Abstract

In this paper we investigate how the method of convex projections for image restoration behaves in the presence of noise. We also introduce and test a new noise-smoothing procedure in which the restored image is forced to lie within a certain L2 distance of the noisy data. We show that, in the presence of noise, restoration by convex projections is superior to the Gerchberg-Papoulis method.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Gerchberg, Opt. Acta 21, 709 (1974).
    [CrossRef]
  2. A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
    [CrossRef]
  3. H. Stark, D. Cahana, H. Webb, J. Opt. Soc. Am. 71, 635 (1981).
    [CrossRef]
  4. J. A. Cadzow, IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
  5. H. Stark, D. Cahana, G. J. Habetter, Opt. Lett. 6, 259 (1981).
    [CrossRef] [PubMed]
  6. D. C. Youla, IEEE Trans. Circuits Syst. CAS-25, 694 (1978).
    [CrossRef]
  7. L. M. Bregman, Dokl. Akad. Nauk SSSR 162, 487 (1965).
  8. L. G. Gubin, B. T. Polyak, E. V. Raik, USSR Comput. Math. Math. Phys. 7, 1 (1967).
    [CrossRef]
  9. A. Lent, H. Tuy, J. Math. Anal. Appl. 83, 554 (1981).
    [CrossRef]
  10. D. C. Youla, H. Webb, IEEE Trans. Med. Imaging, MI-1, 81 (1982).
    [CrossRef]
  11. M. I. Sezan, H. Stark, IEEE Trans. Med. Imaging MI-1, 95 (1982).
    [CrossRef]
  12. Z. Opial, Bull. Am. Math. Soc. 72, 591 (1967).
    [CrossRef]
  13. H. Stark, J. W. Woods, I. Paul, R. Hingorani, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237 (1981).
    [CrossRef]
  14. M. H. Hayes, J. S. Lim, A. V. Oppenheim, IEEE Trans. Acoust., Speech Signal Process. ASSP-28, 672 (1980).
    [CrossRef]
  15. A. Levi, H. Stark, J. Opt. Soc. Am., 73, 810 (1983).
    [CrossRef]

1983 (1)

1982 (2)

D. C. Youla, H. Webb, IEEE Trans. Med. Imaging, MI-1, 81 (1982).
[CrossRef]

M. I. Sezan, H. Stark, IEEE Trans. Med. Imaging MI-1, 95 (1982).
[CrossRef]

1981 (4)

H. Stark, J. W. Woods, I. Paul, R. Hingorani, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237 (1981).
[CrossRef]

H. Stark, D. Cahana, H. Webb, J. Opt. Soc. Am. 71, 635 (1981).
[CrossRef]

H. Stark, D. Cahana, G. J. Habetter, Opt. Lett. 6, 259 (1981).
[CrossRef] [PubMed]

A. Lent, H. Tuy, J. Math. Anal. Appl. 83, 554 (1981).
[CrossRef]

1980 (1)

M. H. Hayes, J. S. Lim, A. V. Oppenheim, IEEE Trans. Acoust., Speech Signal Process. ASSP-28, 672 (1980).
[CrossRef]

1978 (2)

D. C. Youla, IEEE Trans. Circuits Syst. CAS-25, 694 (1978).
[CrossRef]

J. A. Cadzow, IEEE Trans. Circuits Syst. CAS-25, 74 (1978).

1975 (1)

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

1967 (2)

L. G. Gubin, B. T. Polyak, E. V. Raik, USSR Comput. Math. Math. Phys. 7, 1 (1967).
[CrossRef]

Z. Opial, Bull. Am. Math. Soc. 72, 591 (1967).
[CrossRef]

1965 (1)

L. M. Bregman, Dokl. Akad. Nauk SSSR 162, 487 (1965).

Bregman, L. M.

L. M. Bregman, Dokl. Akad. Nauk SSSR 162, 487 (1965).

Cadzow, J. A.

J. A. Cadzow, IEEE Trans. Circuits Syst. CAS-25, 74 (1978).

Cahana, D.

Gerchberg, R. W.

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

Gubin, L. G.

L. G. Gubin, B. T. Polyak, E. V. Raik, USSR Comput. Math. Math. Phys. 7, 1 (1967).
[CrossRef]

Habetter, G. J.

Hayes, M. H.

M. H. Hayes, J. S. Lim, A. V. Oppenheim, IEEE Trans. Acoust., Speech Signal Process. ASSP-28, 672 (1980).
[CrossRef]

Hingorani, R.

H. Stark, J. W. Woods, I. Paul, R. Hingorani, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237 (1981).
[CrossRef]

Lent, A.

A. Lent, H. Tuy, J. Math. Anal. Appl. 83, 554 (1981).
[CrossRef]

Levi, A.

Lim, J. S.

M. H. Hayes, J. S. Lim, A. V. Oppenheim, IEEE Trans. Acoust., Speech Signal Process. ASSP-28, 672 (1980).
[CrossRef]

Opial, Z.

Z. Opial, Bull. Am. Math. Soc. 72, 591 (1967).
[CrossRef]

Oppenheim, A. V.

M. H. Hayes, J. S. Lim, A. V. Oppenheim, IEEE Trans. Acoust., Speech Signal Process. ASSP-28, 672 (1980).
[CrossRef]

Papoulis, A.

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

Paul, I.

H. Stark, J. W. Woods, I. Paul, R. Hingorani, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237 (1981).
[CrossRef]

Polyak, B. T.

L. G. Gubin, B. T. Polyak, E. V. Raik, USSR Comput. Math. Math. Phys. 7, 1 (1967).
[CrossRef]

Raik, E. V.

L. G. Gubin, B. T. Polyak, E. V. Raik, USSR Comput. Math. Math. Phys. 7, 1 (1967).
[CrossRef]

Sezan, M. I.

M. I. Sezan, H. Stark, IEEE Trans. Med. Imaging MI-1, 95 (1982).
[CrossRef]

Stark, H.

A. Levi, H. Stark, J. Opt. Soc. Am., 73, 810 (1983).
[CrossRef]

M. I. Sezan, H. Stark, IEEE Trans. Med. Imaging MI-1, 95 (1982).
[CrossRef]

H. Stark, D. Cahana, H. Webb, J. Opt. Soc. Am. 71, 635 (1981).
[CrossRef]

H. Stark, D. Cahana, G. J. Habetter, Opt. Lett. 6, 259 (1981).
[CrossRef] [PubMed]

H. Stark, J. W. Woods, I. Paul, R. Hingorani, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237 (1981).
[CrossRef]

Tuy, H.

A. Lent, H. Tuy, J. Math. Anal. Appl. 83, 554 (1981).
[CrossRef]

Webb, H.

D. C. Youla, H. Webb, IEEE Trans. Med. Imaging, MI-1, 81 (1982).
[CrossRef]

H. Stark, D. Cahana, H. Webb, J. Opt. Soc. Am. 71, 635 (1981).
[CrossRef]

Woods, J. W.

H. Stark, J. W. Woods, I. Paul, R. Hingorani, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237 (1981).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, IEEE Trans. Med. Imaging, MI-1, 81 (1982).
[CrossRef]

D. C. Youla, IEEE Trans. Circuits Syst. CAS-25, 694 (1978).
[CrossRef]

Bull. Am. Math. Soc. (1)

Z. Opial, Bull. Am. Math. Soc. 72, 591 (1967).
[CrossRef]

Dokl. Akad. Nauk SSSR (1)

L. M. Bregman, Dokl. Akad. Nauk SSSR 162, 487 (1965).

IEEE Trans. Acoust. Speech Signal Process. (1)

H. Stark, J. W. Woods, I. Paul, R. Hingorani, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 237 (1981).
[CrossRef]

IEEE Trans. Acoust., Speech Signal Process. (1)

M. H. Hayes, J. S. Lim, A. V. Oppenheim, IEEE Trans. Acoust., Speech Signal Process. ASSP-28, 672 (1980).
[CrossRef]

IEEE Trans. Circuits Syst. (3)

D. C. Youla, IEEE Trans. Circuits Syst. CAS-25, 694 (1978).
[CrossRef]

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

J. A. Cadzow, IEEE Trans. Circuits Syst. CAS-25, 74 (1978).

IEEE Trans. Med. Imaging (2)

D. C. Youla, H. Webb, IEEE Trans. Med. Imaging, MI-1, 81 (1982).
[CrossRef]

M. I. Sezan, H. Stark, IEEE Trans. Med. Imaging MI-1, 95 (1982).
[CrossRef]

J. Math. Anal. Appl. (1)

A. Lent, H. Tuy, J. Math. Anal. Appl. 83, 554 (1981).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Acta (1)

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

Opt. Lett. (1)

USSR Comput. Math. Math. Phys. (1)

L. G. Gubin, B. T. Polyak, E. V. Raik, USSR Comput. Math. Math. Phys. 7, 1 (1967).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Object: three nested rectangles of brightness 1.0, 0.8, and 0.4 (smallest to largest, respectively).

Fig. 2
Fig. 2

Fourier data of the noise-free object shown in Fig. 1. It is assumed that the Fourier data are known only over a 90° cone.

Fig. 3
Fig. 3

Same as Fig. 2 except the object has been corrupted by moderate noise (SNR = 20 dB).

Fig. 4
Fig. 4

Same as Fig. 2 except the object has been corrupted by heavy noise (SNR = 5 dB).

Fig. 5
Fig. 5

Performance of G-P, unirelax1, and unirelax2 in restoring an image from very noisy data (SNR = 5 dB).

Fig. 6
Fig. 6

Performance of G-P, unirelax1, and unirelax2 in restoring an image from moderately noisy data (SNR = 20 dB).

Fig. 7
Fig. 7

Performance of relax1 vs unirelaxi1 when the data are moderately noisy. relaxi1 which outperformed unirelaxi in a noise-free setting is now outperformed for the same λ when the data are noisy (SNR = 20 dB).

Fig. 8
Fig. 8

Same as Fig. 7 except the data are contaminated by heavy noise (SNR = 5 dB).

Fig. 9
Fig. 9

Performance of unirelax2 vs relax2 in moderate noise.

Fig. 10
Fig. 10

Performance of unirelax1 vs unirelax3; example of where reordering of operators results in insignificant performance changes.

Fig. 11
Fig. 11

Performance of relax2 vs relax4; example of where reordering of operators results in significant performance changes.

Fig. 12
Fig. 12

Performance of relax3 in restoring an image from noise-free data, moderately noisy data (SNR = 15 dB), and very noisy data (SNR = 5 dB).

Fig. 13
Fig. 13

Restoration of the object. Clockwise from upper LEFT: original object; noisy object (SNR = 20 dB); reconstruction using the G-P algorithm; reconstruction using unirelax1.

Tables (2)

Tables Icon

Table I Summary of A Priori Assumed Constraints

Tables Icon

Table II Summary of Results

Equations (39)

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

Ω | f ( x , y ) | 2 dxdy < ,
( f , g ) Ω f ( x , y ) g * ( x , y ) dxdy ,
f [ ( f , f ) ] 1 / 2 .
f k + 1 = P m P m 1 P 1 f k ,
f k + 1 = T m T m 1 T 1 f k ,
T i = I + λ i ( P i I ) , 0 < λ i < 2 .
P 1 f = { f , ( x , y ) S , 0 , ( x , y ) S .
P 2 f { G ( u , υ ) , ( u , υ ) , F ( u , υ ) , ( u , υ ) ,
Ω | f ( x , y ) | 2 dxdy E ρ 2 ,
P 3 f = { 0 , f 1 < 0 , f 1 + , E 1 + E , ( E / E + ) 1 / 2 f 1 + , E 1 + > E ,
E 1 + Ω ( f 1 + ) 2 dxdy .
P 4 f = { f 1 , f 1 0 , 0 , otherwise .
P 5 f = { a , f < a , f , a f b , b , f > b .
G ( u , υ ) = { G ( u , υ ) + n ( u , υ ) , ( u , υ ) 0 , ( u , υ ) ,
X { 1 , ( u , υ ) , 0 , ( u , υ ) .
X H ( u , υ ) G ( u , υ ) σ .
X F 3 G = μ X F 1 + ( 1 μ ) X F 2 G = μ ( X F 1 G ) + ( 1 μ ) ( X F 2 G ) μ X F 1 G + ( 1 μ ) X F 2 G ( triangle inequality ) μ σ + ( 1 μ ) σ = σ .
f n f 2 = Ω ( f 1 f n ) 2 dxdy + Ω f 2 2 dxdy 0 .
P 6 f = { F 1 , X F 1 G < σ , G + σ X F 1 G X F 1 G + ( 1 X ) F 1 , X F 1 G σ .
X X C ( θ ) = { 1 , over all θ within the cone over which the data are given , 0 , otherwise .
e k = 100 f f k f .
f 2 | f ( ξ , η ) | 2 d ξ d η ,
f h Ω 2 = f 1 h Ω 2 + f 2 Ω 2 .
minimize F 1 H c 2
H G 2 σ 2 ,
minimize F 1 H 2
H G 2 σ 2 .
L F 1 H 2 + ω H G 2 ω ( σ 2 α 2 ) = 0 ,
F 1 F 1 r + j F l i , G G r + j G i and H H r + j H i ,
F 1 r H r 2 + F 1 i H i 2 + ω H r G 4 2 + ω H i G i 2 ω ( σ 2 α 2 ) = 0 .
L H r = 0 , L H i = 0
H = F 1 + ω G ω + 1 .
minimize ω ( ω 1 + ω ) 2 F 1 G 2 ,
H = F 1
ω = σ 1 σ 1 .
H = G + σ ( X F 1 G ) F 1 G .
H ( u , υ ) = { F 1 ( u , υ ) , X F 1 G c < σ G ( u , υ ) + σ X F 1 ( u , υ ) G ( u , υ ) X F 1 G c + ( 1 X ) F ( u , υ ) , X F 1 G c > σ ,
SNR = 10 log 10 f f ̅ 2 σ n 2 ( dB ) ,
σ n 2 = f f ̅ 2 10 ( SNR / 10 ) .

Metrics