Abstract

A two-step iterative algorithm is described for designing pupil functions for synthesis of bipolar point spread functions in a two-channel incoherent spatial filtering system. The first step uses the method of projections onto convex sets to determine an appropriate bias for the system pupil functions. This bias determines the complementary point spread functions and therefore the magnitude of the corresponding coherent spread functions. The phase of the coherent spread functions is obtained in the second step via iterative application of a magnitude constraint and a finite support constraint. With both magnitude and phase of the coherent spread functions established, the corresponding pupil functions are realized via Fourier transformation. Results are presented for a 2-D bandpass filter and compared with analytically obtained results. The comparison suggests that the iterative algorithm yields bias functions having less energy than bias functions determined by previously proposed analytic methods.

© 1989 Optical Society of America

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References

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  1. G. L. Rogers, Incoherent Optical Processing (Wiley, New York, 1977).
  2. A. W. Lohmann, “Matched Filtering with Self-Luminous Objects,” Appl. Opt. 7, 561 (1968).
    [CrossRef]
  3. S. Lowenthal, A. Werts, “Filtrage des frequences spatiales en lumiere incoherenta a l’aide d’hologrammes,” C.R. Acad. Sci. Ser. B 266, 542 (1968).
  4. W. T. Rhodes, “Incoherent Spatial Filtering,” Opt. Eng. 19, 323 (1980).
    [CrossRef]
  5. W. T. Rhodes, A. A. Sawchuk, “Incoherent Optical Processing,” in Optical Information Processing, S. H. Lee, Ed. (Springer-Verlag, New York, 1981), p. 69.
    [CrossRef]
  6. S. Lowenthal, P. Chavel, “Noise Problems in Optical Image Processing,” in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesen, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), p. 45.
  7. P. Chavel, S. Lowenthal, “Incoherent Optical-Image Processing Using Synthetic Hologram,” J. Opt. Soc. Am. 66, 14 (1976).
    [CrossRef]
  8. D. Gorlitz, F. Lanzl, “Methods of Zero-Order Non-Coherent Filtering,” Opt. Commun. 20, 68 (1977).
    [CrossRef]
  9. B. Braunecker, R. Hauck, “Grey Level on Axis Computer Holograms for Incoherent Image Processing,” Opt. Commun. 20, 234 (1977).
    [CrossRef]
  10. W. T. Rhodes, “Bipolar Pointspread Function Synthesis by Phase Switching,” Appl. Opt. 16, 265 (1977).
    [CrossRef] [PubMed]
  11. W. T. Rhodes, “Temporal Frequency Carriers in Noncoherent Optical Processing,” in Proceedings, 1978 International Optical Computing Conference, p. 163.
  12. A. W. Lohmann, “Incoherent Optical Processing of Complex Data,” Appl. Opt. 16, 261 (1977).
    [CrossRef] [PubMed]
  13. W. Stoner, “Incoherent Optical Processing Via Spatially Offset Pupil Masks,” Appl. Opt. 17, 2454 (1978).
    [CrossRef] [PubMed]
  14. A. W. Lohmann, W. T. Rhodes, “Two-Pupil Synthesis of Optical Transfer Functions,” Appl. Opt. 17, 1141 (1978).
    [CrossRef] [PubMed]
  15. J. N. Mait, “Pupil Function Optimization for Bipolar Incoherent Spatial Filtering,” Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA (June1985).
  16. J. N. Mait, “Pupil Function Design for Bipolar Incoherent Spatial Filtering,” J. Opt. Soc. Am. A 3, 1826 (1986).
    [CrossRef]
  17. J. N. Mait, “Existence Conditions for Two-Pupil Synthesis of Bipolar Incoherent Pointspread Functions,” J. Opt. Soc. Am. A 3, 437 (1986).
    [CrossRef]
  18. J. N. Mait, J. D. Vaaler, “Necessary and Sufficient Conditions for Bipolar Incoherent Spatial Filtering,” J. Opt. Soc. Am. A 6, 147 (1989).
    [CrossRef]
  19. D. C. Youla, H. Webb, “Image Restoration by the Method of Convex Projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81 (1982).
    [CrossRef]
  20. J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 147 (1981).
  21. J. N. Mait, W. T. Rhodes, “Two-Pupil Synthesis of Optical Transfer Functions. 2: Pupil Function Relationships,” Appl. Opt. 25, 2003 (1986).
    [CrossRef] [PubMed]
  22. A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41 (1963).
    [CrossRef]
  23. J. N. Mait, W. T. Rhodes, “Iterative Design of Pupil Functions for Bipolar Incoherent Spatial Filtering,” Proc. Soc. Photo-Opt. Instrum. Eng. 292, 66 (1981).
  24. W. Lukosz, “Properties of Linear Low-Pass Filters for Nonnegative Signals,” J. Opt. Soc. Am. 52, 827 (1962).
    [CrossRef]
  25. D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1984).
  26. A. Levi, H. Stark, “Signal Restoration from Phase by Projection onto Convex Sets,” J. Opt. Soc. Am. 73, 810 (1983).
    [CrossRef]
  27. R. G. Bartle, The Elements of Real Analysis (Wiley, New York, 1976).
  28. R. H. T. Bates, “Fourier Phase Problems are Uniquely Solvable in more than One Dimension. I: Underlying Theory,” Optik 61, 247 (1982).
  29. J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the Support of an Object from the Support of Its Autocorrelation,” J. Opt. Soc. Am. 72, 610 (1982).
    [CrossRef]
  30. 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 (1982).
    [CrossRef]
  31. E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).
  32. H. L. Royden, Real Analysis (Macmillan, New York, 1968).
  33. A. A. G. Requicha, “The Zeros of Entire Functions: Theory and Engineering Applications,” Proc. IEEE 68, 308 (1980).
    [CrossRef]

1989 (1)

1986 (3)

1983 (1)

1982 (4)

J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the Support of an Object from the Support of Its Autocorrelation,” J. Opt. Soc. Am. 72, 610 (1982).
[CrossRef]

D. C. Youla, H. Webb, “Image Restoration by the Method of Convex Projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81 (1982).
[CrossRef]

R. H. T. Bates, “Fourier Phase Problems are Uniquely Solvable in more than One Dimension. I: Underlying Theory,” Optik 61, 247 (1982).

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 (1982).
[CrossRef]

1981 (2)

J. N. Mait, W. T. Rhodes, “Iterative Design of Pupil Functions for Bipolar Incoherent Spatial Filtering,” Proc. Soc. Photo-Opt. Instrum. Eng. 292, 66 (1981).

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 147 (1981).

1980 (2)

W. T. Rhodes, “Incoherent Spatial Filtering,” Opt. Eng. 19, 323 (1980).
[CrossRef]

A. A. G. Requicha, “The Zeros of Entire Functions: Theory and Engineering Applications,” Proc. IEEE 68, 308 (1980).
[CrossRef]

1978 (2)

1977 (4)

A. W. Lohmann, “Incoherent Optical Processing of Complex Data,” Appl. Opt. 16, 261 (1977).
[CrossRef] [PubMed]

D. Gorlitz, F. Lanzl, “Methods of Zero-Order Non-Coherent Filtering,” Opt. Commun. 20, 68 (1977).
[CrossRef]

B. Braunecker, R. Hauck, “Grey Level on Axis Computer Holograms for Incoherent Image Processing,” Opt. Commun. 20, 234 (1977).
[CrossRef]

W. T. Rhodes, “Bipolar Pointspread Function Synthesis by Phase Switching,” Appl. Opt. 16, 265 (1977).
[CrossRef] [PubMed]

1976 (1)

1968 (2)

S. Lowenthal, A. Werts, “Filtrage des frequences spatiales en lumiere incoherenta a l’aide d’hologrammes,” C.R. Acad. Sci. Ser. B 266, 542 (1968).

A. W. Lohmann, “Matched Filtering with Self-Luminous Objects,” Appl. Opt. 7, 561 (1968).
[CrossRef]

1963 (1)

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41 (1963).
[CrossRef]

1962 (1)

Bartle, R. G.

R. G. Bartle, The Elements of Real Analysis (Wiley, New York, 1976).

Bates, R. H. T.

R. H. T. Bates, “Fourier Phase Problems are Uniquely Solvable in more than One Dimension. I: Underlying Theory,” Optik 61, 247 (1982).

Braunecker, B.

B. Braunecker, R. Hauck, “Grey Level on Axis Computer Holograms for Incoherent Image Processing,” Opt. Commun. 20, 234 (1977).
[CrossRef]

Chavel, P.

P. Chavel, S. Lowenthal, “Incoherent Optical-Image Processing Using Synthetic Hologram,” J. Opt. Soc. Am. 66, 14 (1976).
[CrossRef]

S. Lowenthal, P. Chavel, “Noise Problems in Optical Image Processing,” in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesen, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), p. 45.

Crimmins, T. R.

Fienup, J. R.

J. R. Fienup, T. R. Crimmins, W. Holsztynski, “Reconstruction of the Support of an Object from the Support of Its Autocorrelation,” J. Opt. Soc. Am. 72, 610 (1982).
[CrossRef]

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 147 (1981).

Gorlitz, D.

D. Gorlitz, F. Lanzl, “Methods of Zero-Order Non-Coherent Filtering,” Opt. Commun. 20, 68 (1977).
[CrossRef]

Hauck, R.

B. Braunecker, R. Hauck, “Grey Level on Axis Computer Holograms for Incoherent Image Processing,” Opt. Commun. 20, 234 (1977).
[CrossRef]

Hayes, M. H.

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 (1982).
[CrossRef]

Holsztynski, W.

Kreyszig, E.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).

Lanzl, F.

D. Gorlitz, F. Lanzl, “Methods of Zero-Order Non-Coherent Filtering,” Opt. Commun. 20, 68 (1977).
[CrossRef]

Levi, A.

Lohmann, A. W.

Lowenthal, S.

P. Chavel, S. Lowenthal, “Incoherent Optical-Image Processing Using Synthetic Hologram,” J. Opt. Soc. Am. 66, 14 (1976).
[CrossRef]

S. Lowenthal, A. Werts, “Filtrage des frequences spatiales en lumiere incoherenta a l’aide d’hologrammes,” C.R. Acad. Sci. Ser. B 266, 542 (1968).

S. Lowenthal, P. Chavel, “Noise Problems in Optical Image Processing,” in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesen, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), p. 45.

Luenberger, D. G.

D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1984).

Lukosz, W.

Mait, J. N.

Requicha, A. A. G.

A. A. G. Requicha, “The Zeros of Entire Functions: Theory and Engineering Applications,” Proc. IEEE 68, 308 (1980).
[CrossRef]

Rhodes, W. T.

J. N. Mait, W. T. Rhodes, “Two-Pupil Synthesis of Optical Transfer Functions. 2: Pupil Function Relationships,” Appl. Opt. 25, 2003 (1986).
[CrossRef] [PubMed]

J. N. Mait, W. T. Rhodes, “Iterative Design of Pupil Functions for Bipolar Incoherent Spatial Filtering,” Proc. Soc. Photo-Opt. Instrum. Eng. 292, 66 (1981).

W. T. Rhodes, “Incoherent Spatial Filtering,” Opt. Eng. 19, 323 (1980).
[CrossRef]

A. W. Lohmann, W. T. Rhodes, “Two-Pupil Synthesis of Optical Transfer Functions,” Appl. Opt. 17, 1141 (1978).
[CrossRef] [PubMed]

W. T. Rhodes, “Bipolar Pointspread Function Synthesis by Phase Switching,” Appl. Opt. 16, 265 (1977).
[CrossRef] [PubMed]

W. T. Rhodes, A. A. Sawchuk, “Incoherent Optical Processing,” in Optical Information Processing, S. H. Lee, Ed. (Springer-Verlag, New York, 1981), p. 69.
[CrossRef]

W. T. Rhodes, “Temporal Frequency Carriers in Noncoherent Optical Processing,” in Proceedings, 1978 International Optical Computing Conference, p. 163.

Rogers, G. L.

G. L. Rogers, Incoherent Optical Processing (Wiley, New York, 1977).

Royden, H. L.

H. L. Royden, Real Analysis (Macmillan, New York, 1968).

Sawchuk, A. A.

W. T. Rhodes, A. A. Sawchuk, “Incoherent Optical Processing,” in Optical Information Processing, S. H. Lee, Ed. (Springer-Verlag, New York, 1981), p. 69.
[CrossRef]

Stark, H.

Stoner, W.

Vaaler, J. D.

Walther, A.

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41 (1963).
[CrossRef]

Webb, H.

D. C. Youla, H. Webb, “Image Restoration by the Method of Convex Projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81 (1982).
[CrossRef]

Werts, A.

S. Lowenthal, A. Werts, “Filtrage des frequences spatiales en lumiere incoherenta a l’aide d’hologrammes,” C.R. Acad. Sci. Ser. B 266, 542 (1968).

Youla, D. C.

D. C. Youla, H. Webb, “Image Restoration by the Method of Convex Projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81 (1982).
[CrossRef]

Appl. Opt. (6)

C.R. Acad. Sci. Ser. B (1)

S. Lowenthal, A. Werts, “Filtrage des frequences spatiales en lumiere incoherenta a l’aide d’hologrammes,” C.R. Acad. Sci. Ser. B 266, 542 (1968).

IEEE Trans. Acoust. Speech Signal Process. (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 (1982).
[CrossRef]

IEEE Trans. Med. Imaging (1)

D. C. Youla, H. Webb, “Image Restoration by the Method of Convex Projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81 (1982).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Acta (1)

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41 (1963).
[CrossRef]

Opt. Commun. (2)

D. Gorlitz, F. Lanzl, “Methods of Zero-Order Non-Coherent Filtering,” Opt. Commun. 20, 68 (1977).
[CrossRef]

B. Braunecker, R. Hauck, “Grey Level on Axis Computer Holograms for Incoherent Image Processing,” Opt. Commun. 20, 234 (1977).
[CrossRef]

Opt. Eng. (1)

W. T. Rhodes, “Incoherent Spatial Filtering,” Opt. Eng. 19, 323 (1980).
[CrossRef]

Optik (1)

R. H. T. Bates, “Fourier Phase Problems are Uniquely Solvable in more than One Dimension. I: Underlying Theory,” Optik 61, 247 (1982).

Proc. IEEE (1)

A. A. G. Requicha, “The Zeros of Entire Functions: Theory and Engineering Applications,” Proc. IEEE 68, 308 (1980).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

J. N. Mait, W. T. Rhodes, “Iterative Design of Pupil Functions for Bipolar Incoherent Spatial Filtering,” Proc. Soc. Photo-Opt. Instrum. Eng. 292, 66 (1981).

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 147 (1981).

Other (9)

D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1984).

R. G. Bartle, The Elements of Real Analysis (Wiley, New York, 1976).

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).

H. L. Royden, Real Analysis (Macmillan, New York, 1968).

W. T. Rhodes, A. A. Sawchuk, “Incoherent Optical Processing,” in Optical Information Processing, S. H. Lee, Ed. (Springer-Verlag, New York, 1981), p. 69.
[CrossRef]

S. Lowenthal, P. Chavel, “Noise Problems in Optical Image Processing,” in Applications of Holography and Optical Data Processing, E. Marom, A. A. Friesen, E. Wiener-Avnear, Eds. (Pergamon, New York, 1977), p. 45.

G. L. Rogers, Incoherent Optical Processing (Wiley, New York, 1977).

W. T. Rhodes, “Temporal Frequency Carriers in Noncoherent Optical Processing,” in Proceedings, 1978 International Optical Computing Conference, p. 163.

J. N. Mait, “Pupil Function Optimization for Bipolar Incoherent Spatial Filtering,” Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA (June1985).

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Figures (25)

Fig. 1
Fig. 1

Single-channel incoherent spatial filtering system: O denotes the object plane; P, the pupil plane; and I, the image plane. The focal length of the lens is f and the distance between the planes is 2f.

Fig. 2
Fig. 2

(a) Bipolar PSF f(x) = 2κ sinc2(u1x) cos(2πu0x) and (b) its associated OTF F(u).

Fig. 3
Fig. 3

The PSFs (a) f+(x) and (b) f(x).

Fig. 4
Fig. 4

Pupil functions (a) P+(u) and (b) P(u).

Fig. 5
Fig. 5

Example of bipolar image plus image bias.

Fig. 6
Fig. 6

Minimum bias PSFs (a) fMB+(x) and (b) fMB−(x).

Fig. 7
Fig. 7

Lukosz bounds from Ref. 24.

Fig. 8
Fig. 8

Perspective view of a 2-D bipolar function f(x,y) = 2κ sinc2(u1x) sinc2(υ1y) cos(2πu0x).

Fig. 9
Fig. 9

Effects of sampling |f(x,y)|. Cross-sectional view of (a) the Fourier transform of |f(x,y)|, (b) the discrete Fourier transform of |f(x,y)|, (c) the aliasing error; FT{|f(x,y)|} − DFT{|f(x,y)|}.

Fig. 10
Fig. 10

Constraint bandwidths used in bias design. The dashed line is the Fourier transform of f(x,y) and the solid line, the Fourier transform of |f(x,y)|.

Fig. 11
Fig. 11

Results of the bias design algorithm G30(u,0) for (a) Bψ = Bψ1, (b) Bψ = Bψ2, (c) Bψ = Bψ3.

Fig. 12
Fig. 12

(a) E, (b) SBRe, and (c) D vs iteration for bias design algorithm.

Fig. 13
Fig. 13

For the constraint bandwidth Bψ2, a comparison of the minimum average bias (solid line) and the results of the bias design algorithm (dashed line).

Fig. 14
Fig. 14

SBRe vs iteration for different values of relaxation parameter r.

Fig. 15
Fig. 15

Comparison of SBRe for the original bias design algorithm and the second modification of the algorithm.

Fig. 16
Fig. 16

(a) Cross-sectional view of the second bipolar function used to test the bias design algorithm. (b) Comparison of the analytically constructed bias to the results of the bias design algorithm. (c) As in (b) except the comparison is between the spectra. [Note that the analytic solution is represented by the dashed line in (b), but by the solid line in (c).]

Fig. 17
Fig. 17

(a) Unacceptable bipolar function and (b) the results produced by the bias design algorithm.

Fig. 18
Fig. 18

Analytically determined pupil functions for (a) ψPM(x,y) for Bψ1, (b) ψMA(x,y) for Bψ2, and (c) ψME(x,y) for Bψ2.

Fig. 19
Fig. 19

Positive OTF F+(u,υ) for the constraint bandwidth Bψ3.

Fig. 20
Fig. 20

Support of the pupil function P+(u,υ) for the constraint bandwidth Bψ3.

Fig. 21
Fig. 21

(a) Ez and (b) Es vs iteration for the error reduction algorithm and the combined input–output/error reduction algorithm with r = 0.5.

Fig. 22
Fig. 22

Results of the pupil design algorithm using (a) the combined input–output reduction algorithm with r = 0.5, (b) the error reduction algorithm, and (c) results in (b) but rotated by π rad and with π rad added to the phase.

Fig. 23
Fig. 23

Error between the desired OTF and OTFs resulting from the pupil design algorithms.

Fig. 24
Fig. 24

(a) The magnitude and (b) phase of the pupil function P+(u,υ) resulting from the combined input–output/error reduction algorithm.

Fig. 25
Fig. 25

Ez vs iteration for the combined input–output/error reduction algorithm with r = 1.0.

Equations (77)

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i ( x ) = o ( ζ ) f ( x ζ ) d ζ = o ( x ) * f ( x ) ,
f ( x ) = | p ( x ) | 2 .
f ( x ) = f + ( x ) f ( x ) ,
f + ( x ) = | p + ( x ) | 2 ,
f ( x ) = | p ( x ) | 2 .
f + ( x ) = ( 1 / 2 ) [ f ( x ) + ψ ( x ) ] ,
f ( x ) = ( 1 / 2 ) [ f ( x ) + ψ ( x ) ] ,
ψ ( x ) = f + ( x ) + f ( x ) .
f ( x ) = 2 κ sinc 2 ( u 1 x ) cos ( 2 π u 0 x )
ψ ( x ) = 2 κ sinc 2 ( u 1 x ) { 1 + ( 1 / 2 ) cos [ 2 π ( 2 u 0 ) x ] } ,
f + ( x ) = κ / 2 sinc 2 ( u 1 x ) { 1 + 2 cos ( 2 π u 0 x ) + ( 1 / 2 ) cos [ 2 π ( 2 u 0 ) x ] } ; f ( x ) = κ / 2 sinc 2 ( u 1 x ) { 1 2 cos ( 2 π u 0 x ) + ( 1 / 2 ) cos [ 2 π ( 2 u 0 ) x ] } .
p + ( x ) = ( κ / 8 ) 1 / 2 sinc ( u 1 x ) [ 1 + 2 cos ( 2 π u 0 x ) ] , p ( x ) = ( κ / 8 ) 1 / 2 sinc ( u 1 x ) [ 1 2 cos ( 2 π u 0 x ) ] ,
i + ( x ) = o ( x ) * f + ( x ) = o ( x ) * ( 1 / 2 ) [ f ( x ) + ψ ( x ) ] = ( 1 / 2 ) [ o ( x ) * f ( x ) + o ( x ) * ψ ( x ) ] ,
i ( x ) = ( 1 / 2 ) [ o ( x ) * f ( x ) + o ( x ) * ψ ( x ) ] ,
i s ( x ) = i + ( x ) i ( x ) .
SBR p ( x ) = | o ( x ) * f ( x ) | o ( x ) * ψ ( x ) .
SBR a = | o ( x ) * f ( x ) | d x o ( x ) * ψ ( x ) d x = | o ( x ) * f ( x ) | d x O ( 0 ) ψ ( 0 ) ,
SBR e = | o ( x ) * f ( x ) | 2 d x [ o ( x ) * ψ ( x ) ] 2 d x = | O ( u ) · F ( u ) | 2 d u | O ( u ) · ψ ( u ) | 2 d u .
o ( x ζ ) ψ ( ζ ) o ( x ζ ) | f ( ζ ) | 0 for all ζ , x .
o ( x ζ ) ψ ( ζ ) d ζ o ( x ζ ) | f ( ζ ) | d ζ 0 for all x ,
o ( x ) * ψ ( x ) o ( x ) * | f ( x ) | for all x .
| o ( x ) * f ( x ) | o ( x ) * | f ( x ) | > | o ( x ) * f ( x ) | o ( x ) * ψ ( x ) ,
[ o ( x ) * ψ ( x ) ] d x > [ o ( x ) * | f ( x ) | ] d x ;
| o ( x ) * ψ ( x ) | 2 d x > | o ( x ) * | f ( x ) | | 2 d x ,
f MB + ( x ) = ( 1 / 2 ) [ f ( x ) + ψ MB ( x ) ] ,
f MB ( x ) = ( 1 / 2 ) [ f ( x ) + ψ MB ( x ) ] ,
ψ MB ( x ) = | f ( x ) | .
ψ ( x ) = 2 κ a 0 sinc 2 ( u 1 x ) ,
ψ PM ( x ) = 2 κ sinc 2 ( u 1 x ) .
ψ ( x ) = 2 κ a 0 sinc 2 ( u 1 x / n ) ,
n = { int [ u 1 / B ψ ] + 1 , u 1 m B ψ , u 1 / B ψ , u 1 = m B ψ ,
ψ ( x ) = κ sinc 2 ( u 1 x ) { a 0 + 2 a 2 cos [ 2 π ( 2 u 0 ) x ] } .
f + ( x ) = ( κ / 2 ) sinc 2 ( u 1 x ) { a 0 + 2 cos ( 2 π u 0 x ) + 2 a 2 cos [ 2 π ( 2 u 0 ) x ] } ,
f ( x ) = ( κ / 2 ) sinc 2 ( u 1 x ) { a 0 2 cos ( 2 π u 0 x ) + 2 a 2 cos [ 2 π ( 2 u 0 ) x ] } .
F + ( u ) = ( κ / 2 ) [ a 0 tri ( u u 1 ) + tri ( u u 0 u 1 ) + a 2 tri ( u 2 u 0 u 1 ) + tri ( u + u 0 u 1 ) + a 2 tri ( u + 2 u 0 u 1 ) ] ,
F ( u ) = ( κ / 2 ) [ a 0 tri ( u u 1 ) tri ( u u 0 u 1 ) + a 2 tri ( u 2 u 0 u 1 ) tri ( u + u 0 u 1 ) + a 2 tri ( u + 2 u 0 u 1 ) ] ,
tri ( u ) = { 1 | u | , | u | 1 0 , | u | > 1 .
( 1 2 b + 2 b cos θ ) 2 = 1 + 2 ( 2 1 2 b b ) cos θ + 2 b cos 2 θ ,
a 2 = a 0 + a 0 2 2 4 ,
g n ( x ) = P f P b g n 1 ( x ) ,
P f g ( x ) = { g ( x ) , g ( x ) | f ( x ) | , | f ( x ) | , g ( x ) < | f ( x ) | ,
P b g ( x ) = g ( x ) * 2 B ψ sinc ( 2 B ψ x ) ,
g 0 ( x ) = ψ MB ( x ) .
ψ ( x ) g n ( x ) ψ ( x ) g n 1 ( x ) ,
T = I + r ( P I ) .
SBR e = | f ( x ) | 2 d x | g n ( x ) | 2 d x .
E = | G n ( u ) P b G n ( u ) | 2 d u | P b G n ( u ) | 2 d u ,
D = | | f ( x ) | g n ( x ) | 2 d x | f ( x ) | 2 d x .
g n ( x ) = { P b g n 1 ( x ) , P b g n 1 ( x ) g n 1 ( x ) , g n 1 ( x ) , P b g n 1 ( x ) < g n 1 ( x ) .
g n ( x ) = { P b g n 1 ( x ) , g n 1 ( x ) P b g n 1 ( x ) , g n 1 ( x ) , | f ( x ) | P b g n 1 ( x ) < g n 1 ( x ) , | f ( x ) | , P b g n 1 ( x ) < | f ( x ) | .
f ( x , y ) = sinc 2 ( u 1 x ) [ 1 ( u 1 x ) 2 ] [ 1 ( u 1 x / 2 ) 2 ] sinc 2 ( u 1 y ) ,
ψ ( x , y ) = sinc 2 ( u 1 x ) [ 1 + ( u 1 x ) 4 / 4 ] [ 1 ( u 1 x ) 2 ] 2 [ 1 ( u 1 x / 2 ) 2 ] 2 sinc 2 ( u 1 y ) .
| p + ( x ) | = ( 1 / 2 ) [ f ( x ) + ψ ( x ) ] ,
| p ( x ) | = ( 1 / 2 ) [ f ( x ) + ψ ( x ) ] .
P s P n ( u ) = { P n ( u ) , u Δ , 0 , u Δ ,
P z p n ( x ) = z ( x ) exp [ j θ n ( x ) ] ,
p n + 1 ( x ) p n ( x ) p n ( x ) p n 1 ( x ) .
lim n p n ( x ) = p ( x ) .
T s P n ( u ) = { P n ( u ) , u Δ , P n 1 ( u ) r P n ( u ) , u Δ .
E z = | | p + ( x ) | | p n ( x ) | | 2 d x | p + ( x ) | 2 d x ,
E s = u Δ | P n ( u ) | 2 d u | P n ( u ) | 2 d u .
inf g C h g = h P h .
g 1 = | g ( x ) | d x ,
( g , h ) = g ( x ) h * ( x ) d x ,
g 2 = ( g , g ) < .
P f g ( x ) = { g ( x ) , g ( x ) | f ( x ) | , | f ( x ) | , g ( x ) < | f ( x ) | ,
P b g ( x ) = g ( x ) * 2 B ψ sinc ( 2 B ψ x ) .
P b f h = P f P b h = h .
P b f g P f P b g .
g = P g + P g ,
P b g = g P b g .
P f g ( x ) = g ( x ) r f ( x ) [ g ( x ) | f ( x ) | ] , = | f ( x ) | + [ 1 r f ( x ) ] [ g ( x ) | f ( x ) | ] ,
r f ( x ) = { 0 , g ( x ) | f ( x ) | , 1 , g ( x ) < | f ( x ) | .
( w ) lim n g n = g , g C b f , g n = P f P b g n 1
= | f ( x ) | + [ 1 r f ( x ) ] [ P b g n 1 ( x ) | f ( x ) | ] ,
g 0 = ψ MB .
lim n ( h , g n ) = ( h , g ) .

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