Abstract

A multiresolution (multiscale) analysis based on wavelet transform is applied to the problem of optical phase retrieval from the intensity measured in the in-line geometry (lens-free). The transport-of-intensity equation and the Fresnel diffraction integral are approximated in terms of a wavelet basis. A solution to the phase retrieval problem can be efficiently found in both cases using the multiresolution concept. Due to the hierarchical nature of wavelet spaces, wavelets are well suited to multiresolution methods that contain multigrid algorithms. Appropriate wavelet bases for the best solution approximation are discussed. The proposed approach reduces the computational complexity and accelerates the convergence of the solution. It is robust and reliable, and successful on both simulated and experimental images obtained with hard x rays.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
    [CrossRef]
  15. T. E. Gureyev, "Composite techniques for phase retrieval in the Fresnel region," Opt. Commun. 220, 49-58 (2003).
    [CrossRef]
  16. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
    [CrossRef]
  17. A. Brandt, "A guide to multigrid development," in Multigrid Methods (Springer-Verlag, 1982).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2004 (2)

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, "Optical phase retrieval by use of first Born- and Rytov-type approximations," Appl. Opt. 43, 2418-2430 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

2003 (2)

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

T. E. Gureyev, "Composite techniques for phase retrieval in the Fresnel region," Opt. Commun. 220, 49-58 (2003).
[CrossRef]

1999 (1)

T. E. Gureyev, "Transport of intensity equation for beams in an arbitrary state of temporal and spatial coherence," Optik (Stuttgart) 110, 263-266 (1999).

1997 (1)

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

1996 (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

1995 (1)

1990 (2)

1986 (1)

1983 (1)

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

1948 (1)

D. Gabor, "A new microscopic principle," Nature (London) 161, 777-778 (1948).
[CrossRef]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Blu, T.

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Brandt, A.

A. Brandt, "A guide to multigrid development," in Multigrid Methods (Springer-Verlag, 1982).
[CrossRef]

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Davis, T. J.

Fienup, J. R.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

Gabor, D.

D. Gabor, "A new microscopic principle," Nature (London) 161, 777-778 (1948).
[CrossRef]

Gao, D.

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gureyev, T. E.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, "Optical phase retrieval by use of first Born- and Rytov-type approximations," Appl. Opt. 43, 2418-2430 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, "Composite techniques for phase retrieval in the Fresnel region," Opt. Commun. 220, 49-58 (2003).
[CrossRef]

T. E. Gureyev, "Transport of intensity equation for beams in an arbitrary state of temporal and spatial coherence," Optik (Stuttgart) 110, 263-266 (1999).

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1932-1941 (1995).
[CrossRef]

Henson, V. E.

V. E. Henson, "Multigrid methods for nonlinear problems: an overview," Lawrence Livermore National Laboratory Tech. Rep. UCRL-JC-150259 (2003).

Kovacevic, J.

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice Hall, 1995).

Latto, A.

A. Latto, H. L. Resnikoff, and E. Tenenbaum, "The evaluation of connection coefficients of compactly supported wavelets," in Proceedings of the French-USA Workshop on Wavelets and Turbulence, Y.Maday, ed. (Springer-Verlag, Princeton University, 1991).

Liebling, M.

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

Mayo, S. C.

Millane, R. P.

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1932-1941 (1995).
[CrossRef]

Paganin, D.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Paganin, D. M.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

Pogany, A.

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, "Optical phase retrieval by use of first Born- and Rytov-type approximations," Appl. Opt. 43, 2418-2430 (2004).
[CrossRef] [PubMed]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

Resnikoff, H. L.

H. L. Resnikoff and R. O. Wells, Wavelet Analysis (Springer-Verlag, 1998).
[CrossRef]

A. Latto, H. L. Resnikoff, and E. Tenenbaum, "The evaluation of connection coefficients of compactly supported wavelets," in Proceedings of the French-USA Workshop on Wavelets and Turbulence, Y.Maday, ed. (Springer-Verlag, Princeton University, 1991).

Roberts, A.

Roddier, F.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Teague, M. R.

Tenenbaum, E.

A. Latto, H. L. Resnikoff, and E. Tenenbaum, "The evaluation of connection coefficients of compactly supported wavelets," in Proceedings of the French-USA Workshop on Wavelets and Turbulence, Y.Maday, ed. (Springer-Verlag, Princeton University, 1991).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

Unser, M.

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

Vetterli, M.

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice Hall, 1995).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).
[CrossRef]

Wackerman, C. C.

Wells, R. O.

H. L. Resnikoff and R. O. Wells, Wavelet Analysis (Springer-Verlag, 1998).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, "Optical phase retrieval by use of first Born- and Rytov-type approximations," Appl. Opt. 43, 2418-2430 (2004).
[CrossRef] [PubMed]

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Appl. Opt. (3)

IEEE Trans. Image Process. (1)

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nature (London) (1)

D. Gabor, "A new microscopic principle," Nature (London) 161, 777-778 (1948).
[CrossRef]

Opt. Commun. (2)

T. E. Gureyev, "Composite techniques for phase retrieval in the Fresnel region," Opt. Commun. 220, 49-58 (2003).
[CrossRef]

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004).
[CrossRef]

Optik (Stuttgart) (2)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

T. E. Gureyev, "Transport of intensity equation for beams in an arbitrary state of temporal and spatial coherence," Optik (Stuttgart) 110, 263-266 (1999).

Phys. Rev. Lett. (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, "Quantitative phase imaging using hard x-rays," Phys. Rev. Lett. 77, 2961-2964 (1996).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997).
[CrossRef]

Other (10)

A. Latto, H. L. Resnikoff, and E. Tenenbaum, "The evaluation of connection coefficients of compactly supported wavelets," in Proceedings of the French-USA Workshop on Wavelets and Turbulence, Y.Maday, ed. (Springer-Verlag, Princeton University, 1991).

C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics, 2002).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

A. Brandt, "A guide to multigrid development," in Multigrid Methods (Springer-Verlag, 1982).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++. The Art of Scientific Computing (Cambridge U. Press, 2002).

V. E. Henson, "Multigrid methods for nonlinear problems: an overview," Lawrence Livermore National Laboratory Tech. Rep. UCRL-JC-150259 (2003).

H. L. Resnikoff and R. O. Wells, Wavelet Analysis (Springer-Verlag, 1998).
[CrossRef]

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice Hall, 1995).

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Figures (11)

Fig. 1
Fig. 1

Schematic representation of the diffraction geometry.

Fig. 2
Fig. 2

Image used in the simulations as a phase profile.

Fig. 3
Fig. 3

Simulated diffraction image of the object with the phase profile shown in Fig. 2 and a nonlinear absorption profile (see text). The simulation parameters were maximum phase amplitude S max = 1 , wavelength λ = 0.1 nm , and distance z = 1 m .

Fig. 4
Fig. 4

Diagram illustrating the dependence of the TIE retrieval error on the resolution scale when the diffraction image is taken in the Fresnel region.

Fig. 5
Fig. 5

Phase profile retrieved from Fig. 3 using the TIE procedure.

Fig. 6
Fig. 6

Comparison of the convergence rates of the three algorithms used in the phase reconstruction from Fig. 3. Curve 1 corresponds to the ER algorithm. Curve 2 corresponds to a composite ER + Born -type algorithm. Curve 3 corresponds to the W-FAS algorithm.

Fig. 7
Fig. 7

Retrieved phase profiles that correspond to curves 1–3 in Fig. 6, respectively: (a) ER algorithm, (b) composite ER + Born -type algorithm, (c) W-FAS algorithm.

Fig. 8
Fig. 8

Convergence rate of the W-FAS algorithm for the object with the phase-only profile (Fig. 2), i.e., no amplitude modulation and zero initial guess.

Fig. 9
Fig. 9

SEM picture of the sample used in the experiment.

Fig. 10
Fig. 10

Images of the sample used in the experiment: (a) x-ray diffraction image, (b) phase profile retrieved using TIE procedure, (c) refined phase profile obtained using W-FAS procedure.

Fig. 11
Fig. 11

Line profile of the retrieved phase S in the middle of the letters “ph.”

Equations (41)

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

φ j n ( x ) = 2 j 2 φ ( 2 j x n )
ψ j n ( x ) = 2 j 2 ψ ( 2 j x n ) ,
φ ( x ) = k = 0 2 g 1 g k φ ( 2 x k ) ,
ψ ( x ) = k = 1 2 g 2 ( 1 ) k g k + 1 φ ( 2 x + k ) ,
f J ( x ) = n f J n ( a ) φ J n ( x ) .
f J ( x ) = n f G n ( a ) φ G n ( x ) + j = G J 1 n f j n ( d ) ψ j n .
f j n ( a ) = f ( x ) φ j n ( x ) d x ,
f j n ( d ) = f ( x ) ψ j n ( x ) d x ,
f G ( a ) = H G J f J ( a ) .
I ( x , y ) I 0 ( x , y ) = r k [ I 0 ( x , y ) S ( x , y ) ] ,
Ω φ ( x , y ) Δ I ( x , y ) d x d y = r k D 2 Ω I 0 ( x , y ) φ ( x , y ) S ( x , y ) d x d y ,
Δ I ( x , y ) = 2 J n m Δ I n m φ J n ( x ) φ J m ( y ) ,
I 0 ( x , y ) = 2 J n m I n m φ J n ( x ) φ J m ( y ) ,
S ( x , y ) = 2 J n m S n m φ J n ( x ) φ J m ( y ) ,
Δ I v w = r k h 2 n m k l ( Λ n l , v l 0 , 1 , 1 Λ m k , w k 0 , 0 , 0 + Λ n l , v l 0 , 0 , 0 Λ m k , w k 0 , 1 , 1 ) I l k S n m ,
Λ n , m 0 , 0 , 0 = φ J 0 ( x ) φ J n ( x ) φ J m ( x ) d x ,
Λ n , m 0 , 1 , 1 = φ J 0 ( x ) φ J n ( x ) φ J m ( x ) d x ,
Δ I J = A J S J ,
A J = [ δ m w l Λ n l , v l 0 , 1 , 1 I l w + δ n v k Λ m k , w k 0 , 1 , 1 I v k ]
A J = I 0 [ δ m w Λ n v 1 , 1 + δ n v Λ m w 1 , 1 ] ,
Λ n 1 , 1 = φ J 0 ( x ) φ J n ( x ) d x .
Δ I j = A j S j ,
Q ( x , y ) = ( U P ) ( x , y ) ,
I ( x , y ) = Q ( x , y ) Q * ( x , y ) ,
Q ( x ) = ( U p ) ( x ) .
Q J n = m U J m P J , n m ,
p J n = p ( x y ) φ J n ( x ) φ J 0 ( y ) d x d y .
Q G n ( a ) = m U G m ( a ) p G , n m ( a ) + m U G m ( d ) p G , n m ( d ) ,
P G n ( d ) = p ( x y ) φ G n ( x ) ψ G 0 ( y ) d x d y .
Φ ( q ) = { 1 , for π q π 0 , elsewhere } .
Q G n m = k l U G k l p G , n k p G , m l .
I ̃ G n m = Q G n m Q G n m * .
I G n m = I ̃ G n m + I G n m ( r e s ) ,
A j ( U j ) = I ̃ j
A j ( U j ) = A j ( V j ) + Δ I j .
H n = { 2 , for q n < π 2 1 , for q n = π 2 0 , elsewhere } .
f j k l = F 1 ( f ̃ a b W a W b ) ( 2 J j k , 2 J j l ) ,
f n m = F 1 ( f ̃ a b W a W b ) ( n , m ) ,
E = 100 % × S S 0 S 0 ,
C = ( I max I min ) ( I max + I min )
K = I I 0 I 0

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