## Phase retrieval with transverse translation diversity: a nonlinear optimization approach

Optics Express, Vol. 16, Issue 10, pp. 7264-7278 (2008)

http://dx.doi.org/10.1364/OE.16.007264

Acrobat PDF (322 KB)

### Abstract

We develop and test a nonlinear optimization algorithm for solving the problem of phase retrieval with transverse translation diversity, where the diverse far-field intensity measurements are taken after translating the object relative to a known illumination pattern. Analytical expressions for the gradient of a squared-error metric with respect to the object, illumination and translations allow joint optimization of the object and system parameters. This approach achieves superior reconstructions, with respect to a previously reported technique [H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004)], when the system parameters are inaccurately known or in the presence of noise. Applicability of this method for samples that are smaller than the illumination pattern is explored.

© 2008 Optical Society of America

## 1. Introduction

*e.g.*lenses, mirrors or holographic optical elements) are substituted by computational image reconstruction [1–9

1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

*a priori*information about the object (object support constraint and/or nonnegativity constraint), to computationally retrieve the phase of the field at the detector plane by using an iterative phase retrieval algorithm [1

1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

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

5. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature (London) **400**, 342–344 (1999). [CrossRef]

6. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B **68**, 140101 (2003). [CrossRef]

8. H. N. Chapman*et al.*, “High-resolution *ab initio* three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A **23**, 1179–1200 (2006). [CrossRef]

13. S. G. Podorov, K. M. Pavlov, and D. M. Paganin, “A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging,” Opt. Express **15**, 9954–9962 (2007). [CrossRef] [PubMed]

14. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express **15**, 17592–17612 (2007). [CrossRef] [PubMed]

14. M. Guizar-Sicairos and J. R. Fienup, “Holography with extended reference by autocorrelation linear differential operation,” Opt. Express **15**, 17592–17612 (2007). [CrossRef] [PubMed]

1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

15. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A **3**, 1897–1907 (1986). [CrossRef]

2. J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A **4**, 118–123 (1987). [CrossRef]

7. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express **14**, 498–508 (2006). [CrossRef] [PubMed]

19. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. **93**, 023903 (2004). [CrossRef] [PubMed]

20. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. **85**, 4795–4797 (2004). [CrossRef]

21. J. M. Rodenburg, A. C. Hurst, and A. G. Cullis, “Transmission microscopy without lenses for objects of unlimited size,” Ultramicroscopy **107**, 227–231 (2007). [CrossRef]

22. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. **98**, 034801 (2007). [CrossRef] [PubMed]

*a priori*. It was shown that an iterative phase retrieval technique, named ptychographical iterative engine (PIE), could be used in this case to increase the field of view (FOV) of the reconstruction, make the algorithms more robust, and achieve superior reconstructions. The success of this technique relies, however, on an accurate knowledge of the illumination pattern and the transverse displacements of the object. In practice, these parameters might be inaccurately known due, for example, to the limited precision of the translating stages and inaccurate or incomplete knowledge of features of the aperture (or focusing optics) that generates the illuminating beam [23

23. H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy **103**, 153–164 (2005). [CrossRef] [PubMed]

## 2. Object translation diversity

30. M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE **5896**, 58960G (2005). [CrossRef]

32. H. R. Ingleby and D. R. McGaughey “Parallel multiframe blind deconvolution using wavelength diversity,” Proc. SPIE **5562**, 58–64 (2004). [CrossRef]

*u*,

*v*). The phase retrieval problem then reduces to finding the object field,

*f*(

*x*,

*y*), given a sampled measurement of the Fourier intensity,

*I*(

*u*,

*v*)=|

*F*(

*u*,

*v*)|

^{2}. We approximate the continuous Fourier transform by the discrete Fourier transform,

*N*and

*M*are the array dimensions, (

*x*,

*y*) and (

*u*,

*v*) are integer pixel coordinates in object and Fourier domain respectively, and we have dropped unimportant multiplicative scaling factors. Throughout the remainder of this manuscript, upper-case letters will refer to the discrete Fourier transforms of their lower-case counterparts as given by Eq. (1).

33. W. Hoppe, “Beugung im inhomogenen Primärstrahlwellenfeld. III. Amplituden-und Phasenbestimmung bei unperiodischen Objekten,” Acta Crystallogr. Sect. A **25**, 508–514 (1969). [CrossRef]

34. J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy viaWigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A **339**, 521–553 (1992). [CrossRef]

35. N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. **37**, 6219–6226 (1998). [CrossRef]

19. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. **93**, 023903 (2004). [CrossRef] [PubMed]

*o*(

*x*,

*y*) and

*p*(

*x*,

*y*) as the object transmittance and illumination fields, respectively, the

*n*-th resultant field is then given by

*o*(

*x*,

*y*) that generates the fields

*f*(

_{n}*x*,

*y*) that agree with all the intensity measurements

*I*(

_{n}*u*,

*v*), where the illumination

*p*(

*x*,

*y*) and the coordinate translations (

*x*,

_{n}*y*) are assumed known

_{n}*a priori*.

*p*(

*x*,

*y*) as an illumination function, this formulation does not put any constraints on the form or nature of

*p*(

*x*,

*y*) so that it may be given by either a complex-valued field that originates from the diffraction of an aperture, a focusing lens, or be the transmittance of an aperture itself.

19. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. **93**, 023903 (2004). [CrossRef] [PubMed]

*f*(

_{n}*x*,

*y*), have no overlap with one another, the task is reduced to solving the conventional phase retrieval problem several times with no diversity. It is then important to have a substantial amount of overlap between the different field realizations. This constrains the problem very robustly and dramatically increases the success of phase retrieval. It is also worth noting that because a translation in all fields will produce a linear phase in the Fourier domain that cannot be detected, the technique is not sensitive to the absolute position of either the object or the illumination function. This implies that only the relative displacements of the object and illumination are important, and the technique is insensitive to whether it is the object or the illumination that moves [23

23. H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy **103**, 153–164 (2005). [CrossRef] [PubMed]

8. H. N. Chapman*et al.*, “High-resolution *ab initio* three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A **23**, 1179–1200 (2006). [CrossRef]

15. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A **3**, 1897–1907 (1986). [CrossRef]

*f*(

*x*,

*y*) and

*f**(-

*x*,-

*y*) [where (*) denotes complex conjugation] have the same Fourier intensity pattern since

*f*(

*x*,

*y*) and

*f**(-

*x*,-

*y*) inside the support. The fact that this stagnation mode has more energy outside the support than the actual solution identifies it as a local minima stagnation mode.

*f*(

*x*,

*y*), amplitude shown in Fig. 1(a), was obtained by multiplying an extended object [amplitude and phase shown in Figs. 2(a) and 2(b), respectively] by a real-valued circular aperture of 50 pixel radius, shown in Fig. 2(c). The simulated Fourier intensity data,

*I*(

*u*,

*v*), was obtained by Fourier transforming the field,

*f*(

*x*,

*y*), and taking the squared modulus of the result. Making the extent of the field,

*f*(

*x*,

*y*), less than half of the entire array ensures that the intensity will be Nyquist sampled. The square-root of the Fourier intensity was then used along with the object support (support constraint) as constraints for the phase retrieval algorithm. Because the image is complex-valued, a nonnegativity constraint cannot be used. A sequence of 45 iterations of the hybrid input-output algorithm (HIO) [1

**21**, 2758–2769 (1982). [CrossRef] [PubMed]

*β*=0.7, followed by 5 of the error-reduction algorithm were used, totaling 1000 iterations (20 sets of 50 iterations each). Because the support is centrosymmetric and the image is complex-valued, this reconstruction is particularly difficult for single-measurement phase retrieval.

*f*(

*x*,

*y*). Five of the 10 reconstructions exhibited a mild twin-image problem, such as that shown in Fig. 1(c). Notice that although the twin image dominates the reconstruction shown in Fig. 1(c), and the image is recognizable, faint features of the upright image are also discernable, for example the three coral branches on the upper-right part of the image. For the remaining two reconstructions, a faithful image of the object shown in Fig. 1(a) was obtained, with no discernable twin-image problem.

*S*is the set of points where the reconstruction,

*f̂*(

*x*,

*y*), satisfies the support constraint. The two aforementioned reconstructions with no visually discernable twin-image problem had a final support error of

*E*=0.0026 and

_{S}*E*=0.0060, respectively. For the five reconstructions exhibiting a mild twin-image problem,

_{S}*E*=0.0034, 0.0074, 0.0076, 0.0092, 0.0105, respectively. Notice that on average

_{S}*E*is an indicator of a mild twin-image stagnation problem. Finally, for the three reconstructions with a pronounced twin-image problem, the final support error was

_{S}*E*=0.0150, 0.0162, and 0.0192, respectively. A severe twin-image problem will have large value of

_{S}*E*, which identifies it as a stagnated partially-reconstructed image. If the final value of

_{S}*E*for a given reconstruction is high, then the reconstruction should be started again with a different starting guess. Alternatively there are techniques that can be used to escape stagnation or combine different reconstructions to arrive at an improved composite that can be used for further iterations [15

_{S}15. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A **3**, 1897–1907 (1986). [CrossRef]

*f*(

_{n}*x*,

*y*)=

*o*(

*x*-

*x*,

_{n}*y*-

*y*)

_{n}*p*(

*x*,

*y*) and

*f*

^{*}

_{n}(-

*x*,-

*y*)=

*o**(-

*x*-

*x*,-

_{n}*y*-

*y*)

_{n}*p**(-

*x*,-

*y*) have the same Fourier magnitude, we can differentiate them from the sign of the object shift, (

*x*,

_{n}*y*), between two diversity images if the two fields overlap. Thus translation diversity eliminates the ambiguity between the image and its twin, making the simultaneous appearance of features of the regular and twin image in the reconstruction much less likely.

_{n}## 3. Ptychographical iterative engine

**93**, 023903 (2004). [CrossRef] [PubMed]

*o*(

*x*,

*y*) to sequentially match all of the Fourier intensity measurements. For a single iteration, the current object estimate,

*ô*(

_{n}*x*,

*y*), is multiplied by a translated version of the known illumination function, to obtain an estimate of the

*n*-th diversity field,

*ĥ*(

_{n}*x*,

*y*) is computed:

*n*-th diversity position) is imposed while preserving the phase

*G*(

_{n}*u*,

*v*) is used to update the object estimate by

*p*(

*x*+

*x*,

_{n}*y*+

*y*),

_{n}**21**, 2758–2769 (1982). [CrossRef] [PubMed]

*f̂*(

*x*+

*x*,

_{n}*y*+

*y*), given by

_{n}*R*and

*I*indicate the real and imaginary part respectively (

*f̂*=

*f̂*+

^{R}*if̂*).

^{I}*e.g.*conjugate gradient, are superior to steepest descent, this result indicates that it should be possible to improve on the PIE.

*p*(

*x*,

*y*) with a 50 pixel radius, shown in Fig. 2(c). The resulting field was then Fourier transformed and the squared-magnitude of the result was used as the measured data. This procedure was repeated for 12 different positions of the aperture, (

*x*,

_{n}*y*), indicated by circles in Fig. 2(d), thus obtaining twelve two-dimensional far-field intensity measurements,

_{n}*I*(

_{n}*u*,

*v*). The aperture was shifted in intervals of 14 and 35 pixels in the

*x*and

*y*directions, respectively.

*β*=1 and

*α*=0.01, was used for reconstruction of

*o*(

*x*,

*y*) assuming perfect knowledge of

*p*(

*x*,

*y*) and (

*x*,

_{n}*y*). The algorithm was initialized with

_{n}*ô*(

*x*,

*y*) as a real-valued, uniform function. The amplitude and phase of the reconstruction after 200 iterations are shown in Figs. 2(e) and 2(f), respectively. Notice that a very good estimation of the original object is obtained within the support sampled by

*p*(

*x*,

*y*).

*x*,

_{n}*y*) and the aperture function

_{n}*p*(

*x*,

*y*).

*a priori*known information,

*i.e.*

*p*(

*x*,

*y*) and (

*x*,

_{n}*y*). The error in the translation parameters can be characterized by the global-shift-invariant root-mean-squared error (RMSE) of the translations, Δ

_{n}*r*, defined by

*x*,

_{n}*y*) and (

_{n}*x̂*,

_{n}*ŷ*) are the true and estimated object translations, respectively, for the

_{n}*n*-th diversity image. We introduced minimization with respect to global additive constants (

*a*and

*b*) to account for the fact that the reconstruction is unaffected by a global displacement of all the fields. In other words, since the reconstruction procedure is not affected by the absolute position of the object (only the translations are important), then addition of a constant to every displacement (in either

*x*or

*y*) should not increase our measure of the shift error. Upon minimizing Δ

*r*with respect to

*a*and

*b*we find

*r*=0.8 pixel. The amplitude and phase of the reconstruction with 500 iterations of the PIE algorithm is shown in Figs. 3(a) and 3(b), respectively. Notice that the quality of the image has degraded significantly.

## 4. Nonlinear optimization approach

*a priori*information, finite pixel size, transverse coherence, detector misalignments) and optimize over inaccurately known parameters as well, provided that the gradients of the error metric with respect to the unknown parameters can be computed efficiently.

*δ*is a small constant that prevents problems in the gradient computation where

*I*is close to zero,

_{n}*γ*is a real-valued constant, and

*W*(

_{n}*u*,

*v*) is a weighting function that can be used to emphasize regions with high SNR or set to zero for regions with poor SNR or where no signal was measured. This can be used, for example, to eliminate the effects of a beam stop or dead detector pixels by setting

*W*(

_{n}*u*,

*v*)=0 for those pixels. For the simulation results shown in this paper we used a uniform unity weighting function for all pixels.

*ε*with respect to the real and imaginary parts of the object,

*ô*(

*x*,

*y*)=

*ô*

_{R}(

*x*,

*y*)+

*iô*

_{I}(

*x*,

*y*), is obtained by computing the expression

*p̂*(

*x*,

*y*) can be computed in a similar fashion

*x̂*,

_{n}*ŷ*) is given by

_{n}*ô*(

*x*,

*y*) (25 iterations for this reconstruction) and following with joint estimation of

*ô*(

*x*,

*y*),

*p̂*(

*x*,

*y*) and (

*x̂*,

_{n}*ŷ*) in subsequent iterations. We define one iteration of the algorithm to be every time the gradient needs to be computed. The amplitude and phase of the object reconstruction after 500 iterations (with

_{n}*γ*=0.5) are shown in Figs. 3(d) and 3(e) respectively, where a substantial improvement over the PIE reconstruction is evident. The amplitude of the final estimate of

*p̂*(

*x*,

*y*) is shown in Fig. 3(f). Although for estimation of the object (and for the PIE), as few as 3 or 4 diversity measurements are typically sufficient for proper convergence of the algorithm, we observed that as many as 10 measurements are sometimes necessary to refine an inaccurate initial estimate for the illumination pattern and the translations. This is to be expected because the illumination pattern is optimized on a point-by-point basis and the number of free parameters is significantly increased.

37. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. **36**, 8352–8357 (1997). [CrossRef]

*x*′,

*y*′) and a complex-valued constant

*ρ*for each diversity field. Because a reconstruction that is translated and multiplied by a constant is still considered successful, this minimization procedure is important to make

_{n}*E*invariant to these operations.

*E*, as given in Eq. (22), requires finding the peak of the cross-correlation of

*f̂*and

_{n}*f*to within a small fraction of a pixel [37

_{n}37. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. **36**, 8352–8357 (1997). [CrossRef]

38. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. **33**, 156–158 (2008). [CrossRef] [PubMed]

38. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. **33**, 156–158 (2008). [CrossRef] [PubMed]

*E*versus iteration number for the reconstruction with PIE and nonlinear optimization for the case of inaccurate knowledge of both

*p̂*(

*x*,

*y*) and (

*x̂*,

_{n}*ŷ*). Notice that while the PIE stagnates after a few iterations, the nonlinear optimization approach continues making progress in the reconstruction while refining the estimation of

_{n}*ô*(

*x*,

*y*),

*p̂*(

*x*,

*y*) and (

*x̂*,

_{n}*ŷ*). The shift error, Δ

_{n}_{r}, is shown versus iteration number in Fig. 3(h). During the first 25 iterations this error remains constant because we optimize only over

*ô*(

*x*,

*y*), subsequent iterations show a gradual improvement of the estimation of (

*x̂*,

_{n}*ŷ*).

_{n}*ô*(

_{n}*x*,

*y*) is updated, noise artifacts are introduced by the

*n*-th diversity image. In contrast, nonlinear optimization improves the object estimation on every iteration by using all of the intensity measurements simultaneously, which makes it more robust to noise in the detected intensity patterns. To demonstrate that effect, we performed a computer simulation with noisy data, but with perfectly known

*p*(

*x*,

*y*) and (

*x*,

_{n}*y*).

_{n}^{4}photons on their brightest pixel, we applied poisson distributed noise. Figure 4(e) shows a cut through a sample intensity pattern before and after noise was applied. Notice that after about 20 pixels from the brightest pixel, the intensity falls below an average of 1 photon/pixel; this results in the loss of high frequency components and we thus expect the reconstruction to have a reduced resolution.

*E*, is obtained with the nonlinear optimization approach over the PIE.

## 5. Object contained within the illumination pattern

*p*(

*x*,

*y*) and diversity is introduced by translating the object within the extent of

*p*(

*x*,

*y*) and measuring the resulting far-field intensity patterns. It is then important to investigate whether in this situation translation diversity would work as well as it does with objects that are larger than the aperture.

*p*(

*x*,

*y*) for each of the five positions that were used for reconstruction.

*p*(

*x*,

*y*) and not just the object width as would be for a single-measurement image reconstruction. Faster convergence was observed by translating the object close to the edge of the aperture.

*p*(

*x*,

*y*), indicated by a dashed line in Fig. 5(b), was used for this reconstruction. Additional to the support constraint we used a nonnegativity constraint during the algorithm iterations. In this reconstruction the twin image dominates, but because the support is centrosymmetric, the reconstruction is susceptible to the twin-image problem and faint features of the upright image can also be observed.

## 6. Conclusion

**93**, 023903 (2004). [CrossRef] [PubMed]

20. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. **85**, 4795–4797 (2004). [CrossRef]

*ô*(

*x*,

*y*),

*p̂*(

*x*,

*y*) and (

*x̂*,

_{n}*ŷ*) respectively], so that the initial estimate of the parameters is refined as the reconstruction progresses. We have shown that this optimization approach leads to better reconstructions when the system is inaccurately characterized. In particular, the ability to optimize over the object translations, relaxes the required setup stability and accuracy of calibration for the translating stages in an experimental setup.

_{n}*ô*(

*x*,

*y*),

*p̂*(

*x*,

*y*) and (

*x̂*,

_{n}*ŷ*)], the nonlinear optimization approach is more robust in the presence of inaccurate system parameters and yields reduced noise artifacts. For imaging simulations we have observed comparable convergence for the PIE and nonlinear optimization if the illumination and translations are known accurately. Furthermore, the number of diversity images required for proper convergence was found to be increased when the system parameters are known inaccurately and we jointly optimize over

_{n}*ô*(

*x*,

*y*),

*p̂*(

*x*,

*y*) and (

*x̂*,

_{n}*ŷ*).

_{n}*p*(

*x*,

*y*) and no additional information about the object is introduced by the translation.

## References and links

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2. | J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A |

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30. | M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi-aperture systems utilizing individual sub-aperture control,” Proc. SPIE |

31. | B. J. Thelen, M. F. Reiley, and R. G. Paxman, “Fine-resolution multispectral imaging using wavelength diversity,” in |

32. | H. R. Ingleby and D. R. McGaughey “Parallel multiframe blind deconvolution using wavelength diversity,” Proc. SPIE |

33. | W. Hoppe, “Beugung im inhomogenen Primärstrahlwellenfeld. III. Amplituden-und Phasenbestimmung bei unperiodischen Objekten,” Acta Crystallogr. Sect. A |

34. | J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy viaWigner-distribution deconvolution,” Phil. Trans. R. Soc. Lond. A |

35. | N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. |

36. | S. T. Thurman, The Institute of Optics, University of Rochester, Rochester, New York, USA, R. T. DeRosa and J. R. Fienup are preparing a manuscript to be called “Amplitude metrics for field retrieval with hard-edge and uniformly-illuminated apertures.” |

37. | J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. |

38. | M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. |

39. | Image modified from an original courtesy of Universidad Autónoma de Nuevo León, Mexico. |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

(110.7440) Imaging systems : X-ray imaging

(340.7480) X-ray optics : X-rays, soft x-rays, extreme ultraviolet (EUV)

(110.3200) Imaging systems : Inverse scattering

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 27, 2008

Manuscript Accepted: April 24, 2008

Published: May 5, 2008

**Virtual Issues**

Vol. 3, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Manuel Guizar-Sicairos and James R. Fienup, "Phase retrieval with transverse translation diversity: a nonlinear
optimization approach," Opt. Express **16**, 7264-7278 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7264

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### References

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- Image modified from an original courtesy of Universidad Autónoma de Nuevo León, Mexico.

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