Abstract

We demonstrate a nonlinear diffusion (NLD) regularization method to solve the transport of intensity equation (TIE). A novel NLD regularization function is proposed to enforce piecewise-constant priors and to remove low-frequency artifacts in the TIE solution.

© 2012 Optical Society of America

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References

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  1. M. R. Teague, J. Opt. Soc. Am. 73, 1434 (1983).
    [CrossRef]
  2. M. Fernández-Guasti, J. L. Jiménez, F. Granados-Agustín, and A. Cornejo-Rodríguez, J. Opt. Soc. Am. A 20, 1629 (2003).
    [CrossRef]
  3. N. Streibl, Opt. Commun. 49, 6 (1984).
    [CrossRef]
  4. J. Petruccelli, L. Tian, and G. Barbastathis, in Biomedical Optics (Optical Society of America, 2012), p. BSu3A.69.
  5. E. D. Barone-Nugent, A. Barty, and K. A. Nugent, J. Microsc. 206, 194 (2002).
    [CrossRef]
  6. L. Waller, S. S. Kou, C. J. R. Sheppard, and G. Barbastathis, Opt. Express 18, 22817 (2010).
    [CrossRef]
  7. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
    [CrossRef]
  8. K. Ishizuka and B. Allman, J. Electron Microsc. 54, 191 (2005).
    [CrossRef]
  9. L. Waller, L. Tian, and G. Barbastathis, Opt. Express 18, 12552 (2010).
    [CrossRef]
  10. L. Waller, M. Tsang, S. Ponda, S. Y. Yang, and G. Barbastathis, Opt. Express 19, 2805 (2011).
    [CrossRef]
  11. L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, Opt. Express 20, 8296 (2012).
    [CrossRef]
  12. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
    [CrossRef]
  13. T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1942 (1995).
    [CrossRef]
  14. C. J. R. Sheppard, J. Opt. Soc. Am. A 21, 828 (2004).
    [CrossRef]
  15. O. Scherzer and J. Weickert, J. Math. Imaging Vis. 12, 43 (2000).
    [CrossRef]
  16. P. Perona and J. Malik, IEEE Trans. Pattern Anal. 12, 629 (1990).
    [CrossRef]
  17. J. Weickert, Lect. Notes Comput. Sci. 1252, 1 (1997).
    [CrossRef]
  18. D. Strong and T. Chan, Tech. Rep. 96–41 (UCLA Math Department, 1996).
  19. J. Weickert, B. Romeny, and M. Viergever, IEEE Trans. Image Process. 7, 398 (1998).
    [CrossRef]
  20. J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
    [CrossRef]

2012 (1)

2011 (1)

2010 (2)

2007 (1)

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

2005 (1)

K. Ishizuka and B. Allman, J. Electron Microsc. 54, 191 (2005).
[CrossRef]

2004 (2)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

C. J. R. Sheppard, J. Opt. Soc. Am. A 21, 828 (2004).
[CrossRef]

2003 (1)

2002 (1)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, J. Microsc. 206, 194 (2002).
[CrossRef]

2000 (1)

O. Scherzer and J. Weickert, J. Math. Imaging Vis. 12, 43 (2000).
[CrossRef]

1998 (1)

J. Weickert, B. Romeny, and M. Viergever, IEEE Trans. Image Process. 7, 398 (1998).
[CrossRef]

1997 (1)

J. Weickert, Lect. Notes Comput. Sci. 1252, 1 (1997).
[CrossRef]

1996 (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

1995 (1)

1990 (1)

P. Perona and J. Malik, IEEE Trans. Pattern Anal. 12, 629 (1990).
[CrossRef]

1984 (1)

N. Streibl, Opt. Commun. 49, 6 (1984).
[CrossRef]

1983 (1)

Allman, B.

K. Ishizuka and B. Allman, J. Electron Microsc. 54, 191 (2005).
[CrossRef]

Barbastathis, G.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, J. Microsc. 206, 194 (2002).
[CrossRef]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, J. Microsc. 206, 194 (2002).
[CrossRef]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Chan, T.

D. Strong and T. Chan, Tech. Rep. 96–41 (UCLA Math Department, 1996).

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Cornejo-Rodríguez, A.

Fernández-Guasti, M.

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Granados-Agustín, F.

Gureyev, T. E.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1942 (1995).
[CrossRef]

Ishizuka, K.

K. Ishizuka and B. Allman, J. Electron Microsc. 54, 191 (2005).
[CrossRef]

Jiménez, J. L.

Kou, S. S.

Lee, J.

Malik, J.

P. Perona and J. Malik, IEEE Trans. Pattern Anal. 12, 629 (1990).
[CrossRef]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, J. Microsc. 206, 194 (2002).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, J. Opt. Soc. Am. A 12, 1942 (1995).
[CrossRef]

Oh, S. B.

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Perona, P.

P. Perona and J. Malik, IEEE Trans. Pattern Anal. 12, 629 (1990).
[CrossRef]

Petruccelli, J.

J. Petruccelli, L. Tian, and G. Barbastathis, in Biomedical Optics (Optical Society of America, 2012), p. BSu3A.69.

Ponda, S.

Roberts, A.

Romeny, B.

J. Weickert, B. Romeny, and M. Viergever, IEEE Trans. Image Process. 7, 398 (1998).
[CrossRef]

Scherzer, O.

O. Scherzer and J. Weickert, J. Math. Imaging Vis. 12, 43 (2000).
[CrossRef]

Sheppard, C. J. R.

Streibl, N.

N. Streibl, Opt. Commun. 49, 6 (1984).
[CrossRef]

Strong, D.

D. Strong and T. Chan, Tech. Rep. 96–41 (UCLA Math Department, 1996).

Teague, M. R.

Tian, L.

L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, Opt. Express 20, 8296 (2012).
[CrossRef]

L. Waller, L. Tian, and G. Barbastathis, Opt. Express 18, 12552 (2010).
[CrossRef]

J. Petruccelli, L. Tian, and G. Barbastathis, in Biomedical Optics (Optical Society of America, 2012), p. BSu3A.69.

Tsang, M.

Viergever, M.

J. Weickert, B. Romeny, and M. Viergever, IEEE Trans. Image Process. 7, 398 (1998).
[CrossRef]

Waller, L.

Weickert, J.

O. Scherzer and J. Weickert, J. Math. Imaging Vis. 12, 43 (2000).
[CrossRef]

J. Weickert, B. Romeny, and M. Viergever, IEEE Trans. Image Process. 7, 398 (1998).
[CrossRef]

J. Weickert, Lect. Notes Comput. Sci. 1252, 1 (1997).
[CrossRef]

Yang, S. Y.

IEEE Trans. Image Process. (2)

J. Weickert, B. Romeny, and M. Viergever, IEEE Trans. Image Process. 7, 398 (1998).
[CrossRef]

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

IEEE Trans. Pattern Anal. (1)

P. Perona and J. Malik, IEEE Trans. Pattern Anal. 12, 629 (1990).
[CrossRef]

J. Electron Microsc. (1)

K. Ishizuka and B. Allman, J. Electron Microsc. 54, 191 (2005).
[CrossRef]

J. Math. Imaging Vis. (1)

O. Scherzer and J. Weickert, J. Math. Imaging Vis. 12, 43 (2000).
[CrossRef]

J. Microsc. (2)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, J. Microsc. 206, 194 (2002).
[CrossRef]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, J. Microsc. 214, 51 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Lect. Notes Comput. Sci. (1)

J. Weickert, Lect. Notes Comput. Sci. 1252, 1 (1997).
[CrossRef]

Opt. Commun. (1)

N. Streibl, Opt. Commun. 49, 6 (1984).
[CrossRef]

Opt. Express (4)

Phys. Rev. Lett. (1)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996).
[CrossRef]

Other (2)

D. Strong and T. Chan, Tech. Rep. 96–41 (UCLA Math Department, 1996).

J. Petruccelli, L. Tian, and G. Barbastathis, in Biomedical Optics (Optical Society of America, 2012), p. BSu3A.69.

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Figures (3)

Fig. 1.
Fig. 1.

(a) NLD regularizing function and (b) magnitude of the flux as a function of the normalized magnitude of the gradient of the phase.

Fig. 2.
Fig. 2.

(a) Original phase, (b) intensity derivative, (c) direct, (d) Tikhonov-regularized, (e) TV-regularized, (f) Weickert function regularized, and (g) hybrid function regularized solutions. (f) Phase cross sections taken along the dashed-line in (a).

Fig. 3.
Fig. 3.

Average per-pixel RMS error versus SNR.

Equations (4)

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

k I 0 I ( x , y , z ) z = 2 ϕ ( x , y ) ,
minimize 1 2 f A ϕ 2 + τ Ψ ( | ϕ | ) d x d y .
ϕ ( x , y ; t ) t = τ · ( Ψ ( | ϕ | ) | ϕ | ϕ ) .
F H ( s ) = 1 exp ( 3.86 / s 12 ) .

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