fig. 7 (a = 15).

Fig. 9
Fig. 9

Image of the object in Fig. 7 (a = 30).

Tables (1)

Tables Icon

Table 1 Numbers of Iterations Settled with Our Criterion for Each Harmonic of the Objects in Examples 1 and 2

Equations (29)

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i ( x , y ) = + s ( x ξ , y η ) o ( ξ , η ) d ξ d η ,
F { i } ( ν 1 , ν 2 ) = F { s } ( ν 1 , ν 2 ) F { o } ( ν 1 , ν 2 ) ,
F { g } ( ω , ϕ ) = 0 2 π 0 + g ( r , ϕ ) exp [ i 2 π ω r cos ( θ ϕ ) ] r d r d θ .
S ( ω ) = 0 + r s ( r ) J 0 ( 2 π ω r ) d r .
o ( r , θ ) = N = + O N ( r ) exp ( i N θ ) , 0 r 1 , 0 θ 2 π ; i ( r , θ ) = N = + I N ( r ) exp ( i N θ ) , 0 r + , 0 θ 2 π .
F { o } ( ω , ϕ ) = 2 π N = + ( i ) N Õ N ( ω ) exp ( i N ϕ ) , F { i } ( ω , ϕ ) = 2 π N = + ( i ) N Ĩ N ( ω ) exp ( i N ϕ ) , for 0 ω + , 0 ϕ 2 π ,
Õ N ( ω ) = 0 1 r O N ( r ) J N ( 2 π ω r ) d r ,
I N ( r ) = 4 π 2 0 + ω Ĩ N ( ω ) J N ( 2 π ω r ) d ω .
I N ( r ) = 8 π 3 0 + ω S ( ω ) Õ N ( ω ) J N ( 2 π ω r ) d ω .
I N ( r ) = 0 1 S N ( ρ , r ) O N ( ρ ) d ρ ( Ŝ N O N ) ( r ) ,
i ( x ¯ ) = R s a ( x ¯ y ¯ ) o ( y ¯ ) d y ¯ ( Ŝ a o ) ( x ¯ ) ,
s a ( x ¯ ) = P a exp ( 2 π i ν ¯ x ¯ ) d ν ¯ .
S N , a ( ρ , r ) = 4 a 2 0 1 ω ρ J N ( 2 π ν M r ω ) J N ( 2 π ν M ρ ω ) d ω .
μ k , l ψ k , l ( x ¯ ) = ( Ŝ a ψ k , l ) ( x ¯ ) ,
I k , l = μ k , l O k , l
o ( x ¯ ) = k , l O k , l ψ k , l ( x ¯ ) , T i ( x ¯ ) = k , l I k , l ψ k , l ( x ¯ ) , x ¯ in R .
ψ ( r , θ ) = N = + R N ( r ) exp ( i N θ ) ,
γ N , n R N , n ( r ) = 0 1 J N ( 2 π ν M ρ r ) R N , n ( ρ ) ρ d ρ ,
μ N , n = 4 a 2 | γ N , n | 2 .
μ N , n R N , n ( r ) = ( Ŝ N , a R N , n ) ( r ) .
I N ( r ) = n = 0 + i N , n R N , n ( r ) , O N ( r ) = n = 0 + o N , n R N , n ( r ) .
i N , n = μ N , n o N , n .
( K ̂ N , a o ) ( x ) = 0 1 { δ ( x y ) S N , a ( x , y ) } o ( y ) d y ,
I N ( r ) = O N ( r ) ( K ̂ N , a O N ) ( r ) for r in [ 0 , 1 ]
O N ( l ) ( r ) = I N ( r ) + [ K ̂ N , a O N ( l 1 ) ] ( r ) = I N ( r ) + j = 1 l 1 ( K ̂ N , a j I N ) ( r ) , O N ( 1 ) ( r ) = I N ( r ) ,
O N ( l ) ( r ) = n = 0 + o N , n q N , n ( l ) R N , n ( r ) ,
q N , n ( l ) = 1 ( 1 μ N , n ) l .
lim l + q N , n ( l ) = 1 for each N , n ,
e N ( l ) = k = 1 m { O N ( l ) ( k Δ r ) O N ( l 1 ) ( k Δ r ) } 2 / k = 1 m { O N ( 1 ) ( k Δ r ) } 2 ,

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