## Experimental compressive phase space tomography |

Optics Express, Vol. 20, Issue 8, pp. 8296-8308 (2012)

http://dx.doi.org/10.1364/OE.20.008296

Acrobat PDF (859 KB)

### Abstract

Phase space tomography estimates correlation functions entirely from snapshots in the evolution of the wave function along a time or space variable. In contrast, traditional interferometric methods require measurement of multiple two–point correlations. However, as in every tomographic formulation, undersampling poses a severe limitation. Here we present the first, to our knowledge, experimental demonstration of compressive reconstruction of the classical optical correlation function, *i.e.* the mutual intensity function. Our compressive algorithm makes explicit use of the physically justifiable assumption of a low–entropy source (or state.) Since the source was directly accessible in our classical experiment, we were able to compare the compressive estimate of the mutual intensity to an independent ground–truth estimate from the van Cittert–Zernike theorem and verify substantial quantitative improvements in the reconstruction.

© 2012 OSA

## 1. Introduction

*i.e.*, those with a time–independent Hamiltonian) [2]. Classical mutual intensity expresses the joint statistics between two points on a wavefront, and it is traditionally measured using interferometry: two sheared versions of a field are overlapped in a Young, Mach–Zehnder, or rotational shear [3

3. K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A **3**, 94–100 (1986). [CrossRef]

4. D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the Fresnel domain,” Appl. Opt. **38**, 1332–1342 (1999). [CrossRef]

6. K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. **68**, 2261–2264 (1992). [CrossRef] [PubMed]

8. C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A **22**, 1691–1700 (2005). [CrossRef]

9. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. **18**, 2041–2043 (1993). [CrossRef] [PubMed]

10. K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A **40**, 2847–2849 (1989). [CrossRef] [PubMed]

13. C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London) **386**, 150–153 (1997). [CrossRef]

*n*points in space, a standard implementation would require at least

*n*

^{2}data points.

14. E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory **52**, 489–509 (2006). [CrossRef]

16. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory **52**, 1289–1306 (2006). [CrossRef]

17. E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Found. Comput. Math. **9**, 717–772 (2009). [CrossRef]

18. E. J. Candès and T. Tao, “The power of convex relaxation: near-optimal matrix completion,” IEEE Trans. Inform. Theory **56**, 2053–2080 (2010). [CrossRef]

19. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. **105**, 150401 (2010). [CrossRef]

6. K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. **68**, 2261–2264 (1992). [CrossRef] [PubMed]

7. M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. **72**, 1137–1140 (1994). [CrossRef] [PubMed]

*i.e.*, tomographic projections in Wigner space) and utilize propagation along the optical axis to rotate the Wigner space between projections. The difference lies in the fact that in QST the state is recovered via successive applications of the Pauli dimensionality–reducing operator, and there is no need to evolve the state. Nevertheless, both approaches lead to the same Hermitian LRMR problem, as long as the assumption of a quasi–pure unknown state is satisfied. In [20

20. D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory **57**, 1548–1566 (2011). [CrossRef]

*n*and rank

*r*requires only

*O*(

*rn*ln

*n*) to

*O*(

*rn*ln

^{2}

*n*) data points. A similar LRMR method was also used to recover the complex amplitude of an unknown object under known illumination [21–23

23. Y. Shechtman, Y. C. Eldar, A. Szameit, and M. Segev, “Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing,” Opt. Express **19**, 14807–14822 (2011). [CrossRef] [PubMed]

*i.e.*coherent modes [24

24. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. **72**, 343–351 (1982). [CrossRef]

19. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. **105**, 150401 (2010). [CrossRef]

*i.e.*a nearly pure quantum state in the quantum analogue. This assumption is valid for lasers, synchrotron and table–top X–ray sources [25

25. D. Pelliccia, A. Y. Nikulin, H. O. Moser, and K. A. Nugent, “Experimental characterization of the coherence properties of hard x-ray sources,” Opt. Express **19**, 8073–8078 (2011). [CrossRef] [PubMed]

*i.e.*the measured energy is approximately evenly spread between modes [20

20. D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory **57**, 1548–1566 (2011). [CrossRef]

19. D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. **105**, 150401 (2010). [CrossRef]

## 2. Theory and method

*g*(

*x*).

*i.e.*the intensity, after propagation by distance

*z*, is [1] This can be expressed in operator form as where

*P*denotes the free–space propagation operator that combines both the quadratic phase and Fourier transform operations in Eq. (2), tr(·) computes the trace, and

*x*

_{o}denotes the lateral coordinate at the observation plane. By changing variables

*x*= (

*x*

_{1}+

*x*

_{2})/2,

*x*′ =

*x*

_{1}–

*x*

_{2}and Fourier transforming the mutual intensity with respect to

*x*we obtain the Ambiguity Function (AF) [26

26. K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. **44**, 323–326 (1983). [CrossRef]

28. J. Tu, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E **55**, 1946–1949 (1997). [CrossRef]

6. K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. **68**, 2261–2264 (1992). [CrossRef] [PubMed]

8. C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A **22**, 1691–1700 (2005). [CrossRef]

26. K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. **44**, 323–326 (1983). [CrossRef]

28. J. Tu, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E **55**, 1946–1949 (1997). [CrossRef]

*Ĩ*is the Fourier transform of the vector of measured intensities with respect to

*x*

_{o}. Thus, radial slices of the AF may be obtained from Fourier transforming the vectors of intensities measured at corresponding propagation distances, and from the AF the mutual intensity can be recovered by an additional inverse Fourier transform, subject to sufficient sampling.

*T*upon the mutual intensity

*J*, followed by Fourier transform , and adding measurement noise

*e*as

*i.e.*the set of eigenvectors, is also known as coherent modes in optical coherence theory, whereas the whole process is known as coherent mode decomposition [24

24. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. **72**, 343–351 (1982). [CrossRef]

*physically*meaningful priors:

*(1)*existence of the coherent modes [24

24. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. **72**, 343–351 (1982). [CrossRef]

*(2)*sparse representation of the partially coherent field in terms of coherent modes.

*λ*and the estimated mutual intensity as

_{i}*J*̂, the method can be written as Direct rank minimization is NP–hard; however, it can be accomplished by solving instead a proxy problem: convex minimization of the “nuclear norm” (

*ℓ*

_{1}norm) of the matrix

*J*[17

17. E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Found. Comput. Math. **9**, 717–772 (2009). [CrossRef]

*σ*= |

_{i}*λ*|, ||

_{i}*J*̂||

_{*}= ∑

_{i}*σ*. This problem is convex and a number of numerical solvers can be applied to solve it. In our implementation, we used the singular value thresholding (SVT) method [30]. The output estimate after each iteration of SVT typically has a sub-normalized total energy,

_{i}*i.e.*∑

_{i}*λ*< 1; we compensated for this by renormalizing at the end of each iteration [19

_{i}**105**, 150401 (2010). [CrossRef]

## 3. Numerical simulations

31. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. **72**, 923–928 (1982). [CrossRef]

*σ*determines the spatial extent of the source, and

_{I}*σ*is proportional to the coherence length and determines the number of coherent modes in the input source. The eigenvalues of GSMS are never zero (analytical solution given in [31

_{c}31. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. **72**, 923–928 (1982). [CrossRef]

*r*as the first

*r*modes containing the 99% of the total energy.

*σ*= 17 and

_{I}*σ*= 13 (rank

_{c}*r*= 6). Intensities are calculated at 40 different axial distances and the coverage in Ambiguity space is shown in Fig. 1(b). We simulate the case where data from both the near field and the far field are missing due to the finite range of camera scanning motion allowed in the actual experiment. The missing cone around the

*u*′-axis is due to missing data from near field, while the data missing from far field results in the missing cone around the

*x*′-axis. Both cones have an apex angle of 20 degrees.

31. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. **72**, 923–928 (1982). [CrossRef]

*σ*= 36 and

_{I}*σ*= 18 (rank

_{c}*r*= 9). We generate noisy data with different signal-to-noise ratio (SNR) from both an additive random Gaussian noise model and a Poisson noise model. However, we emphasize that the reconstruction algorithm does not make use of the noise statistics. For each SNR level, we repeat the simulation 100 times with different random noise terms, and then record the average relative mean-square-error (MSE) from the LRMR reconstruction. The ratio between the number of samples taken from the intensity measurements and the rank

*r*of the input mutual intensity matrix determines the oversampling rate [22]. This rate is plotted versus relative MSE for different SNR cases in Fig. 4. For good performance, the required oversampling rate is at least 5–6 (the theoretical oversampling rate is on the order of ln(256) = 5.5 according to [20

20. D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory **57**, 1548–1566 (2011). [CrossRef]

## 4. Experimental result

*μ*m (0.014″) is placed immediately after the LED and one focal length (75 mm) to the left of a cylindrical lens. One focal length to the right of the lens, we place the second single slit of width 457.2

*μ*m (0.018″), which is used as a one–dimensional (1D) object.

*z*–distances downstream from the object, as described in the theory. We measured the intensities at 20

*z*–distances, ranging from 18.2mm to 467.2mm, to the right of the object. The data are given in Fig. 6. Each 1D intensity measurement consists of 512 samples, captured by a CMOS sensor with 12

*μ*m pixel size. The dimension of the unknown mutual intensity matrix to be recovered is 512 × 512. Since only intensities at positive

*z*,

*i.e.*downstream from the object, are accessible, we can only fill up the top right and bottom left quadrants of Ambiguity space. The other two quadrants are filled symmetrically,

*i.e.*assuming that if the field propagating to the right of the object were phase conjugated with respect to the axial variable

*z*, it would yield the correct field to the left of the object,

*i.e.*negative

*z*[8

8. C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A **22**, 1691–1700 (2005). [CrossRef]

13. C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London) **386**, 150–153 (1997). [CrossRef]

*u*′–axis is approximately 17.4 degrees, and the one around the

*x*′–axis is approximately 28.6 degrees. The number of measurements is only 7.8% of the total number of entries in the unknown mutual intensity matrix.

*μ*m (38 pixels), which agrees with the actual width of the slit. The imaginary part is orders of magnitude smaller than the real part.

**105**, 150401 (2010). [CrossRef]

## 5. Discussion

*n*–dimensional unit sphere, whereas we simply utilized free space propagation. The phase masks described in [21] to implement optimal sampling are outside the scope of the present work.

## Acknowledgments

## References and links

1. | L. Mandel and E. Wolf, |

2. | K. Blum, |

3. | K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A |

4. | D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the Fresnel domain,” Appl. Opt. |

5. | J. W. Goodman, |

6. | K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. |

7. | M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. |

8. | C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A |

9. | M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett. |

10. | K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A |

11. | D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. |

12. | U. Leonhardt, “Quantum–state tomography and discrete Wigner function,” Phys. Rev. Lett. |

13. | C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London) |

14. | E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory |

15. | E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math. |

16. | D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory |

17. | E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Found. Comput. Math. |

18. | E. J. Candès and T. Tao, “The power of convex relaxation: near-optimal matrix completion,” IEEE Trans. Inform. Theory |

19. | D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. |

20. | D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory |

21. | E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” ArXiv: 1109.4499v1 (2011). |

22. | E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” ArXiv: 1109.0573 (2011). |

23. | Y. Shechtman, Y. C. Eldar, A. Szameit, and M. Segev, “Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing,” Opt. Express |

24. | E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. |

25. | D. Pelliccia, A. Y. Nikulin, H. O. Moser, and K. A. Nugent, “Experimental characterization of the coherence properties of hard x-ray sources,” Opt. Express |

26. | K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. |

27. | K.-H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta. |

28. | J. Tu, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E |

29. | E. J. Candès and Y. Plan, “Matrix completion with noise,” ArXiv: 0903.3131 (2009). |

30. | J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” ArXiv: 0810.3286 (2008). |

31. | A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. |

32. | A. C. Kak and M. Slaney, |

33. | M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. |

34. | M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A |

35. | A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. |

36. | M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(100.6950) Image processing : Tomographic image processing

(050.5082) Diffraction and gratings : Phase space in wave options

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: January 13, 2012

Revised Manuscript: February 24, 2012

Manuscript Accepted: March 20, 2012

Published: March 26, 2012

**Virtual Issues**

April 9, 2012 *Spotlight on Optics*

**Citation**

Lei Tian, Justin Lee, Se Baek Oh, and George Barbastathis, "Experimental compressive phase space tomography," Opt. Express **20**, 8296-8308 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8296

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### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
- K. Blum, Density Matrix Theory and Applications (Plenum Press, 1981).
- K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A3, 94–100 (1986). [CrossRef]
- D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the Fresnel domain,” Appl. Opt.38, 1332–1342 (1999). [CrossRef]
- J. W. Goodman, Statistical Optics (Wiley-Interscience, 2000).
- K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett.68, 2261–2264 (1992). [CrossRef] [PubMed]
- M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994). [CrossRef] [PubMed]
- C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A22, 1691–1700 (2005). [CrossRef]
- M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, “Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses,” Opt. Lett.18, 2041–2043 (1993). [CrossRef] [PubMed]
- K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A40, 2847–2849 (1989). [CrossRef] [PubMed]
- D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett.70, 1244–1247 (1993). [CrossRef] [PubMed]
- U. Leonhardt, “Quantum–state tomography and discrete Wigner function,” Phys. Rev. Lett.74, 4101–4105 (1995). [CrossRef] [PubMed]
- C. Kurtsiefer, T. Pfau, and J. Mlynek, “Measurement of the Wigner function of an ensemble of Helium atoms,” Nature (London)386, 150–153 (1997). [CrossRef]
- E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory52, 489–509 (2006). [CrossRef]
- E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math.59, 1207–1223 (2006). [CrossRef]
- D. L. Donoho, “Compressed sensing,” IEEE Trans. Inform. Theory52, 1289–1306 (2006). [CrossRef]
- E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Found. Comput. Math.9, 717–772 (2009). [CrossRef]
- E. J. Candès and T. Tao, “The power of convex relaxation: near-optimal matrix completion,” IEEE Trans. Inform. Theory56, 2053–2080 (2010). [CrossRef]
- D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett.105, 150401 (2010). [CrossRef]
- D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” IEEE Trans. Inf. Theory57, 1548–1566 (2011). [CrossRef]
- E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: exact and stable signal recovery from magnitude measurements via convex programming,” ArXiv: 1109.4499v1 (2011).
- E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” ArXiv: 1109.0573 (2011).
- Y. Shechtman, Y. C. Eldar, A. Szameit, and M. Segev, “Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing,” Opt. Express19, 14807–14822 (2011). [CrossRef] [PubMed]
- E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am.72, 343–351 (1982). [CrossRef]
- D. Pelliccia, A. Y. Nikulin, H. O. Moser, and K. A. Nugent, “Experimental characterization of the coherence properties of hard x-ray sources,” Opt. Express19, 8073–8078 (2011). [CrossRef] [PubMed]
- K.-H. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun.44, 323–326 (1983). [CrossRef]
- K.-H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta.31, 213–223 (1984). [CrossRef]
- J. Tu, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E55, 1946–1949 (1997). [CrossRef]
- E. J. Candès and Y. Plan, “Matrix completion with noise,” ArXiv: 0903.3131 (2009).
- J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” ArXiv: 0810.3286 (2008).
- A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am.72, 923–928 (1982). [CrossRef]
- A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001). [CrossRef]
- M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun.25, 26–30 (1978). [CrossRef]
- M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A3, 1227–1238 (1986). [CrossRef]
- A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am.72, 1538–1544 (1982). [CrossRef]
- M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A1, 711–715 (1984). [CrossRef]

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