## Phase and amplitude imaging from noisy images by Kalman filtering |

Optics Express, Vol. 19, Issue 3, pp. 2805-2814 (2011)

http://dx.doi.org/10.1364/OE.19.002805

Acrobat PDF (1967 KB)

### Abstract

We propose and demonstrate a computational method for complex-field imaging from many noisy intensity images with varying defocus, using an extended complex Kalman filter. The technique offers dynamic smoothing of noisy measurements and is recursive rather than iterative, so is suitable for adaptive measurements. The Kalman filter provides near-optimal results in very low-light situations and may be adapted to propagation through turbulent, scattering, or nonlinear media.

© 2011 Optical Society of America

## 1. Introduction

1. W. Bragg, “Elimination of the unwanted image in diffraction microscopy,” Nature **167**, 190–191 (1951). [CrossRef] [PubMed]

2. C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. A **21**(5), 828–831 (2004). [CrossRef]

*any*complex transfer function will introduce phase contrast. Defocus is a popular way to introduce a complex-valued transfer function, because it is pure-phase (no absorption) and simple to implement - one need only move the camera along the optical axis between image captures (see setup in Fig. 1). Since intensity of a complex-field is proportional to amplitude squared, the measurement process is nonlinear.

3. D Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D **6**, L6–L9 (1973). [CrossRef]

*i.e.*far field), and alternately bounces between the two domains, updating an estimate of the complex-field at each step with measured or a priori information [5

5. R. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. A **66**, 961–964 (1976). [CrossRef]

8. G. Yang, B. Dong, B. Gu, J. Zhuang, and O. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. **33**(2), 209–218 (1994). [CrossRef] [PubMed]

14. H. Stark, Y. Yang, and Y. Yang, *Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics*. (John Wiley & Sons, 1998). [PubMed]

15. T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics **4**(3), 188–193 (2010). [CrossRef]

17. R. Horstmeyer, S. Oh, and R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express **18**(21), 22545–22555 (2010). [CrossRef] [PubMed]

18. A. Devaney and R. Childlaw, “On the uniqueness question in the problem of phase retrieval from intensity measurements,” J. Opt. Soc. Am. A **68**1352–1354 (1978). [CrossRef]

19. J. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. **32**1737–1746 (1993). [CrossRef] [PubMed]

16. S. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express **16**(5), 3484–3489 (2008). [CrossRef] [PubMed]

*i.e.*a stack of defocused images) [19

19. J. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. **32**1737–1746 (1993). [CrossRef] [PubMed]

22. G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. **30**(8), 833–835 (2005). [CrossRef] [PubMed]

23. H. Campbell, S. Zhang, A. Greenaway, and S. Restaino, “Generalized phase diversity for wave-front sensing,” Opt. Lett. **29**(23), 2707–2709 (2004). [CrossRef] [PubMed]

8. G. Yang, B. Dong, B. Gu, J. Zhuang, and O. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. **33**(2), 209–218 (1994). [CrossRef] [PubMed]

24. B. Dean and C. Bowers, “Diversity selection for phase-diverse phase retrieval,” J. Opt. Soc. Am. A **20**(8), 1490–1504 (2003). [CrossRef]

25. R. Gonsalves, “Phase retrieval by differential intensity measurements,” J. Opt. Soc. Am. A **4**, 166–170 (1987). [CrossRef]

31. J. Miao, D. Sayre, and H. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A **15**(6), 1662–1669 (1998). [CrossRef]

26. W. Southwell, “Wave-front analyzer using a maximum likelihood algorithm,” J. Opt. Soc. Am. **67**(3), 396–399 (1977). [CrossRef]

27. R. Gonsalves, “Small-phase solution to the phase-retrieval problem,” Opt. Lett. **26**(10), 684–685 (2001). [CrossRef]

28. T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. **231**(1–6), 53–70 (2004). [CrossRef]

29. J. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. **32**(12), 1617–1619 (2007). [CrossRef] [PubMed]

30. D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. **206**(1), 33–40 (2002). [CrossRef] [PubMed]

32. S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. **207**, 79–96 (2002). [CrossRef] [PubMed]

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

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

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

36. 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]

37. L. Waller, Y. Luo, S. Y. Yang, and G. Barbastathis, “Transport of intensity phase imaging in a volume holographic microscope,” Opt. Lett. **35**(17), 2961–2963 (2010). [CrossRef] [PubMed]

39. 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]

*Δz*, with a single model.

41. R. Paxman, T. Schulz, and J. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A **9**(7), 1072–1085 (1992). [CrossRef]

42. R. Paxman and J. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A **5**(6), 914–923 (1988). [CrossRef]

43. D. Lee, M. Roggemann, B. Welsh, and E. Crosby, “Evaluation of least-squares phase-diversity technique for space telescope wave-front sensing,” Appl. Opt. **36**(35), 9186–9197 (1997). [CrossRef]

43. D. Lee, M. Roggemann, B. Welsh, and E. Crosby, “Evaluation of least-squares phase-diversity technique for space telescope wave-front sensing,” Appl. Opt. **36**(35), 9186–9197 (1997). [CrossRef]

44. M. Roggemann, D. Tyler, and M. Bilmont, “Linear reconstruction of compensated images: theory and experimental results,” Appl. Opt. **31**(35), 7429–7441 (1992). [CrossRef] [PubMed]

45. M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. **35**(10), 4556–4567 (2008). [CrossRef] [PubMed]

*phase estimation accuracy is fundamentally limited by the defocus distances chosen and noise levels*. The Cramer-Rao bound (CRB) is an information theory metric describing the minimum achievable error of

*any*estimator, with no guarantee that this ideal estimator exists [46

46. D. Lee, M. Roggemann, and B. Welsh, “Cramér-Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A **16**(5), 1005–1015 (1999). [CrossRef]

*efficient*. Unfortunately, the bound itself is object-dependent, implying that the optimal measurement set is

*object-dependent*and only heuristic conclusions may be made. Generally, larger distances provide better diffraction contrast [24

24. B. Dean and C. Bowers, “Diversity selection for phase-diverse phase retrieval,” J. Opt. Soc. Am. A **20**(8), 1490–1504 (2003). [CrossRef]

32. S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. **207**, 79–96 (2002). [CrossRef] [PubMed]

47. L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E **63**(3), 037602 (2001). [CrossRef]

48. R. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. **82**(1), 35–45 (1960). [CrossRef]

*efficient*algorithm for linear Gaussian systems, guaranteeing optimal estimation. This means that the noise of the result approaches the fundamental limit of the noise in a single image divided by the total number of images used. Here we use a complex Extended Kalman filter (EKF), augmenting the nominal KF to handle nonlinear and complex-valued systems [49]. The filter simultaneously estimates complex-field and filters out noise from the intensity measurements, so is robust to very high noise levels and is able to provide the near-optimal result.

## 2. Theory

*A*(

*x*,

*y*,

*z*

_{0}) = |

*A*(

*x*,

*y*,

*z*

_{0})|

*e*

^{iϕ(x,y,z0)}at a distance

*z*

_{0}from the camera, where

*ϕ*(

*x*,

*y*,

*z*

_{0}) is the phase. We capture a sequence of

*N*noisy intensity measurements,

*I*(

*x*,

*y*,

*z*) = |

_{n}*A*(

*x*,

*y*,

*z*)|

_{n}^{2}+

*υ**at various distances*

_{n}*z*

_{0},

*z*

_{1}, ...

*z*separated by

_{N}*Δz*, where

*υ**describes the noise at each pixel. We assume here that the field propagates through a linear medium and obeys the homogeneous paraxial wave equation where*

_{n}*λ*is the wavelength of illumination and ∇

_{⊥}is the gradient operator in the lateral (

*x*,

*y*) dimensions only. We assume that the probability distribution of measured intensity at each pixel is independent and Poisson distributed: where

*γ*is the photon count detected by the camera. We would like to find the

*conditional*probability distribution of

*A*(

*x*,

*y*,

*z*) given measurements of

_{n}*I*(

*x*,

*y*,

*z*). Because the statistics of

_{n}*I*(

*x*,

*y*,

*z*) are non-Gaussian and depend nonlinearly on

_{n}*A*(

*x*,

*y*,

*z*), the estimation problem is nonlinear, and it is computationally expensive to solve for the exact conditional probability distribution

_{n}*P*[

*A*(

*x*,

*y*,

*z*

_{0})|

*I*(

*x*,

*y*,

*z*)]. One way of obtaining a near-optimal estimate is to use the EKF followed by back-propagation, also called the Rauch-Tung-Streibel (RTS) smoother [50

_{n}50. J. Crassidis and J. Junkins, *Optimal Estimation of Dynamic Systems* (Chapman & Hall, 2004). [CrossRef]

52. D. Simon, *Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches* (Wiley-Interscience, 2006). [CrossRef]

**(**

*a**z*), and do the same for the pixels of the intensity image that constitutes the measurement,

*η**:*

_{n}*M*is the number of pixels in each dimension (

*x*,

*y*). The evolution of the discretized complex-field is

*d*

**/**

*a**dz*=

**, where**

*La***is a matrix describing the evolution of the complex-field along the optical axis, determined by Eq. (1). Where the measurement scheme involves other complex transfer functions,**

*L***should be modified accordingly.**

*L***(**

*a**z*) plus noise:

_{n}*denotes the transpose and diag*

^{T}**is a diagonal matrix with the diagonal vector given by**

*u***.**

*u***(**

*â**z*

_{0}) and initial covariance matrices

**(**

*Q**z*

_{0}) and

**(**

*P**z*

_{0}) defined as

**and**

*Q***are discretized coherence functions. At each subsequent step in**

*P**z*, we forward-propagate both the estimate and the covariance matrices to obtain estimates of their values at the next image along the optical axis:

*η**at each step, we update the estimate and covariance matrices by defining a state vector as*

_{n}52. D. Simon, *Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches* (Wiley-Interscience, 2006). [CrossRef]

*I*is the identity matrix.

*K**is called the Kalman gain matrix and given by where c.c. denotes complex conjugate. Since the measurement noise covariance matrix*

_{n}

*R**depends on*

_{n}**(**

*a**z*), we use the estimated

_{n}**(**

*â**z*) to calculate

_{n}

*R**at each step. The propagation and Kalman-filter update are repeated for each*

_{n}*z*-step until

*z*is obtained, it can be numerically back-propagated.

_{N}7. J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

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

## 3. Implementation

*M*

^{2}, where

*M*is the number of pixels in one dimension. This state vector will have a covariance matrix of size 2

*M*

^{4}, which is currently an unrealizable memory requirement for typical image sizes on standard computers. Thus, we suggest below two methods for reducing computational complexity, but point out that the use of 64 bit personal computers and better memory management will significantly relax the computational requirements of this technique in the future. Processing speed-up was achieved by using parallel processing on a Graphics Processing Unit (GPU), though the main computational limitation was memory.

### 3.1. Computational method 1. Compressive storage

**matrix in that domain can be derived. For example, smooth phase distributions are well represented by the low coefficients in the Fourier domain. If Fourier coefficients are used as state variables, the full information may be conserved by a smaller state vector. As with compressive sensing, the information need not be confined to the low frequencies, as long as it is sparse. Another example could be wavefront aberrations in a microscope, which can often be described by just a few Zernike polynomials, whose coefficients could make up the state vector.**

*L*### 3.2. Computational method 2. Block processing

## 4. Simulations

*μ*m steps over a total distance of 50

*μ*m with wavelength 532nm. The intensity data was then corrupted by Poisson noise such that each pixel detected an average of

*γ*= 0.998 photons, giving a signal-to-noise ratio (SNR) of

*z*= 0 to

*z*= 50

*μ*m, adding a noisy intensity measurement at each step to refine the dynamic estimate. We start here with a zero initial guess and find that the error in both the phase and amplitude of the estimate decreases as more images are added (see Fig. 4(c)). Error is defined as the average root-mean-squared (RMS) error across all pixels.

*Δz*= 50

*μ*m) and is very susceptible to noise, particularly in low frequencies (Fig. 5(b)). Higher order TIE [36

36. 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]

*order TIE is used (similar to [35*

^{st}35. M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. **46**(33), 7978–7981 (2007). [CrossRef] [PubMed]

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

*all*measurement planes simultaneously, then replaces measured amplitude and backpropagates all planes to the object plane, the new estimate becoming the average of all these estimates. This method explicitly accounts for noise and therefore has significantly lower error than the standard iterative algorithm (Fig. 5(e)). However, our Kalman estimation method provides the lowest error for this particular data set (Fig. 5(f)).

*Δz*= 0.2 waves,

*γ*= 200 and keep the 18x18 lowest Fourier coefficients as state variables. Results are shown in Fig. 6, along with the error maps for both amplitude and phase.

## 5. Experimental results

*λ*= 532nm) and a motion stage for moving the camera axially to obtain a stack of defocused images (SNR 700). A test phase object was electron beam etched into PMMA substrate, having 190nm trenches. A microscopic 4

*f*system (see Fig. 1) magnified and relayed the field at the object plane to the camera plane. Intensity images were collected at 50 axial planes separated by 2

*μ*m, with the final image in focus. Some collected images are shown in Fig. 7(a). Here, the sharp edges of the phase object make the Fourier compression scheme invalid and block processing is used instead (50x50 pixel blocks). Recovered amplitude and phase are shown in Fig. 7(b,c) and Media 1 shows the evolution of the estimation process. For verification, we used an atomic force microscope (AFM) to independently measure the surface profile (Fig. 7(d)). The dark outline of the phase object lettering in our reconstruction may be due to lost light at sharp edges, but seems to also appear to some degree in the AFM results, as evidenced by the overshoot at the edges of the M in Fig. 7(e).

## 6. Conclusion

**matrices for propagation through inhomogeneous or nonlinear media.**

*L*## Acknowledgments

## References and links

1. | W. Bragg, “Elimination of the unwanted image in diffraction microscopy,” Nature |

2. | C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. A |

3. | D Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D |

4. | R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik |

5. | R. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. A |

6. | J. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. |

7. | J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

8. | G. Yang, B. Dong, B. Gu, J. Zhuang, and O. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. |

9. | J Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng |

10. | R. Rolleston and N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. |

11. | R. Piestun and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. |

12. | Z. Zalevsky, D. Mendlovic, and R. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. |

13. | J. Dominguez-Caballero, S. Takahashi, S. J. Lee, and G. Barbastathis, “Design and fabrication of computer generated holograms for Fresnel domain lithography,” In OSA DH and 3D Imaging, paper DWB3 (2009). |

14. | H. Stark, Y. Yang, and Y. Yang, |

15. | T. D. Gerke and R. Piestun, “Aperiodic volume optics,” Nat. Photonics |

16. | S. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express |

17. | R. Horstmeyer, S. Oh, and R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express |

18. | A. Devaney and R. Childlaw, “On the uniqueness question in the problem of phase retrieval from intensity measurements,” J. Opt. Soc. Am. A |

19. | J. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. |

20. | L. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. |

21. | Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, “Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm,” Opt. Express |

22. | G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. |

23. | H. Campbell, S. Zhang, A. Greenaway, and S. Restaino, “Generalized phase diversity for wave-front sensing,” Opt. Lett. |

24. | B. Dean and C. Bowers, “Diversity selection for phase-diverse phase retrieval,” J. Opt. Soc. Am. A |

25. | R. Gonsalves, “Phase retrieval by differential intensity measurements,” J. Opt. Soc. Am. A |

26. | W. Southwell, “Wave-front analyzer using a maximum likelihood algorithm,” J. Opt. Soc. Am. |

27. | R. Gonsalves, “Small-phase solution to the phase-retrieval problem,” Opt. Lett. |

28. | T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. |

29. | J. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. |

30. | D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. |

31. | J. Miao, D. Sayre, and H. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A |

32. | S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. |

33. | M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A |

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

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

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

37. | L. Waller, Y. Luo, S. Y. Yang, and G. Barbastathis, “Transport of intensity phase imaging in a volume holographic microscope,” Opt. Lett. |

38. | 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. |

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

40. | R Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. |

41. | R. Paxman, T. Schulz, and J. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A |

42. | R. Paxman and J. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A |

43. | D. Lee, M. Roggemann, B. Welsh, and E. Crosby, “Evaluation of least-squares phase-diversity technique for space telescope wave-front sensing,” Appl. Opt. |

44. | M. Roggemann, D. Tyler, and M. Bilmont, “Linear reconstruction of compensated images: theory and experimental results,” Appl. Opt. |

45. | M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. |

46. | D. Lee, M. Roggemann, and B. Welsh, “Cramér-Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A |

47. | L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E |

48. | R. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. |

49. | G. Welch and G. Bishop, |

50. | J. Crassidis and J. Junkins, |

51. | H. Van Trees, |

52. | D. Simon, |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

**ToC Category:**

Image Processing

**History**

Original Manuscript: December 13, 2010

Revised Manuscript: January 20, 2011

Manuscript Accepted: January 24, 2011

Published: January 31, 2011

**Citation**

Laura Waller, Mankei Tsang, Sameera Ponda, Se Young Yang, and George Barbastathis, "Phase and amplitude imaging from noisy images by Kalman filtering," Opt. Express **19**, 2805-2814 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2805

Sort: Year | Journal | Reset

### References

- W. Bragg, “Elimination of the unwanted image in diffraction microscopy,” Nature 167, 190–191 (1951). [CrossRef] [PubMed]
- C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. A 21(5), 828–831 (2004). [CrossRef]
- D. Misell, “A method for the solution of the phase problem in electron microscopy,” J. Phys. D 6, L6–L9 (1973). [CrossRef]
- R. Gerchberg, and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).
- R. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. A 66, 961–964 (1976). [CrossRef]
- J. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978). [CrossRef] [PubMed]
- J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
- G. Yang, B. Dong, B. Gu, J. Zhuang, and O. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33(2), 209–218 (1994). [CrossRef] [PubMed]
- J. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19(3), 291–305 (1980).
- R. Rolleston, and N. George, “Image reconstruction from partial Fresnel zone information,” Appl. Opt. 25(2), 178–183 (1986). [CrossRef] [PubMed]
- R. Piestun, and J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19(11), 771–773 (1994). [CrossRef] [PubMed]
- Z. Zalevsky, D. Mendlovic, and R. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996). [CrossRef] [PubMed]
- J. Dominguez-Caballero, S. Takahashi, S. J. Lee, and G. Barbastathis, “Design and fabrication of computer generated holograms for Fresnel domain lithography,” In OSA DH and 3D Imaging, paper DWB3 (2009).
- H. Stark, Y. Yang, and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics. (John Wiley & Sons, 1998). [PubMed]
- T. D. Gerke, and R. Piestun, “Aperiodic volume optics,” Nat. Photonics 4(3), 188–193 (2010). [CrossRef]
- S. Pavani, and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express 16(5), 3484–3489 (2008). [CrossRef] [PubMed]
- R. Horstmeyer, S. Oh, and R. Raskar, “Iterative aperture mask design in phase space using a rank constraint,” Opt. Express 18(21), 22545–22555 (2010). [CrossRef] [PubMed]
- A. Devaney, and R. Childlaw, “On the uniqueness question in the problem of phase retrieval from intensity measurements,” J. Opt. Soc. Am. A 68, 1352–1354 (1978). [CrossRef]
- J. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993). [CrossRef] [PubMed]
- L. Allen, and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1–4), 65–75 (2001). [CrossRef]
- Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, “Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm,” Opt. Express 11(24), 3234–3241 (2003). [CrossRef] [PubMed]
- G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30(8), 833–835 (2005). [CrossRef] [PubMed]
- H. Campbell, S. Zhang, A. Greenaway, and S. Restaino, “Generalized phase diversity for wave-front sensing,” Opt. Lett. 29(23), 2707–2709 (2004). [CrossRef] [PubMed]
- B. Dean, and C. Bowers, “Diversity selection for phase-diverse phase retrieval,” J. Opt. Soc. Am. A 20(8), 1490–1504 (2003). [CrossRef]
- R. Gonsalves, “Phase retrieval by differential intensity measurements,” J. Opt. Soc. Am. A 4, 166–170 (1987). [CrossRef]
- W. Southwell, “Wave-front analyzer using a maximum likelihood algorithm,” J. Opt. Soc. Am. 67(3), 396–399 (1977). [CrossRef]
- R. Gonsalves, “Small-phase solution to the phase-retrieval problem,” Opt. Lett. 26(10), 684–685 (2001). [CrossRef]
- T. Gureyev, A. Pogany, D. Paganin, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004). [CrossRef]
- J. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32(12), 1617–1619 (2007). [CrossRef] [PubMed]
- D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002). [CrossRef] [PubMed]
- J. Miao, D. Sayre, and H. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15(6), 1662–1669 (1998). [CrossRef]
- S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging,” J. Microsc. 207, 79–96 (2002). [CrossRef] [PubMed]
- M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983). [CrossRef]
- D. Paganin, A. Barty, P. J. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004). [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. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]
- L. Waller, Y. Luo, S. Y. Yang, and G. Barbastathis, “Transport of intensity phase imaging in a volume holographic microscope,” Opt. Lett. 35(17), 2961–2963 (2010). [CrossRef] [PubMed]
- 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. 35(3), 447–449 (2010). [CrossRef] [PubMed]
- 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]
- R. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
- R. Paxman, T. Schulz, and J. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9(7), 1072–1085 (1992). [CrossRef]
- R. Paxman, and J. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5(6), 914–923 (1988). [CrossRef]
- D. Lee, M. Roggemann, B. Welsh, and E. Crosby, “Evaluation of least-squares phase-diversity technique for space telescope wave-front sensing,” Appl. Opt. 36(35), 9186–9197 (1997). [CrossRef]
- M. Roggemann, D. Tyler, and M. Bilmont, “Linear reconstruction of compensated images: theory and experimental results,” Appl. Opt. 31(35), 7429–7441 (1992). [CrossRef] [PubMed]
- M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35(10), 4556–4567 (2008). [CrossRef] [PubMed]
- D. Lee, M. Roggemann, and B. Welsh, “Cramér-Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A 16(5), 1005–1015 (1999). [CrossRef]
- L. Allen, H. Faulkner, K. Nugent, M. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(3), 037602 (2001). [CrossRef]
- R. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82(1), 35–45 (1960). [CrossRef]
- G. Welch, and G. Bishop, An introduction to the Kalman filter, University of North Carolina at Chapel Hill, Chapel Hill, NC (1995).
- J. Crassidis, and J. Junkins, Optimal Estimation of Dynamic Systems (Chapman & Hall, 2004). [CrossRef]
- H. Van Trees, Detection, Estimation, and Modulation Theory (Krieger, 1992).
- D. Simon, Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches (Wiley-Interscience, 2006). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.