Abstract

A probabilistic model is introduced to describe error diffusion (a method of binarization). This model is solved for general error-diffusion coefficients. The solution leads to an analytic error expression that is minimized to optimize the quality of binarization. It is shown that this procedure is useful in the production of computer-generated holograms. Simulations are presented to verify both the theoretical development and the optimization procedure.

© 1988 Optical Society of America

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References

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  1. H. S. Watkins, J. S. Moore, “A survey of color graphics printing,” IEEE Spectrum 21, 26–37 (1984).
  2. T. S. Perry, P. Wallich, eds., “Displays (Applications Topic),” IEEE Spectrum 22, 52–73 (1985).
  3. J. C. Stoffel, J. F. Moreland, “A survey of electron techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
    [CrossRef]
  4. R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).
  5. R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).
  6. J. R. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comput Graph. Image Process. 5, 13–40 (1976).
    [CrossRef]
  7. J. G. Kim, G. Kim, “Design of optimal filters for error-feedback quantization of monochrome pictures,” Inf. Sci. 39, 285–298 (1986).
    [CrossRef]
  8. J. A. G. Hale, “Dot spacing modulation for the production of pseudo grey pictures,” Proc. Soc. Inf. Disp. 17, 63–74 (1976).
  9. R. Eschbach, R. Hauck, “Analytic description of the 1-D error diffusion technique for halftoning,” Opt. Commun. 52, 165–168 (1984).
    [CrossRef]
  10. R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
    [CrossRef]
  11. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  12. S. H. Algie, “Resolution and tonal continuity in bilevel printed picture continuity,” Comput. Vision Graph. Image Process. 24, 329–346 (1983).
    [CrossRef]
  13. W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), p. 121.
  14. J. P. Allebach, “Representation-related errors in binary digital holograms: a unified analysis,” Appl. Opt. 20, 290–299 (1981).
    [CrossRef] [PubMed]
  15. G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-generated holograms: carrier of polar geometry,” Appl. Opt. 24, 777–784 (1985).
    [CrossRef] [PubMed]
  16. J. Richards, P. Vermeulen, E. Barnard, D. P. Casasent, “Parallel holographic generation of multiple Hough transform slices,” submitted to Appl. Opt.
  17. M. H. Kuhn, W. Menhardt, I. C. Carlsen, “Real time interactive NMR image synthesis,” IEEE Trans. Med. Imag. MI-4, 160–164 (1985).
    [CrossRef]
  18. C. Billotet-Hoffmann, O. Bryngdahl, “Optical pseudocolor encoding using adaptive electronic halftoning,” Opt. Commun. 45, 327–330 (1983).
    [CrossRef]
  19. D. Kermisch, P. G. Roetling, “Fourier spectrum of halftone images,” J. Opt. Soc. Am. 65, 716–723 (1975).
    [CrossRef]
  20. M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
    [CrossRef]
  21. J. P. Allebach, B. Liu, “Minimax spectrum shaping with a bandwidth constraint,” Appl. Opt. 14, 3062–3072 (1975).
    [CrossRef] [PubMed]

1987 (1)

1986 (2)

M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

J. G. Kim, G. Kim, “Design of optimal filters for error-feedback quantization of monochrome pictures,” Inf. Sci. 39, 285–298 (1986).
[CrossRef]

1985 (3)

T. S. Perry, P. Wallich, eds., “Displays (Applications Topic),” IEEE Spectrum 22, 52–73 (1985).

G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-generated holograms: carrier of polar geometry,” Appl. Opt. 24, 777–784 (1985).
[CrossRef] [PubMed]

M. H. Kuhn, W. Menhardt, I. C. Carlsen, “Real time interactive NMR image synthesis,” IEEE Trans. Med. Imag. MI-4, 160–164 (1985).
[CrossRef]

1984 (3)

R. Eschbach, R. Hauck, “Analytic description of the 1-D error diffusion technique for halftoning,” Opt. Commun. 52, 165–168 (1984).
[CrossRef]

R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
[CrossRef]

H. S. Watkins, J. S. Moore, “A survey of color graphics printing,” IEEE Spectrum 21, 26–37 (1984).

1983 (2)

C. Billotet-Hoffmann, O. Bryngdahl, “Optical pseudocolor encoding using adaptive electronic halftoning,” Opt. Commun. 45, 327–330 (1983).
[CrossRef]

S. H. Algie, “Resolution and tonal continuity in bilevel printed picture continuity,” Comput. Vision Graph. Image Process. 24, 329–346 (1983).
[CrossRef]

1981 (2)

J. P. Allebach, “Representation-related errors in binary digital holograms: a unified analysis,” Appl. Opt. 20, 290–299 (1981).
[CrossRef] [PubMed]

J. C. Stoffel, J. F. Moreland, “A survey of electron techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
[CrossRef]

1976 (3)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

J. R. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comput Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

J. A. G. Hale, “Dot spacing modulation for the production of pseudo grey pictures,” Proc. Soc. Inf. Disp. 17, 63–74 (1976).

1975 (2)

Algie, S. H.

S. H. Algie, “Resolution and tonal continuity in bilevel printed picture continuity,” Comput. Vision Graph. Image Process. 24, 329–346 (1983).
[CrossRef]

Allebach, J. P.

Barnard, E.

J. Richards, P. Vermeulen, E. Barnard, D. P. Casasent, “Parallel holographic generation of multiple Hough transform slices,” submitted to Appl. Opt.

Billotet-Hoffmann, C.

C. Billotet-Hoffmann, O. Bryngdahl, “Optical pseudocolor encoding using adaptive electronic halftoning,” Opt. Commun. 45, 327–330 (1983).
[CrossRef]

Broja, M.

M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

Bryngdahl, O.

M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

G. Neugebauer, R. Hauck, O. Bryngdahl, “Computer-generated holograms: carrier of polar geometry,” Appl. Opt. 24, 777–784 (1985).
[CrossRef] [PubMed]

R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
[CrossRef]

C. Billotet-Hoffmann, O. Bryngdahl, “Optical pseudocolor encoding using adaptive electronic halftoning,” Opt. Commun. 45, 327–330 (1983).
[CrossRef]

Carlsen, I. C.

M. H. Kuhn, W. Menhardt, I. C. Carlsen, “Real time interactive NMR image synthesis,” IEEE Trans. Med. Imag. MI-4, 160–164 (1985).
[CrossRef]

Casasent, D. P.

J. Richards, P. Vermeulen, E. Barnard, D. P. Casasent, “Parallel holographic generation of multiple Hough transform slices,” submitted to Appl. Opt.

Eschbach, R.

M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

R. Eschbach, R. Hauck, “Analytic description of the 1-D error diffusion technique for halftoning,” Opt. Commun. 52, 165–168 (1984).
[CrossRef]

Floyd, R. W.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

Hale, J. A. G.

J. A. G. Hale, “Dot spacing modulation for the production of pseudo grey pictures,” Proc. Soc. Inf. Disp. 17, 63–74 (1976).

Hauck, R.

Jarvis, J. R.

J. R. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comput Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

Judice, C. N.

J. R. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comput Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

Kermisch, D.

Kim, G.

J. G. Kim, G. Kim, “Design of optimal filters for error-feedback quantization of monochrome pictures,” Inf. Sci. 39, 285–298 (1986).
[CrossRef]

Kim, J. G.

J. G. Kim, G. Kim, “Design of optimal filters for error-feedback quantization of monochrome pictures,” Inf. Sci. 39, 285–298 (1986).
[CrossRef]

Kuhn, M. H.

M. H. Kuhn, W. Menhardt, I. C. Carlsen, “Real time interactive NMR image synthesis,” IEEE Trans. Med. Imag. MI-4, 160–164 (1985).
[CrossRef]

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), p. 121.

Liu, B.

Menhardt, W.

M. H. Kuhn, W. Menhardt, I. C. Carlsen, “Real time interactive NMR image synthesis,” IEEE Trans. Med. Imag. MI-4, 160–164 (1985).
[CrossRef]

Moore, J. S.

H. S. Watkins, J. S. Moore, “A survey of color graphics printing,” IEEE Spectrum 21, 26–37 (1984).

Moreland, J. F.

J. C. Stoffel, J. F. Moreland, “A survey of electron techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
[CrossRef]

Neugebauer, G.

Ninke, W. H.

J. R. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comput Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

Richards, J.

J. Richards, P. Vermeulen, E. Barnard, D. P. Casasent, “Parallel holographic generation of multiple Hough transform slices,” submitted to Appl. Opt.

Roetling, P. G.

Seldowitz, M. A.

Steinberg, L.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

Stoffel, J. C.

J. C. Stoffel, J. F. Moreland, “A survey of electron techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
[CrossRef]

Sweeney, D. W.

Ulichney, R.

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).

Vermeulen, P.

J. Richards, P. Vermeulen, E. Barnard, D. P. Casasent, “Parallel holographic generation of multiple Hough transform slices,” submitted to Appl. Opt.

Watkins, H. S.

H. S. Watkins, J. S. Moore, “A survey of color graphics printing,” IEEE Spectrum 21, 26–37 (1984).

Appl. Opt. (4)

Comput Graph. Image Process. (1)

J. R. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous tone pictures on bilevel displays,” Comput Graph. Image Process. 5, 13–40 (1976).
[CrossRef]

Comput. Vision Graph. Image Process. (1)

S. H. Algie, “Resolution and tonal continuity in bilevel printed picture continuity,” Comput. Vision Graph. Image Process. 24, 329–346 (1983).
[CrossRef]

IEEE Spectrum (2)

H. S. Watkins, J. S. Moore, “A survey of color graphics printing,” IEEE Spectrum 21, 26–37 (1984).

T. S. Perry, P. Wallich, eds., “Displays (Applications Topic),” IEEE Spectrum 22, 52–73 (1985).

IEEE Trans. Commun. (1)

J. C. Stoffel, J. F. Moreland, “A survey of electron techniques for pictorial image reproduction,” IEEE Trans. Commun. C-29, 1898–1925 (1981).
[CrossRef]

IEEE Trans. Med. Imag. (1)

M. H. Kuhn, W. Menhardt, I. C. Carlsen, “Real time interactive NMR image synthesis,” IEEE Trans. Med. Imag. MI-4, 160–164 (1985).
[CrossRef]

Inf. Sci. (1)

J. G. Kim, G. Kim, “Design of optimal filters for error-feedback quantization of monochrome pictures,” Inf. Sci. 39, 285–298 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (3)

R. Eschbach, R. Hauck, “Analytic description of the 1-D error diffusion technique for halftoning,” Opt. Commun. 52, 165–168 (1984).
[CrossRef]

M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
[CrossRef]

C. Billotet-Hoffmann, O. Bryngdahl, “Optical pseudocolor encoding using adaptive electronic halftoning,” Opt. Commun. 45, 327–330 (1983).
[CrossRef]

Proc. Soc. Inf. Disp. (2)

J. A. G. Hale, “Dot spacing modulation for the production of pseudo grey pictures,” Proc. Soc. Inf. Disp. 17, 63–74 (1976).

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Disp. 17, 78–84 (1976).

Other (3)

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), p. 121.

J. Richards, P. Vermeulen, E. Barnard, D. P. Casasent, “Parallel holographic generation of multiple Hough transform slices,” submitted to Appl. Opt.

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Figures (13)

Fig. 1
Fig. 1

Input–output relationship for direct quantization.

Fig. 2
Fig. 2

(a) Basic 1-D error-diffusion procedure. (b) 2-D error diffusion coefficients used by Floyd and Steinberg.4

Fig. 3
Fig. 3

Graphical derivation of the pdf’s of the error sums s(n).

Fig. 4
Fig. 4

Variance of E(r) for error diffusion (■) and direct quantization (□)

Fig. 5
Fig. 5

Variance of E(r) when nearest-neighbor correlations are taken into account. Symbols are as in Fig. 4.

Fig. 6
Fig. 6

Diffusion coefficients β(a, b) corresponding to the algorithm depicted in Fig. 2(b).

Fig. 7
Fig. 7

Derivation of the pdf of s(1, 0) in the 2-D case.

Fig. 8
Fig. 8

2-D error variances for (left) β(0) = β(1) = 0.5 and (right) β(0) = 1, β(1) = 0.

Fig. 9
Fig. 9

Effects of 1-D quantization (bold solid curves, exact; dashed curves, after error diffusion) on (a) a random function and (c) a piecewise-constant function. The errors (■) arising from error diffusion for the same functions are shown in (b) and (d), respectively. The theoretical curve shown for the error-diffused error (dashed curves) is 2σ(r), with σ(r) defined by Eq. (10).

Fig. 10
Fig. 10

2-D quantization of a rectangular block. (a) The original image, (b) the image with the error diffused from left to right [β(1, 0) = 1], (c) the image with error diffused from left to right and from top to bottom [β(1, 0) = β(0, 1) = 1/2], and (d) the result of direct binanzation.

Fig. 11
Fig. 11

2-D quantization of an outdoor photograph. The various outputs were obtained as in Fig. 10.

Fig. 12
Fig. 12

(a) Optimized diffusion coefficients obtained when the indicated four coefficients are allowed to be nonzero and (b) resulting rendition of the outdoor scene.

Fig. 13
Fig. 13

Functions involved in Eq. (C4).

Equations (51)

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g ( n ) = s ( n 1 ) + f ( n )
h ( n ) = θ [ g ( n ) T ] ,
s ( n ) = g ( n ) h ( n ) .
e ( n ) = f ( n ) h ( n ) .
s ( n ) = s ( n 1 ) + e ( n ) .
s ( 0 ) = { f ( 0 ) if f ( 0 ) < T f ( 0 ) 1 if f ( 0 ) T .
E ( r ) = n = 0 N 1 α r ( n ) e ( n ) ,
σ 2 ( r ) = | E ( r ) E ( r ) | 2 ,
E ( r ) = T 1 2
σ 2 ( r ) = 1 12 [ 1 + 4 ( N 1 ) sin 2 ( π r / N ) ] .
E ( r ) = N ( T 1 2 ) δ r 0
σ 2 ( r ) = 1 12 N .
g ( n , m ) = a , b β ( a , b ) s ( n a , m b ) + f ( n , m ) .
s ( n , m ) [ T 1 , T ] ,
β ( a , b ) s ( n a , m b ) { [ ( T 1 ) β ( a , b ) , T β ( a , b ) ] if β ( a , b ) > 0 [ T β ( a , b ) , ( T 1 ) β ( a , b ) ] if β ( a , b ) < 0 [ L ( a , b ) , H ( a , b ) ] .
k ( n , m ) [ a , b L ( a , b ) , a , b H ( a , b ) ] .
a , b L ( a , b ) T 1
a , b H ( a , b ) T .
s ( n , m ) = T 1 2
[ s ( n , m ) s ( n , m ) ] 2 = 1 / 12 ,
σ 2 ( r n , r m ) = 1 / 12 { N 2 + ( β 0 2 + β 1 2 ) N ( N 1 ) + 2 β 0 β 1 ( N 1 ) 2 × cos [ 2 π ( r n r m ) N ] 2 N ( N 1 ) × [ β 0 cos ( 2 π r m N ) + β 1 cos ( 2 π r n N ) ] } .
t ( x , y ) = n , m f ( n , m ) rect ( x n d d , y m a d ) ,
I err ( u , υ ) = sin c 2 ( u d , υ d ) | E ( u d N , υ d N ) | 2 ,
= u min u max υ min υ max I err ( u , υ ) d u d υ .
β ( a , b ) = 0
a , b | β ( a , b ) | 1.
E ( r ) = n = 0 N 1 α r ( n ) e ( n ) .
E ( r ) = n = 0 N 1 α r ( n ) [ s ( n ) s ( n 1 ) ] = n = 0 N 1 s ( n ) [ α r ( n ) α r ( n + 1 ) ] ,
E ( r ) = n = 0 N 1 s ( n ) [ α r ( n ) α r ( n + 1 ) ] = s ( 0 ) = T 1 2 .
σ 2 ( r ) = | n = 0 N 1 s ( n ) [ α r ( n ) α r ( n + 1 ) ] n = 0 N 1 s ( n ) [ α r ( n ) α r ( n + 1 ) ] | 2 = | n = 0 N 1 s ( n ) [ α r ( n ) α r ( n + 1 ) ] | 2 = n = 0 N 1 s 2 ( n ) | α r ( n ) α r ( n + 1 ) | 2 + n = 0 N 1 m = 0 , m n N 1 s ( n ) s ( m ) × [ α r ( n ) α r ( n + 1 ) ] [ α r ( m ) α r ( m + 1 ) ] * .
s ( n ) s ( m ) = 0 for n m .
σ 2 ( r ) = n = 0 N 1 s 2 ( n ) | α r ( n ) α r ( n + 1 ) | 2 = 1 12 n = 0 N 1 | [ α r ( n ) α r ( n + 1 ) ] | 2 = 1 12 [ 1 + 4 ( N 1 ) sin 2 ( π r N ) ] ,
E ( r ) = n = 0 N 1 α r ( n ) e ( n ) = N ( 2 T 1 ) 2 δ r 0 ,
σ 2 ( r ) = | n = 0 N 1 [ e ( n ) e ( n ) ] α r ( n ) | 2 = N 12 ,
s ( n ) s ( m ) = 1 12 ( δ n m + μ δ n , m + 1 + μ δ n , m 1 ) ,
Δ = 2 n = 0 N 2 s ( n ) s ( n + 1 ) Re { α r ( n ) α r * ( n + 1 ) + α r ( n + 1 ) × α r * ( n + 2 ) α r ( n + 1 ) α r * ( n + 1 ) α r ( n ) α r * ( n + 2 ) } = 2 μ 12 [ ( 2 N 3 ) cos ( 2 π r N ) ( N 1 ) ( N 2 ) cos ( 4 π r N ) ] ,
s ( n , m ) = { g ( n , m ) if g ( n , m ) < T g ( n , m ) 1 if g ( n , m ) > T .
p S ( s ) = p G ( g = s ) = p G ( g = s ) + p G ( g = s + 1 ) = p G ( g = s + 1 ) if s < T if s [ T 1 , T ] if s > T .
g ( n , m ) = f ( n , m ) + k ( n , m ) .
p S ( s ) = p K ( k ) p F ( s k ) d k if s < T 1 = p K ( k ) [ p F ( s k ) + p F ( s k + 1 ) ] d k if s [ T 1 , T ] = p K ( k ) p F ( s k + 1 ) d k if s > T .
p [ s ( n , m ) ] = { 0 if s ( n , m ) [ T 1 , T ] 1 if s ( n , m ) [ T 1 , T ]
s ( n , m ) = e ( n , m ) + a b β ( a , b ) s ( n a , m b ) .
E ( r n , r m ) = n , m = 0 N 1 α r n , r m ( n , m ) e ( n , m ) = n , m = 0 N 1 α r n , r m ( n , m ) × [ s ( n , m ) a b β ( a , b ) s ( n a , m b ) ] ,
E ( r n , r m ) = n , m = 0 N 1 s ( n , m ) [ α r n , r m ( n , m ) a b β ( a , b ) α r n , r m ( n + a , m + b ) ] = 2 T 1 2 ( N 2 δ r n 0 δ r m 0 a b β ( a , b ) × exp [ j 2 π r n max ( 0 , a ) N ] × exp [ j 2 π r m max ( 0 , b ) N ] × { 1 exp [ j 2 π r n ( N a ) / N ] } { 1 exp [ j 2 π r m ( N b ) / N ] } { 1 exp ( j 2 π r n / N ) } { 1 exp [ j 2 π r m / N ] } ) ,
σ 2 ( r n , r m ) = n , m α r n , r m ( n , m ) e ( n , m ) p , q α * r n , r m ( p , q ) e ( p , q ) = n , m s ( n , m ) [ α r n , r m ( n , m ) a , b β ( a , b ) α r n , r m ( n + a , m + b ) ] × p , q s ( p , q ) { α r n , r m ( p , q ) c , d β ( c , d ) α r n , r m ( p + c , q + d ) } * .
σ 2 ( r n , r m ) = ( 1 / 12 ) n , m [ α r n , r m ( n , m ) a , b β ( a , b ) α r n , r m ( n + a , m + b ) ] × [ α r n , r m ( n , m ) c , d β ( c , d ) α r n , r m ( n + c , m + d ) ] * .
σ 2 ( r n , r m ) = ( 1 / 12 ) ( N 2 2 a , b ( N | a | ) ( N | b | ) β ( a , b ) × cos [ 2 π N ( r n a + r m b ) ] + a , b ( N | a | ) ( N | b | ) β ( a , b ) 2 + 2 a , b c > a , d > b [ N χ ( a , c ) ] [ N χ ( b , d ) ] × β ( a , b ) β ( c , d ) cos { 2 π [ r n ( a c ) + r m ( c d ) ] N } ) ,
χ ( a , b ) = max ( 0 , a , b ) + max ( 0 , a , b ) .
B ( a , b ) = { 1 if ( a , b ) = ( 0 , 0 ) β ( a , b ) otherwise .
σ 2 ( r n , r m ) = ( 1 / 12 ) n , m [ a , b B ( a , b ) α r n , r m ( n + a , m + b ) ] × [ c , d B ( c , d ) α r n , r m ( n + c , m + d ) ] * 1 / 12 n , m [ a , b , c , d B ( a , b ) B ( c , d ) α r n , r m ( a c , b d ) ] = N 2 / 12 [ a , b , c , d B ( a + c , b + d ) B ( c , d ) α r n , r m ( a , b ) ] .
σ 2 ( r n , r m ) = ( N 2 / 12 ) ( a , b B ( a , b ) 2 + 2 a , b c > a , d > b B ( a , b ) B ( c , d ) × cos { 2 π N [ r n ( a c ) + r m ( b d ) ] } ) .

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