Abstract

An eigenvalue analysis of the noise-prone image leads to (a) an analysis of the eigenfunctions and eigenvalues of the sin2(x)/x2 kernel; and (b) an expression relating an effective number Neff of degrees of freedom directly to the signal-to-noise ratio σ0/σv. The latter are the variances of object and noise, respectively. For the particular case of incoherent, diffraction-limited imagery, Neff is found to be reduced from its noise-free value, the Shannon number, by the factor (1 − σv/σ0). A maximum number Nmax 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 Nmax of degrees of freedom. However, for the corresponding two-dimensional case, the incoherent value for Nmax is double that of the coherent value.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [Crossref]
  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).
    [Crossref]
  4. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [Crossref] [PubMed]
  5. B. R. Frieden, in Progress in Optics, IX, edited by E. Wolf, (North–Holland, Amsterdam, 1971), p. 311.
    [Crossref]
  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).
  9. F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).
    [Crossref]
  10. H. J. Landau, Acta Math. 117, 37 (1967).
    [Crossref]
  11. R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).
    [Crossref] [PubMed]
  12. A. Consortini and F. Pasqualetti, Opt. Acta 20, 793 (1973).
    [Crossref]
  13. B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
    [Crossref] [PubMed]

1973 (2)

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

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

1972 (1)

1969 (2)

1968 (1)

1967 (1)

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

1956 (1)

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

1955 (1)

Consortini, A.

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

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).

Frieden, B. R.

B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
[Crossref] [PubMed]

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

Gori, F.

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

Guattari, G.

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

Harris, R. W.

Landau, H. J.

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

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).
[Crossref]

Riesz, F.

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

Rushforth, C. K.

Sanderson, R. L.

Streifer, W.

Toraldo di Francia, G.

Acta Math. (1)

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

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Opt. Acta (1)

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

Opt. Commun. (1)

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

Trans. IRE (1)

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

Other (4)

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

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).

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 (4)

Fig. 1
Fig. 1

Eigenvalue γ vs index n for the kernel sin2(x)/x2. Different values of caΩ are assumed. Note the approximate triangular falloff.

Fig. 2
Fig. 2

Eigenfunctions un(x) of the kernel sin2(x)/x2. A value c = 10π is assumed. n = 0 (2) 22.

Fig. 3
Fig. 3

Eigenfunctions as in Fig. 2. n = 24 (2) 46.

Fig. 4
Fig. 4

Self-reproducing objects for different values of c. The dashed carves are cosw0x, with w0 derived from Eq. (22).

Equations (28)

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

i ( x , y ) = Σ s ( x , y ) K ( x - x , y - y ) d x d y ,
γ n u n ( x , y ) = Σ u n ( x , y ) K ( x - x , y - y ) d x d y ,
γ n m γ n + i , m + j ,             i , j 0.
s ( x , y ) = n m a n m u n m ( x , y ) ,
i ( x , y ) = n m b n m u n m ( x , y ) ,
b n m = γ n m a n m .
i ( x , y ) = i ( x , y ) + ν ( x , y ) ,
i ( x , y ) = n m ( b n m + ν n m ) u n m ( x , y ) .
i ( x , y ) = n N m M ( b n m + ν n m ) u n m ( x , y ) = n N m M ( γ n m a n m + ν n m ) u n m ( x , y ) .
= Σ i ( x , y ) - i ( x , y ) 2 d x d y .
= n = N + 1 m = M + 1 γ n m 3 a n m 2 + n = 0 N m = 0 M γ n m ν n m 2 = minimum
= n = N + 1 m = M + 1 γ n m 3 a n m 2 + n = 0 N m = 0 M γ n m ν n m 2 = minimum ,
s ( x ) s * ( x ) = σ 0 2 δ ( x - x ) , ν ( x ) ν * ( x ) = σ ν 2 δ ( x - x ) ,
a n 2 = σ 0 2 / γ n ,             ν n 2 = σ ν 2 / γ n .
σ 0 2 m = 0 γ n 2 - n = 0 N [ σ 0 2 γ n 2 - σ ν 2 ] = minimum
γ [ N ] = σ ν / σ 0 ,
γ n u n ( x ) = - a a u n ( x ) K ( x - x ) d x
K ( x - x ) = sin 2 Ω ( x - x ) π Ω ( x - x ) 2 ,
s ( x ) = n a n u n ( x ) ,             i ( x ) = n b n u n ( x ) , i ( x ) = n N ( b ν + ν n ) u n ( x )
γ n = 1 - π 4 c n , for n 4 c π γ n = 0 , for n 4 c π
N 4 c π ( 1 - σ ν σ 0 ) .
N max = 4 c π = 2 a D λ f .
tan w n a = c / w n a
K ( x - x , y - y ) = sin 2 Ω ( x - x ) 2 π Ω ( x - x ) 2 sin 2 Ω ( y - y ) π Q ( y - y ) 2 , u n m ( x , y ) = u n ( x ) u m ( y ) , γ n m = γ n γ m .
γ [ N ] γ [ M ] = σ ¯ ν / σ ¯ 0 ,
( 1 - π 4 c N ) 2 = σ ¯ ν σ ¯ 0 .
N eff = N 2 = ( 4 c π ) 2 [ 1 - ( σ ¯ ν σ ¯ 0 ) 1 2 ] 2 ;
N max = ( 4 c / π ) 2 .