## Compressive Holography

Optics Express, Vol. 17, Issue 15, pp. 13040-13049 (2009)

http://dx.doi.org/10.1364/OE.17.013040

Acrobat PDF (3418 KB)

### Abstract

Compressive sampling enables signal reconstruction using less than one measurement per reconstructed signal value. Compressive measurement is particularly useful in generating multidimensional images from lower dimensional data. We demonstrate single frame 3D tomography from 2D holographic data.

© 2009 Optical Society of America

## 1. Introduction

1. C. E. Shannon, “Communications in the presence of noise,” in Proc.of the IREv **37**, 10–21 (1949).
[CrossRef]

2. M. Golay, “Multislit spectroscopy,” J. Opt. Soc. Am. **39**, 437–444 (1949).
[CrossRef] [PubMed]

3. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics **59**, 1207–1223 (2006).
[CrossRef]

4. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Transactions on Information Theory **52(12)**, 5406–5425 (2006).
[CrossRef]

5. D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory **52(4)**, 1289–1306 (2006).
[CrossRef]

*et al*. applied compressive sampling algorithms in demonstrating a single pixel camera [6]. We have previously applied these methods with Golay-inspired coded aperture spectroscopy to demonstrate compressive measurement of 3D spectral datacubes from 2D measurements [7

7. M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express **15(21)**, 14013–14027 (2007).
[CrossRef]

8. A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. **47(10)**, B44–B51 (2008).
[CrossRef]

9. D. Gabor, “A new microscopic principle,” Nature **161**, 777–778 (1948).
[CrossRef] [PubMed]

10. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. **52**, 1123–1130 (1962).
[CrossRef]

11. S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip **9(6)**, 777–787 (2009).
[CrossRef]

*et al*. [19

19. W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. **33(9)**, 974–976 (2008).
[CrossRef]

## 2. Compressive sensing background

3. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics **59**, 1207–1223 (2006).
[CrossRef]

4. E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Transactions on Information Theory **52(12)**, 5406–5425 (2006).
[CrossRef]

*N*–

*S*) coefficients that are exactly zero. A matrix

*H*∈R

^{M×N}is said to satisfy S-RIP with constant

*δ*∈(0,1) if, for any

_{S}*S*-sparse

*f*,

*T*denotes the set of indices on which the

*S*-sparse signal is supported, and ‖·‖2 denotes the Euclidean norm. This condition implies that for any

*S*-sparse object to be reconstructed accurately and reliably, the corresponding sub-matrix

*H*of

_{T}*H*composed of

*S*columns of

*H*has to form a nearly isometry transformation. Note that the condition also implies that all the eigenvalues of the Gram matrix of any

*S*-column sub-matrix

*H*are distributed near 1 (in fact in the range of [1-

_{T}*δ*1+

_{S}*δ*]), which consequently ensures that any S-column sub-matrix

_{S}*H*is well-conditioned.

_{T}*µ*

_{1}(

*H*,Ψ) be defined by

*h*and

_{m}*ψ*denote the

_{n}*m*-th row of

*H*and the

*n*-th column of Ψ, respectively. Candés

*et al*. [3

3. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics **59**, 1207–1223 (2006).
[CrossRef]

5. D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory **52(4)**, 1289–1306 (2006).
[CrossRef]

*f*is

*S*-sparse in Ψ, and

*M*satisfies

*θ*‖1=∑

*|*

_{i}*θ*|. S denotes the number of nonzero (or significant) coefficients of

_{i}*f*in the Ψ domain. As clear from Eqn. (3), the smaller

*µ*

_{1}is, the more accurate the reconstruction would be for the same

*M*.

*d*is the

_{i}*i*-th column of the sensing matrix

*D*=

*H*Ψ. This can also be interpreted as the maximum off-diagonal element of the Gram matrix of

*D*, whose columns are normalized. When

*µ*

_{2}(

*D*) is minimal such that

*S*-sparse is necessarily the sparsest solution that satisfies min

*‖*

_{θ}*θ*‖

_{0}, and hence can be obtained by solving Eqn. (4) [23

23. D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via 𡄓1 minimization,” Proceedings of the National Academy of Sciences of the United States of America , **100(5)**, 2197–2202 (2003).
[CrossRef]

*I*with

*I*being an identity matrix (i.e., the canonical basis),

*µ*

_{1}(

*H, I*)=1 for a discrete Fourier transform (DFT) matrix

*H*. Hence, the DFT matrix that generates Fourier samples distributed uniformly at random over the frequency domain satisfies the so-called restricted isometry property with high probability given that

*M*≥

*CS*log

*N*is satisfied. Our Gabor hologram multiplex encoder may be considered as 3D Fourier transform encoder. However, our Fourier samples are limited to a certain band volume, which may produce a relatively large µ1, larger than 1 in general. Holographic measurements using illumination with multiple wavelengths and/or angles can provide the Fourier samples over a larger band volume improving

*µ*

_{1}and, consequently, the RIP for our Gabor hologram multiplex encoder.

## 3. Theory and Methods

*A*and a 3D object with scattering density

*η*(

*x*′,

*y*′,

*z*′). A 2D detector array records the irradiance

*E*is defined under the Born approximation as

*h*is the product of exp(

*ikz*) representing the phase delay at a distance

*z*[24

24. R. E. Blahut, *Theory of Remote Image Formation*, (Cambridge University Press, 2004).
[CrossRef]

*E*(

*x,y*)|

^{2}produces the autocorrelation of the Fourier transform of the field

*E*in the Fourier domain. In Eqn. (7), the term |

*A*|

^{2}is simply a constant, and hence the effect of |

*A*|

^{2}can be removed by eliminating the DC term (the term at the origin) from the Fourier transform of the interference irradiance measurements

*I*(

*x,y*). Also, we may assume that

*A*is 1 without loss of generality. Then, we may proceed with

*A**

*E*(

*x,y*)+

*AE**(

*x,y*)+|

*E*(

*x,y*)|

^{2}=2Re{

*E*(

*x,y*)}+|

*E*(

*x,y*)|2=2Re{

*E*(

*x,y*)}+

*e*(

*x,y*). If we neglect the nonlinearity caused by |

*E*(

*x,y*)|

^{2}and regard

*e*(

*x,y*) as model error, then Eqn. (7) represents a linear mapping between the object scattering density and measurement data.

*η*′

_{m}_{1}

*m*′

_{2}

*l*=

*η*(

*m*′

_{1}Δ,

*m*′

_{2}Δ,

*l*Δ

*). Note that the terms in the square bracket form the 2D discrete Fourier transform of*

_{z}*η*. Also note that the last exponential term in the last line of Eqn. (9) also forms the inverse 2D Fourier transform in conjunction with the summations over

*m*

_{1}and

*m*

_{2}. Hence, this equation can simply be written as

*η*̂ denotes the Fourier transform of

*η*, and

*𝓕*

^{−1}denotes the inverse Fourier transform operator. Equation (10) is called a multislice approximation [26

26. D. M. Paganin, *Coherent X-ray Optics*, (Oxford Science Publications, 2006).
[CrossRef]

*η*. The 2D slice can be interpreted as a surface patch of the Ewald

*k*-sphere as in traditional diffraction tomography [27

27. A. C. Kak and M. Slaney, *Principle of Computerized Tomographic Imaging*, (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

*η*or the 3D spatial scattering density

*η*, the multiscale approximation requires no such interpolation and is less sensitive to errors that would result from 3D interpolation.

*N*and

_{x}*N*denote the numbers of pixels of the detector in the

_{y}*x*-direction and

*y*-direction, respectively. With these definitions, Eqn. (10) may be rewritten as

*B*=bldiag(

*F*, ⋯,

_{2D},F_{2D}*F*) with

_{2D}*F*being the matrix representing the 2D DFT whose size is (

_{2D}*N*×

_{x}*N*)×(

_{y}*N*×

_{x}*N*) and “bldiag” denoting the block diagonal matrix,

_{y}*Q*=[

*P*

_{1}

*P*

_{2}⋯

*PN*] with

_{z}*P*at the intersection of the row

_{l}*m*and the column

_{1}*m*

_{2}, and

*G*represents the 2D inverse DFT matrix. As discussed earlier, ignoring the non-linearity of Eqn. (10) caused by the squared field term e, the Gabor hologram measurement may be algebraically written as

_{2D}*H*=[

_{ij}*G*]

_{2D}QB*, and*

_{i j}*e*and

*n*denote vectorized |

*E*(

*x,y*)|

^{2}and additive noise, respectively. The nonlinear term is traditionally removed using off-axis holography. As demonstrated below, however, this term may be eliminated algorithmically from Gabor data using decompressive inference. We discuss the effects of

*e*in Sec. 5.

*H*(e.g., see Eqn. (12)) in holography are complex valued. In contrast, the typical optical imaging modality provides only nonnegative values because the only observable are the intensities. Such nonnegativity constraints confine the measurement basis space to be only the nonnegative orthant, which limits the angles between the measurement basis vectors (the columns of

*H*) and effectively makes the coherence values large. On the other hand, interferometric imaging modalities such as holography allow the measurement vectors to reside in the entire space as opposed to the nonnegative orthant by allowing for negative values in the measurement basis vectors. Such negative values may enable us to design a sensing system with a small coherence value. This advantage with holography for compressive sensing remains valid in Gabor holography because although the

*f*and

*g*may be assumed to be real, the elements of the multiplex encoder

*H*are still complex valued. We also note that in other holography such as the Leith-Upatnieks holography [25], the

*f*and

*g*as well as

*H*are, in general, all complex valued.

*f*from Eqn. (12), we briefly consider the spatial resolution expected in the reconstruction and a sparsity constraint to enable decompressive inference. Optical measurement over a finite aperture is bandlimited. The band volume is the support in the 3D Fourier space of

*η*(

*x*′,

*y*′,

*z*′) for sampling

*E*(

*x,y*) over a finite aperture. Spatial resolution in imaging systems is assumed to be inversely proportional to the limits of the band volume, which yields transverse resolution Δ

*=*

_{x}*λ z/D*for objects at range

*z*observed over aperture

*D*and the axial resolution is Δ

*=*

_{z}*λ*(2

*z*)

^{2}/

*D*

^{2}[28

28. D. J. Brady, *Optical Imaging and Spectroscopy*, (Wiley, 2009).
[CrossRef]

*D*for a Gabor hologram observed in the Fresnel diffraction zone is determined by size of the object feature observed. A feature of cross section w produces a diffraction pattern with cross section

*D*

*≈λz/w*. This implies that Δ

*≈*

_{x}*w*and Δ

*≈4*

_{z}*w*

^{2}/

*λ*. The dependence of axial resolution on feature size is the result of the “missing cone” in the paraxial band volume [28

28. D. J. Brady, *Optical Imaging and Spectroscopy*, (Wiley, 2009).
[CrossRef]

*f*may be assumed to be sparse or by enforcing a sparsity constraint on the total variation, as defined by Rudin

*et al*[29

29. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D **60(1–4)**, 259–268 (1992).
[CrossRef]

*f*. We choose the second approach here and estimate

*f*as

*f*‖

*is defined by*

_{TV}*f*denotes a 2D plane of the 3D object datacube. We adapt the two-step iterative shrinkage/thresholding algorithm (TwIST) [30

_{k}30. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration,”, IEEE Transactions on Image Processing **16**, 2992–3004 (2007).
[CrossRef] [PubMed]

## 4. Experimental results

*taraxacum*) with a collimated, spatially filtered Helium-Neon laser of 632.8 nm wavelength. One object is placed 1.5 cm away from the detector array, and the other dandelion is placed 5.5 cm away from the detector array. Figures 3(b) and 3(c) are photographs of the two seed parachutes. The illumination and scattered fields were captured in the Gabor hologram shown in Fig. 3(a). Figure 3(d) is the 3D datacube estimated from the Gabor recording by the TV-minimization algorithm. As the reconstruction shows, the stem and the petals, representing the high-frequency features in the image, are reconstructed well. In addition, the distance between the detector plane and the first parachute and the distance between the two parachutes are also accurately estimated. We notice that there are still some errors in some reconstruction planes including the planes in which the two seed parachutes. We conjecture that the errors are mainly reconstruction errors and the effects of noise, and they occur because of the rather insufficient number of measurements. The reconstruction errors can be suppressed by exploiting the phase-shifting holography that can increase the effective SNR and the effective number of measurements [31]. The reconstruction error in the plane of

*z*=0 is explained in Sec. 5. Figure 3(e) shows the backpropagated (or digital refocusing) field that is obtained by digitally backpropagating the hologram using the propagation kernel

*h*in Eqn. (8) [25]. In contrast to the reconstruction in Fig. 3(d), the backpropagated field shows messy planes full of out-of-focus features obscuring the object features of the two parachutes.

## 5. Discussion

*e*(

*x,y*) on the reconstruction, we consider the behavior of the algorithm. The algorithm determines how much portion of the measurement (or measurement estimate) should be placed in which object plane based on the amount of correlation that the diffraction patterns of the measurement and the diffraction pattern that the estimate at a particular object plane would produce. The squared field

*e*is inclined to produce the diffraction patterns that bear little correlation with all of the object planes. Consequently, most of the reconstruction error caused by

*e*tends to remain in the plane of

*z*=0. In this way, we may effectively isolate most of the errors that result from the squared field term. Since these errors are concentrated in the plane of

*z*=0 (measurement plane), such reconstruction values are not part of the object scattering density

*η*by definition. Therefore, we may effectively remove most of the errors that would result from the squared field (zero-order) term in the reconstructions.

*E**(

*x,y*) on the reconstruction. In conventional digital holography, the effect of the conjugate term on reconstructions is called the twin image problem [25]. Considering that

*e*would behave as discussed above, our forward model including the conjugated field

*E**(

*x,y*) is approximated well to be linear. In our approach, the effect of the conjugate term on the reconstruction can be numerically eliminated by confining our estimate domain to only the one side of the measurement plane (i.e.,

*z*≥0). Our decompressive inference method directly reconstructs the 3D volume object

*η*rather than the field at a distance which is a superposition of all the object planes each of which is at a different focus. This implies that the virtual object

*η** is placed to the other side of the measurement plane (i.e.,

*z*≤0) by our inference method. This aspect of our decompressive inference allows a separation of the real and the virtual objects. Hence, by confining the reconstruction domain to be the region in which

*z*≥0, we can readily resolve the twin image problem in the reconstruction. A similar philosophy has successfully been applied to 2D holographic reconstruction to remove the twin image problem [32

32. T. Latychevskaia and H. W. Fink, “Solution to the Twin Image Problem in Holography,” Phys. Rev. Lett. **98**, 233901 (2007).
[CrossRef] [PubMed]

*µ*m transverse resolution and 0.8 cm axial resolution reconstructed from a single 2D (512×512) hologram. This demonstrates the main advantages of compressive holography, i.e. that holograms naturally encode high quality multiplex data and that decompressive inference can infer multidimensional objects from lower dimensional data. Extensions of compressive holography may use off-axis encoding to filter nonlinear terms and multispectral illumination to increase the band volume and improve axial resolution. It would be also useful to combine our approach with phase-shifting digital holography.

## Acknowledgments

## References and links

1. | C. E. Shannon, “Communications in the presence of noise,” in Proc.of the IREv |

2. | M. Golay, “Multislit spectroscopy,” J. Opt. Soc. Am. |

3. | E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics |

4. | E. J. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Transactions on Information Theory |

5. | D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory |

6. | D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in |

7. | M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express |

8. | A. Wagadarikar, R. John, R. Willett, and D. J. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. |

9. | D. Gabor, “A new microscopic principle,” Nature |

10. | E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. |

11. | S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab on a Chip |

12. | W. Jueptner and U. Schnars, |

13. | T. C. Poon, |

14. | M. E. Brezinski, |

15. | D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science |

16. | J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3d microscopy and gene expression studies,” Science |

17. | A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” Journal of Mathematical Physics |

18. | A. J. Devaney, “Geophysical diffraction tomography,” IEEE Transactions on Geoscience and Remote Sensing |

19. | W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. |

20. | I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. |

21. | R. Baraniuk and P. Steeghs, “Compressive radar imaging,” IEEE Radar Conference, pp. 128–133, April, (2007). |

22. | L. Li, W. Zhang, and F. Li, “Compressive diffraction tomography for Weakly Scattering,” Submitted to IEEE Trans. on Geosciences and Remote Sensing (2009). |

23. | D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via 𡄓1 minimization,” Proceedings of the National Academy of Sciences of the United States of America , |

24. | R. E. Blahut, |

25. | J. W. Goodman, |

26. | D. M. Paganin, |

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

28. | D. J. Brady, |

29. | L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D |

30. | J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration,”, IEEE Transactions on Image Processing |

31. | V. Micó, J. García, Z. Zalevsky, and B. Javidi, “Phase-shifting Gabor holography,” Opt. Express |

32. | T. Latychevskaia and H. W. Fink, “Solution to the Twin Image Problem in Holography,” Phys. Rev. Lett. |

**OCIS Codes**

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

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: May 18, 2009

Revised Manuscript: July 2, 2009

Manuscript Accepted: July 5, 2009

Published: July 15, 2009

**Citation**

David J. Brady, Kerkil Choi, Daniel L. Marks, Ryoichi Horisaki, and Sehoon Lim, "Compressive Holography," Opt. Express **17**, 13040-13049 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13040

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### References

- C. E. Shannon, "Communications in the presence of noise," Proc. IREv 37, 10-21 (1949). [CrossRef]
- M. Golay, "Multislit spectroscopy," J. Opt. Soc. Am. 39,437-444 (1949). [CrossRef] [PubMed]
- E. J. Candes, J. K. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59,1207-1223 (2006). [CrossRef]
- E. J. Candes and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?," IEEE Transactions on Information Theory 52(12), 5406-5425 (2006). [CrossRef]
- D. L. Donoho, "Compressed sensing," IEEE Trans. Info. Theory 52(4), 1289-1306 (2006). [CrossRef]
- D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera architecture using optical-domain compression," in Computational Imaging IV, 6065, (San Jose, CA, USA), p. 606509, SPIE, 2006.
- M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, "Single-shot compressive spectral imaging with a dual-disperser architecture," Opt. Express 15(21), 14013-14027 (2007). [CrossRef]
- A. Wagadarikar, R. John, R. Willett, and D. J. Brady, "Single disperser design for coded aperture snapshot spectral imaging," Appl. Opt. 47(10), B44-B51 (2008). [CrossRef]
- D. Gabor, "A new microscopic principle," Nature 161,777-778 (1948). [CrossRef] [PubMed]
- E. N. Leith and J. Upatnieks, "Reconstructed wavefronts and communication theory," J. Opt. Soc. Am. 52,1123-1130 (1962). [CrossRef]
- S. Seo, T. W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, "Lensfree holographic imaging for on-chip cytometry and diagnostics," Lab on a Chip 9(6), 777-787 (2009). [CrossRef]
- W. Jueptner and U. Schnars, Digital Holography, (New York, Springer-Verlag, Berlin Heidelberg, 2005).
- T. C. Poon, Digital holography and three-dimensional display, (New York; London: Springer, 2006). [CrossRef]
- M. E. Brezinski, Optical coherence tomography. (Amsterdam; Boston: Academic Press, 2006).
- D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, and R. B. Brady, "Visible cone-beam tomography with a lensless interferometric camera," Science 284(5423), 2164-2166 (1999). [CrossRef]
- J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, and J. Hecksher-Sorensen, R. Baldock, and D. Davidson, "Optical projection tomography as a tool for 3d microscopy and gene expression studies," Science 296(5567), 541-545 (2002). [CrossRef]
- A. J. Devaney, "Nonuniqueness in the inverse scattering problem," J. Math. Phys. 19(7), 1526-1531 (1978). [CrossRef]
- A. J. Devaney, "Geophysical diffraction tomography," IEEE Trans. Geoscie. Remote Sens. GE-22, 3-13 (1984). [CrossRef]
- W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, "Terahertz imaging with compressed sensing and phase retrieval," Opt. Lett. 33(9), 974-976 (2008). [CrossRef]
- I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, "Image reconstruction only by phase data in phaseshifting digital holography," Appl. Opt. 45(5), 975-983 (2006). [CrossRef]
- R. Baraniuk and P. Steeghs, "Compressive radar imaging," IEEE Radar Conference, pp. 128-133, April, (2007).
- L. Li, W. Zhang, F. Li, "Compressive diffraction tomography for Weakly Scattering," Submitted to IEEE Trans. on Geoscie. Remote Sens. (2009).
- D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via _1 minimization," Proc. Nat. Acad. Scie. U. S. A. 100(5), 2197-2202 (2003). [CrossRef]
- R. E. Blahut, Theory of Remote Image Formation (Cambridge University Press, 2004). [CrossRef]
- J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).
- D. M. Paganin, Coherent X-ray Optics (Oxford Science Publications, 2006). [CrossRef]
- A. C. Kak and M. Slaney, Principle of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001). [CrossRef]
- D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009). [CrossRef]
- L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60(1-4), 259-268 (1992). [CrossRef]
- J. M. Bioucas-Dias and M. A. T. Figueiredo, "A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration," IEEE Trans. Image Proc. 16, 2992-3004 (2007). [CrossRef] [PubMed]
- V. Mic’o, J. Garc’ıa, Z. Zalevsky, and B. Javidi, "Phase-shifting Gabor holography," Opt. Express 17, 1492-1494 (2009).
- T. Latychevskaia and H. W. Fink, "Solution to the Twin Image Problem in Holography," Phys. Rev. Lett. 98, 233901 (2007). [CrossRef] [PubMed]

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