Abstract

An analysis of the error diffusion procedure is presented that is based on the terminology of filter theory. It is demonstrated that the error diffusion procedure is a powerful means to avoid signal error caused by a nonlinear system. An appropriate filter design method is described. The theoretical results are applied to treat picture binarization as well as quantization and coding in diffractive optics–digital holography.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. R. Schroeder, “Image from computers,” IEEE Spectrum 6, 66–78 (1969).
    [CrossRef]
  2. R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).
  3. R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
    [CrossRef]
  4. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
    [CrossRef]
  5. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
    [CrossRef]
  6. E. Barnard, “Optical error diffusion for computer-generated holograms,” J. Opt. Soc. Am. A 5, 1803–1817 (1988).
    [CrossRef]
  7. R. Eschbach, R. Hauck, “Analytic description on the 1-D error diffusion technique for halftoning,” Opt. Commun. 52, 165–168 (1984).
    [CrossRef]
  8. J. G. Kim, G. Kim, “Design of optimal filters for error-feedback quantization of monochrome pictures,” Inf. Sci. (NY) 39, 285–298 (1986).
  9. M. Broja, R. Eschbach, O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1986).
    [CrossRef]
  10. R. Mrusek, M. Broja, O. Bryngdahl, “Halftoning by carrier and spectrum control,” Opt. Commun. 75, 375–380 (1990).
    [CrossRef]
  11. O. Bryngdahl, F. Wyrowski, “Digital holography: computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28.
    [CrossRef]
  12. F. Wyrowski, “Diffraction efficiency of analog and quantized digital amplitude holograms: analysis and manipulation,” J. Opt. Soc. Am. A 7, 383–393 (1990).
    [CrossRef]
  13. D. E. Knuth, “Digital halftones by dot diffusion,” ACM Trans. Graph. 6, 245–273 (1987).
    [CrossRef]
  14. R. Eschbach, “Pulse-density modulation on rastered media: combining pulse-density modulation and error diffusion,” J. Opt. Soc. Am. A 7, 708–716 (1990).
    [CrossRef]
  15. 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]
  16. E. Barnard, P. Vermeulen, D. P. Casasent, “Optical correlation CGHs with modulated error diffusion,” Appl. Opt. 28, 5358–5362 (1989).
    [CrossRef] [PubMed]
  17. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1059–1064 (1988).
    [CrossRef]
  18. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Error-diffusion algorithm in phase synthesis and retrieval techniques,” submitted to Opt. Lett.

1990 (3)

1989 (2)

E. Barnard, P. Vermeulen, D. P. Casasent, “Optical correlation CGHs with modulated error diffusion,” Appl. Opt. 28, 5358–5362 (1989).
[CrossRef] [PubMed]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

1988 (3)

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

E. Barnard, “Optical error diffusion for computer-generated holograms,” J. Opt. Soc. Am. A 5, 1803–1817 (1988).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1059–1064 (1988).
[CrossRef]

1987 (2)

1986 (2)

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

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

1984 (2)

R. Eschbach, R. Hauck, “Analytic description on 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]

1976 (1)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).

1969 (1)

M. R. Schroeder, “Image from computers,” IEEE Spectrum 6, 66–78 (1969).
[CrossRef]

Allebach, J. P.

Barnard, E.

Broja, M.

R. Mrusek, M. Broja, O. Bryngdahl, “Halftoning by carrier and spectrum control,” Opt. Commun. 75, 375–380 (1990).
[CrossRef]

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

Bryngdahl, O.

R. Mrusek, M. Broja, O. Bryngdahl, “Halftoning by carrier and spectrum control,” Opt. Commun. 75, 375–380 (1990).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1059–1064 (1988).
[CrossRef]

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

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

O. Bryngdahl, F. Wyrowski, “Digital holography: computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28.
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Error-diffusion algorithm in phase synthesis and retrieval techniques,” submitted to Opt. Lett.

Casasent, D. P.

Eschbach, R.

R. Eschbach, “Pulse-density modulation on rastered media: combining pulse-density modulation and error diffusion,” J. Opt. Soc. Am. A 7, 708–716 (1990).
[CrossRef]

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

R. Eschbach, R. Hauck, “Analytic description on 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 greyscale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).

Hauck, R.

R. Eschbach, R. Hauck, “Analytic description on 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]

Kim, G.

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

Kim, J. G.

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

Knuth, D. E.

D. E. Knuth, “Digital halftones by dot diffusion,” ACM Trans. Graph. 6, 245–273 (1987).
[CrossRef]

Mrusek, R.

R. Mrusek, M. Broja, O. Bryngdahl, “Halftoning by carrier and spectrum control,” Opt. Commun. 75, 375–380 (1990).
[CrossRef]

Schroeder, M. R.

M. R. Schroeder, “Image from computers,” IEEE Spectrum 6, 66–78 (1969).
[CrossRef]

Seldowitz, M. A.

Steinberg, L.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).

Sweeney, D. W.

Vermeulen, P.

Weissbach, S.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Error-diffusion algorithm in phase synthesis and retrieval techniques,” submitted to Opt. Lett.

Wyrowski, F.

F. Wyrowski, “Diffraction efficiency of analog and quantized digital amplitude holograms: analysis and manipulation,” J. Opt. Soc. Am. A 7, 383–393 (1990).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1059–1064 (1988).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Error-diffusion algorithm in phase synthesis and retrieval techniques,” submitted to Opt. Lett.

O. Bryngdahl, F. Wyrowski, “Digital holography: computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28.
[CrossRef]

ACM Trans. Graph. (1)

D. E. Knuth, “Digital halftones by dot diffusion,” ACM Trans. Graph. 6, 245–273 (1987).
[CrossRef]

Appl. Opt. (2)

IEEE Spectrum (1)

M. R. Schroeder, “Image from computers,” IEEE Spectrum 6, 66–78 (1969).
[CrossRef]

Inf. Sci. (NY) (1)

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

J. Opt. Soc. Am. A (5)

Opt. Commun. (5)

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

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[CrossRef]

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

R. Mrusek, M. Broja, O. Bryngdahl, “Halftoning by carrier and spectrum control,” Opt. Commun. 75, 375–380 (1990).
[CrossRef]

Proc. Soc. Inf. Disp. (1)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Disp. 17, 75–77 (1976).

Other (2)

O. Bryngdahl, F. Wyrowski, “Digital holography: computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28.
[CrossRef]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Error-diffusion algorithm in phase synthesis and retrieval techniques,” submitted to Opt. Lett.

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 (19)

Fig. 1
Fig. 1

Notations used to describe the influence of a nonlinear system on (a) the original signal f (x) and (b) the adjusted signal fc(x). In (b) the adjustment c(x) and the error ec(x) are combined to output e ˜ (x).

Fig. 2
Fig. 2

(a) Coordinate system used in the space domain; (b) illustration of a part of a diffusion matrix; (c) the diffusion of errors by elements of this matrix.

Fig. 3
Fig. 3

Period of the magnitude of the filter H(u) for the diffusion matrix introduced by Floyd and Steinberg.2

Fig. 4
Fig. 4

Magnitudes of (a) a MAE filter with d(1) = −1 and ED filters with (b) d(1) = −1; (c) d(1) = j.

Fig. 5
Fig. 5

Illustration of the dependence of two real-valued diffusion weights d(1) and d(2) on the zero u0 of the resulting filter (see third row in Table II).

Fig. 6
Fig. 6

Magnitudes of one-dimensional ED-filters caused by different choices of two real-valued diffusion weights (see Table I): (a) u0 = 0; (b) u0 = 1/8; (c) u0 = 1/4.

Fig. 7
Fig. 7

Nonlinearity used in the one-dimensional simulation experiments that are described in Subsection III.A.3.

Fig. 8
Fig. 8

(a) Test signal; (b) influence of the nonlinearity sketched in Fig. 7 on the signal; (c) adjustment of the signal that is due to ED; (d) influence of the nonlinearity on the adjusted signal.

Fig. 9
Fig. 9

Spectra that correspond to the distributions shown in Figs. 8(a), 8(b), and 8(d). The SNR’s are 25 in (b) and 1800 in (c).

Fig. 10
Fig. 10

Illustration of the influence of the nonlinearity sketched in Fig. 7 on a second test signal in the Fourier domain: (a) spectrum of the signal; (b) disturbed spectrum; (c) spectrum resulting from ED. The SNR’s are 16 in (b) and 500 in (c).

Fig. 11
Fig. 11

(a) One-dimensional ED in the êφ direction, which is suitable for a two-dimensional application; (b) restriction of the êφ direction because of a sampling grid.

Fig. 12
Fig. 12

Magnitude of two-dimensional filters with a zero at u0 = (¼, ⅛) for three different diffusion angles: (a) φ = 0°; (b) φ = 45°; (c) φ = 116.57° (see Table III).

Fig. 13
Fig. 13

Resulting magnitude of filters when two ED filters with orthogonal directions are combined: (a) φ = 0° and u0 = 0; (b) φ = 45° and u0 = 0; (c) φ = 45° and u0 = (1/4, 1/4) (see Tables IV and V).

Fig. 14
Fig. 14

Illustration of the coordinate system used for the derivation of Eqs. (42).

Fig. 15
Fig. 15

Illustrations of the application of ED in picture binarization. Employment of (a) and (b) the original Floyd–Steinberg diffusion matrix (see the filter in Fig. 3) and (c) and (d) the diffusion weights d (1, 0) = d (0, 1) = −1 and d (1, 1) = 1 [see the filter in Fig. 13(a)]. The binary pictures are shown in (a) and (c) and the corresponding spectra (logarithmic scale and eliminated dc peak) in (b) and (d).

Fig. 16
Fig. 16

(a) Part of an amplitude hologram binarized with ED;(b) the corresponding calculated diffraction pattern for d(−1, 1) = −1 (see Table VI).

Fig. 17
Fig. 17

(a) Part of an amplitude hologram binarized with ED;(b) the corresponding calculated diffraction pattern for d(1, 1) = 1 (see Table VI).

Fig. 18
Fig. 18

(a) Part of an amplitude hologram binarized with error diffusion; (b) the corresponding calculated diffraction pattern for d(1, 1) = 1, d(−1, 1) = −1, and d(0, 2) = −1 (see Table VI).

Fig. 19
Fig. 19

Calculated diffraction patterns of diffractive phase elements generated with ED for (a) an on-axis signal; (b) an off-axis signal (see Table VII).

Tables (7)

Tables Icon

Table I Characteristic Values of the Filters Shown in Fig. 6

Tables Icon

Table II Diffusion Weights for Real-Valued Signals Dependent on u0

Tables Icon

Table III Characteristic Values of the Filters Depicted in Fig. 12 with u0 = (1/4, 1/8) and Different Integers m′

Tables Icon

Table IV Characteristic Values of Examples of Hφ(u) and Hϑ(u)

Tables Icon

Table V Nonzero Diffusion Weights of Filters H(u) = Hφ(u)Hϑ(u) Depicted in Fig. 13 with the Parameters of Table IV

Tables Icon

Table VI Diffusion Weights and SNR’s for the Binarization of Amplitude Holograms[u0(1/4, 1/4)]

Tables Icon

Table VII Diffusion Weights and Parameters for the Calculation of Diffractive Phase Elements

Equations (85)

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

g ( x ) = S f ( x ) .
e ( x ) = g ( x ) - f ( x ) .
E ( u ) = F e ( x ) = G ( u ) - F ( u ) .
f c ( x ) = f ( x ) + c ( x ) .
f c ( D ) f ( D ) .
g c ( x ) = S f c ( x ) = f c ( x ) + e c ( x ) ,
e ˜ ( x ) = c ( x ) + e c ( x ) .
g c ( x ) = f ( x ) + e ˜ ( x ) ;
G c ( u ) = F ( u ) ,             u F ,
E ˜ ( u ) = 0 ,             u F .
E ˜ ( u ) = S ( u ) H ( u ) ,
E ˜ ( u ) = C ( u ) + E c ( u ) .
C ( u ) = D 1 ( u ) E c ( u ) + D 2 ( u ) E ˜ ( u ) ,
E ˜ ( u ) = E c ( u ) 1 + D 1 ( u ) 1 - D 2 ( u ) .
H ( u ) = 1 + D 1 ( u ) 1 - D 2 ( u ) .
c ( x ) = d 1 ( x ) e c ( x - x ) d x d y + d 2 ( x ) e ˜ ( x - x ) d x d y .
d i ( x ) = 0             for x 0 and y = 0 or y < 0 ,
H ( u ) = 0 ,             u F ,
H ( u 0 ) = 0 ,             u 0 center of F ,
d i ( x ) = m d i ( m ) δ ( x - m Δ , y - n Δ ) ,
c ( x ) = m 1 d 1 ( m 1 ) e ˜ ( x - m 1 Δ , y - n 1 Δ ) + m 2 d 2 ( m 2 ) e c ( x - m 2 Δ , y - n 2 Δ ) .
d i ( m ) = 0             for m 0 and n = 0 or n < 0.
D i ( u ) = m d i ( m ) exp [ 2 π j Δ ( m u + n v ) ] ,
- ½ ( 1 / Δ , 1 / Δ ) u ½ ( 1 / Δ , 1 / Δ ) .
H ( u ) = 1 1 - D 2 ( u )
H ( u ) = 1 + D 1 ( u )
m d ( m ) exp [ j β ( m ) ] = - 1 ,
d ( m ¯ ) = - exp [ - j β ( m ¯ ) ] { 1 + m m ¯ d ( m ) exp [ j β ( m ) ] } .
H ( u ) = 1 + d ( m ¯ ) exp [ 2 π j m ¯ u Δ + j δ ( m ¯ ) ] ,
d ( m ¯ ) = 1 ,
δ ( m ¯ ) = - 2 π m ¯ u 0 Δ + π .
| d H d u ( u 0 ) | = 2 π m ¯ Δ .
H ( u ) = 2 sin [ π ( u - u 0 ) Δ ] .
m d ( m ) = - 1
m d ( m ) ( - 1 ) m = - 1
H ( u ) = 1 + d ( 1 ) exp ( 2 π j u Δ ) + d ( 2 ) exp ( 4 π j u Δ )
d ( 1 ) = - 2 cos ( 2 π u 0 Δ ) ,
d ( 2 ) = 1 ,
H ( u ) = 2 cos ( 2 π u Δ ) - cos ( 2 π u 0 Δ ) .
| d H d u ( u 0 ) | = 4 π Δ sin ( 2 π u 0 Δ ) ,
SNR = F F ( u ) 2 d u F ( α G ( u ) - F ( u ) ) 2 d u ,
α = F F ( u ) G ( u ) d u F G ( u ) 2 d u .
m d ( m ) exp [ j β ( m ) ] = - 1 ,
φ [ 0 0 , 180 0 )
φ = 90 0 ( 1 - sgn [ m ] ) + arctan [ n / m ]
d ( x ) = k d ( k ) δ ( x - k m Δ , y - k n Δ ) ,
d ( k ) = d ( k m , k n ) .
H ( u ) = 1 + D ( u ) = 1 + k d ( k ) exp [ 2 π j k Δ ( m u + n v ) ] .
m u + n v = r 0 ,
H ( u 0 ) = 1 + k d ( k ) exp ( 2 π j m r 0 Δ ) .
Δ φ = Δ [ ( m ) 2 + ( n ) 2 ] 1 / 2 ,
H ( u ) = H φ ( u ) H ϑ ( u ) ,
m ϑ = - n φ ,             n ϑ = m φ .
Q ( φ 1 , φ 2 ) = [ α ( φ 1 ) / α ( φ 2 ) ] 2 ,
Q ( φ 1 , φ 2 ) = [ α ( φ 1 ) β ( φ 2 ) α ( φ 2 ) β ( φ 2 ) ] 2 5 - 3 sin 2 ( 2 φ ¯ 1 ) 5 - 3 sin 2 ( 2 φ ¯ 2 )
φ ¯ i = mod ( φ i + 45 0 , 90 0 ) - 45 0 ,
S f ( x ) = { 1 , f ( x ) > 0.5 0 , f ( x ) 0.5 ,
g ( x ) = S f ( x ) = exp [ i Φ ( x ) ] ,
d ( 1 ) exp ( j γ ) + d ( 2 ) exp ( j 2 γ ) = - 1
1 + d ( 1 ) cos ( γ ) + d ( 2 ) cos ( 2 γ ) = 0 ,
d ( 1 ) sin ( γ ) + d ( 2 ) sin ( 2 γ ) = 0.
d ( 1 ) = - 2 d ( 2 ) cos ( 2 π u 0 Δ ) .
1 - 2 d ( 2 ) cos 2 ( γ ) + d ( 2 ) cos ( 2 γ ) = 0.
d ( 2 ) = 1.
H ( u ) = 1 + d ( 1 ) exp ( j γ ) ,
H ¯ ( u ) = 1 + 2 d ( 1 ) exp ( j γ ) + [ d ( 1 ) ] 2 exp ( j 2 γ ) .
d ¯ ( 1 ) = 2 d ( 1 ) ,
d ¯ ( 2 ) = [ d ( 1 ) ] 2 .
H ( u ) = 1 + d ( 1 ) exp [ j γ ( u m + v n ) ] + d ( 2 ) exp [ j 2 γ ( u m + v n ) ] ,
H ( u ) = m 4 π j Δ exp [ j γ ( u m + u n ) ] × { exp [ j γ ( u m + v n ) ] - cos ( 2 π r 0 Δ ) } ,
e ^ φ = m [ ( m ) 2 + ( n ) 2 ] - 1 / 2 .
[ H ( u 0 ) · e ^ φ ] = - 4 π j Δ [ ( m ) 2 + ( n ) 2 ] 1 / 2 sin ( 2 π r 0 Δ ) ,
u = t e ^ φ ,
m u + n v = [ ( m ) 2 + ( n ) 2 ] 1 / 2 t .
H φ ( u ) = 1 + D φ ( u ) = 1 + d φ ( 1 ) exp [ j γ ( u m + v n ) ] + d φ ( 2 ) exp [ j 2 γ ( u m + v n ) ] ,
H ϑ ( u ) = 1 + D ϑ ( u ) = 1 + d ϑ ( 1 ) exp [ j γ ( - u n + v m ) ] + d φ ( 2 ) exp [ j 2 γ ( - u n + v m ) ] .
H ( u ) = H φ ( u ) H ϑ ( u ) = 1 + D ( u ) = 1 + D φ ( u ) + D ϑ ( u ) + d φ ( 1 ) d ϑ ( 1 ) exp { j γ [ u ( m - n ) + v ( m + n ) ] } + d φ ( 1 ) d ϑ ( 2 ) exp { j γ [ u ( m - 2 n ) + v ( 2 m + n ) ] } + d φ ( 2 ) d ϑ ( 1 ) exp { j γ [ n ( 2 m - n ) + v ( m + 2 n ) ] } + d φ ( 2 ) d ϑ ( 2 ) exp { j 2 γ [ n ( m - n ) + v ( m + n ) ] } .
d ( m , n ) = d φ ( 1 ) ,
d ( - n , m ) = d ϑ ( 1 ) ,
d ( 2 m , n ) = d φ ( 2 ) ,
d ( - 2 n , 2 m ) = d ϑ ( 2 ) ,
d ( m - n , m + n ) = d φ ( 1 ) d ϑ ( 1 ) ,
d ( m - 2 n , 2 m + n ) = d φ ( 1 ) d φ ( 2 ) ,
d ( 2 m - n , m + 2 n ) = d φ ( 2 ) d ϑ ( 1 ) ,
d ( 2 m - 2 n , 2 m + 2 n ) = d φ ( 2 ) d ϑ ( 2 ) .

Metrics