Abstract

We comment on the jitter model proposed by Stark [ J. Opt. Soc. Am. 5, 700 ( 1977)] to explain the spectral properties of a nonoverlapping grain structure (NOGS), and we discuss the points of concordance and disparity between Stark’s work and our own. A more general formulation for the two-dimensional jitter model is established on a theoretical basis. This different approach permits deeper insight into the interpretation of jitter model spectral properties, as confirmed by our experimental results. A one-dimensional model generated by a Poisson process is used to establish the NOGS.

© 1979 Optical Society of America

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References

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  1. H. Stark, J. Opt. Soc. Am.,  5, 700 (1977).
    [Crossref]
  2. O. R. White and M. Y. Cha, Solar Phys.,  31, 23 (1973).
    [Crossref]
  3. C. Aime, “A morphological interpretation of the spatial power-spectrum of the solar granulation,” Astron. and Astrophys.,  67, (1978).
  4. E. Parzen, Stochastic processes, (Holden-Day, New York, 1962) pp. 160–186.
  5. E. A. Trabka, J. Opt. Soc. Am.,  61, 800 (1971).
    [Crossref]
  6. P. E. Castro, J. H. B. Kemperman, and E. A. Trabka, J. Opt. Soc. Am.,  63, 820 (1973).
    [Crossref]
  7. Y. W. Lee, Statistical theory of communication (Wiley, New York, 1960) pp. 209–215, 240–248.
  8. J. L. Lawson and G. E. Uhlenbeck, Threshold signals, M.I.T. Rad. Lab. Series, 24, (McGraw-Hill, New York, 1950).
  9. R. Bracewell, The Fourier transform and its applications, (McGraw-Hill, New York1965) pp. 86, 238.
  10. A. Papoulis, Probability, variables and Stochastic processes, (McGraw-Hill, New York, 1965) p. 287.

1978 (1)

C. Aime, “A morphological interpretation of the spatial power-spectrum of the solar granulation,” Astron. and Astrophys.,  67, (1978).

1977 (1)

H. Stark, J. Opt. Soc. Am.,  5, 700 (1977).
[Crossref]

1973 (2)

1971 (1)

Aime, C.

C. Aime, “A morphological interpretation of the spatial power-spectrum of the solar granulation,” Astron. and Astrophys.,  67, (1978).

Bracewell, R.

R. Bracewell, The Fourier transform and its applications, (McGraw-Hill, New York1965) pp. 86, 238.

Castro, P. E.

Cha, M. Y.

O. R. White and M. Y. Cha, Solar Phys.,  31, 23 (1973).
[Crossref]

Kemperman, J. H. B.

Lawson, J. L.

J. L. Lawson and G. E. Uhlenbeck, Threshold signals, M.I.T. Rad. Lab. Series, 24, (McGraw-Hill, New York, 1950).

Lee, Y. W.

Y. W. Lee, Statistical theory of communication (Wiley, New York, 1960) pp. 209–215, 240–248.

Papoulis, A.

A. Papoulis, Probability, variables and Stochastic processes, (McGraw-Hill, New York, 1965) p. 287.

Parzen, E.

E. Parzen, Stochastic processes, (Holden-Day, New York, 1962) pp. 160–186.

Stark, H.

H. Stark, J. Opt. Soc. Am.,  5, 700 (1977).
[Crossref]

Trabka, E. A.

Uhlenbeck, G. E.

J. L. Lawson and G. E. Uhlenbeck, Threshold signals, M.I.T. Rad. Lab. Series, 24, (McGraw-Hill, New York, 1950).

White, O. R.

O. R. White and M. Y. Cha, Solar Phys.,  31, 23 (1973).
[Crossref]

Astron. and Astrophys. (1)

C. Aime, “A morphological interpretation of the spatial power-spectrum of the solar granulation,” Astron. and Astrophys.,  67, (1978).

J. Opt. Soc. Am. (3)

Solar Phys. (1)

O. R. White and M. Y. Cha, Solar Phys.,  31, 23 (1973).
[Crossref]

Other (5)

E. Parzen, Stochastic processes, (Holden-Day, New York, 1962) pp. 160–186.

Y. W. Lee, Statistical theory of communication (Wiley, New York, 1960) pp. 209–215, 240–248.

J. L. Lawson and G. E. Uhlenbeck, Threshold signals, M.I.T. Rad. Lab. Series, 24, (McGraw-Hill, New York, 1950).

R. Bracewell, The Fourier transform and its applications, (McGraw-Hill, New York1965) pp. 86, 238.

A. Papoulis, Probability, variables and Stochastic processes, (McGraw-Hill, New York, 1965) p. 287.

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Figures (7)

FIG. 1
FIG. 1

(a) Jittered array of grains with p() constant inside squares of side D − Δ. (b) Corresponding diffraction pattern showing the central light depletion zone and the Dirac delta functions enlarged by the analyzing spectral window.

FIG. 2
FIG. 2

Diffraction pattern obtained from the random distribution of grains in Fig. 3 of Stark’s paper.

FIG. 3
FIG. 3

(a) Jittered array of grains with p() constant inside squares of side D. (b) Corresponding diffraction pattern. The central light depletion zone is still present but is narrower than in Fig. 1(b). The Dirac delta functions around the center have disappeared.

FIG. 4
FIG. 4

(a) Section along the u′ axis of the theoretical two-dimensional power-spectrum W(u′,v′) of expression (6) (full line). Radial cut of the Airy function S, the term 1 – sinc2 (D − Δ)u′and the position of the Dirac comb III(u′,0) are represented by dashed lines. U m is the value given by Stark for the width of the light depletion zone, (b) Theoretical curves showing W(u′,0) for different sizes of the “confinement square” limiting the pdf. These curves show that the light depletion zone persists even when the grains overlap. The curve corresponding to a confinement square of side D is shown experimentally in Fig. 3(b).

FIG. 5
FIG. 5

Autocorrelation functions of: (a) Sequence of Poisson distributed pulses; (b) Pulse sequence corresponding to a “renewal counting process” with a constant dead-time Δ; (c) Sequence of rectangles generated from the sequence of pulses in (b) where each pulse is replaced by a rectangle of width Δ.

FIG. 6
FIG. 6

Theoretical curves for the power-spectrum WΔ(f) of a nonoverlapping sequence of rectangles. The sinc2 Δf term is shown by a dotted line.

FIG. 7
FIG. 7

Diffraction pattern of a totally random array of circular grains (Poisson statistics). There is no light depletion zone in the center.

Equations (20)

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D ( r ) = i j B ij ( r α i j i j ) .
W ( f ) = S ( f ) S x ( f ) | φ ̂ ( f ) | 2 + I I I _ ( b f ) S x ( f ) | φ ̂ ( f ) | 2 ,
W ( f ) = S ( f ) [ 1 | φ ̂ ( f ) | 2 + I I I ( d f ) | φ ̂ ( f ) | 2 ] ,
S ( f ) = S ( u , υ ) = [ 2 J 1 ( π f Δ ) / π f Δ ] 2
p ( ) = 1 ( D Δ ) 2 rect ( ϕ D Δ ) rect ( ψ D Δ ) ,
W ( f ) = S ( u , υ ) [ 1 sinc 2 ( D Δ ) u sinc 2 ( D Δ ) υ + I I I ( u d , υ d ) sinc 2 ( D Δ ) u sinc 2 ( D Δ ) υ ] .
W = S ( 1 A ) + S I I I A ,
C p ( ρ ) = λ 2 + λ δ ( ρ ) ,
| X i + 1 X i | Δ .
T Δ ( x ) = i δ ( x X i ) * rect ( x Δ ) .
C Δ ( ρ ) = C ( ρ ) * Λ ( ρ / Δ ) ,
C ( ρ ) = lim T 1 2 T T + T F ( t ) F ( t + ρ ) d t = x 1 x 2 P ξ 1 ξ 2 ( x 1 , x 2 ; ρ ) d x 1 d x 2 ,
C ( ρ ) = x 0 2 P ξ 1 ξ 2 ( x 0 , x 0 ; ρ ) = x 0 2 P ξ 1 ( x 0 ) P ξ 2 | ξ 1 ( x 0 | x 0 ; ρ ) ,
C ( 0 ) = A 2 λ e λ Δ d τ | A A d τ = 1 = λ e λ Δ A | A = λ e λ Δ δ ( 0 ) .
C 0 < | ρ | Δ ( ρ ) = 0 .
P ξ 2 | ξ 1 ( x 0 | x 0 ; ρ ) = P ξ 1 ( x 0 ) = P ξ 2 ( x 0 ) ,
C | ρ | > Δ ( ρ ) = AA λ e λ Δ d τ λ e λ Δ d τ | A A d τ = 1 = λ 2 e 2 λ Δ .
C ( ρ ) = { λ e λ Δ δ ( ρ ) , ρ = 0 0 , 0 < | ρ | Δ λ 2 e 2 λ Δ , | ρ | > Δ .
C Δ ( ρ ) = { λ e λ Δ δ ( ρ ) + λ 2 e 2 λ Δ × [ 1 rect ( ρ / 2 Δ ) ] } * Λ ( ρ / Δ ) .
W Δ ( f ) = λ Δ 2 e λ Δ [ λ e λ Δ δ ( f ) + 1 2 λ Δ e λ Δ sinc 2 f Δ ] sinc 2 f Δ .