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 〈θ<sub><i>j</i></sub>-θ¯|<sup><i>K</i></sup>.〉 be a minimum. Subscript <i>j</i> denotes the spatial frequency ω<sub><i>j</i></sub> at which the unknown object spectrum is to be restored, θ¯ denotes the optimum restoration by this criterion, and <i>K</i> is any positive number at the user’s discretion. In general, such processing is nonlinear and requires the use of an electronic computer. Processor θ¯ uses the presence of known, Markov-image statistics to enhance the restoration quality and permits the image-forming phenomenon to obey an arbitrary law <i>I</i><sub>j</sub> = £(τ<sub><i>j</i></sub>, θ<sub>j</sub>, <i>N<sub>j</sub></i>). Here, τ<sub><i>j</i></sub> denotes the intrinsic system characteristic (usually the optical transfer function), and <i>N<sub>j</sub></i> represents a noise function. When restored values θ¯<sub><i>j</i></sub>, <i>j</i>= 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 θ¯<i><sub>j</sub></i> is found to be the root of a finite polynomial. When the particular value <i>K</i>=2 is used, the root θ¯<i><sub>j</sub></i> is known analytically. Particular restorations θ¯<i><sub>j</sub></i> 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.

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  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, Sr., J. Opt. Soc. Am. 54, 931 (1964).
  3. C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).
  4. G. J. Buck and J. J. Gustincic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).
  5. B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).
  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).
  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, Jr. 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 meansquare criterion. See, e.g., M. Zakai, IEEE Trans. Inform. Theory IT-10, 94 (1964).
  10. J. A. Eyer, J. Opt. Soc. Am. 48, 938 (1958).
  11. D. H. Kelly, J. Opt. Soc. Am. 50, 269 (1960).
  12. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), p. 43.
  13. J. L. Harris, Sr., J. Opt. Soc. Am. 54, 606 (1964).
  14. This may be checked by use of Eq. (1Oa) to form the expectation of any function ƒ(ס′). By the sifting property of the delta function, thls expectation becomes the usual formula for the case of discrete statistics.

Barnes, C. W.

C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).

Buck, G. J.

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

Davenport, Jr., W. B.

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

Eyer, J. A.

J. A. Eyer, J. Opt. Soc. Am. 48, 938 (1958).

Frieden, B. R.

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

Gustincic, J. J.

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

Harris, J. L.

J. L. Harris, Sr., J. Opt. Soc. Am. 54, 606 (1964).

Harris, R. W.

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).

Harris, Sr., J. L.

J. L. Harris, Sr., J. Opt. Soc. Am. 54, 931 (1964).

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.

D. H. Kelly, J. Opt. Soc. Am. 50, 269 (1960).

Linfoot, E. H.

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

Root, W. L.

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

Rushforth, C. K.

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).

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 meansquare criterion. See, e.g., M. Zakai, IEEE Trans. Inform. Theory IT-10, 94 (1964).

Other

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

J. L. Harris, Sr., J. Opt. Soc. Am. 54, 931 (1964).

C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).

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

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

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).

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

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

J. A. Eyer, J. Opt. Soc. Am. 48, 938 (1958).

D. H. Kelly, J. Opt. Soc. Am. 50, 269 (1960).

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

J. L. Harris, Sr., J. Opt. Soc. Am. 54, 606 (1964).

This may be checked by use of Eq. (1Oa) to form the expectation of any function ƒ(ס′). By the sifting property of the delta function, thls expectation becomes the usual formula for the case of discrete statistics.

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