## Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter - theory and applications |

Optics Express, Vol. 21, Issue 5, pp. 5346-5362 (2013)

http://dx.doi.org/10.1364/OE.21.005346

Acrobat PDF (2677 KB)

### 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

## 1. Introduction

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]

2. S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy **83**(1-2), 67–73 (2000). [CrossRef] [PubMed]

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]

4. 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]

5. 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]

6. M. Reed Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. **73**(11), 1434–1441 (1983). [CrossRef]

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

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]

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

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

10. A. M. Zysk, R. W. Schoonover, P. S. Carney, and M. A. Anastasio, “Transport of intensity and spectrum for partially coherent fields,” Opt. Lett. **35**(13), 2239–2241 (2010). [CrossRef] [PubMed]

11. S. S. Gorthi and E. Schonbrun, “Phase imaging flow cytometry using a focus-stack collecting microscope,” Opt. Lett. **37**(4), 707–709 (2012). [CrossRef] [PubMed]

12. L. Waller, S. S. Kou, C. J. R. Sheppard, and G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express **18**(22), 22817–22825 (2010). [CrossRef] [PubMed]

13. M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy **102**(1), 37–49 (2004). [CrossRef] [PubMed]

15. 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]

13. M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy **102**(1), 37–49 (2004). [CrossRef] [PubMed]

15. 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]

14. 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]

16. M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. **46**(33), 7978–7981 (2007). [CrossRef] [PubMed]

15. 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]

17. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express **18**(12), 12552–12561 (2010). [CrossRef] [PubMed]

17. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express **18**(12), 12552–12561 (2010). [CrossRef] [PubMed]

18. R. Bie, X.-H. Yuan, M. Zhao, and L. Zhang, “Method for estimating the axial intensity derivative in the TIE with higher order intensity derivatives and noise suppression,” Opt. Express **20**(7), 8186–8191 (2012). [CrossRef] [PubMed]

19. B. Xue, S. Zheng, L. Cui, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express **19**(21), 20244–20250 (2011). [CrossRef] [PubMed]

20. S. Zheng, B. Xue, W. Xue, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express **20**(2), 972–985 (2012). [CrossRef] [PubMed]

18. R. Bie, X.-H. Yuan, M. Zhao, and L. Zhang, “Method for estimating the axial intensity derivative in the TIE with higher order intensity derivatives and noise suppression,” Opt. Express **20**(7), 8186–8191 (2012). [CrossRef] [PubMed]

20. S. Zheng, B. Xue, W. Xue, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express **20**(2), 972–985 (2012). [CrossRef] [PubMed]

## 2. Problem formulation

### 2.1 Transport of intensity equation

6. M. Reed Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. **73**(11), 1434–1441 (1983). [CrossRef]

13. M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy **102**(1), 37–49 (2004). [CrossRef] [PubMed]

14. 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]

*compromise*is made where

14. 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]

21. 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,” Ultramicroscopy **106**(10), 914–924 (2006). [CrossRef] [PubMed]

### 2.1 Multiple-plane schemes for derivative estimation

*high-order finite difference*method was first used by Ishizuka and Allman [15

**54**(3), 191–197 (2005). [CrossRef] [PubMed]

*et al.*[17

17. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express **18**(12), 12552–12561 (2010). [CrossRef] [PubMed]

22. 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.

19. B. Xue, S. Zheng, L. Cui, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express **19**(21), 20244–20250 (2011). [CrossRef] [PubMed]

*et al.*[17

**18**(12), 12552–12561 (2010). [CrossRef] [PubMed]

*least-squares fitting*method to estimate the first order derivative. Because the least-squares fitting process weights all data equally, and the high-order polynomial fitting conserves the non-linear component of the original data, so both the higher order and noise effect can be treated simultaneously.

### 2.2 Main connections with Savitzky-Golay filters

*Observation 1*: The high-order finite difference method corresponds to the SGDF with degree

*Observation 2*: The noise-reduction finite difference method corresponds to the SGDF with degree 1.

*Observation 3*: The higher order finite difference with noise-reduction method corresponds to the SGDF with degree

19. B. Xue, S. Zheng, L. Cui, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express **19**(21), 20244–20250 (2011). [CrossRef] [PubMed]

20. S. Zheng, B. Xue, W. Xue, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express **20**(2), 972–985 (2012). [CrossRef] [PubMed]

25. 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]

### 2.3 Property of Savitzky-Golay differentiation filters

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

16. M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. **46**(33), 7978–7981 (2007). [CrossRef] [PubMed]

*m*

^{th}degree SGDF is an unbiased estimator of the derivative, and can achieve the Cramer-Rao lower bound if the signal can be perfectly modeled by an

*m*-order polynomial and the observed data are with independent Gaussian white noise [26

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

## 3. Applications of SGDF in phase retrieval by TIE

### 3.1 Derivative estimation using adaptive-degree SGDF

**18**(12), 12552–12561 (2010). [CrossRef] [PubMed]

27. 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]

28. 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]

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

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

### 3.2 Phase retrieval by optimal frequency selection

**214**(1), 51–61 (2004). [CrossRef] [PubMed]

28. 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]

**18**(12), 12552–12561 (2010). [CrossRef] [PubMed]

27. 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]

28. 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]

*low-pass filter that is implicit in the SGDF*. Now, reconsidering Fig. 1(b), which indicates the effect of low-pass filter decreases with increasing degree of SGDF, one may think that a higher-degree SGDF gives a more intact phase (with wider range of spatial frequencies can be accurately recovered) than a lower degree one. However, the higher degree SGDF results in larger NRR than the one assumes a smaller order. In another word, smaller order filters always give estimates of the low-frequency phase with higher SNR, but they suffer from the problem of not having enough response for high frequency phase variations. The above discussion reaffirmed that using a single degree of SGDF cannot offer the optimal solution and a tradeoff between phase information at high and low spatial frequencies is necessary.

*correct reconstruction*’ or ‘

*reliable retrieval*’ is necessary.

*reliable retrieval*. The 0.3dB-points for filters with different

*m*and

*n*, Eq. (26) should be adequate for most applications.

*m*

^{th}degree SGDF. For

*m*

^{th}order SGDF

*m*

^{th}degree SGDF. The filter can either be applied to the phase obtained by TIE or together with the inverse Laplacian when solving the TIE in Fourier domain. A flowchart schematic of the whole procedure, which we call as optimal frequency selection (OFS) is shown in Fig. 3 . Note usually the final reconstructed phase is not necessarily a composite of all possible orders because of the sampling effect and the limited physically significant frequency of the object. Therefore, we recommend the summation of Eq. (28) begins from

## 4. Simulations

## 5. Experiments

*f*system - two lenses of focal length

*f*= 25 mm separated by the distance 2

*f*, and the distance from the object to the first lens is

*f*. The camera is set on a translation stage in order to modify the defocus distance. The phase object under test is a geometry pattern etched on PMMA substrate, which is also shown in Fig. 8(a). Fifty-one images were captured with an equal separation of 50μm (Media 2). Some data samples from the intensity stack are shown in Fig. 8(b). All these images are recorded by a monochrome CCD imaging device (The Imaging Source DMK 41AU02, 4.65μm pixel size) and digitally processed using MATLAB.

31. Q. Weijuan, C. O. Choo, Y. Yingjie, and A. Asundi, “Microlens characterization by digital holographic microscopy with physical spherical phase compensation,” Appl. Opt. **49**(33), 6448–6454 (2010). [CrossRef] [PubMed]

31. Q. Weijuan, C. O. Choo, Y. Yingjie, and A. Asundi, “Microlens characterization by digital holographic microscopy with physical spherical phase compensation,” Appl. Opt. **49**(33), 6448–6454 (2010). [CrossRef] [PubMed]

32. W. Qu, C. O. Choo, V. R. Singh, Y. Yingjie, and A. Asundi, “Quasi-physical phase compensation in digital holographic microscopy,” J. Opt. Soc. Am. A **26**(9), 2005–2011 (2009). [CrossRef] [PubMed]

## 6. Conclusions and discussions

## Appendix A: Proof of Observation 3

## Acknowledgments

## References and links

1. | E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. |

2. | S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy |

3. | T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun. |

4. | G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. |

5. | 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. |

6. | M. Reed Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. |

7. | N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. |

8. | A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. |

9. | D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. |

10. | A. M. Zysk, R. W. Schoonover, P. S. Carney, and M. A. Anastasio, “Transport of intensity and spectrum for partially coherent fields,” Opt. Lett. |

11. | S. S. Gorthi and E. Schonbrun, “Phase imaging flow cytometry using a focus-stack collecting microscope,” Opt. Lett. |

12. | L. Waller, S. S. Kou, C. J. R. Sheppard, and G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express |

13. | M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy |

14. | D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. |

15. | K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) |

16. | M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. |

17. | L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express |

18. | R. Bie, X.-H. Yuan, M. Zhao, and L. Zhang, “Method for estimating the axial intensity derivative in the TIE with higher order intensity derivatives and noise suppression,” Opt. Express |

19. | B. Xue, S. Zheng, L. Cui, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express |

20. | S. Zheng, B. Xue, W. Xue, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express |

21. | 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,” Ultramicroscopy |

22. | 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. |

23. | S. J. Orfanidis, |

24. | A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares Procedures,” Anal. Chem. |

25. | P. A. Gorry, “General least-squares smoothing and differentiation of nonuniformly spaced data by the convolution method,” Anal. Chem. |

26. | J. Luo, K. Ying, P. He, and J. Bai, “Properties of Savitzky–Golay digital differentiators,” Digit. Signal Process. |

27. | T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. |

28. | K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. |

29. | P. Barak, “Smoothing and differentiation by an adaptive-degree polynomial filter,” Anal. Chem. |

30. | J. M. Cowley, |

31. | Q. Weijuan, C. O. Choo, Y. Yingjie, and A. Asundi, “Microlens characterization by digital holographic microscopy with physical spherical phase compensation,” Appl. Opt. |

32. | W. Qu, C. O. Choo, V. R. Singh, Y. Yingjie, and A. Asundi, “Quasi-physical phase compensation in digital holographic microscopy,” J. Opt. Soc. Am. A |

33. | S. S. Kou, L. Waller, G. Barbastathis, and C. J. R. Sheppard, “Transport-of-intensity approach to differential interference contrast (TI-DIC) microscopy for quantitative phase imaging,” Opt. Lett. |

34. | L. Tian, J. C. Petruccelli, and G. Barbastathis, “Nonlinear diffusion regularization for transport of intensity phase imaging,” Opt. Lett. |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 9, 2013

Revised Manuscript: February 8, 2013

Manuscript Accepted: February 13, 2013

Published: February 25, 2013

**Citation**

Chao Zuo, Qian Chen, Yingjie Yu, and Anand Asundi, "Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter - theory and applications," Opt. Express **21**, 5346-5362 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5346

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### References

- 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]
- T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun.147(4-6), 229–232 (1998). [CrossRef]
- 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]
- 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]
- M. Reed Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am.73(11), 1434–1441 (1983). [CrossRef]
- N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49(1), 6–10 (1984). [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]
- A. M. Zysk, R. W. Schoonover, P. S. Carney, and M. A. Anastasio, “Transport of intensity and spectrum for partially coherent fields,” Opt. Lett.35(13), 2239–2241 (2010). [CrossRef] [PubMed]
- S. S. Gorthi and E. Schonbrun, “Phase imaging flow cytometry using a focus-stack collecting microscope,” Opt. Lett.37(4), 707–709 (2012). [CrossRef] [PubMed]
- L. Waller, S. S. Kou, C. J. R. Sheppard, and G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express18(22), 22817–22825 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt.46(33), 7978–7981 (2007). [CrossRef] [PubMed]
- L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express18(12), 12552–12561 (2010). [CrossRef] [PubMed]
- R. Bie, X.-H. Yuan, M. Zhao, and L. Zhang, “Method for estimating the axial intensity derivative in the TIE with higher order intensity derivatives and noise suppression,” Opt. Express20(7), 8186–8191 (2012). [CrossRef] [PubMed]
- B. Xue, S. Zheng, L. Cui, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express19(21), 20244–20250 (2011). [CrossRef] [PubMed]
- S. Zheng, B. Xue, W. Xue, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express20(2), 972–985 (2012). [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]
- 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).
- 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]
- J. Luo, K. Ying, P. He, and J. Bai, “Properties of Savitzky–Golay digital differentiators,” Digit. Signal Process.15(2), 122–136 (2005). [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]
- P. Barak, “Smoothing and differentiation by an adaptive-degree polynomial filter,” Anal. Chem.67(17), 2758–2762 (1995). [CrossRef]
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