Abstract

In reconstructing an object function F(r) from finitely many noisy linear-functional values F(r)Gn(r)dr we face the problem that finite data, noisy or not, are insufficient to specify F(r) uniquely. Estimates based on the finite data may succeed in recovering broad features of F(r), but may fail to resolve important detail. Linear and nonlinear, model-based data extrapolation procedures can be used to improve resolution, but at the cost of sensitivity to noise. To estimate linear-functional values of F(r) that have not been measured from those that have been, we need to employ prior information about the object F(r), such as support information or, more generally, estimates of the overall profile of F(r). One way to do this is through minimum-weighted-norm (MWN) estimation, with the prior information used to determine the weights. The MWN approach extends the Gerchberg–Papoulis band-limited extrapolation method and is closely related to matched-filter linear detection, the approximation of the Wiener filter, and to iterative Shannon-entropy-maximization algorithms. Nonlinear versions of the MWN method extend the noniterative, Burg, maximum-entropy spectral-estimation procedure.

© 2006 Optical Society of America

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  1. R. W. Gerchberg, "Super-restoration through error energy reduction," Opt. Acta 21, 709-720 (1974).
    [Crossref]
  2. A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
    [Crossref]
  3. J. Burg, "Maximum entropy spectral analysis," presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City, Oklahoma, July 1967.
  4. J. Burg, "The relationship between maximum entropy spectra and maximum likelihood spectra," Geophysics 37, 375-376 (1972).
    [Crossref]
  5. J. Burg, "Maximum Entropy Spectral Analysis," Ph.D. thesis (Stanford University, Stanford, California,1975).
  6. M. Bertero, "Sampling theory, resolution limits and inversion methods," in Inverse Problems in Scattering and Imaging, M.Bertero and E.R.Pike, eds. (Malvern Physics Series, Adam Hilger, IOP Publishing, 1992), pp. 71-94.
  7. C. L. Byrne and R. M. Fitzgerald, "A unifying model for spectrum estimation," presented at the Rome Air Development Center Workshop on Spectrum Estimation, Griffiss Air Force Base, Rome, New York, October 3-5, 1979.
  8. H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets and Optics (Wiley, 1998).
    [PubMed]
  9. H. Cox, "Resolving power and sensitivity to mismatch of optimum array processors," J. Acoust. Soc. Am. 54, 771-785 (1973).
    [Crossref]
  10. C. L. Byrne and R. M. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933-940 (1982).
    [Crossref]
  11. C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, and A. M. Darling, "Image restoration and resolution enhancement," J. Opt. Soc. Am. 73, 1481-1487 (1983).
    [Crossref]
  12. T. Poggio and S. Smale, "The mathematics of learning: dealing with data," Not. Am. Math. Soc. 50, 537-544 (2003).
  13. R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471-481 (1970).
    [Crossref] [PubMed]
  14. C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. IP-2, 96-103 (1993).
    [Crossref]
  15. C. L. Byrne, "Erratum and addendum to 'Iterative image reconstruction algorithms based on cross-entropy minimization'," IEEE Trans. Image Process. IP-4, 225-226 (1995).
  16. C. L. Byrne and M. A. Fiddy, "Estimation of continuous object distributions from Fourier magnitude measurements," J. Opt. Soc. Am. A 4, 412-417 (1987).
    [Crossref]
  17. M. A. Fiddy, "The phase retrieval problem," in Inverse Optics, A. Devaney, ed., Proc. SPIE 413, 176-181 (1983).
  18. J. C. Dainty and M. A. Fiddy, "The essential role of prior knowledge in phase retrieval," Opt. Acta 31, 325-330 (1984).
    [Crossref]
  19. J. Fienup, "Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint," J. Opt. Soc. Am. A 4, 118-123 (1987).
    [Crossref]
  20. R. Lane, "Recovery of complex images from Fourier magnitude," Opt. Commun. 63, 6-10 (1987).
    [Crossref]
  21. J. Cederquist, J. Fienup, C. Wackerman, S. Robinson, and D. Kryskowski, "Wave-front phase estimation from Fourier intensity measurements," J. Opt. Soc. Am. A 6, 1020-1026 (1989).
    [Crossref]
  22. J. Fienup, "Space object imaging through the turbulent atmosphere," Opt. Eng. (Bellingham) 18, 529-534 (1979).
  23. C.-W. Liao, M. A. Fiddy, and C. L. Byrne, "Imaging from the zero locations of far-field intensity data," J. Opt. Soc. Am. A 14, 3155-3161 (1997).
    [Crossref]
  24. H. M. Shieh, M. E. Testorf, C. L. Byrne, and M. A. Fiddy, "Iterative image reconstruction using prior knowledge," submitted to J. Opt. Soc. Am. A.
  25. C. L. Byrne, Signal Processing: A Mathematical Approach (AK Peters, 2005).
  26. C. L. Byrne and M. A. Fiddy, "Images as power spectra; reconstruction as Wiener filter approximation," Inverse Probl. 4, 399-409 (1988).
    [Crossref]
  27. C. L. Byrne and R. M. Fitzgerald, "Spectral estimators that extend the maximum entropy and maximum likelihood methods," SIAM J. Appl. Math. 44, 425-442 (1984).
    [Crossref]

2003 (1)

T. Poggio and S. Smale, "The mathematics of learning: dealing with data," Not. Am. Math. Soc. 50, 537-544 (2003).

1997 (1)

1995 (1)

C. L. Byrne, "Erratum and addendum to 'Iterative image reconstruction algorithms based on cross-entropy minimization'," IEEE Trans. Image Process. IP-4, 225-226 (1995).

1993 (1)

C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. IP-2, 96-103 (1993).
[Crossref]

1989 (1)

1988 (1)

C. L. Byrne and M. A. Fiddy, "Images as power spectra; reconstruction as Wiener filter approximation," Inverse Probl. 4, 399-409 (1988).
[Crossref]

1987 (3)

C. L. Byrne and M. A. Fiddy, "Estimation of continuous object distributions from Fourier magnitude measurements," J. Opt. Soc. Am. A 4, 412-417 (1987).
[Crossref]

J. Fienup, "Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint," J. Opt. Soc. Am. A 4, 118-123 (1987).
[Crossref]

R. Lane, "Recovery of complex images from Fourier magnitude," Opt. Commun. 63, 6-10 (1987).
[Crossref]

1984 (2)

J. C. Dainty and M. A. Fiddy, "The essential role of prior knowledge in phase retrieval," Opt. Acta 31, 325-330 (1984).
[Crossref]

C. L. Byrne and R. M. Fitzgerald, "Spectral estimators that extend the maximum entropy and maximum likelihood methods," SIAM J. Appl. Math. 44, 425-442 (1984).
[Crossref]

1983 (1)

1982 (1)

C. L. Byrne and R. M. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933-940 (1982).
[Crossref]

1979 (1)

J. Fienup, "Space object imaging through the turbulent atmosphere," Opt. Eng. (Bellingham) 18, 529-534 (1979).

1975 (1)

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[Crossref]

1974 (1)

R. W. Gerchberg, "Super-restoration through error energy reduction," Opt. Acta 21, 709-720 (1974).
[Crossref]

1973 (1)

H. Cox, "Resolving power and sensitivity to mismatch of optimum array processors," J. Acoust. Soc. Am. 54, 771-785 (1973).
[Crossref]

1972 (1)

J. Burg, "The relationship between maximum entropy spectra and maximum likelihood spectra," Geophysics 37, 375-376 (1972).
[Crossref]

1970 (1)

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[Crossref] [PubMed]

Bender, R.

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[Crossref] [PubMed]

Bertero, M.

M. Bertero, "Sampling theory, resolution limits and inversion methods," in Inverse Problems in Scattering and Imaging, M.Bertero and E.R.Pike, eds. (Malvern Physics Series, Adam Hilger, IOP Publishing, 1992), pp. 71-94.

Burg, J.

J. Burg, "The relationship between maximum entropy spectra and maximum likelihood spectra," Geophysics 37, 375-376 (1972).
[Crossref]

J. Burg, "Maximum Entropy Spectral Analysis," Ph.D. thesis (Stanford University, Stanford, California,1975).

J. Burg, "Maximum entropy spectral analysis," presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City, Oklahoma, July 1967.

Byrne, C. L.

C.-W. Liao, M. A. Fiddy, and C. L. Byrne, "Imaging from the zero locations of far-field intensity data," J. Opt. Soc. Am. A 14, 3155-3161 (1997).
[Crossref]

C. L. Byrne, "Erratum and addendum to 'Iterative image reconstruction algorithms based on cross-entropy minimization'," IEEE Trans. Image Process. IP-4, 225-226 (1995).

C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. IP-2, 96-103 (1993).
[Crossref]

C. L. Byrne and M. A. Fiddy, "Images as power spectra; reconstruction as Wiener filter approximation," Inverse Probl. 4, 399-409 (1988).
[Crossref]

C. L. Byrne and M. A. Fiddy, "Estimation of continuous object distributions from Fourier magnitude measurements," J. Opt. Soc. Am. A 4, 412-417 (1987).
[Crossref]

C. L. Byrne and R. M. Fitzgerald, "Spectral estimators that extend the maximum entropy and maximum likelihood methods," SIAM J. Appl. Math. 44, 425-442 (1984).
[Crossref]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, and A. M. Darling, "Image restoration and resolution enhancement," J. Opt. Soc. Am. 73, 1481-1487 (1983).
[Crossref]

C. L. Byrne and R. M. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933-940 (1982).
[Crossref]

C. L. Byrne and R. M. Fitzgerald, "A unifying model for spectrum estimation," presented at the Rome Air Development Center Workshop on Spectrum Estimation, Griffiss Air Force Base, Rome, New York, October 3-5, 1979.

C. L. Byrne, Signal Processing: A Mathematical Approach (AK Peters, 2005).

H. M. Shieh, M. E. Testorf, C. L. Byrne, and M. A. Fiddy, "Iterative image reconstruction using prior knowledge," submitted to J. Opt. Soc. Am. A.

Cederquist, J.

Cox, H.

H. Cox, "Resolving power and sensitivity to mismatch of optimum array processors," J. Acoust. Soc. Am. 54, 771-785 (1973).
[Crossref]

Dainty, J. C.

J. C. Dainty and M. A. Fiddy, "The essential role of prior knowledge in phase retrieval," Opt. Acta 31, 325-330 (1984).
[Crossref]

Darling, A. M.

Fiddy, M. A.

C.-W. Liao, M. A. Fiddy, and C. L. Byrne, "Imaging from the zero locations of far-field intensity data," J. Opt. Soc. Am. A 14, 3155-3161 (1997).
[Crossref]

C. L. Byrne and M. A. Fiddy, "Images as power spectra; reconstruction as Wiener filter approximation," Inverse Probl. 4, 399-409 (1988).
[Crossref]

C. L. Byrne and M. A. Fiddy, "Estimation of continuous object distributions from Fourier magnitude measurements," J. Opt. Soc. Am. A 4, 412-417 (1987).
[Crossref]

J. C. Dainty and M. A. Fiddy, "The essential role of prior knowledge in phase retrieval," Opt. Acta 31, 325-330 (1984).
[Crossref]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, and A. M. Darling, "Image restoration and resolution enhancement," J. Opt. Soc. Am. 73, 1481-1487 (1983).
[Crossref]

M. A. Fiddy, "The phase retrieval problem," in Inverse Optics, A. Devaney, ed., Proc. SPIE 413, 176-181 (1983).

H. M. Shieh, M. E. Testorf, C. L. Byrne, and M. A. Fiddy, "Iterative image reconstruction using prior knowledge," submitted to J. Opt. Soc. Am. A.

Fienup, J.

Fitzgerald, R. M.

C. L. Byrne and R. M. Fitzgerald, "Spectral estimators that extend the maximum entropy and maximum likelihood methods," SIAM J. Appl. Math. 44, 425-442 (1984).
[Crossref]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, and A. M. Darling, "Image restoration and resolution enhancement," J. Opt. Soc. Am. 73, 1481-1487 (1983).
[Crossref]

C. L. Byrne and R. M. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933-940 (1982).
[Crossref]

C. L. Byrne and R. M. Fitzgerald, "A unifying model for spectrum estimation," presented at the Rome Air Development Center Workshop on Spectrum Estimation, Griffiss Air Force Base, Rome, New York, October 3-5, 1979.

Gerchberg, R. W.

R. W. Gerchberg, "Super-restoration through error energy reduction," Opt. Acta 21, 709-720 (1974).
[Crossref]

Gordon, R.

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[Crossref] [PubMed]

Hall, T. J.

Herman, G. T.

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[Crossref] [PubMed]

Kryskowski, D.

Lane, R.

R. Lane, "Recovery of complex images from Fourier magnitude," Opt. Commun. 63, 6-10 (1987).
[Crossref]

Liao, C.-W.

Papoulis, A.

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[Crossref]

Poggio, T.

T. Poggio and S. Smale, "The mathematics of learning: dealing with data," Not. Am. Math. Soc. 50, 537-544 (2003).

Robinson, S.

Shieh, H. M.

H. M. Shieh, M. E. Testorf, C. L. Byrne, and M. A. Fiddy, "Iterative image reconstruction using prior knowledge," submitted to J. Opt. Soc. Am. A.

Smale, S.

T. Poggio and S. Smale, "The mathematics of learning: dealing with data," Not. Am. Math. Soc. 50, 537-544 (2003).

Stark, H.

H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets and Optics (Wiley, 1998).
[PubMed]

Testorf, M. E.

H. M. Shieh, M. E. Testorf, C. L. Byrne, and M. A. Fiddy, "Iterative image reconstruction using prior knowledge," submitted to J. Opt. Soc. Am. A.

Wackerman, C.

Yang, Y.

H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets and Optics (Wiley, 1998).
[PubMed]

Geophysics (1)

J. Burg, "The relationship between maximum entropy spectra and maximum likelihood spectra," Geophysics 37, 375-376 (1972).
[Crossref]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[Crossref]

IEEE Trans. Image Process. (2)

C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. IP-2, 96-103 (1993).
[Crossref]

C. L. Byrne, "Erratum and addendum to 'Iterative image reconstruction algorithms based on cross-entropy minimization'," IEEE Trans. Image Process. IP-4, 225-226 (1995).

Inverse Probl. (1)

C. L. Byrne and M. A. Fiddy, "Images as power spectra; reconstruction as Wiener filter approximation," Inverse Probl. 4, 399-409 (1988).
[Crossref]

J. Acoust. Soc. Am. (1)

H. Cox, "Resolving power and sensitivity to mismatch of optimum array processors," J. Acoust. Soc. Am. 54, 771-785 (1973).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Theor. Biol. (1)

R. Gordon, R. Bender, and G. T. Herman, "Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography," J. Theor. Biol. 29, 471-481 (1970).
[Crossref] [PubMed]

Not. Am. Math. Soc. (1)

T. Poggio and S. Smale, "The mathematics of learning: dealing with data," Not. Am. Math. Soc. 50, 537-544 (2003).

Opt. Acta (2)

R. W. Gerchberg, "Super-restoration through error energy reduction," Opt. Acta 21, 709-720 (1974).
[Crossref]

J. C. Dainty and M. A. Fiddy, "The essential role of prior knowledge in phase retrieval," Opt. Acta 31, 325-330 (1984).
[Crossref]

Opt. Commun. (1)

R. Lane, "Recovery of complex images from Fourier magnitude," Opt. Commun. 63, 6-10 (1987).
[Crossref]

Opt. Eng. (Bellingham) (1)

J. Fienup, "Space object imaging through the turbulent atmosphere," Opt. Eng. (Bellingham) 18, 529-534 (1979).

SIAM J. Appl. Math. (2)

C. L. Byrne and R. M. Fitzgerald, "Spectral estimators that extend the maximum entropy and maximum likelihood methods," SIAM J. Appl. Math. 44, 425-442 (1984).
[Crossref]

C. L. Byrne and R. M. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933-940 (1982).
[Crossref]

Other (8)

J. Burg, "Maximum entropy spectral analysis," presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City, Oklahoma, July 1967.

J. Burg, "Maximum Entropy Spectral Analysis," Ph.D. thesis (Stanford University, Stanford, California,1975).

M. Bertero, "Sampling theory, resolution limits and inversion methods," in Inverse Problems in Scattering and Imaging, M.Bertero and E.R.Pike, eds. (Malvern Physics Series, Adam Hilger, IOP Publishing, 1992), pp. 71-94.

C. L. Byrne and R. M. Fitzgerald, "A unifying model for spectrum estimation," presented at the Rome Air Development Center Workshop on Spectrum Estimation, Griffiss Air Force Base, Rome, New York, October 3-5, 1979.

H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets and Optics (Wiley, 1998).
[PubMed]

M. A. Fiddy, "The phase retrieval problem," in Inverse Optics, A. Devaney, ed., Proc. SPIE 413, 176-181 (1983).

H. M. Shieh, M. E. Testorf, C. L. Byrne, and M. A. Fiddy, "Iterative image reconstruction using prior knowledge," submitted to J. Opt. Soc. Am. A.

C. L. Byrne, Signal Processing: A Mathematical Approach (AK Peters, 2005).

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Figures (9)

Fig. 1
Fig. 1

DFT estimation from noiseless data.

Fig. 2
Fig. 2

MDFT estimate with Ω = 1.8 from noiseless data (a) in spatial domain, (b) in spectrum domain.

Fig. 3
Fig. 3

MDFT estimate with Ω = 0.9 from noiseless data (a) in spatial domain, (b) in spectrum domain.

Fig. 4
Fig. 4

MDFT estimate with Ω = 0.9 from noiseless data (the true support is between 7 π 8 and 3 π 8 (a) in spatial domain, (b) in spectrum domain.

Fig. 5
Fig. 5

Eigenvalues of the matrix B (a) corresponding to Ω = 1.8 , (b) corresponding to Ω = 0.9 .

Fig. 6
Fig. 6

Example showing the absolute values of U n ( r ) for n = 1 , 2,…,15, where their corresponding eigenvalues are (a) 1.00, (b) 9.99 × 10 1 , (c) 9.82 × 10 1 , (d) 8.29 × 10 1 , (e) 4.02 × 10 1 , (f) 7.82 × 10 2 , (g) 6.83 × 10 3 , (h) 3.52 × 10 4 , (i) 1.21 × 10 5 , (j) 2.91 × 10 7 , (k) 4.85 × 10 9 , (l) 5.53 × 10 11 , (m) 4.12 × 10 13 , (n) 1.72 × 10 15 , (o) 4.28 × 10 18 . Two vertical dashed lines indicate the boundaries of the prior support.

Fig. 7
Fig. 7

PDFT estimate with Ω = 0.7 (smaller than the true support) from noiseless data (a) in spatial domain, (b) in spectrum domain.

Fig. 8
Fig. 8

PDFT estimate with Ω = 0.9 from noisy data ( ϵ = 0.001 ) (a) in spatial domain, (b) in spectrum domain.

Fig. 9
Fig. 9

PDFT estimate with Ω = 0.9 from noisy data ( ϵ = 0.1 ) (a) in spatial domain, (b) in spectrum domain.

Equations (42)

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

d n = F ( r ) G n ( r ) ¯ d r ,
H ( r ) = F MN ( r ) = n = 1 N a n G n ( r ) ,
d m = n = 1 N a n G n ( r ) G m ( r ) ¯ d r ,
d m = 1 2 π F ( r ) exp ( i x m r ) d r = f ( x m ) ,
f ( x ) = 1 2 π F ( r ) exp ( i x r ) d r ,
F MN ( r ) = n = 1 N a n exp ( i x n r ) ,
f ( x m ) = n = 1 N a n exp [ i ( x n x m ) r ] d r 2 π ,
F DFT ( r ) = n = 1 N f ( n ) exp ( i n r ) ,
F ( r ) = m = f ( m ) exp ( i m r ) .
H ( r ) = χ Ω ( r ) n = M M a n exp ( i n r ) ,
f ( m ) = n = M M a n sin [ Ω ( m n ) ] π ( m n ) ,
f MDFT ( x ) = n = M M a n sin [ Ω ( x n ) ] π ( x n ) ,
Ω Ω F ( r ) n = M M a n exp ( i n r ) 2 d r .
F PDFT ( r ) = P ( r ) n = M M b n exp ( i n r ) ,
f ( m ) = n = M M b n p ( m n ) ,
p ( x ) = 1 2 π π π P ( r ) exp ( i x r ) d r .
f PDFT ( x ) = n = M M b n p ( x n ) .
π π F ( r ) P ( r ) n = M M b n exp ( i n r ) 2 P ( r ) 1 d r ,
H ( r ) log H ( r ) P ( r ) + P ( r ) H ( r ) d r
F MCE ( r ) = P ( r ) exp [ n = M M c n exp ( i n r ) ] ,
f ( m ) = a b H ( r ) exp ( i m r ) d r 2 π ,
B = n = 1 2 M + 1 λ n u n ( u n ) ,
B 1 = n = 1 2 M + 1 λ n 1 u n ( u n ) .
U n ( r ) = m = M M u m n exp ( i m r ) .
π π U k ( r ) U n ( r ) ¯ d r = 0 ,
π π U n ( r ) U n ( r ) ¯ d r = π π U n ( r ) 2 d r = 1 .
Ω Ω U n ( r ) 2 d r = ( u n ) Q u n = λ n .
F DFT ( r ) = n = 1 2 M + 1 [ ( u n ) d ] U n ( r ) .
F MDFT ( r ) = n = 1 2 M + 1 λ n 1 [ ( u n ) d ] U n ( r ) .
( u n ) d = 1 2 π π π F ( r ) U n ( r ) ¯ d r ,
z = γ s + q
b = 1 e ( ω ) Q 1 e ( ω ) Q 1 e ( ω ) ,
γ ̂ = b z = 1 e ( ω ) Q 1 e ( ω ) e ( ω ) Q 1 z .
r z ( n ) = r s ( n ) + r q ( n )
R z ( ω ) = R s ( ω ) + R q ( ω )
H ( ω ) = k = h k e i k ω ,
y n = k = h k z n k .
R z ( ω ) H ( ω ) = R s ( ω ) ,
H ( ω ) = R s ( ω ) R z ( ω ) .
v n = k = K L f k z n k .
π π H ( ω ) k = K L f k e i k ω 2 R z ( ω ) d ω .
r s ( m ) = k = K L f k r z ( m k ) , for K m L .

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