Abstract

We show how a deconvolution can be performed from the significant structures of an image in the wavelet space. These significant structures are computed from the wavelet coefficients of the data. The wavelet-transform algorithm that is used is a new one based on the fast Fourier transform. This approach is first studied for interferometric images, simulations are done, and then the method is generalized to any kind of data.

© 1994 Optical Society of America

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References

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  1. H. C. Andrews, B. R. Hunt, “Preliminary concepts in image restoration,” in Digital Image Restoration, A. V. Oppenheim, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 113–116.
  2. A. Lannes, S. Roques, “Resolution and robustness in image processing: a new regularization principle,” J. Opt. Soc. Am. A 4, 189–199 (1987).
    [CrossRef]
  3. A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).
  4. H. C. Andrews, B. R. Hunt, “Linear algebraic restoration,” in Digital Image Restoration, A. V. Oppenheim, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 147–186.
  5. B. R. Frieden, “Image enhancement and restoration,” Vol. 6 of Springer Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 177–249.
  6. J. Cohen, “Tests of the photometric accuracy of image restoration using the maximum entropy algorithm,” Astrophys. J. 101, 734–737 (1991).
  7. T. J. Cornwell, “Deconvolution for real and synthetic apertures,” in Astronomical Data Analysis Software and System I, D. M. Worral, C. Biemesderfer, J. Barnes, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 1992), pp. 163–169.
  8. C. H. Chui, Wavelet Analysis and Its Application (Academic, San Diego, Calif., 1992).
  9. I. Daubechies, “Orthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, 909–996 (1988).
    [CrossRef]
  10. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
    [CrossRef]
  11. Y. Meyer, “Orthonormal wavelets,” in Wavelets, J. M. Combes, A. Grossmann, Ph. Tchanitchian, eds. (Springer-Verlag, Berlin, 1989), pp. 21–37.
    [CrossRef]
  12. G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai, Wavelets and Their Applications (Jones and Bartlett, Cambridge, Mass., 1992).
  13. S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [CrossRef]
  14. M. J. T. Smith, T. P. Barnwell, “Exact reconstruction technique for tree structured subband coders,” IEEE Trans. Acoust. Speech Signal Process. 34, 434–441 (1988).
    [CrossRef]
  15. A. Cohen, I. Daubechies, J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 45, 485–560 (1992).
    [CrossRef]
  16. J. C. Feauveau, “Analyse multirésolution par ondelettes non orthogonales et bancs de filtres numériques,” Ph.D. dissertation (Université Paris-Sud, Orsay, France, 1990).
  17. A. Labeyrie, “Stellar interferometry methods,” Ann. Rev. Astron. Astrophys. 16, 77–102 (1978).
    [CrossRef]
  18. J. M. Beckers, “Interferometric imaging with the very large telescope,” J. Opt. 22, 73–83 (1991).
    [CrossRef]
  19. R. Narayan, R. Nityanananda, “Maximum entropy image restoration in astronomy,” Ann. Rev. Astron. Astrophys. 24, 127–170 (1986).
    [CrossRef]
  20. J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).
  21. B. P. Wakker, U. J. Schwarz, “The multi-resolution clean and its application to the short-spacing problem in interferometry,” Astron. Astrophys. 200, 312–322 (1988).
  22. J.-L. Starck, A. Bijaoui, “Wavelets and multiresolution clean,” in High Resolution Imaging by Interferometry II, J. R. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1992), pp. 853–861.
  23. J.-L. Starck, A. Bijaoui, “Deconvolution from the pyramidal structure,” in Wavelet Analysis and Applications, Y. Meyer, S. Rogues, eds. (Edition Frontière, Gyf-sur-Yvette, France, 1992), pp. 447–450.
  24. L. B. Lucy, “An iteration technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  25. P. H. Van Cittert, Z. Physik 69, 298 (1931).
    [CrossRef]
  26. J.-L. Starck, A. Bijaoui, B. Lopez, “Reconstruction of images of two evolved stars from speckle interferometry observations by the wavelet transform,” in Symposium on Very High Angular Resolution Imaging, J. Davis, R. D. E. Key, eds. (Reidel, Dordrecht, The Netherlands, to be published).

1992 (1)

A. Cohen, I. Daubechies, J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

1991 (2)

J. M. Beckers, “Interferometric imaging with the very large telescope,” J. Opt. 22, 73–83 (1991).
[CrossRef]

J. Cohen, “Tests of the photometric accuracy of image restoration using the maximum entropy algorithm,” Astrophys. J. 101, 734–737 (1991).

1989 (1)

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

1988 (3)

M. J. T. Smith, T. P. Barnwell, “Exact reconstruction technique for tree structured subband coders,” IEEE Trans. Acoust. Speech Signal Process. 34, 434–441 (1988).
[CrossRef]

I. Daubechies, “Orthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, 909–996 (1988).
[CrossRef]

B. P. Wakker, U. J. Schwarz, “The multi-resolution clean and its application to the short-spacing problem in interferometry,” Astron. Astrophys. 200, 312–322 (1988).

1987 (1)

1986 (1)

R. Narayan, R. Nityanananda, “Maximum entropy image restoration in astronomy,” Ann. Rev. Astron. Astrophys. 24, 127–170 (1986).
[CrossRef]

1978 (1)

A. Labeyrie, “Stellar interferometry methods,” Ann. Rev. Astron. Astrophys. 16, 77–102 (1978).
[CrossRef]

1974 (2)

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).

L. B. Lucy, “An iteration technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

1931 (1)

P. H. Van Cittert, Z. Physik 69, 298 (1931).
[CrossRef]

Andrews, H. C.

H. C. Andrews, B. R. Hunt, “Preliminary concepts in image restoration,” in Digital Image Restoration, A. V. Oppenheim, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 113–116.

H. C. Andrews, B. R. Hunt, “Linear algebraic restoration,” in Digital Image Restoration, A. V. Oppenheim, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 147–186.

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Barnwell, T. P.

M. J. T. Smith, T. P. Barnwell, “Exact reconstruction technique for tree structured subband coders,” IEEE Trans. Acoust. Speech Signal Process. 34, 434–441 (1988).
[CrossRef]

Beckers, J. M.

J. M. Beckers, “Interferometric imaging with the very large telescope,” J. Opt. 22, 73–83 (1991).
[CrossRef]

Beylkin, G.

G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai, Wavelets and Their Applications (Jones and Bartlett, Cambridge, Mass., 1992).

Bijaoui, A.

J.-L. Starck, A. Bijaoui, “Wavelets and multiresolution clean,” in High Resolution Imaging by Interferometry II, J. R. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1992), pp. 853–861.

J.-L. Starck, A. Bijaoui, “Deconvolution from the pyramidal structure,” in Wavelet Analysis and Applications, Y. Meyer, S. Rogues, eds. (Edition Frontière, Gyf-sur-Yvette, France, 1992), pp. 447–450.

J.-L. Starck, A. Bijaoui, B. Lopez, “Reconstruction of images of two evolved stars from speckle interferometry observations by the wavelet transform,” in Symposium on Very High Angular Resolution Imaging, J. Davis, R. D. E. Key, eds. (Reidel, Dordrecht, The Netherlands, to be published).

Chui, C. H.

C. H. Chui, Wavelet Analysis and Its Application (Academic, San Diego, Calif., 1992).

Cohen, A.

A. Cohen, I. Daubechies, J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

Cohen, J.

J. Cohen, “Tests of the photometric accuracy of image restoration using the maximum entropy algorithm,” Astrophys. J. 101, 734–737 (1991).

Coifman, R.

G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai, Wavelets and Their Applications (Jones and Bartlett, Cambridge, Mass., 1992).

Cornwell, T. J.

T. J. Cornwell, “Deconvolution for real and synthetic apertures,” in Astronomical Data Analysis Software and System I, D. M. Worral, C. Biemesderfer, J. Barnes, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 1992), pp. 163–169.

Daubechies, I.

A. Cohen, I. Daubechies, J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

I. Daubechies, “Orthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, 909–996 (1988).
[CrossRef]

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai, Wavelets and Their Applications (Jones and Bartlett, Cambridge, Mass., 1992).

Feauveau, J. C.

A. Cohen, I. Daubechies, J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

J. C. Feauveau, “Analyse multirésolution par ondelettes non orthogonales et bancs de filtres numériques,” Ph.D. dissertation (Université Paris-Sud, Orsay, France, 1990).

Frieden, B. R.

B. R. Frieden, “Image enhancement and restoration,” Vol. 6 of Springer Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 177–249.

Högbom, J. A.

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).

Hunt, B. R.

H. C. Andrews, B. R. Hunt, “Linear algebraic restoration,” in Digital Image Restoration, A. V. Oppenheim, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 147–186.

H. C. Andrews, B. R. Hunt, “Preliminary concepts in image restoration,” in Digital Image Restoration, A. V. Oppenheim, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 113–116.

Labeyrie, A.

A. Labeyrie, “Stellar interferometry methods,” Ann. Rev. Astron. Astrophys. 16, 77–102 (1978).
[CrossRef]

Lannes, A.

Lopez, B.

J.-L. Starck, A. Bijaoui, B. Lopez, “Reconstruction of images of two evolved stars from speckle interferometry observations by the wavelet transform,” in Symposium on Very High Angular Resolution Imaging, J. Davis, R. D. E. Key, eds. (Reidel, Dordrecht, The Netherlands, to be published).

Lucy, L. B.

L. B. Lucy, “An iteration technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Mallat, S.

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai, Wavelets and Their Applications (Jones and Bartlett, Cambridge, Mass., 1992).

Meyer, Y.

G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai, Wavelets and Their Applications (Jones and Bartlett, Cambridge, Mass., 1992).

Y. Meyer, “Orthonormal wavelets,” in Wavelets, J. M. Combes, A. Grossmann, Ph. Tchanitchian, eds. (Springer-Verlag, Berlin, 1989), pp. 21–37.
[CrossRef]

Narayan, R.

R. Narayan, R. Nityanananda, “Maximum entropy image restoration in astronomy,” Ann. Rev. Astron. Astrophys. 24, 127–170 (1986).
[CrossRef]

Nityanananda, R.

R. Narayan, R. Nityanananda, “Maximum entropy image restoration in astronomy,” Ann. Rev. Astron. Astrophys. 24, 127–170 (1986).
[CrossRef]

Raphael, L.

G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai, Wavelets and Their Applications (Jones and Bartlett, Cambridge, Mass., 1992).

Roques, S.

Ruskai, M. B.

G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai, Wavelets and Their Applications (Jones and Bartlett, Cambridge, Mass., 1992).

Schwarz, U. J.

B. P. Wakker, U. J. Schwarz, “The multi-resolution clean and its application to the short-spacing problem in interferometry,” Astron. Astrophys. 200, 312–322 (1988).

Smith, M. J. T.

M. J. T. Smith, T. P. Barnwell, “Exact reconstruction technique for tree structured subband coders,” IEEE Trans. Acoust. Speech Signal Process. 34, 434–441 (1988).
[CrossRef]

Starck, J.-L.

J.-L. Starck, A. Bijaoui, “Wavelets and multiresolution clean,” in High Resolution Imaging by Interferometry II, J. R. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1992), pp. 853–861.

J.-L. Starck, A. Bijaoui, “Deconvolution from the pyramidal structure,” in Wavelet Analysis and Applications, Y. Meyer, S. Rogues, eds. (Edition Frontière, Gyf-sur-Yvette, France, 1992), pp. 447–450.

J.-L. Starck, A. Bijaoui, B. Lopez, “Reconstruction of images of two evolved stars from speckle interferometry observations by the wavelet transform,” in Symposium on Very High Angular Resolution Imaging, J. Davis, R. D. E. Key, eds. (Reidel, Dordrecht, The Netherlands, to be published).

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Van Cittert, P. H.

P. H. Van Cittert, Z. Physik 69, 298 (1931).
[CrossRef]

Wakker, B. P.

B. P. Wakker, U. J. Schwarz, “The multi-resolution clean and its application to the short-spacing problem in interferometry,” Astron. Astrophys. 200, 312–322 (1988).

Ann. Rev. Astron. Astrophys. (2)

A. Labeyrie, “Stellar interferometry methods,” Ann. Rev. Astron. Astrophys. 16, 77–102 (1978).
[CrossRef]

R. Narayan, R. Nityanananda, “Maximum entropy image restoration in astronomy,” Ann. Rev. Astron. Astrophys. 24, 127–170 (1986).
[CrossRef]

Astron. Astrophys. (2)

J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. 15, 417–426 (1974).

B. P. Wakker, U. J. Schwarz, “The multi-resolution clean and its application to the short-spacing problem in interferometry,” Astron. Astrophys. 200, 312–322 (1988).

Astron. J. (1)

L. B. Lucy, “An iteration technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Astrophys. J. (1)

J. Cohen, “Tests of the photometric accuracy of image restoration using the maximum entropy algorithm,” Astrophys. J. 101, 734–737 (1991).

Comm. Pure Appl. Math. (2)

A. Cohen, I. Daubechies, J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

I. Daubechies, “Orthogonal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, 909–996 (1988).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

M. J. T. Smith, T. P. Barnwell, “Exact reconstruction technique for tree structured subband coders,” IEEE Trans. Acoust. Speech Signal Process. 34, 434–441 (1988).
[CrossRef]

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

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

J. Opt. (1)

J. M. Beckers, “Interferometric imaging with the very large telescope,” J. Opt. 22, 73–83 (1991).
[CrossRef]

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

Z. Physik (1)

P. H. Van Cittert, Z. Physik 69, 298 (1931).
[CrossRef]

Other (13)

J.-L. Starck, A. Bijaoui, B. Lopez, “Reconstruction of images of two evolved stars from speckle interferometry observations by the wavelet transform,” in Symposium on Very High Angular Resolution Imaging, J. Davis, R. D. E. Key, eds. (Reidel, Dordrecht, The Netherlands, to be published).

J.-L. Starck, A. Bijaoui, “Wavelets and multiresolution clean,” in High Resolution Imaging by Interferometry II, J. R. Beckers, F. Merkle, eds. (European Southern Observatory, Garching, Germany, 1992), pp. 853–861.

J.-L. Starck, A. Bijaoui, “Deconvolution from the pyramidal structure,” in Wavelet Analysis and Applications, Y. Meyer, S. Rogues, eds. (Edition Frontière, Gyf-sur-Yvette, France, 1992), pp. 447–450.

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston, Washington, D.C., 1977).

H. C. Andrews, B. R. Hunt, “Linear algebraic restoration,” in Digital Image Restoration, A. V. Oppenheim, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 147–186.

B. R. Frieden, “Image enhancement and restoration,” Vol. 6 of Springer Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 177–249.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

Y. Meyer, “Orthonormal wavelets,” in Wavelets, J. M. Combes, A. Grossmann, Ph. Tchanitchian, eds. (Springer-Verlag, Berlin, 1989), pp. 21–37.
[CrossRef]

G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai, Wavelets and Their Applications (Jones and Bartlett, Cambridge, Mass., 1992).

J. C. Feauveau, “Analyse multirésolution par ondelettes non orthogonales et bancs de filtres numériques,” Ph.D. dissertation (Université Paris-Sud, Orsay, France, 1990).

H. C. Andrews, B. R. Hunt, “Preliminary concepts in image restoration,” in Digital Image Restoration, A. V. Oppenheim, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 113–116.

T. J. Cornwell, “Deconvolution for real and synthetic apertures,” in Astronomical Data Analysis Software and System I, D. M. Worral, C. Biemesderfer, J. Barnes, eds. (Astronomical Society of the Pacific, San Francisco, Calif., 1992), pp. 163–169.

C. H. Chui, Wavelet Analysis and Its Application (Academic, San Diego, Calif., 1992).

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Figures (8)

Fig. 1
Fig. 1

Top, interpolation function ϕ ̂; bottom, wavelet ψ ̂.

Fig. 2
Fig. 2

Top, filter h ̂; bottom, filter g ̂.

Fig. 3
Fig. 3

Image of the galaxy NGC2997.

Fig. 4
Fig. 4

Wavelet transform of the galaxy NGC2997.

Fig. 5
Fig. 5

Restoration of two point sources on an extended background. (a) The original object, (b) The simulated image found by convolving (a) with a Gaussian PSF of FWHM equal to three pixels and adding noise. Restorations with (c) maximum entropy, (d) Lucy, and (e) cleanmethods and (f) MRC with regularization.

Fig. 6
Fig. 6

Simulated object and UV plane coverage.

Fig. 7
Fig. 7

Restoration of a simulated object.

Fig. 8
Fig. 8

Restoration of a galaxy. Upper left, original object, upper right, object convolved with a PSF, bottom left, restoration with Lucy and 50 iterations; bottom right, restoration from the significant structure in the wavelet space.

Equations (53)

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

I ( x , y ) = + O ( X , Y ) P ( x X , y Y ) d X d Y ;
Î ( u , υ ) = Ô ( u , υ ) P ̂ ( u , υ ) .
c j ( k ) = f ( x ) , 2 j ϕ ( 2 j x k ) ;
1 2 ϕ ( x 2 ) = n h ( n ) ϕ ( x n ) ,
ϕ ̂ ( 2 ν ) = ĥ ( ν ) ϕ ̂ ( ν ) ,
ĥ ( ν ) = n h ( n ) exp ( 2 π n ν ) .
c j + 1 ( k ) = n h ( n 2 k ) c j ( n ) .
1 2 ψ ( x 2 ) = n g ( n ) ϕ ( x n ) ,
ψ ̂ ( 2 ν ) = ĝ ( ν ) ϕ ̂ ( ν ) .
w j + 1 ( k ) = n g ( n 2 k ) c j ( n ) .
c j ( k ) = 2 l [ c j + 1 ( l ) h ( k + 2 l ) + w j + 1 ( l ) g ( k + 2 l ) ] .
ĥ ( ν + 1 2 ) h ̂ ( ν ) + ĝ ( ν + 1 2 ) g ̂ ( ν ) = 0 .
ĥ ( ν ) h ̂ ( ν ) + ĝ ( ν ) g ̂ ( ν ) = 1
ĝ ( ν ) = exp ( 2 π ν ) ĥ * ( ν + ½ ) ,
h ̂ ( ν ) = ĥ * ( ν ) ,
g ̂ ( ν ) = ĝ * ( ν ) ,
| ĥ ( ν ) | 2 + | ĥ ( ν + ½ ) | 2 = 1 .
ĝ ( ν ) = exp ( 2 π ν ) h ̂ * ( ν + ½ ) ,
g ̂ ( ν ) = exp ( 2 π ν ) ĥ * ( ν + ½ ) ,
ĥ ( ν ) h ̂ ( ν ) + ĥ * ( ν + ½ ) h ̂ * ( ν + ½ ) = 1 .
c 1 ( k ) = f ( x ) , 1 2 ϕ ( x 2 k ) ,
ĥ ( ν ) = { ϕ ̂ ( 2 ν ) ϕ ̂ ( ν ) if | ν | < ν c 0 if ν c | ν | < 1 / 2 ,
ν , n ĥ ( ν + n ) = ĥ ( ν ) ,
ĉ j + 1 ( ν ) = ĉ j ( ν ) ĥ ( 2 j ν ) .
w j + 1 ( k ) = f ( x ) , 2 ( j + 1 ) ψ [ 2 ( j + 1 ) x k ] ,
ŵ j + 1 ( ν ) = ĉ j ( ν ) ĝ ( 2 j ν ) ,
ĝ ( ν ) = { ψ ̂ ( 2 ν ) ϕ ̂ ( ν ) if | ν | < ν c 1 if ν c | ν | < 1 / 2 ,
ν , n ĝ ( ν + n ) = ĝ ( ν ) .
B ̂ l ( ν ) = ( sin π ν π ν ) l + 1 .
ϕ ̂ ( ν ) = 3 / 2 B 3 ( 4 ν ) .
ϕ ( x ) = 3 8 ( sin π x 4 π x 4 ) 4 .
ψ ̂ ( 2 ν ) = ϕ ̂ ( ν ) ϕ ̂ ( 2 ν ) ,
ĝ ( ν ) = 1 ĥ ( ν ) ;
ĉ 0 ( ν ) = ĉ n p ( ν ) + j ŵ j ( ν ) .
ĉ j + 1 = ĥ ( 2 j ν ) ĉ j ( ν ) ,
ŵ j + 1 = ĝ ( 2 j ν ) ĉ j ( ν ) ,
p ̂ h ( 2 j ν ) | ĉ j + 1 ( ν ) ĥ ( 2 j ν ) ĉ j ( ν ) | 2 + p ̂ g ( 2 j ν ) × | ŵ j + 1 ( ν ) ĝ ( 2 j ν ) ĉ j ( ν ) | 2
ĉ j ( ν ) = ĉ j + 1 ( ν ) h ̂ ( 2 j ν ) + ŵ j + 1 ( ν ) g ̂ ( 2 j ν ) ,
h ̂ ( ν ) = p ̂ h ( ν ) ĥ * ( ν ) p ̂ h ( ν ) | ĥ ( ν ) | 2 + p ̂ g ( ν ) | ĝ ( ν ) | 2 ,
g ̂ ( ν ) = p ̂ g ( ν ) ĝ * ( ν ) p ̂ h ( ν ) | ĥ ( ν ) | 2 + p ̂ g ( ν ) | ĝ ( ν ) | 2 .
ĝ ( ν ) = [ 1 | ĥ ( ν ) | 2 ] 1 / 2 ,
| ψ ̂ ( 2 ν ) | 2 = | ϕ ̂ 2 ( ν ) | 2 | ϕ ̂ 2 ( 2 ν ) | 2
P ̂ ( u , υ ) Ô ( u , υ ) = V m ( u , υ ) ,
ŵ j ( I ) ( u , υ ) = ŵ j ( P ) Ô ( u , υ ) ,
δ j = { A j , 1 δ ( x x j , 1 , y y j , 1 ) , A j , 2 δ ( x x j , 2 , y y j , 2 ) , A j , n j δ ( x x j , n j , y y j , n j ) } ,
w j ( E ) ( x , y ) = δ j w j ( B ) ( x , y ) + w j ( R ) ( x , y ) = k A j , k w j ( B ) ( x x j , k , y y j , k ) + w j ( R ) ( x , y ) ,
δ j ( n + 1 ) = { A j , 1 ( n + 1 ) δ ( x x j , 1 , y y j , 1 ) } , W δ ( n + 1 ) = { δ 1 ( n + 1 ) , δ 2 ( n + 1 ) , } .
w j ( C ) = δ j ( n ) w j ( B ) .
r ̂ ( n ) = p [ V Ô ( n ) ] ,
| w j ( I ) ( x , y ) | > k σ N , j ,
w j ( Õ ) = w ( I ) if w j ( I ) > k σ N , j ,
w j ( Õ n + 1 ) = { w j ( Õ n ) + w j ( I ) w j ( Õ n ) if w j ( I ) > k σ N , j w j ( Õ n ) + w j ( Õ n 1 ) · w j ( Õ n ) if w j ( I ) k σ N , j and w j ( Õ 1 ) = w j ( Õ 0 ) .
{ w ( I ) . w j ( O n ) } s 2 { w ( I ) } s 2 < .

Metrics