Abstract

It has been traditional to constrain image processing to linear operations upon the image. This is a realistic limitation of analog processing. In this paper, we find the optimum restoration of a noisy image by the criterion that expectation 〈 Oj-O¯jK〉 be a minimum. Subscript j denotes the spatial frequency ωj at which the unknown object spectrum O¯ is to be restored, O¯ denotes the optimum restoration by this criterion, and K is any positive number at the user’s discretion. In general, such processing is nonlinear and requires the use of an electronic computer. Processor O¯ uses the presence of known, Markov-image statistics to enhance the restoration quality and permits the image-forming phenomenon to obey an arbitrary law Ij = (τj, Oj, Nj). Here, τj denotes the intrinsic system characteristic (usually the optical transfer function), and Nj represents a noise function. When restored values O¯j, j=1, 2, ⋯, are used as inputs to the band-unlimited restoration procedure (derived in a previous paper), the latter is optimized for the presence of noise. The optimum O¯j is found to be the root of a finite polynomial. When the particular value K=2 is used, the root O¯j is known analytically. Particular restorations O¯j are found for the case of additive, independent, gaussian detection noise and a white object region. These restorations are graphically compared with that due to conventional, linear processing.

© 1968 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Wolter, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), Ch. V, Sec. 4.6.
  2. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
    [CrossRef]
  3. C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).
    [CrossRef]
  4. G. J. Buck and J. J. Gustincic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).
    [CrossRef]
  5. B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).
    [CrossRef]
  6. The serious effect of noise upon restorations due to the band-unlimited method is discussed in Ref. 4 and extensively in C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
    [CrossRef]
  7. See, e.g., J.M. Wozencraft and I.M. Jacobs, Principles of Communication Engineering (John Wiley & Sons, Inc., New York, 1965), p. 585.
  8. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958).
  9. It is now known that the Wiener filter is the optimum processor for a much wider class of error criteria than the mean-square criterion. See, e.g., M. Zakai, IEEE Trans. Inform. Theory IT-10, 94 (1964).
    [CrossRef]
  10. J. A. Eyer, J. Opt. Soc. Am. 48, 938 (1958).
    [CrossRef]
  11. D. H. Kelly, J. Opt. Soc. Am. 50, 269 (1960).
    [CrossRef]
  12. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), p. 43.
  13. J. L. Harris, J. Opt. Soc. Am. 54, 606 (1964).
    [CrossRef]
  14. This may be checked by use of Eq. (10a) to form the expectation of any function f(O′). By the sifting property of the delta function, this expectation becomes the usual formula for the case of discrete statistics.

1968 (1)

1967 (2)

B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).
[CrossRef]

G. J. Buck and J. J. Gustincic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).
[CrossRef]

1966 (1)

1964 (3)

It is now known that the Wiener filter is the optimum processor for a much wider class of error criteria than the mean-square criterion. See, e.g., M. Zakai, IEEE Trans. Inform. Theory IT-10, 94 (1964).
[CrossRef]

J. L. Harris, J. Opt. Soc. Am. 54, 606 (1964).
[CrossRef]

J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
[CrossRef]

1960 (1)

1958 (1)

Barnes, C. W.

Buck, G. J.

G. J. Buck and J. J. Gustincic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).
[CrossRef]

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958).

Eyer, J. A.

Frieden, B. R.

Gustincic, J. J.

G. J. Buck and J. J. Gustincic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).
[CrossRef]

Harris, J. L.

Harris, R. W.

Jacobs, I.M.

See, e.g., J.M. Wozencraft and I.M. Jacobs, Principles of Communication Engineering (John Wiley & Sons, Inc., New York, 1965), p. 585.

Kelly, D. H.

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), p. 43.

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958).

Rushforth, C. K.

Wolter, H.

H. Wolter, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), Ch. V, Sec. 4.6.

Wozencraft, J.M.

See, e.g., J.M. Wozencraft and I.M. Jacobs, Principles of Communication Engineering (John Wiley & Sons, Inc., New York, 1965), p. 585.

Zakai, M.

It is now known that the Wiener filter is the optimum processor for a much wider class of error criteria than the mean-square criterion. See, e.g., M. Zakai, IEEE Trans. Inform. Theory IT-10, 94 (1964).
[CrossRef]

IEEE Trans. Antennas Propagation (1)

G. J. Buck and J. J. Gustincic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).
[CrossRef]

IEEE Trans. Inform. Theory (1)

It is now known that the Wiener filter is the optimum processor for a much wider class of error criteria than the mean-square criterion. See, e.g., M. Zakai, IEEE Trans. Inform. Theory IT-10, 94 (1964).
[CrossRef]

J. Opt. Soc. Am. (7)

Other (5)

This may be checked by use of Eq. (10a) to form the expectation of any function f(O′). By the sifting property of the delta function, this expectation becomes the usual formula for the case of discrete statistics.

H. Wolter, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), Ch. V, Sec. 4.6.

See, e.g., J.M. Wozencraft and I.M. Jacobs, Principles of Communication Engineering (John Wiley & Sons, Inc., New York, 1965), p. 585.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958).

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), p. 43.

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 (1)

Fig. 1
Fig. 1

Classical vs optimal processing in the case of a constant, or white, object spectrum over M+1 adjacent frequencies. A noisy image was generated upon an electronic computer according to law (4) for linear image formation with additive noise. The latter was made to be gaussian random by use of the computer’s random-number generator and an associated subroutine. The noisy image was independently processed with the classical formula [Eq. (9)], and with the optimal formula [Eq. (8a)] with successive values M=0, 1, 5, 10, and 50. The actual object, O j = 1.0, is plotted as a dashed line for comparison with each restoration. Graphical indications are that (a) optimal processing offers quantitative improvement over classical processing, and (b) optimal processing improves as the object foreknowledge (in this case, M) increases. The processor for M=0 is a close relative of the Wiener filter.

Equations (13)

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

j O j - O ¯ j K = min .
O j = O ¯ j ( I )
I j = ( τ j , O j , N j ) ,
I j = τ j O j + N j .
i , i K / 2 ( K / 2 i ) ( K / 2 i ) × ( - 1 ) i + i i q ( i , i ) ( I ) O ¯ j ( I ) i - 1 O ¯ j * ( I ) i = 0 ,
q ( i , i ) ( I ) = - d τ d O J ( τ , O , I ) - 1 × O j K / 2 - i O j * K / 2 - i p τ ( τ ) p O ( O ) p N ( τ , O , I ) .
O ¯ j ( I ) = q ( 0 , 1 ) ( I ) / q ( 1 , 1 ) ( I ) .
I j = ( τ j N j ) O j
O ¯ j ( I ) = σ N 2 O j + σ O j 2 k = 0 M τ j - k * I j - k σ N 2 + σ O j 2 k = 0 M τ j - k 2 .
j = 2 σ N 2 σ O j 2 / [ σ N 2 + σ O j 2 k = 0 M τ j - k 2 ] .
O ˆ j = I j / τ j
p O ( O ) = l = 1 L P l δ ( O - O l ) ,
l = 1 L P l = 1.