Abstract

Superresolution algorithms have demonstrated impressive image-restoration results in the space domain. We consider the limits on superresolution performance in terms of usable bandwidth of the restored frequency spectrum. On the basis of a characterization of the spectral extrapolation errors (viz., null objects), we derive an expression for an approximate bound on accurate bandwidth extension for the general class of superresolution algorithms that incorporate a priori assumptions of a nonnegative, space-limited object. It is shown that accurate bandwidth extension is inversely related to the spatial extent of the object and the noise level in the image. For superresolution of sampled data, we present preliminary results relating bandwidth extrapolation to the difference between sampling rate and the discrete optical cutoff frequency. Simulation results are presented that substantiate the derived bandwidth extrapolation bounds.

© 1993 Optical Society of America

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References

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    [Crossref]
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  4. M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis I. The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
    [Crossref]
  5. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. IX, pp. 311–407.
    [Crossref]
  6. T. J. Holmes, “Expectation-maximization restoration of band-limited, truncated point-process intensities with application in microscopy,” J. Opt. Soc. Am. A 6, 1006–1014 (1989).
    [Crossref]
  7. E. L. Kosarev, “Shannon’s superresolution limit for signal recovery,” Inv. Probl. 6, 55–76 (1990).
    [Crossref]
  8. B. R. Hunt, “Imagery super-resolution: emerging prospects,” in Implementations II, F. T. Luk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 600–608 (1991).
  9. A. V. Oppenheim, R. W. Shafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  10. B. R. Hunt, P. Sementilli, “Description of a Poisson imagery super-resolution algorithm,” in Astronomical Data Analysis Software and Systems, D. Worral, C. Biemesderfer, J. Barnes, eds. (Astronomy Society of the Pacific, San Francisco, Calif., 1992).
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    [Crossref]
  12. H. H. Barrett, J. N. Aarsvold, T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” in 11th International Conference on Information Processing in Medical Imaging, D. A. Ortendahl, J. Llacer, eds. (Wiley-Liss, New York, 1991), pp. 221–226.
  13. R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
    [Crossref]
  14. T. M. Apostol, Mathematical Analysis (Addison-Wesley, Reading, Mass., 1957).
  15. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1976).
  16. A. Rosenfeld, A. C. Kak, Digital Picture Processing, (Academic, Orlando, Fla., 1982), Vol. 1.
  17. D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).
  18. C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 417–430 (1992).
    [Crossref]
  19. B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-22, 805–812 (1973).
    [Crossref]
  20. M. I. Sezan, H. Stark, “Image restoration by convex projections in the presence of noise,” Appl. Opt. 22, 2781–2789 (1983).
    [Crossref] [PubMed]
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    [Crossref]
  22. D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
    [Crossref]
  23. E. Levitan, G. T. Herman, “A Maximum a posterioriprobability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
    [Crossref]
  24. T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. Imaging 8, 194–202 (1989).
    [Crossref] [PubMed]
  25. D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 367–383 (1992).
    [Crossref]
  26. P. J. Sementilli, “Suppression of artifacts in super-resolved imagery,” Ph.D. dissertation (U. of Arizona, Tucson, Ariz., 1993).
  27. E. S. Meinel, “Origins of linear and nonlinear recursive restoration algorithms,” J. Opt. Soc. Am. A 3, 787–799 (1986).
    [Crossref]
  28. The proof for the case of N→ ∞ was suggested by an anonymous reviewer.

1992 (1)

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 367–383 (1992).
[Crossref]

1990 (1)

E. L. Kosarev, “Shannon’s superresolution limit for signal recovery,” Inv. Probl. 6, 55–76 (1990).
[Crossref]

1989 (2)

T. J. Holmes, “Expectation-maximization restoration of band-limited, truncated point-process intensities with application in microscopy,” J. Opt. Soc. Am. A 6, 1006–1014 (1989).
[Crossref]

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. Imaging 8, 194–202 (1989).
[Crossref] [PubMed]

1987 (1)

E. Levitan, G. T. Herman, “A Maximum a posterioriprobability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
[Crossref]

1986 (1)

1984 (1)

R. Prost, R. Goutte, “Discrete constrained iterative de-convolution algorithms with optimized rate of convergence,” Signal Process. 7, 209–230 (1984).
[Crossref]

1983 (1)

1982 (1)

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis I. The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
[Crossref]

1981 (1)

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

1973 (1)

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-22, 805–812 (1973).
[Crossref]

1968 (1)

1964 (1)

1955 (1)

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).

Aarsvold, J. N.

H. H. Barrett, J. N. Aarsvold, T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” in 11th International Conference on Information Processing in Medical Imaging, D. A. Ortendahl, J. Llacer, eds. (Wiley-Liss, New York, 1991), pp. 221–226.

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Apostol, T. M.

T. M. Apostol, Mathematical Analysis (Addison-Wesley, Reading, Mass., 1957).

Barrett, H. H.

H. H. Barrett, J. N. Aarsvold, T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” in 11th International Conference on Information Processing in Medical Imaging, D. A. Ortendahl, J. Llacer, eds. (Wiley-Liss, New York, 1991), pp. 221–226.

Bertero, M.

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis I. The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
[Crossref]

Frieden, B. R.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. IX, pp. 311–407.
[Crossref]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).

Geman, D.

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 367–383 (1992).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Goutte, R.

R. Prost, R. Goutte, “Discrete constrained iterative de-convolution algorithms with optimized rate of convergence,” Signal Process. 7, 209–230 (1984).
[Crossref]

Harris, J. L.

Harris, R. W.

Hebert, T.

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. Imaging 8, 194–202 (1989).
[Crossref] [PubMed]

Herman, G. T.

E. Levitan, G. T. Herman, “A Maximum a posterioriprobability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
[Crossref]

Holmes, T. J.

Hunt, B. R.

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-22, 805–812 (1973).
[Crossref]

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1976).

B. R. Hunt, “Imagery super-resolution: emerging prospects,” in Implementations II, F. T. Luk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 600–608 (1991).

B. R. Hunt, P. Sementilli, “Description of a Poisson imagery super-resolution algorithm,” in Astronomical Data Analysis Software and Systems, D. Worral, C. Biemesderfer, J. Barnes, eds. (Astronomy Society of the Pacific, San Francisco, Calif., 1992).

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, (Academic, Orlando, Fla., 1982), Vol. 1.

Kosarev, E. L.

E. L. Kosarev, “Shannon’s superresolution limit for signal recovery,” Inv. Probl. 6, 55–76 (1990).
[Crossref]

Leahy, R.

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. Imaging 8, 194–202 (1989).
[Crossref] [PubMed]

Levitan, E.

E. Levitan, G. T. Herman, “A Maximum a posterioriprobability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
[Crossref]

Matson, C. L.

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 417–430 (1992).
[Crossref]

Meinel, E. S.

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

Miller, M. I.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
[Crossref]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Shafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Pike, E. R.

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis I. The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
[Crossref]

Prost, R.

R. Prost, R. Goutte, “Discrete constrained iterative de-convolution algorithms with optimized rate of convergence,” Signal Process. 7, 209–230 (1984).
[Crossref]

Reynolds, G.

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 367–383 (1992).
[Crossref]

Richards, M. A.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

Roney, T. J.

H. H. Barrett, J. N. Aarsvold, T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” in 11th International Conference on Information Processing in Medical Imaging, D. A. Ortendahl, J. Llacer, eds. (Wiley-Liss, New York, 1991), pp. 221–226.

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, (Academic, Orlando, Fla., 1982), Vol. 1.

Rushforth, C. K.

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

Sementilli, P.

B. R. Hunt, P. Sementilli, “Description of a Poisson imagery super-resolution algorithm,” in Astronomical Data Analysis Software and Systems, D. Worral, C. Biemesderfer, J. Barnes, eds. (Astronomy Society of the Pacific, San Francisco, Calif., 1992).

Sementilli, P. J.

P. J. Sementilli, “Suppression of artifacts in super-resolved imagery,” Ph.D. dissertation (U. of Arizona, Tucson, Ariz., 1993).

Sezan, M. I.

Shafer, R. W.

A. V. Oppenheim, R. W. Shafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Snyder, D. L.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
[Crossref]

Stark, H.

Toraldo di Francia, G.

Appl. Opt. (1)

IEEE Trans. Comput. (1)

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comput. C-22, 805–812 (1973).
[Crossref]

IEEE Trans. Med. Imaging (2)

E. Levitan, G. T. Herman, “A Maximum a posterioriprobability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
[Crossref]

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. Imaging 8, 194–202 (1989).
[Crossref] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 367–383 (1992).
[Crossref]

Inv. Probl. (1)

E. L. Kosarev, “Shannon’s superresolution limit for signal recovery,” Inv. Probl. 6, 55–76 (1990).
[Crossref]

J. Inst. Elect. Eng. (1)

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis I. The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
[Crossref]

Proc. IEEE (1)

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

Signal Process. (1)

R. Prost, R. Goutte, “Discrete constrained iterative de-convolution algorithms with optimized rate of convergence,” Signal Process. 7, 209–230 (1984).
[Crossref]

Other (13)

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
[Crossref]

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 417–430 (1992).
[Crossref]

P. J. Sementilli, “Suppression of artifacts in super-resolved imagery,” Ph.D. dissertation (U. of Arizona, Tucson, Ariz., 1993).

The proof for the case of N→ ∞ was suggested by an anonymous reviewer.

T. M. Apostol, Mathematical Analysis (Addison-Wesley, Reading, Mass., 1957).

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1976).

A. Rosenfeld, A. C. Kak, Digital Picture Processing, (Academic, Orlando, Fla., 1982), Vol. 1.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

H. H. Barrett, J. N. Aarsvold, T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” in 11th International Conference on Information Processing in Medical Imaging, D. A. Ortendahl, J. Llacer, eds. (Wiley-Liss, New York, 1991), pp. 221–226.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. IX, pp. 311–407.
[Crossref]

B. R. Hunt, “Imagery super-resolution: emerging prospects,” in Implementations II, F. T. Luk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 600–608 (1991).

A. V. Oppenheim, R. W. Shafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

B. R. Hunt, P. Sementilli, “Description of a Poisson imagery super-resolution algorithm,” in Astronomical Data Analysis Software and Systems, D. Worral, C. Biemesderfer, J. Barnes, eds. (Astronomy Society of the Pacific, San Francisco, Calif., 1992).

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Figures (6)

Fig. 1
Fig. 1

(a) Two-point-source object imaged through a slit and the restoration achieved by the Poisson MAP super-resolution algorithm, (b) five-point-source object imaged through the same slit and the restoration achieved by the Poisson MAP superresolution algorithm.

Fig. 2
Fig. 2

(a) Object spectrum and restored spectrum associated with Fig. 1(a), (b) object spectrum and restored spectrum associated with Fig. 1(b).

Fig. 3
Fig. 3

Example of null object imaging. If ω0ωc, then the image of f(x) + k(x) is nearly identical to the image of f(x) alone.

Fig. 4
Fig. 4

Relationship between the noise power spectrum and the null object spectrum. The noise spectrum is constant at level σn2. The null object spectrum K(ω) is analytic and hence may increase smoothly until the bandwidth extension tolerance T is exceeded.

Fig. 5
Fig. 5

Illustration of estimate (18) by Eq. (19).

Fig. 6
Fig. 6

True (dashed curves) and restored (solid curves) spectra are shown for a four-point-source object imaged through apertures of diameter 16, 20, 22, and 28. (The dotted–dashed curves represent the image spectrum.) It is shown that as diameter (and hence ωc) increases, spectral restoration improves.

Tables (1)

Tables Icon

Table 1 Comparison of Theoretically Predicted Bandwidth Extension with Simulation (ωc = 0.1 π)

Equations (77)

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

g ( x ) = a b h ( x - ξ ) f ( ξ ) d ξ = h ( x ) * f ( x ) ,
g ( x ) = h ( x ) * f ( x ) + n ( x ) .
G ( ω ) = H ( ω ) F ( ω ) + N ( ω ) .
h ( x ) = sin π c x π c x = sinc ( c x ) ,
h ( x ) = sinc 2 ( c x ) .
F ^ ( ω ) - F ( ω ) T             ω [ 0 , ω e ] ,
f ^ ( x ) = f ( x ) + k ( x ) .
k ( x ) L 2 { a , b } ,
f ( x ) + k ( x ) 0             x [ a , b ] .
f ^ ( x ) = 0             x χ 2 ,
k ( x ) = 0             x max ( χ 2 , χ 2 ) .
f ^ ( x ) * h ( x ) - g ( x ) 2 = f ^ ( x ) * h ( x ) - f ( x ) * h ( x ) - n ( x ) 2 = k ( x ) * h ( x ) - n ( x ) 2 < ɛ ,
k ( x ) * h ( x ) 2 < n ( x ) 2 + ,
K ( ω ) = 0             ω ω c .
K ( ω ) = K ( 0 ) + ω d K ( 0 ) d ω + ω 2 2 d 2 K ( 0 ) d ω 2 + .
k ( x ) 0             x [ a , b ] .
| a b h ( ξ ) cos ( ω ξ + ϕ ) d ξ | < ,             ω ω 0 .
k ( x ) = cos ω 0 x .
f ( x ) * h ( x ) - f ^ ( x ) * h ( x ) 2 = f ( x ) * h ( x ) - [ f ( x ) + k ( x ) ] * h ( x ) 2 = f ( x ) * h ( x ) - [ f ( x ) + cos ( ω 0 x ) ] * h ( x ) 2 = cos ( ω 0 x ) * h ( x ) 2 = a b h ( ξ ) cos [ ω 0 ( x - ξ ) ] d ξ 2 < .
| a b h ( ξ ) m ( ξ ) cos ( ω ξ + ϕ ) d ξ | < ,             ω ω 0 .
| a b [ h ( ξ ) m ( ξ ) ] cos ( ω ξ + ϕ ) d ξ | < ,             ω ω 0 .
k ( x ) = m ( x ) cos ω 0 x ,
m ( x ) = 0 ,             x > χ 2 .
f ( x ) + m ( x ) cos ω 0 x 0 ,             x [ a , b ] .
m ( x ) f ( x ) ,             x [ a , b ] .
k ( x ) = i = 1 N m i ( x ) cos ( ω i x + ϕ i ) ,             ω i ω 0 , i = 1 , , N ,
f ( x ) * h ( x ) - f ^ ( x ) * h ( x ) 2 < .
K ( ω ) 2 σ n 2 ,             ω [ 0 , ω c ] ,
K ( ω ) = i = 1 N M i ( ω ± ω i ) ,
K ( ω ) 2 = | Δ | ω - Δ - ω c | σ n 2 .
S ( ω ) = { sin π [ ( ω - ω 1 ) / ( ω 2 - ω 1 ) ] if ω [ ω 1 , ω 2 ] 0 otherwise .
m ( x ) = { A sin π [ ( x + χ / 2 ) / χ ] if x ( - χ / 2 , χ / 2 ) 0 otherwise ,
m ( x ) = A cos π ( x χ ) rect ( x χ ) ,             x [ a , b ] ,
K ( ω ) = A χ 2 sinc [ χ ( ω ± 1 2 χ ± ω 0 ) ] .
K ˜ ( ω ) = { A χ 2 sinc [ 2 χ 3 ( ω ± ω 0 ) ] if ω ( ω 0 - 3 2 χ , ω 0 + 3 2 χ ) K ( ω ) otherwise ,
A = 2 σ n C 1 χ ,
K ˜ ( ω c ) 2 = σ n 2 .
ω 0 = ω c + 3 2 χ sinc - 1 C 1 .
K ˜ ( ω ) = σ n C 1 sinc [ 2 χ 3 ( ω - ω c ) - sinc - 1 C 1 ]
K ˜ ( ω e ) = T .
ω e - ω c = 3 2 χ [ sinc - 1 C 1 - sinc - 1 ( C 1 T σ n ) ] ,
ω e = ω c + 3 2 χ b ( T , σ n 2 ) ,
T σ n C 1 ,
σ n < T σ n C 1 1 < T σ n 1 C 1 ,
g = Hf + n , f i 0 ,             i ,
ω 0 * = π - 1.5 χ π ,
k ˜ ( ω * ) 2 = T 2 sinc 2 [ 2 χ 3 ( ω * - ω 0 * ) ] .
ω * = π - 1.5 χ [ sinc - 1 ( σ n T ) + π ] .
χ * = 1.5 [ sinc - 1 ( σ n / T ) + π ] π - ω c .
f ( x ) = i = 1 N a i δ ( x - x i ) ,
p ( f g ) = x = 1 M [ f ( x ) * h ( x ) ] g ( x ) exp [ - h ( x ) * f ( x ) ] g ( x ) ! p ( f ) ,
f ^ n + 1 ( x ) = f ¯ ( x ) exp { [ g ( x ) f ^ n ( x ) * h ( x ) - 1 ] * h ( x ) } ,
f ¯ ( x ) = f ^ n ( x ) , f ^ 0 ( x ) = g ( x ) .
c ( x ) = g ( x ) f ^ n ( x ) * h ( x ) - 1.
f ¯ ( x ) exp [ c ( x ) * h ( x ) ]
f ^ ( x ) = f ( x ) + k ( x )
k ( x ) = i = 1 N m i ( x ) cos ( ω i x + ϕ i ) ,             ω i ω 0 , i = 1 , , N ,
f ( x ) + k ( x ) 0 ,             x ,
k ( x ) = 0 ,             x > χ 2
k ( x ) * h ( x ) < ,
k ( x ) = i = 1 N m i ( x ) cos ( ω i x + ϕ i ) ,             ω i ω 0 , i = 1 , , N ,
| a b h ( ξ ) k ( ξ ) d ξ | = | a b h ( ξ ) i = 1 N m i ( ξ ) cos ( ω i ξ + ϕ i ) d ξ | = | i = 1 N a b h ( ξ ) m i ( ξ ) cos ( ω i ξ + ϕ i ) d ξ | i = 1 N | a b h ( ξ ) m i ( ξ ) cos ( ω i ξ + ϕ i ) d ξ | .
| a b h ( ξ ) m i ( ξ ) cos ( ω i ξ + ϕ i ) d ξ | < ,             ω i ω 0 .
| a b h ( ξ ) k ( ξ ) d ξ | i = 1 N | a b h ( ξ ) m i ( ξ ) cos ( ω i ξ + ϕ i ) d ξ | < N = ,
m i ( x ) = 0 , x > χ 2 , | i = 1 N m i ( x ) | f ( x ) , x ,
K ( z ) = n = 0 a n ( z - z ) n
S = { s : d d z K ( z ) z = s = 0 }
s i < s j ,             where i < j ,             i , j are integers .
K ( z ) = n = 0 a n z n = n = 1 a n z n .
K ( z ) = M ( z - s 1 ) .
M ( z - s 1 ) m ( x ) exp ( j s 1 x ) ,
k ( x ) = Re [ m ( x ) exp ( j s i x ) ] = m ( x ) cos ( ω 1 x + ϕ 1 ) .
k ( x ) = i = 1 N - 1 m i ( x ) cos ( ω i x + ϕ i ) .
K ( z ) = n = 1 N a n z n = n = 1 N - 1 a n z n + a N z N .
k ( x ) = - 1 { n = 1 N - 1 a n z n } + - 1 { a N z N } .
k ( x ) = n = 1 N - 1 m i ( x ) cos ( ω i x + ϕ i ) + m N ( x ) cos ( ω N x + ϕ N ) = i = 1 N m i ( x ) cos ( ω i x + ϕ i ) ,
k ( x ) = i = 1 m i cos ( i ω 0 x ) ,             x [ a , b ] .

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