Abstract

We consider the problem of reconstructing either a one-dimensional or a two-dimensional signal from its Fourier intensity and the Fourier intensity of another signal that is related to the first by the addition of a known reference signal. Several theorems are given that give conditions under which a unique reconstruction is possible, and a recursive algorithm is provided that allows for the reconstruction of the signal from the pair of Fourier intensities.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
    [CrossRef]
  2. M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
    [CrossRef]
  3. M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,” J. Opt. Soc. Am. 73, 1427–1433 (1983).
    [CrossRef]
  4. P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from signed Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
    [CrossRef]
  5. R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron microscopy,” Optik 34, 275–284 (1971).
  6. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  7. D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. test calculations,” J. Phys. D 6, 2200–2216 (1973).
    [CrossRef]
  8. D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. sources of error,” J. Phys. D 6, 2217–2225 (1973).
    [CrossRef]
  9. R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in 1980 International Optical Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980).
    [CrossRef]
  10. R. A. Gonsalves, “Phase retrieval by differential intensity measurements,” J. Opt. Soc. Am. A 4, 166–170 (1987).
    [CrossRef]
  11. N. Nakajima, “Phase retrieval from two intensity measurements using the Fourier series expansion,” J. Opt. Soc. Am. A 4, 154–158 (1987).
    [CrossRef]
  12. N. Nakajima, “Phase retrieval using the logarithmic Hilbert transform and the Fourier-series expansion,” J. Opt. Soc. Am. A 5, 257–262 (1988).
    [CrossRef]
  13. S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
    [CrossRef]
  14. G. N. Ramachandran, R. Srinivasan, Fourier Methods in Crystallography (Wiley-Interscience, New York, 1970).
  15. J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
    [CrossRef]
  16. J. E. Marsden, Basic Complex Analysis (Freeman, San Francisco, Calif., 1973).
  17. L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
    [CrossRef]
  18. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  19. We say that a sequence x(n) is nonsymmetric if it does not have even or odd symmetry, i.e., if x(n) ≠ x(n0− n) and x(n) ≠ − x(n0− n) for all integer values of n0. This is equivalent to the constraint that x(n) not be a linear phase sequence.
  20. A. H. Greenaway, “Proposal for phase recovery from a single intensity distribution,” Opt. Lett. 1, 10–12 (1977).
    [CrossRef] [PubMed]
  21. C. L. Mehta, “New approach to the phase problem in optical coherence theory,” J. Opt. Soc. Am. 58, 1233–1234 (1968).
    [CrossRef]
  22. J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. 73, 1421–1426 (1983).
    [CrossRef]
  23. M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
    [CrossRef] [PubMed]

1988 (1)

1987 (2)

1983 (5)

J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. 73, 1421–1426 (1983).
[CrossRef]

M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,” J. Opt. Soc. Am. 73, 1427–1433 (1983).
[CrossRef]

M. A. Fiddy, B. J. Brames, J. C. Dainty, “Enforcing irreducibility for phase retrieval in two dimensions,” Opt. Lett. 8, 96–98 (1983).
[CrossRef] [PubMed]

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from signed Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

1982 (2)

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

1981 (1)

L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

1979 (1)

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

1977 (1)

1973 (2)

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. test calculations,” J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. sources of error,” J. Phys. D 6, 2217–2225 (1973).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1971 (1)

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron microscopy,” Optik 34, 275–284 (1971).

1968 (1)

Boucher, R. H.

R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in 1980 International Optical Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980).
[CrossRef]

Brames, B. J.

Dainty, J. C.

Fiddy, M. A.

Fienup, J. R.

J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. 73, 1421–1426 (1983).
[CrossRef]

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron microscopy,” Optik 34, 275–284 (1971).

Gonsalves, R. A.

Greenaway, A. H.

Hayes, M. H.

M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,” J. Opt. Soc. Am. 73, 1427–1433 (1983).
[CrossRef]

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from signed Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

Lim, J. S.

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from signed Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

Marsden, J. E.

J. E. Marsden, Basic Complex Analysis (Freeman, San Francisco, Calif., 1973).

McClellan, J. H.

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Mehta, C. L.

Misell, D. L.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. test calculations,” J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. sources of error,” J. Phys. D 6, 2217–2225 (1973).
[CrossRef]

Nakajima, N.

Nawab, S. H.

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

Oppenheim, A. V.

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from signed Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Quatieri, T. F.

M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,” J. Opt. Soc. Am. 73, 1427–1433 (1983).
[CrossRef]

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

Ramachandran, G. N.

G. N. Ramachandran, R. Srinivasan, Fourier Methods in Crystallography (Wiley-Interscience, New York, 1970).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron microscopy,” Optik 34, 275–284 (1971).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Srinivasan, R.

G. N. Ramachandran, R. Srinivasan, Fourier Methods in Crystallography (Wiley-Interscience, New York, 1970).

Taylor, L. S.

L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

Van Hove, P. L.

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from signed Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (3)

P. L. Van Hove, M. H. Hayes, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from signed Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1286–1293 (1983).
[CrossRef]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

S. H. Nawab, T. F. Quatieri, J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 986–998 (1983).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

L. S. Taylor, “The phase retrieval problem,” IEEE Trans. Antennas Propag. AP-29, 386–391 (1981).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

J. Phys. D (2)

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. test calculations,” J. Phys. D 6, 2200–2216 (1973).
[CrossRef]

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. sources of error,” J. Phys. D 6, 2217–2225 (1973).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, “Space object imaging through the turbulent atmosphere,” Opt. Eng. 18, 529–534 (1979).
[CrossRef]

Opt. Lett. (2)

Optik (2)

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron microscopy,” Optik 34, 275–284 (1971).

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. IEEE (1)

M. H. Hayes, J. H. McClellan, “Reducible polynomials in more than one variable,” Proc. IEEE 70, 197–198 (1982).
[CrossRef]

Other (5)

G. N. Ramachandran, R. Srinivasan, Fourier Methods in Crystallography (Wiley-Interscience, New York, 1970).

R. H. Boucher, “Convergence of algorithms for phase retrieval from two intensity distributions,” in 1980 International Optical Computing Conference (Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 130–141 (1980).
[CrossRef]

J. E. Marsden, Basic Complex Analysis (Freeman, San Francisco, Calif., 1973).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

We say that a sequence x(n) is nonsymmetric if it does not have even or odd symmetry, i.e., if x(n) ≠ x(n0− n) and x(n) ≠ − x(n0− n) for all integer values of n0. This is equivalent to the constraint that x(n) not be a linear phase sequence.

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 (2)

Fig. 1
Fig. 1

Region of support of an image (cross-hatched area), the region of the 2-D dimensional plane corresponding to the off-axis holography condition (Region I), and the region for boundary-value determination (Region II).

Fig. 2
Fig. 2

Initial values of x (m, n) computed in the 2-D reconstruction algorithm (hatched region) for a point-source reference located at (m0, n0).

Equations (87)

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

r x ( k ) = n = x ( n ) x ( n + k ) .
r x ( k ) = 1 2 N m = 0 2 N 1 | X ( m ) | 2 exp ( j π N k m ) .
y ( n ) = x ( n ) + h ( n ) ,
y ( n ) = x ( n ) + A δ ( n n 0 ) .
| Y ( e j ω ) | 2 = | X ( e j ω ) | 2 + A 2 + 2 A | X ( e j ω ) | cos [ ϕ ( ω ) + n 0 ω ] ,
y ( n ) = x ( n ) + A δ ( n n 0 ) ,
| Y ( e j ω ) | 2 = | X ( e j ω ) | 2 + A 2 + 2 A | X ( e j ω ) | cos [ ϕ ( ω ) + n 0 ω ] ,
| ( e j ω ) | 2 = | X ( e j ω ) | 2 + A 2 + 2 A | X ( e j ω ) | cos [ ϕ ( ω ) + n 0 ω ] ,
cos [ ϕ ( ω ) + n 0 ω ] = cos [ ϕ ( ω ) + n 0 ω ] .
ϕ ( ω ) + n 0 ω = ± [ ϕ ( ω ) + n 0 ω ] + 2 π k ,
ϕ ( ω ) = ϕ ( ω ) ,
ϕ ( ω ) = ϕ ( ω ) 2 n 0 ω .
r y ( k ) = r x ( k ) + A 2 δ ( k ) + A x ( k + n 0 ) + A x ( n 0 k ) ,
H lp ( e j ω ) = A ( e j ω ) exp ( j n 0 ω ) ,
| Y ( e j ω ) | 2 = | X ( e j ω ) | 2 + A ( e j ω ) 2 + 2 A ( e j ω ) | X ( e j ω ) | cos [ ϕ ( ω ) + n 0 ω ] .
H lp ( e j ω ) = A ( e j ω ) exp [ j ( n 0 ω π / 2 ) ] ,
y ( n ) = x ( n ) + A δ ( n n 0 ) ,
h ( n ) = a ( n ) * h lp ( n ) ,
x ( n ) = a ( n ) * u ( n ) ,
y ( n ) = x ( n ) + h ( n ) = a ( n ) * [ u ( n ) + h lp ( n ) ]
( n ) = x ( n ) + h ( n ) = a ( n ) * [ u ( 2 n 0 n ) + h lp ( n ) ] ,
y ( n ) = x ( n ) + h ( n ) .
H ( z ) = A ( z ) H lp ( z ) ,
| X 1 ( e j ω ) | = | X 2 ( e j ω ) |
| X 1 ( e j ω ) + H ( e j ω ) | = | X 2 ( e j ω ) + H ( e j ω ) | ,
X 1 ( z ) X 1 ( z 1 ) = X 2 ( z ) X 2 ( z 1 ) ,
{ X 1 ( z ) H ( z ) } { X 1 ( z 1 ) + H ( z 1 ) } = { X 2 ( z ) + H ( z ) } { X 2 ( z 1 ) + H ( z 1 ) } .
{ X 1 ( z ) X 2 ( z ) } H ( z 1 ) = { X 2 ( z 1 ) X 1 ( z 1 ) } H ( z ) .
X 1 ( z ) = k ( z ) X 2 ( z ) ,
X 2 ( z 1 ) = k ( z ) X 1 ( z 1 ) .
X 2 ( z ) [ k ( z ) 1 ] H ( z 1 ) = X 1 ( z 1 ) [ k ( z ) 1 ] H ( z ) .
X 2 ( z ) = X 1 ( z 1 ) H ( z ) H ( z 1 ) .
X 2 ( z ) = ± z 2 n 0 X 1 ( Z 1 ) A ( z ) A ( z 1 ) .
X ( z ) = ( 1 + 2 z 1 ) ( 1 + 3 z 1 ) ( 1 + 4 z 1 ) ,
H ( z ) = z 1 ( 1 + 2 z 1 ) .
X ( z ) = ( 1 + 2 z 1 ) ( 3 + z 1 ) ( 4 + z 1 ) .
X ( z ) + H ( z ) = ( 1 + 2 z 1 ) [ ( 1 + 3 z 1 ) ( 1 + 4 z 1 ) + z 1 ] = ( 1 + 2 z 1 ) ( 1 + 8 z 1 + 12 z 2 )
X ( z ) + H ( z ) = ( 1 + 2 z 1 ) [ ( 3 + z 1 ) ( 4 + z 1 ) + z 1 ] = ( 1 + 2 z 1 ) ( 12 + 8 z 1 + z 2 ) .
r x ( k , l ) = m = n = x ( m , n ) x ( m + k , n + l ) .
r x ( k , l ) = 1 4 M N m = 0 2 M 1 n = 0 2 N 1 | X ( m , n ) | 2 exp ( j π M k m + j π N ln ) .
X ( z 1 , z 2 ) = k = 1 N X k ( z 1 , z 2 ) ,
X ( z 1 , z 2 ) k = 1 N X k ( z 1 , z 2 ) .
y ( m , n ) = x ( m , n ) + h ( m , n ) ,
y ( m , n ) = x ( m , n ) + A δ ( m m 0 , n n 0 ) ,
| Y [ exp ( j ω 1 ) , exp ( j ω 2 ) ] | 2 = | X [ exp ( j ω 1 , exp ( j ω 2 ) | 2 + A 2 + 2 A | X [ exp ( j ω 1 ) , exp ( j ω 2 ) ] | × cos [ ϕ ( ω 1 , ω 2 ) + m 0 ω 1 + n 0 ω 2 ] ,
y ( m , n ) = x ( m , n ) + A δ ( m m 0 , n n 0 ) ,
H ( z 1 , z 2 ) = ± z 1 2 m 0 z 2 2 n 0 H ( z 1 1 , z 2 1 )
y ( m , n ) = x ( m , n ) + A δ ( m m 0 , n n 0 ) ,
y ( m , n ) = x ( m , n ) + h ( m , n ) .
H ( z 1 , z 2 ) = A ( z 1 , z 2 ) H lp ( z 1 , z 2 ) ,
X ( z 1 , z 2 ) = H ( z 1 , z 2 ) U ( z 1 , z 2 ) ,
r y ( k ) = r x ( k ) + r h ( k ) + h ( k ) * x ( k ) + h ( k ) * x ( k ) .
h ( n ) = A δ ( n n 0 ) ,
r yxh ( k ) = A x ( n 0 + k ) + A x ( n 0 k ) ,
r yxh ( k ) = r y ( k ) r x ( k ) r h ( k )
r h ( k ) = A 2 δ ( n )
x ( n 0 + k ) = 1 A r yxh ( k ) , k > n 0
x ( n ) = 1 A r yxh ( n n 0 ) , n > 2 n 0 .
A x ( 2 n 0 n ) + A x ( n ) = r yxh ( n 0 n ) ,
x ( n ) = 1 x ( N 1 ) [ r x ( N 1 n ) i = 0 n 1 x ( i ) x ( N 1 n + i ) ] .
h ( n ) = h ( N 1 n ) .
h ( n ) = i = 1 M A i δ ( n n i ) ,
r yxh ( k ) = i = 1 M A i x ( n i + k ) + i = 1 M A i x ( n i k ) ,
r h ( k ) = i = 1 M j = 1 M A i A j δ ( k n i + n j )
n 1 + n M < N 1 ,
r yxh ( k ) = i = 1 M A i x ( n i + k ) , k > n M .
x ( n 1 + k ) = 1 A 1 [ r yxh ( k ) i = 2 M A i x ( n i + k ) ] , k > n M
x ( n ) = 1 A 1 [ r yxh ( n n 1 ) i = 2 M A i x ( n + n i n 1 ) ] , n > n 1 + n M .
A 1 x ( n 1 + n M n ) + A M x ( n ) = r yxh ( n M n ) i = 2 M A i x ( n M + n i n ) i = 1 M 1 A i x ( n M + n i + n ) ,
x ( n ) = 1 x ( N 1 ) [ r x ( N 1 n ) i = 0 n 1 x ( i ) x ( N 1 n + i ) ] .
x = [ 3 , 2 , 1 , 1 , 4 ] ,
y = [ 3 , 0 , 1 , 1 , 4 ] .
r x = [ 31 , 1 , 9 , 5 , 12 ] , r y = [ 27 , 3 , 7 , 3 , 21 ] .
r yxh = [ 8 , 4 , 2 , 8 , 0 ] ,
x ( n ) = 1 2 r yxh ( n 1 ) , n = 3 , 4 .
x ( 3 ) = 1 , x ( 4 ) = 4 .
2 x ( 2 ) + 2 x ( 0 ) = r yxh ( 1 )
r x ( 3 ) = x ( 0 ) x ( 3 ) + x ( 1 ) x ( 2 ) ,
h ( m , n ) = A δ ( m m 0 , n n 0 ) .
r y ( k , l ) = r x ( k , l ) + h ( k , l ) * x ( k , l ) + h ( k , l ) * x ( k , l ) + r h ( k , l ) .
r yxh ( k , l ) = A x ( m 0 + k , n 0 + l ) + A x ( m 0 k , n 0 l ) ,
r yxh ( k , l ) = r y ( k , l ) r x ( k , l ) r h ( k , l )
r h ( k , l ) = A 2 δ ( k , l )
x ( m 0 + k , n 0 + l ) = 1 A r yxh ( k , l ) , k > m 0 or l > n 0
x ( m , n ) = 1 A r yxh ( m m 0 , n n 0 ) , m > 2 m 0 or n > 2 n 0 .
A x ( 2 m 0 m , 2 n 0 n ) + A x ( m , n ) = r yxh ( m 0 m , n 0 n ) ,
x ( m , n ) = 1 x ( M 1 , N 1 ) [ r x ( M 1 m , N 1 n ) i = 0 ( i , j ) m j = 0 ( 0 , 0 ) n x ( i , j ) x ( M 1 m + i , N 1 n + j ) ] .

Metrics