Abstract

An eigenvalue analysis of the noise-prone image leads to (a) an analysis of the eigenfunctions and eigenvalues of the sin<sup>2</sup>(<i>x</i>)/<i>x</i><sup>2</sup> kernel; and (b) an expression relating an effective number <i>N</i><sub>eff</sub> of degrees of freedom directly to the signal-to-noise ratio σ<sub>0</sub>/σ<sub><i>v</i></sub>. The latter are the variances of object and noise, respectively. For the particular case of incoherent, diffraction-limited imagery, <i>N</i><sub>eff</sub> is found to be reduced from its noise-free value, the Shannon number, by the factor (1-σ<sub><i>v</i></sub>/σ<sub>0</sub>). A maximum number <i>N</i><sub>max</sub> of degrees of freedom is also defined. Comparing one-dimensional objects illuminated alternatively by coherent and incoherent light, we find they have the same number <i>N</i><sub>max</sub> of degrees of freedom. However, for the corresponding two-dimensional case, the incoherent value for <i>N</i><sub>max</sub> is double that of the coherent value.

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  1. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
  2. G. Toraldo di Francia, Trans. IRE AP-4, 473 (1956).
  3. C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
  4. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
  5. B. R. Frieden, in Progress in Optics, IX, edited by E. Wolf, (North-Holland, Amsterdam, 1971), p. 311.
  6. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  7. F. Riesz and B. S. Nagy, Functional Analysis (Fredelich Angars, New York, 1965).
  8. A. Fedotowsky, thesis (Laval University, Canada) (1972).
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  10. H. J. Landau, Acta Math. 117, 37 (1967).
  11. R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).
  12. A. Consortini and F. Pasqualetti, Opt. Acta 20, 793 (1973).
  13. B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).

Consortini, A.

A. Consortini and F. Pasqualetti, Opt. Acta 20, 793 (1973).

Fedotowsky, A.

A. Fedotowsky, thesis (Laval University, Canada) (1972).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Francia, G. Toraldo di

G. Toraldo di Francia, Trans. IRE AP-4, 473 (1956).

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).

G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).

Frieden, B. R.

B. R. Frieden, in Progress in Optics, IX, edited by E. Wolf, (North-Holland, Amsterdam, 1971), p. 311.

B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).

Gori, F.

F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).

Guattari, G.

F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).

Harris, R. W.

C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).

Landau, H. J.

H. J. Landau, Acta Math. 117, 37 (1967).

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nagy, B. S.

F. Riesz and B. S. Nagy, Functional Analysis (Fredelich Angars, New York, 1965).

Pasqualetti, F.

A. Consortini and F. Pasqualetti, Opt. Acta 20, 793 (1973).

Riesz, F.

F. Riesz and B. S. Nagy, Functional Analysis (Fredelich Angars, New York, 1965).

Rushforth, C. K.

C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).

Sanderson, R. L.

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).

Streifer, W.

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).

Other

G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).

G. Toraldo di Francia, Trans. IRE AP-4, 473 (1956).

C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).

B. R. Frieden, in Progress in Optics, IX, edited by E. Wolf, (North-Holland, Amsterdam, 1971), p. 311.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

F. Riesz and B. S. Nagy, Functional Analysis (Fredelich Angars, New York, 1965).

A. Fedotowsky, thesis (Laval University, Canada) (1972).

F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).

H. J. Landau, Acta Math. 117, 37 (1967).

R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).

A. Consortini and F. Pasqualetti, Opt. Acta 20, 793 (1973).

B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).

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