Abstract

Arrays with good correlation properties are required for coded-aperture imaging, as well as for other applications of two-dimensional signal processing. Since binary arrays with perfect periodic autocorrelation functions exist for only a few sizes, mismatched filtering is discussed. Mismatched filtering entirely suppresses any sidelobes of the periodic autocorrelation function at the expense of a slightly reduced signal-to-noise ratio. New construction methods for binary arrays are presented for which this loss, with respect to periodic or odd-periodic correlation, converges to zero with an increasing array size.

© 1998 Optical Society of America

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References

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  1. E. E. Fenimore, T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337–347 (1978).
    [CrossRef] [PubMed]
  2. A. Busboom, H. D. Schotten, H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. A 14, 1058–1065 (1997).
    [CrossRef]
  3. J. Jedwab, C. Mitchell, “Constructing new perfect binary arrays,” Electron. Lett. 24, 650–652 (1988).
    [CrossRef]
  4. P. Wild, “Infinite families of perfect binary arrays,” Electron. Lett. 24, 845–847 (1988).
    [CrossRef]
  5. H. D. Lüke, Korrelationssignale (Springer-Verlag, New York, 1992).
    [CrossRef]
  6. P. Seidler, “Nebenmaximumreduktion bei Korrelationsempfang binär phasencodierter Impulssignale,” Nachrichtentech. Z 29, 154–159 (1976).
  7. V. P. Ipatov, “Total suppression of sidelobes of periodic correlation functions of phase manipulated signals,” Radio Eng. Electron. Phys. 22, 42–47 (1977).
  8. V. P. Ipatov, B. V. Fedorov, “Regular binary sequences with small losses in suppressing sidelobes,” Radio Electron. Commun. Sys. (Radioelektron.) 27, 29–33 (1987).
  9. P. Seidler, “Mismatched filtering for coded aperture imaging with minimum sidelobes,” Electron. Lett. 17, 96–97 (1981).
    [CrossRef]
  10. C. Brown, “Multiplex imaging with multiple-pinhole cameras,” J. Appl. Phys. 45, 1806–1811 (1974).
    [CrossRef]
  11. H. Weiss, “Abbildung mit kodierter Apertur,” NTZ Archiv 3, 329–335 (1981).
  12. M. Antweiler, H. D. Lüke, “Mismatched-Filterung periodischer Binärarrays,” Frequenz 48, 58–65 (1994).
    [CrossRef]
  13. H. D. Lüke, A. Busboom, “Binary arrays with perfect odd-periodic autocorrelation,” Appl. Opt. 36, 6612–6619 (1997).
    [CrossRef]
  14. M. Antweiler, L. Bömer, H. D. Lüke, “Perfect ternary arrays,” IEEE Trans. Inf. Theory 36, 696–705 (1990).
    [CrossRef]
  15. V. P. Ipatov, “Synthesis of a binary periodic signal-filter pair,” Radio Electron. Commun. Syst. (Radioelektron.) 23, 46–51 (1980).
  16. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  17. G. Dueck, “New optimization heuristics—the great deluge algorithm and the record-to-record travel,” J. Comput. Phys. 104, 86–92 (1993).
    [CrossRef]
  18. A. Busboom, H. Elders-Boll, H. D. Schotten, “Combinatorial design of near-optimum masks for coded aperture imaging,” in Proceedings of the 1997 IEEE International Conference on Acoustics, Speech, Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2817–2820.
    [CrossRef]
  19. J. Jedwab, C. J. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
    [CrossRef]
  20. H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
    [CrossRef]
  21. H. D. Lüke, “Binary odd-periodic complementary sequences,” IEEE Trans. Inf. Theory 43, 365–367 (1997).
    [CrossRef]
  22. H. D. Lüke, “Binäre Folgen und Arrays mit optimalen ungeraden Autokorrelationsfunktionen,” Frequenz 48, 213–220 (1994).
    [CrossRef]
  23. H. D. Lüke, A. Busboom , “Mismatched filtering of odd-periodic binary sequences,” IEEE Trans. Aerosp. Electron. Syst. (to be published).
  24. R. M. Rideout, “Coded imaging systems for x-ray astronomy,” Ph.D. dissertation (University of Birmingham, Birmingham, UK, 1995).
  25. M. H. Finger, T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the Nineteenth International Cosmic Ray Conference (Scientific and Technical Information Branch, NASA, Washington, D.C., 1985), pp. 295–298.
  26. U. B. Jayanthi, J. Braga, “Physical implementation of an antimask in URA based coded mask systems,” Nucl. Instr. Meth. A 310, 685–689 (1991).
    [CrossRef]

1997 (3)

1995 (1)

H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
[CrossRef]

1994 (2)

H. D. Lüke, “Binäre Folgen und Arrays mit optimalen ungeraden Autokorrelationsfunktionen,” Frequenz 48, 213–220 (1994).
[CrossRef]

M. Antweiler, H. D. Lüke, “Mismatched-Filterung periodischer Binärarrays,” Frequenz 48, 58–65 (1994).
[CrossRef]

1993 (1)

G. Dueck, “New optimization heuristics—the great deluge algorithm and the record-to-record travel,” J. Comput. Phys. 104, 86–92 (1993).
[CrossRef]

1991 (1)

U. B. Jayanthi, J. Braga, “Physical implementation of an antimask in URA based coded mask systems,” Nucl. Instr. Meth. A 310, 685–689 (1991).
[CrossRef]

1990 (2)

J. Jedwab, C. J. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
[CrossRef]

M. Antweiler, L. Bömer, H. D. Lüke, “Perfect ternary arrays,” IEEE Trans. Inf. Theory 36, 696–705 (1990).
[CrossRef]

1988 (2)

J. Jedwab, C. Mitchell, “Constructing new perfect binary arrays,” Electron. Lett. 24, 650–652 (1988).
[CrossRef]

P. Wild, “Infinite families of perfect binary arrays,” Electron. Lett. 24, 845–847 (1988).
[CrossRef]

1987 (1)

V. P. Ipatov, B. V. Fedorov, “Regular binary sequences with small losses in suppressing sidelobes,” Radio Electron. Commun. Sys. (Radioelektron.) 27, 29–33 (1987).

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1981 (2)

P. Seidler, “Mismatched filtering for coded aperture imaging with minimum sidelobes,” Electron. Lett. 17, 96–97 (1981).
[CrossRef]

H. Weiss, “Abbildung mit kodierter Apertur,” NTZ Archiv 3, 329–335 (1981).

1980 (1)

V. P. Ipatov, “Synthesis of a binary periodic signal-filter pair,” Radio Electron. Commun. Syst. (Radioelektron.) 23, 46–51 (1980).

1978 (1)

1977 (1)

V. P. Ipatov, “Total suppression of sidelobes of periodic correlation functions of phase manipulated signals,” Radio Eng. Electron. Phys. 22, 42–47 (1977).

1976 (1)

P. Seidler, “Nebenmaximumreduktion bei Korrelationsempfang binär phasencodierter Impulssignale,” Nachrichtentech. Z 29, 154–159 (1976).

1974 (1)

C. Brown, “Multiplex imaging with multiple-pinhole cameras,” J. Appl. Phys. 45, 1806–1811 (1974).
[CrossRef]

Antweiler, M.

M. Antweiler, H. D. Lüke, “Mismatched-Filterung periodischer Binärarrays,” Frequenz 48, 58–65 (1994).
[CrossRef]

M. Antweiler, L. Bömer, H. D. Lüke, “Perfect ternary arrays,” IEEE Trans. Inf. Theory 36, 696–705 (1990).
[CrossRef]

Bömer, L.

M. Antweiler, L. Bömer, H. D. Lüke, “Perfect ternary arrays,” IEEE Trans. Inf. Theory 36, 696–705 (1990).
[CrossRef]

Braga, J.

U. B. Jayanthi, J. Braga, “Physical implementation of an antimask in URA based coded mask systems,” Nucl. Instr. Meth. A 310, 685–689 (1991).
[CrossRef]

Brown, C.

C. Brown, “Multiplex imaging with multiple-pinhole cameras,” J. Appl. Phys. 45, 1806–1811 (1974).
[CrossRef]

Busboom, A.

H. D. Lüke, A. Busboom, “Binary arrays with perfect odd-periodic autocorrelation,” Appl. Opt. 36, 6612–6619 (1997).
[CrossRef]

A. Busboom, H. D. Schotten, H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. A 14, 1058–1065 (1997).
[CrossRef]

H. D. Lüke, A. Busboom , “Mismatched filtering of odd-periodic binary sequences,” IEEE Trans. Aerosp. Electron. Syst. (to be published).

A. Busboom, H. Elders-Boll, H. D. Schotten, “Combinatorial design of near-optimum masks for coded aperture imaging,” in Proceedings of the 1997 IEEE International Conference on Acoustics, Speech, Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2817–2820.
[CrossRef]

Cannon, T. M.

Dueck, G.

G. Dueck, “New optimization heuristics—the great deluge algorithm and the record-to-record travel,” J. Comput. Phys. 104, 86–92 (1993).
[CrossRef]

Elders-Boll, H.

A. Busboom, H. D. Schotten, H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. A 14, 1058–1065 (1997).
[CrossRef]

A. Busboom, H. Elders-Boll, H. D. Schotten, “Combinatorial design of near-optimum masks for coded aperture imaging,” in Proceedings of the 1997 IEEE International Conference on Acoustics, Speech, Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2817–2820.
[CrossRef]

Fedorov, B. V.

V. P. Ipatov, B. V. Fedorov, “Regular binary sequences with small losses in suppressing sidelobes,” Radio Electron. Commun. Sys. (Radioelektron.) 27, 29–33 (1987).

Fenimore, E. E.

Finger, M. H.

M. H. Finger, T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the Nineteenth International Cosmic Ray Conference (Scientific and Technical Information Branch, NASA, Washington, D.C., 1985), pp. 295–298.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Ipatov, V. P.

V. P. Ipatov, B. V. Fedorov, “Regular binary sequences with small losses in suppressing sidelobes,” Radio Electron. Commun. Sys. (Radioelektron.) 27, 29–33 (1987).

V. P. Ipatov, “Synthesis of a binary periodic signal-filter pair,” Radio Electron. Commun. Syst. (Radioelektron.) 23, 46–51 (1980).

V. P. Ipatov, “Total suppression of sidelobes of periodic correlation functions of phase manipulated signals,” Radio Eng. Electron. Phys. 22, 42–47 (1977).

Jayanthi, U. B.

U. B. Jayanthi, J. Braga, “Physical implementation of an antimask in URA based coded mask systems,” Nucl. Instr. Meth. A 310, 685–689 (1991).
[CrossRef]

Jedwab, J.

J. Jedwab, C. J. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
[CrossRef]

J. Jedwab, C. Mitchell, “Constructing new perfect binary arrays,” Electron. Lett. 24, 650–652 (1988).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Lüke, H. D.

H. D. Lüke, “Binary odd-periodic complementary sequences,” IEEE Trans. Inf. Theory 43, 365–367 (1997).
[CrossRef]

H. D. Lüke, A. Busboom, “Binary arrays with perfect odd-periodic autocorrelation,” Appl. Opt. 36, 6612–6619 (1997).
[CrossRef]

H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
[CrossRef]

M. Antweiler, H. D. Lüke, “Mismatched-Filterung periodischer Binärarrays,” Frequenz 48, 58–65 (1994).
[CrossRef]

H. D. Lüke, “Binäre Folgen und Arrays mit optimalen ungeraden Autokorrelationsfunktionen,” Frequenz 48, 213–220 (1994).
[CrossRef]

M. Antweiler, L. Bömer, H. D. Lüke, “Perfect ternary arrays,” IEEE Trans. Inf. Theory 36, 696–705 (1990).
[CrossRef]

H. D. Lüke, A. Busboom , “Mismatched filtering of odd-periodic binary sequences,” IEEE Trans. Aerosp. Electron. Syst. (to be published).

H. D. Lüke, Korrelationssignale (Springer-Verlag, New York, 1992).
[CrossRef]

Mitchell, C.

J. Jedwab, C. Mitchell, “Constructing new perfect binary arrays,” Electron. Lett. 24, 650–652 (1988).
[CrossRef]

Mitchell, C. J.

J. Jedwab, C. J. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
[CrossRef]

Prince, T. A.

M. H. Finger, T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the Nineteenth International Cosmic Ray Conference (Scientific and Technical Information Branch, NASA, Washington, D.C., 1985), pp. 295–298.

Rideout, R. M.

R. M. Rideout, “Coded imaging systems for x-ray astronomy,” Ph.D. dissertation (University of Birmingham, Birmingham, UK, 1995).

Schotten, H. D.

A. Busboom, H. D. Schotten, H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. A 14, 1058–1065 (1997).
[CrossRef]

H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
[CrossRef]

A. Busboom, H. Elders-Boll, H. D. Schotten, “Combinatorial design of near-optimum masks for coded aperture imaging,” in Proceedings of the 1997 IEEE International Conference on Acoustics, Speech, Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2817–2820.
[CrossRef]

Seidler, P.

P. Seidler, “Mismatched filtering for coded aperture imaging with minimum sidelobes,” Electron. Lett. 17, 96–97 (1981).
[CrossRef]

P. Seidler, “Nebenmaximumreduktion bei Korrelationsempfang binär phasencodierter Impulssignale,” Nachrichtentech. Z 29, 154–159 (1976).

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Weiss, H.

H. Weiss, “Abbildung mit kodierter Apertur,” NTZ Archiv 3, 329–335 (1981).

Wild, P.

P. Wild, “Infinite families of perfect binary arrays,” Electron. Lett. 24, 845–847 (1988).
[CrossRef]

Appl. Opt. (2)

Electron. Lett. (4)

J. Jedwab, C. J. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
[CrossRef]

J. Jedwab, C. Mitchell, “Constructing new perfect binary arrays,” Electron. Lett. 24, 650–652 (1988).
[CrossRef]

P. Wild, “Infinite families of perfect binary arrays,” Electron. Lett. 24, 845–847 (1988).
[CrossRef]

P. Seidler, “Mismatched filtering for coded aperture imaging with minimum sidelobes,” Electron. Lett. 17, 96–97 (1981).
[CrossRef]

Frequenz (2)

H. D. Lüke, “Binäre Folgen und Arrays mit optimalen ungeraden Autokorrelationsfunktionen,” Frequenz 48, 213–220 (1994).
[CrossRef]

M. Antweiler, H. D. Lüke, “Mismatched-Filterung periodischer Binärarrays,” Frequenz 48, 58–65 (1994).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
[CrossRef]

IEEE Trans. Inf. Theory (2)

H. D. Lüke, “Binary odd-periodic complementary sequences,” IEEE Trans. Inf. Theory 43, 365–367 (1997).
[CrossRef]

M. Antweiler, L. Bömer, H. D. Lüke, “Perfect ternary arrays,” IEEE Trans. Inf. Theory 36, 696–705 (1990).
[CrossRef]

J. Appl. Phys. (1)

C. Brown, “Multiplex imaging with multiple-pinhole cameras,” J. Appl. Phys. 45, 1806–1811 (1974).
[CrossRef]

J. Comput. Phys. (1)

G. Dueck, “New optimization heuristics—the great deluge algorithm and the record-to-record travel,” J. Comput. Phys. 104, 86–92 (1993).
[CrossRef]

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

Nachrichtentech. Z (1)

P. Seidler, “Nebenmaximumreduktion bei Korrelationsempfang binär phasencodierter Impulssignale,” Nachrichtentech. Z 29, 154–159 (1976).

NTZ Archiv (1)

H. Weiss, “Abbildung mit kodierter Apertur,” NTZ Archiv 3, 329–335 (1981).

Nucl. Instr. Meth. A (1)

U. B. Jayanthi, J. Braga, “Physical implementation of an antimask in URA based coded mask systems,” Nucl. Instr. Meth. A 310, 685–689 (1991).
[CrossRef]

Radio Electron. Commun. Sys. (Radioelektron.) (1)

V. P. Ipatov, B. V. Fedorov, “Regular binary sequences with small losses in suppressing sidelobes,” Radio Electron. Commun. Sys. (Radioelektron.) 27, 29–33 (1987).

Radio Electron. Commun. Syst. (Radioelektron.) (1)

V. P. Ipatov, “Synthesis of a binary periodic signal-filter pair,” Radio Electron. Commun. Syst. (Radioelektron.) 23, 46–51 (1980).

Radio Eng. Electron. Phys. (1)

V. P. Ipatov, “Total suppression of sidelobes of periodic correlation functions of phase manipulated signals,” Radio Eng. Electron. Phys. 22, 42–47 (1977).

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (5)

H. D. Lüke, Korrelationssignale (Springer-Verlag, New York, 1992).
[CrossRef]

A. Busboom, H. Elders-Boll, H. D. Schotten, “Combinatorial design of near-optimum masks for coded aperture imaging,” in Proceedings of the 1997 IEEE International Conference on Acoustics, Speech, Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2817–2820.
[CrossRef]

H. D. Lüke, A. Busboom , “Mismatched filtering of odd-periodic binary sequences,” IEEE Trans. Aerosp. Electron. Syst. (to be published).

R. M. Rideout, “Coded imaging systems for x-ray astronomy,” Ph.D. dissertation (University of Birmingham, Birmingham, UK, 1995).

M. H. Finger, T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the Nineteenth International Cosmic Ray Conference (Scientific and Technical Information Branch, NASA, Washington, D.C., 1985), pp. 295–298.

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Figures (3)

Fig. 1
Fig. 1

Filter efficiencies of square (N x × N x ) and almost-square [N x × (N x + 1)] binary arrays with η ≥ 90%. Open circles represent perfect periodic and odd-periodic arrays. Open squares represent periodic mismatched arrays. Crosses represent odd-periodic mismatched arrays. Hatched squares represent periodic and odd-periodic mismatched arrays. The solid curve represents the lower bound [expression (A4)] for almost-perfect odd-periodic arrays.

Fig. 2
Fig. 2

Design of a coded mask and an antimask from a periodic ±1 array.

Fig. 3
Fig. 3

Design of a coded mask and antimask from an odd-periodic ±1 array.

Tables (1)

Tables Icon

Table 1 Square (Nx × Nx) and Almost-Square [Nx × (Nx + 1)] Binary Arrays with a Filter Efficiency of η ≥ 90%a

Equations (46)

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ϕ ˜ sw l ,   k = x = 0 N x - 1 y = 0 N y - 1   s x ,   y w ˜ x + 1 ,   y + k ,
w ˜ x ,   y = w x   mod   N x ,   y   mod   N y .
ϕ ˜ ss l ,   k = 0 ,   l 0 ,   1 , N x - 1 ,     k 0 ,   1 , N y - 1 ,   l ,   k 0 ,   0 .
W ˜ u ,   v = c S ˜ * u ,   v
SNR mismatched = ϕ ˜ sw 2 0 ,   0 N 0 ϕ ˜ ww 0 ,   0 .
SNR matched = ϕ ˜ ss 0 ,   0 N 0 .
η = SNR mismatched SNR matched = ϕ ˜ sw 2 0 ,   0 ϕ ˜ ss 0 ,   0 ϕ ˜ ww 0 ,   0 .
s x ,   y = s 0 n ,     w x ,   y = w 0 n ,
ϕ ˜ sw l ,   k = ϕ ˜ s 0 w 0 m ,
s 0 n = 1   1   1   1   1   - 1   - 1   1   1   - 1   1   - 1 , w 0 n = 2   1   1   2   1   - 1   - 2   1   1   - 2   1   - 1 ,
s x ,   y = | 1 1 1 - 1 1 - 1 - 1 1 1 1 1 - 1 | , w x ,   y = | 2 1 1 - 2 1 - 1 - 2 1 1 2 1 - 1 | ,
ϕ ˜ sw l ,   k = | 16 0 0 0 0 0 0 0 0 0 0 0 | ,
s ˜ x ,   y = s ˜ 1 x ,   y s ˜ 2 x ,   y
ϕ ˜ sw l ,   k = ϕ ˜ s 1 w 1 l ,   k ϕ ˜ s 2 w 2 l ,   k ,
η = ϕ ˜ s 1 w 1 2 0 ,   0 ϕ ˜ s 2 w 2 2 0 ,   0 ϕ ˜ s 1 s 1 0 ,   0 ϕ ˜ w 1 w 1 0 ,   0 ϕ ˜ s 2 s 2 0 ,   0 ϕ ˜ w 2 w 2 0 ,   0 = η 1 η 2 ,
s 2 x ,   y = w 2 x ,   y = | 1 1 1 - 1 | ,
s 1 n = 1   1   1   1   - 1 , w 1 n = 1   1   1   1   - 2 ,
s n = | 1 1 1 1 - 1 1 1 1 1 - 1 1 1 1 1 - 1 1 1 1 1 - 1 - 1 - 1 - 1 - 1 1 | , w n = | 1 1 1 1 - 2 1 1 1 1 - 2 1 1 1 1 - 2 1 1 1 1 - 2 - 2 - 2 - 2 - 2 4 | ,
s l n = n N x - 1 / 2 mod   N x .
ϕ ˜ s l s l m = N x - 1 - 1 if   m = 0 otherwise .
s l n = s l N x - n - s l N x - n if   N x 1   mod   4 if   N x 3   mod   4 , for   n 1 ,   2 , ,   N x - 1 .
s x ,   y = + 1 s l x s l y if   x = 0   or   y = 0 otherwise .
ϕ ˜ ss l ,   k = N x 2 1 5 if   l = k = 0 if   s l ,   k = + 1 if   s l ,   k = - 1 .
w x ,   y = 3 N x - 1 N x - 2 - N x + 1 if   x = y = 0 if   s x ,   y = + 1 if   s x ,   y = - 1 ,
ϕ ˜ sw l ,   k = 2 N x - 1 N x 2 - 1 2   δ l ,   k .
η = 2 N x - 1 2 N x - 1 N x + 1 2 2 N x 2 2 N x 3 + 9 N x - 7 .
w ˆ ( x ,   y ) = ( - 1 ) x / N x + y / N y s ( x   mod   N x ,   y   mod   N y ) ,
ϕ ˆ sw l ,   k = x = 0 N x - 1 y = 0 N y - 1   s x ,   y w ˆ x + l ,   y + k .
ϕ ˆ ss l ,   k = 0 , l 0 ,   1 , ,   N x - 1 ,   k { 0 ,   1 , ,   N y - 1 } ,   l ,   k 0 ,   0 .
ϕ ˆ sw l ,   k = - 1 l / N x + k / N y c 0 if   l     0   mod   N x ,       k     0   mod   N y otherwise .
S D u ,   v = x = 0 2 N x - 1 y = 0 2 N y - 1   s ˆ x ,   y exp - j 2 π xu 2 N x + yv 2 N y
Φ D u ,   v = S D * u ,   v W D u ,   v = 16 c 0 if   u   and   v   odd otherwise .
W D u ,   v = 16 c S D * u ,   v 0 if   u   and   v   odd otherwise .
s ( x ,   y ) = ( - 1 ) x + y   s 1 ( x ,   y ) , w ( x ,   y ) = ( - 1 ) x + y   w 1 ( x ,   y ) ,   N x and   N y odd .
ϕ ˆ sw l ,   k = - 1 l + k   ϕ ˜ s 1 w 1 l ,   k .
s ( x ,   y ) = ( - 1 ) n / N x + n / N y s 0 ( n ) , w ( x ,   y ) = ( - 1 ) n / N x + n / N y w 0 ( n ) , n { 0 ,   1 , ,   N x N y - 1 } , x n   mod   N x ,   y n   mod   N y , gcd ( N x ,   N y ) = 1 ,     N x + N y odd ,
ϕ ˆ sw ( l ,   k ) = ( - 1 ) m / N x + m / N y ϕ ˆ s 0 w 0 ( m ) ,
s x ,   y = | 1 - 1 - 1 - 1 1 1 | ,     w x ,   y = | 1 - 3 - 1 - 2 1 2 |
s ˆ x ,   y = s ˆ 1 x ,   y s ˆ 2 x ,   y
ϕ ˆ sw l ,   k = ϕ ˆ s 1 w 1 l ,   k ϕ ˆ s 2 w 2 l ,   k .
s + ( x , y ) = { 1 if   s ( x , y ) = + 1 0 otherwise , s - ( x , y ) = { 1 if   s ( x , y ) = - 1 0 otherwise .
s 0 n = x n q - 1 / 2 mod   q , n 0 ,   1 , ,   q , q = p a ,       p   odd   prime , a 1 ,   2 , .
x n = 0   2   3   3   1   3 , s 0 n = x 2 n mod   5 , = 0   - 1   - 1   - 1   1   - 1 .
s 0 n = 1     - 1     - 1     - 1     1     - 1 ,
w 0 n = 1     - 2     - 1     - 3     1     - 2 ,
η 1 - 4 N - 1 N 2 ,

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