Abstract

The transport of intensity equation (TIE) is a two-dimensional second order elliptic partial differential equation that must be solved under appropriate boundary conditions. However, the boundary conditions are difficult to obtain in practice. The fast Fourier transform (FFT) based TIE solutions are widely adopted for its speed and simplicity. However, it implies periodic boundary conditions, which lead to significant boundary artifacts when the imposed assumption is violated. In this work, TIE phase retrieval is considered as an inhomogeneous Neumann boundary value problem with the boundary values experimentally measurable around a hard-edged aperture, without any assumption or prior knowledge about the test object and the setup. The analytic integral solution via Green’s function is given, as well as a fast numerical implementation for a rectangular region using the discrete cosine transform. This approach is applicable for the case of non-uniform intensity distribution with no extra effort to extract the boundary values from the intensity derivative signals. Its efficiency and robustness have been verified by several numerical simulations even when the objects are complex and the intensity measurements are noisy. This method promises to be an effective fast TIE solver for quantitative phase imaging applications.

© 2014 Optical Society of America

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2014 (1)

2013 (4)

2012 (1)

2011 (1)

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

2010 (4)

2007 (3)

2006 (2)

A. Talmi and E. N. Ribak, “Wavefront reconstruction from its gradients,” J. Opt. Soc. Am. A 23(2), 288–297 (2006).
[Crossref] [PubMed]

R. Reeves and K. Kubik, “Shift, scaling and derivative properties for the discrete cosine transform,” Signal Process. 86(7), 1597–1603 (2006).
[Crossref]

2005 (1)

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

2004 (2)

V. V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98(2-4), 271–281 (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]

2003 (1)

2002 (2)

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33(5), 411–416 (2002).
[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]

2001 (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
[Crossref]

1998 (2)

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]

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 (2)

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]

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996).
[Crossref]

1995 (3)

1992 (1)

E. Feig and S. Winograd, “Fast algorithms for the discrete cosine transform,” IEEE Trans. Signal Process. 40(9), 2174–2193 (1992).
[Crossref]

1990 (1)

1988 (1)

1984 (1)

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

1983 (1)

1979 (1)

J. Tribolet and R. E. Crochiere, “Frequency domain coding of speech,” IEEE Trans. Acoust. Speech Signal Process. 27(5), 512–530 (1979).
[Crossref]

1974 (1)

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE Trans. Comput. C-23(1), 90–93 (1974).
[Crossref]

Ahmed, N.

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE Trans. Comput. C-23(1), 90–93 (1974).
[Crossref]

Alimi, R.

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
[Crossref]

Allman, B.

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

Altmeyer, S.

Asundi, A.

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]

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

Campbell, C.

Campbell, H. I.

Chen, Q.

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]

Crochiere, R. E.

J. Tribolet and R. E. Crochiere, “Frequency domain coding of speech,” IEEE Trans. Acoust. Speech Signal Process. 27(5), 512–530 (1979).
[Crossref]

De Graef, M.

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33(5), 411–416 (2002).
[Crossref] [PubMed]

Dorrer, C.

Eliezer, S.

Falaggis, K.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

Feig, E.

E. Feig and S. Winograd, “Fast algorithms for the discrete cosine transform,” IEEE Trans. Signal Process. 40(9), 2174–2193 (1992).
[Crossref]

Frank, J.

Greenaway, A. H.

Gureyev, T.

Gureyev, T. E.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[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]

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[Crossref]

Han, I. W.

I. W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232–1237 (1995).
[Crossref]

Ishizuka, K.

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

Kou, S. S.

Kozacki, T.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

Kubik, K.

R. Reeves and K. Kubik, “Shift, scaling and derivative properties for the discrete cosine transform,” Signal Process. 86(7), 1597–1603 (2006).
[Crossref]

Kujawinska, M.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

Martinez-Carranza, J.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).

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]

Natarajan, T.

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE Trans. Comput. C-23(1), 90–93 (1974).
[Crossref]

Nugent, K.

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]

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]

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[Crossref]

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001).
[Crossref]

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]

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]

Paganin, D. M.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

Pavlov, K. M.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

Perelmutter, L.

Petruccelli, J. C.

Pinhasi, S. V.

Qu, W.

Rao, K. R.

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE Trans. Comput. C-23(1), 90–93 (1974).
[Crossref]

Reed Teague, M.

Reeves, R.

R. Reeves and K. Kubik, “Shift, scaling and derivative properties for the discrete cosine transform,” Signal Process. 86(7), 1597–1603 (2006).
[Crossref]

Ribak, E. N.

Roberts, A.

Roddier, F.

Schmalz, J. A.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: Validity of Teague's method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

Sheppard, C. J. R.

Streibl, N.

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

Talmi, A.

Tian, L.

Tribolet, J.

J. Tribolet and R. E. Crochiere, “Frequency domain coding of speech,” IEEE Trans. Acoust. Speech Signal Process. 27(5), 512–530 (1979).
[Crossref]

Volkov, V. V.

V. V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98(2-4), 271–281 (2004).
[Crossref] [PubMed]

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33(5), 411–416 (2002).
[Crossref] [PubMed]

Waller, L.

Wernicke, G.

Winograd, S.

E. Feig and S. Winograd, “Fast algorithms for the discrete cosine transform,” IEEE Trans. Signal Process. 40(9), 2174–2193 (1992).
[Crossref]

Woods, S. C.

Yu, Y.

Zhu, Y.

V. V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98(2-4), 271–281 (2004).
[Crossref] [PubMed]

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33(5), 411–416 (2002).
[Crossref] [PubMed]

Zuegel, J. D.

Zuo, C.

Appl. Opt. (2)

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

J. Tribolet and R. E. Crochiere, “Frequency domain coding of speech,” IEEE Trans. Acoust. Speech Signal Process. 27(5), 512–530 (1979).
[Crossref]

IEEE Trans. Comput. (1)

N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE Trans. Comput. C-23(1), 90–93 (1974).
[Crossref]

IEEE Trans. Signal Process. (1)

E. Feig and S. Winograd, “Fast algorithms for the discrete cosine transform,” IEEE Trans. Signal Process. 40(9), 2174–2193 (1992).
[Crossref]

J. Electron Microsc. (1)

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

J. Microsc. (2)

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Figures (6)

Fig. 1
Fig. 1

Simulation results for the phase function with zero Laplacian. (a) True phase distribution. (b) Longitudinal intensity derivative distribution in the absence of the aperture. (c) Longitudinal intensity derivative distribution with the aperture. (d) Reconstructed phase with the proposed algorithm. (e) Residual error map (RMSE 0.91%). The white dashed box in (a) outlines the aperture edge.

Fig. 2
Fig. 2

Reconstruction results for the zero-Laplacian phase function with non-uniform intensity distribution. (a) Simulated non-uniform intensity distribution. (b) Longitudinal intensity derivative distribution without the aperture. (c) Longitudinal intensity derivative distribution with the aperture. (d) Reconstructed phase with the proposed algorithm (using Algorithm 1 or 3). (e) Residual error map. (f) Residual error when the intensity distribution is assumed to be uniform (using Algorithm 2 or 4). The white dashed box in (a) outlines the aperture edge.

Fig. 3
Fig. 3

Simulation result of complex intensity and phase distribution. (a) True phase map. (b) Simulated intensity map. (c) Longitudinal intensity derivative without the aperture. (d) Longitudinal intensity derivative with the aperture. (e) Enlarged region corresponding to the boxed area of (c). (f) Enlarged region corresponding to the boxed area of (d). The white dashed box in (a) outlines the aperture edge, which defines the integral region Ω ¯ shown in (f).

Fig. 4
Fig. 4

Comparison of different TIE solutions. The first row shows the recovered phase by (a) the proposed algorithm with the aperture, (b) FFT-based algorithm with the aperture, (c) FFT-based algorithm without the aperture, (d) even symmetrization method without the aperture. The second row (e)-(f) shows the corresponding residual errors for different methods.

Fig. 5
Fig. 5

Phase reconstruction results by the proposed method at different noise levels when defocus distance is 10μm. The first row shows the longitudinal intensity derivative distribution with noise stand deviation (a) 0.0001, (b) 0.001, and (c) 0.01. The second row shows the reconstructed phases at different noise levels (d)-(f). The third row shows the corresponding residual error at different noise levels (g)-(i).

Fig. 6
Fig. 6

Phase reconstruction results by the proposed method when the defocus distance is not small. The first row shows the (a) longitudinal intensity derivative distribution, (b) enlarged region corresponding to the boxed area of (a), (c) reconstructed phase map and (d) residual error when the noise standard deviation is 10−3, and the defocus distance is 100μm. The second row (e)-(h) shows same figures when the noise standard deviation is 10−2, and the defocus distance is 500μm.

Tables (6)

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Table 1 Algorithm 1: Solution to TIE using the DCT-based Green's function method under non-uniform intensity distribution ( Iconstant ). (This gives identical results to Algorithm 2 when I=constant ).

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Table 2 Algorithm 2: Solution to TIE using the DCT-based Green's function method under non-uniform intensity distribution ( I=constant ).

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Table 3 Algorithm 3: Alternative to Algorithm 1 based on FFT and even extension (this gives the exactly the same solution as Algorithm 1).

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Table 4 Algorithm 4: Alternative to Algorithm 2 based on FFT and even extension] (this gives the exactly the same solution as Algorithm 2).

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Table 1 RMSEs of the Phases Reconstructed by Different Solutions to the TIE

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Table 2 RMSEs of the Phases Reconstructed by the Proposed Algorithm Under Different Noise Levels and at Difference Defocus Distances

Equations (69)

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k I( r ) z =[ I( r )ϕ( r ) ],
k I z =Iϕ+I 2 ϕ.
ϕ| Ω =g.
I ϕ n | Ω =g.
Ω I( r ) ϕ( r ) n ds= Ω k I( r ) z dr ,
2 I( r ) z dr=0.
Ω I( r ) z dr=0.
I= A Ω I 0 ={ I 0 r Ω ¯ 0others ,
I={ I δ Ω nrΩ I 0 ,rΩ 0others ,
2 f( r ) δ Ω d r= Ω f( r )ds .
k I z = A Ω ( I 0 2 ϕ+ I 0 ϕ )I ϕ n δ Ω .
2 k I( r ) z dr = Ω ¯ k I( r ) z dr= Ω k I( r ) z dr Ω I( r ) ϕ( r ) n ds.
Ω ¯ k I( r ) z dr= Ω k I( r ) z dr Ω I( r ) ϕ( r ) n ds=0,
ψ=Iϕ
k I z = 2 ψ,
ψ n | Ω = I ϕ n | Ω .
ψ( r )= Ω ( I 2 ϕ+Iϕ )G( r, r )d r Ω G( r, r )I( r ) ϕ( r ) n d s ,
2 G( r, r )=δ( r r ) A 1 , G( r, r ) n | Ω =0,
2 k I( r ) z G( r, r )d r = Ω ¯ k I( r ) z G( r, r )d r = Ω [ A Ω ( I 0 2 ϕ+ I 0 ϕ ) I Ω ϕ n δ Ω ] G( r, r )d r = Ω ( I 2 ϕ+Iϕ )G( r, r )d r Ω G( r, r )I( r ) ϕ( r ) n d s .
ψ( r )= Ω ¯ k I( r ) z G( r, r )d r .
( I 1 ψ )= 2 ϕ.
ϕ n | Ω ¯ =0.
ϕ( r )= Ω ¯ ( I 1 ψ )G( r, r )d r ,
G( r, r )= m=0 n=0 Ψ m,n (r) Ψ m,n ( r ) λ m,n .
Ψ m,n (x,y)= a m,n cos( mπ a x )cos( nπ b y ),
a m,n ={ 2 ab ,m,n0 2 ab ,morn=0 ,
λ m,n = π 2 ( m 2 a 2 + n 2 b 2 ).
ψ( r )= m=0 n=0 λ m,n 1 [ a m,n 2 Ω ¯ k I( r ) z cos( mπ a x )cos( nπ b y )d r ]cos( mπ a x )cos( nπ b y ).
S m,n = A m A n x =0 M1 y =0 N1 k I( r ) z cos[ mπ M ( x +0.5 ) ]cos[ nπ N ( y +0.5 ) ] ;
ψ( r )= m=0 M1 n=0 N1 A m A n λ m,n 1 S m,n cos[ mπ M ( x+0.5 ) ]cos[ nπ N ( y+0.5 ) ].
( I 1 ψ )= DCT ( I 1 DCT ψ )= xDCT ( I 1 xDCT ψ )+ yDCT ( I 1 yDCT ψ ),
I( r ) z I + ( r ) I ( r ) 2Δz .
k I z = I 2 ϕ .
Iϕ=ψ+rotη,
2 ϕ=( I 1 ψ )+( I 1 rotη )
f(x)= k=0 F k cos( πk a x ) .
F k = a k 0 a f( x )cos( πk a x )dx ,
a k ={ 2 a ,k0 1 a ,k=0 .
F(k)= A k n=0 N1 f(n)cos[ πk N ( n+0.5 ) ] ;k=0,1,...,N1.
A k ={ 2 N ,k0 1 N ,k=0 .
x(n)= k=0 N1 A k X(k)cos[ πk N ( n+0.5 ) ] ;n=0,1,...,N1.
Ψ k (n)= { A k cos[ πk N ( n+0.5 ) ] } n=1,2,...,N1 ;k=0,1,...,N1.
F( m,n )= A m A n x=0 M1 y=0 N1 f( x,y )cos[ πm M ( x+0.5 ) ]cos[ πn N ( y+0.5 ) ] ;
f( x,y )= m=0 M1 n=0 N1 A m A n F( m,n )cos[ πm M ( x+0.5 ) ]cos[ πn N ( y+0.5 ) ] .
f ¯ (n)={ f(n)n=0,1,...,N1 f(2N1n)n=N,N+1,...,2N1 .
F ¯ (k)= 1 2N n=0 2N1 f ¯ (n) e j( 2πkn 2N ) = 1 2N e j( πk N ) n=0 N1 f(n) cos[ πk N ( n+0.5 ) ],k=0,1,...,2N1.
F(k)= A k 2N e j( πk 2N ) F ¯ (k);k=0,1,...,N1.
F ¯ (k)={ A k 2N e j( πk 2N ) F(k);k=0,1,...,N1 0;k=0 A k 2N e j( πk 2N ) F(2Nk);k=N+1,...,2N1 .
f(n)= f ¯ (n);n=0,1,...,N1,
f (l) (x)= k=0 a k F k Ψ k (l) ( x ) = k=0 a k F k cos (l) ( πk a x ) .
f (n)= k=0 N1 A n ( πk a )F(k)sin[ πk N ( n+0.5 ) ] ;
f (n)= k=0 N1 A n ( πk a ) 2 F(k)cos[ πk N ( n+0.5 ) ] ;
f( x,y ) x = m=0 M1 n=0 N1 A m A n ( πm a )F( m,n )sin[ πm M ( x+0.5 ) ]cos[ πn N ( y+0.5 ) ] ;
2 f( x,y ) x 2 = m=0 M1 n=0 N1 A m A n ( πm a ) 2 F( m,n )cos[ πm M ( x+0.5 ) ]cos[ πn N ( y+0.5 ) ] .
f( x,y ) y = m=0 M1 n=0 N1 A m A n ( πn b )F( m,n )cos[ πm M ( x+0.5 ) ]sin[ πn N ( y+0.5 ) ] ;
2 f( x,y ) y 2 = m=0 M1 n=0 N1 A m A n ( πn b ) 2 F( m,n )cos[ πm M ( x+0.5 ) ]cos[ πn N ( y+0.5 ) ] .
DCT =( xDCT , yDCT ).
DCT f=( xDCT f, yDCT f ),
DCT f= xDCT f x + yDCT f y .
DCT 2 f( x,y )= xDCT 2 f+ yDCT 2 f= m=0 M1 n=0 N1 A m A n [ ( πm a ) 2 + ( πn b ) 2 ]F( m,n )cos[ πm M ( x+0.5 ) ]cos[ πn N ( y+0.5 ) ] ,
DCT 2 f= m=0 M1 n=0 N1 A m A n [ ( πm a ) 2 + ( πn b ) 2 ] 1 F( m,n )cos[ πm M ( x+0.5 ) ]cos[ πn N ( y+0.5 ) ] .
DCT 2 f=DC T 1 λ m,n 1 DCT( f ).
F ¯ (k)={ 2πjk a F(k);k=0,1,...,N 2πj a (k2N)F(k);k=N,...,2N1 .
f (n)= f ¯ (n);n=0,1,...,N1.
F ¯ (k)={ ( 2πk a ) 2 F(k);k=0,1,...,N1 [ 2π( k2N ) a ] 2 F(k);k=N,...,2N1 ,
f (n)= f ¯ (n);n=0,1,...,N1.
xDCT f= xFFT f ¯ ,
yDCT f= yFFT f ¯ ,
DCT 2 f= FFT 2 f ¯ ,

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