Abstract

Several existing strategies for estimating the axial intensity derivative in the transport-of-intensity equation (TIE) from multiple intensity measurements have been unified by the Savitzky-Golay differentiation filter - an equivalent convolution solution for differentiation estimation by least-squares polynomial fitting. The different viewpoint from the digital filter in signal processing not only provides great insight into the behaviors, the shortcomings, and the performance of these existing intensity derivative estimation algorithms, but more important, it also suggests a new way of improving solution strategies by extending the applications of Savitzky-Golay differentiation filter in TIE. Two novel methods for phase retrieval based on TIE are presented - the first by introducing adaptive-degree strategy in spatial domain and the second by selecting optimal spatial frequencies in Fourier domain. Numerical simulations and experiments verify that the second method outperforms the existing methods significantly, showing reliable retrieved phase with both overall contrast and fine phase variations well preserved.

© 2013 OSA

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2012 (4)

2011 (1)

2010 (5)

2009 (1)

2007 (1)

2006 (2)

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett.31(6), 775–777 (2006).
[CrossRef] [PubMed]

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy106(10), 914–924 (2006).
[CrossRef] [PubMed]

2005 (3)

J. Luo, K. Ying, P. He, and J. Bai, “Properties of Savitzky–Golay digital differentiators,” Digit. Signal Process.15(2), 122–136 (2005).
[CrossRef]

P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett.30(5), 468–470 (2005).
[CrossRef] [PubMed]

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo)54(3), 191–197 (2005).
[CrossRef] [PubMed]

2004 (2)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy102(1), 37–49 (2004).
[CrossRef] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc.214(1), 51–61 (2004).
[CrossRef] [PubMed]

2002 (1)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc.206(3), 194–203 (2002).
[CrossRef] [PubMed]

2000 (1)

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

1998 (3)

T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun.147(4-6), 229–232 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett.23(11), 817–819 (1998).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80(12), 2586–2589 (1998).
[CrossRef]

1997 (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133(1-6), 339–346 (1997).
[CrossRef]

1996 (1)

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

1995 (1)

P. Barak, “Smoothing and differentiation by an adaptive-degree polynomial filter,” Anal. Chem.67(17), 2758–2762 (1995).
[CrossRef]

1991 (1)

P. A. Gorry, “General least-squares smoothing and differentiation of nonuniformly spaced data by the convolution method,” Anal. Chem.63(5), 534–536 (1991).
[CrossRef]

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984).
[CrossRef]

1983 (1)

1964 (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares Procedures,” Anal. Chem.36(8), 1627–1639 (1964).
[CrossRef]

Acosta, E.

Allen, L. J.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy106(10), 914–924 (2006).
[CrossRef] [PubMed]

Allman, B.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo)54(3), 191–197 (2005).
[CrossRef] [PubMed]

Anastasio, M. A.

Asundi, A.

Bai, J.

J. Luo, K. Ying, P. He, and J. Bai, “Properties of Savitzky–Golay digital differentiators,” Digit. Signal Process.15(2), 122–136 (2005).
[CrossRef]

Bai, X.

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

Barak, P.

P. Barak, “Smoothing and differentiation by an adaptive-degree polynomial filter,” Anal. Chem.67(17), 2758–2762 (1995).
[CrossRef]

Barbastathis, G.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc.206(3), 194–203 (2002).
[CrossRef] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc.214(1), 51–61 (2004).
[CrossRef] [PubMed]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc.206(3), 194–203 (2002).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett.23(11), 817–819 (1998).
[CrossRef] [PubMed]

Beleggia, M.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy102(1), 37–49 (2004).
[CrossRef] [PubMed]

Bie, R.

Carney, P. S.

Chen, F. R.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy106(10), 914–924 (2006).
[CrossRef] [PubMed]

Choo, C. O.

Colomb, T.

Cookson, D. J.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Cuche, E.

Cui, L.

Dasari, R. R.

Depeursinge, C.

Emery, Y.

Feld, M. S.

Findlay, S. D.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy106(10), 914–924 (2006).
[CrossRef] [PubMed]

Golay, M. J. E.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares Procedures,” Anal. Chem.36(8), 1627–1639 (1964).
[CrossRef]

Gorry, P. A.

P. A. Gorry, “General least-squares smoothing and differentiation of nonuniformly spaced data by the convolution method,” Anal. Chem.63(5), 534–536 (1991).
[CrossRef]

Gorthi, S. S.

Gureyev, T. E.

T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun.147(4-6), 229–232 (1998).
[CrossRef]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133(1-6), 339–346 (1997).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

He, P.

J. Luo, K. Ying, P. He, and J. Bai, “Properties of Savitzky–Golay digital differentiators,” Digit. Signal Process.15(2), 122–136 (2005).
[CrossRef]

Hsieh, W. K.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy106(10), 914–924 (2006).
[CrossRef] [PubMed]

Ikeda, T.

Ishizuka, K.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo)54(3), 191–197 (2005).
[CrossRef] [PubMed]

Kai, J. J.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy106(10), 914–924 (2006).
[CrossRef] [PubMed]

Kou, S. S.

Luo, J.

J. Luo, K. Ying, P. He, and J. Bai, “Properties of Savitzky–Golay digital differentiators,” Digit. Signal Process.15(2), 122–136 (2005).
[CrossRef]

Magistretti, P. J.

Marquet, P.

Martin, A. V.

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy106(10), 914–924 (2006).
[CrossRef] [PubMed]

McCartney, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc.214(1), 51–61 (2004).
[CrossRef] [PubMed]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc.214(1), 51–61 (2004).
[CrossRef] [PubMed]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc.206(3), 194–203 (2002).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80(12), 2586–2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett.23(11), 817–819 (1998).
[CrossRef] [PubMed]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133(1-6), 339–346 (1997).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc.214(1), 51–61 (2004).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80(12), 2586–2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett.23(11), 817–819 (1998).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Petruccelli, J. C.

Popescu, G.

Qu, W.

Rappaz, B.

Reed Teague, M.

Roberts, A.

Savitzky, A.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares Procedures,” Anal. Chem.36(8), 1627–1639 (1964).
[CrossRef]

Schofield, M. A.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy102(1), 37–49 (2004).
[CrossRef] [PubMed]

Schonbrun, E.

Schoonover, R. W.

Sheppard, C. J. R.

Singh, V. R.

Soto, M.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984).
[CrossRef]

Tian, L.

Volkov, V. V.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy102(1), 37–49 (2004).
[CrossRef] [PubMed]

Wall, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

Waller, L.

Weijuan, Q.

Wilkins, S. W.

T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun.147(4-6), 229–232 (1998).
[CrossRef]

Xue, B.

Xue, W.

Ying, K.

J. Luo, K. Ying, P. He, and J. Bai, “Properties of Savitzky–Golay digital differentiators,” Digit. Signal Process.15(2), 122–136 (2005).
[CrossRef]

Yingjie, Y.

Yuan, X.-H.

Zhang, L.

Zhao, M.

Zheng, S.

Zhou, F.

Zhu, Y.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy102(1), 37–49 (2004).
[CrossRef] [PubMed]

Zysk, A. M.

Anal. Chem. (3)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares Procedures,” Anal. Chem.36(8), 1627–1639 (1964).
[CrossRef]

P. A. Gorry, “General least-squares smoothing and differentiation of nonuniformly spaced data by the convolution method,” Anal. Chem.63(5), 534–536 (1991).
[CrossRef]

P. Barak, “Smoothing and differentiation by an adaptive-degree polynomial filter,” Anal. Chem.67(17), 2758–2762 (1995).
[CrossRef]

Appl. Opt. (2)

Digit. Signal Process. (1)

J. Luo, K. Ying, P. He, and J. Bai, “Properties of Savitzky–Golay digital differentiators,” Digit. Signal Process.15(2), 122–136 (2005).
[CrossRef]

J. Electron Microsc. (Tokyo) (1)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo)54(3), 191–197 (2005).
[CrossRef] [PubMed]

J. Microsc. (2)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc.214(1), 51–61 (2004).
[CrossRef] [PubMed]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc.206(3), 194–203 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (3)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133(1-6), 339–346 (1997).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984).
[CrossRef]

T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun.147(4-6), 229–232 (1998).
[CrossRef]

Opt. Express (5)

Opt. Lett. (7)

Phys. Rev. Lett. (2)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80(12), 2586–2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Ultramicroscopy (3)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy102(1), 37–49 (2004).
[CrossRef] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy83(1-2), 67–73 (2000).
[CrossRef] [PubMed]

A. V. Martin, F. R. Chen, W. K. Hsieh, J. J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy106(10), 914–924 (2006).
[CrossRef] [PubMed]

Other (3)

L. N. Trefethen, Finite difference and spectral methods for ordinary and partial differential equations, unpublished text, available at http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/pdetext.html , 1996.

S. J. Orfanidis, Introduction to Signal Processing (Prentice-Hall, Inc., 1995).

J. M. Cowley, Diffraction Physics, 2 ed. (North-Holland Pub. Co, 1993).

Supplementary Material (3)

» Media 1: MOV (1774 KB)     
» Media 2: MOV (511 KB)     
» Media 3: MOV (76 KB)     

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Figures (11)

Fig. 1
Fig. 1

Frequency characteristics of SGDF for n = 15 and various polynomial degrees; (a) frequency response of the SGDF filter; (b) the frequency response of the low-pass filter behind the SGDF filter in Log magnitude.

Fig. 2
Fig. 2

Normalized 0.3 dB cutoff frequencies f c = ω c /π for different data half-length n and polynomial degree m.

Fig. 3
Fig. 3

A flowchart schematic of the OFS algorithm.

Fig. 4
Fig. 4

Phase recovery comparison of the synthetic noisy test data; (a) true phase, (b) traditional TIE using two planes separated by 20μm (RMSE = 0.2552); (c) 1st fixed degree SGDF (RMSE = 0.0128); (d) 31st fixed degree SGDF (RMSE = 0.0651); (e) 7th fixed degree SGDF (RMSE = 0.0092); (f) traditional TIE using two planes separated by 100μm (RMSE = 0.0115); (g) adaptive-degree SGDF (RMSE = 0.0453); (h) OFS (RMSE = 0.0020); the square areas with red dot lines are magnified for clarity.

Fig. 5
Fig. 5

Phase error images for (a) 7th fixed degree SGDF, (b) traditional TIE using two planes separated by 100μm, (c) OFS.

Fig. 6
Fig. 6

(Media 1) The curve of RMSE evolution, shown with corresponding magnified areas.

Fig. 7
Fig. 7

Comparison of the RMSE for the estimated phase at different noise levels. The fixed degree SGDF uses m = 1, 5, 9, 15, 23, and 31 orders.

Fig. 8
Fig. 8

(a) The experimental test setup, and (b) (Media 2) intensity images from the focus stack.

Fig. 9
Fig. 9

Phase recovery comparison of a test phase object; (a) traditional TIE using two images separated by 100μm; (b) 1st fixed degree SGDF; (c) 5th fixed degree SGDF; (d) 15th fixed degree SGDF; (e) 27th fixed degree SGDF; (f) traditional TIE using two images separated by 700μm; (g) adaptive-degree SGDF; (h) OFS (Media 3); the bottom right red squares show the corresponding enlarged regions.

Fig. 10
Fig. 10

Comparison between the phases retrieved from digital holography and TIE; (a) hologram of the same test sample; (b) hologram Spectrum; (c) reconstructed phase from digital holography (the red dot square at bottom right shows a corresponding area of Fig. 9(h)); (d) cross-sections of the phase marked with the red and blue lines in (c).

Fig. 11
Fig. 11

Phase recovery results under significant noise (noise standard deviation 0.01); (a) intensity image with defocus distance 80μm, (b) 7th fixed degree SGDF (RMSE = 0.6016); (c) OFS (RMSE = 0.0616); (d) OFS with Tikhonov-regularization (RMSE = 0.0332).

Tables (1)

Tables Icon

Table 1 Normalized 0.3dB Cutoff Frequencies as a Function of m and n (n = 1~10)

Equations (33)

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k I( r ) z =[ I( r )ϕ( r ) ],
I( r ) z I( r,Δz )I( r,Δz ) 2Δz ,
I( r ) z i=n n a i I( r,iΔz ) Δz ,
a i = i ( 1 ) i+1 ( n! ) 2 ( n+i )!( ni )! .
a i = 3i n( n+1 )( 2n+1 ) .
P m (i)= k=0 m b k i k .
A T Ab= A T I,
b= ( A T A ) 1 A T I=HI.
b 1 = i=n n h 1,i I[n],
h s,i = k=0 m ( 2k+1 ) ( 2n ) ( k ) ( 2n+k+1 ) ( k+1 ) P k n ( i ) P k n,s ( t ),
P k n ( i )= j=0 k ( 1 ) j+k ( j+k ) ( 2j ) ( m+i ) ( j ) ( j! ) 2 ( 2m ) ( j ) ,
P k n,s ( t )= ( d s d x s P k n ( t ) ) x=t .
h 1,i = k=0 m ( 2k+1 ) ( 2n ) ( k ) ( 2n+k+1 ) ( k+1 ) P k n ( i ) P k n,1 ( 0 ).
h 1,i = i ( 1 ) i+1 ( n! ) 2 ( n+i )!( ni )! .
h 1,i = 3i n( n+1 )( 2n+1 ) .
H SG ( e jω )=jω H SG ( e jω ) jω = H ideal ( e jω ) H LP ( e jω ).
k I( r ) I( r ) z = 2 ϕ( r ).
F= ( SS R m 1 SS R m 2 ) / ( m 2 m 1 ) SS R m 2 / ( N m 2 +1 ) ,
SS R m =SS R m1 i=n n P m n ( i )I[i] / i=n n [ P m n ( i ) ] 2 ,m>1.
U( x,0 )=| U( x,0 ) |exp[ iϕ( x ) ]=exp[ iϕ( x )μ( x ) ],
I( x,z )= | U( x,z ) | 2 =12μ( x )Fcos( πλz u 2 )2ϕ( x )Fsin( πλz u 2 ).
I( x ) z i=n n a i I( x,iΔz ) Δz = i=n n a i i=n n a i { 2μ( x )Fcos( iΔzπλ u 2 )+2ϕ( x )Fsin( iΔzπλ u 2 ) } Δz .
F I( x ) z 2Fϕ( x ) Δz i=n n [ a i sin( iΔzπλ u 2 ) ] = 2Fϕ( x ) jΔz H SG ( e jω ) | ω=Δzπλ u 2 ,
F I( x ) z =2πλ u 2 Fϕ( x ),
ϕ ^ ( x )= F 1 { Fϕ( x ) H SG ( e jω ) | ω=Δzπλ u 2 jΔzπλ u 2 }=ϕ( x ) F 1 { H LP ( e jω ) | ω=Δzπλ u 2 },
f c = m0.3 3.5n1 ,n25.
ϕ m ( x )= ϕ ^ m ( x ) F 1 [ H m ( e jω ) | ω=Δzπλ u 2 ]= F 1 { F [ I( x ) z ] m H m ( e jω ) | ω=Δzπλ u 2 Δzπλ u 2 },
ϕ ( x )= ϕ 1 ( x )+ ϕ 3 ( x )+...+ ϕ 2n1 ( x ),
{ min 1 2 a T a s.t.Ba=c ,
[ (n) 0 (1) 0 1 1 0 n 0 (n) 1 (1) 1 0 1 1 n 1 (n) 2 (1) 2 0 1 2 n 2 (n) m (1) m 0 1 m n m ][ a n a 1 a 0 a 1 a n ]=[ 0 1 0 0 ].
[ E B T B 0 ][ a λ L ]=[ 0 c ],
λ L = ( B B T ) 1 c and a= ( ( B B T ) 1 B ) T c.
a= ( ( A T A ) 1 A T ) T c = H T c,

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