Abstract

Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95nm isotropic resolution.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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  33. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X-rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  45. T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22, 867–878 (2015).
    [Crossref] [PubMed]
  46. A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
    [Crossref]
  47. 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]
  48. M. Krenkel, M. Töpperwien, M. Bartels, P. Lingor, D. Schild, and T. Salditt, “X-ray phase contrast tomography from whole organ down to single cells,” Proc. SPIE 9212, 92120R (2014).
    [Crossref]
  49. L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12, 2960–2965 (2004).
    [Crossref] [PubMed]
  50. C. Homann, T. Hohage, J. Hagemann, A.-L. Robisch, and T. Salditt, “Validity of the empty-beam correction in near-field imaging,” Phys. Rev. A 91, 013821 (2015).
    [Crossref]
  51. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E=50-30 000 eV, Z=1-92,” At. Data. Nucl. Data Tables 54, 181–342 (1993).
    [Crossref]
  52. K. A. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24, 536–547 (2007).
    [Crossref]
  53. R. Xu, C.-C. Chen, L. Wu, M. C. Scott, W. Theis, C. Ophus, M. Bartels, Y. Yang, H. Ramezani-Dakhel, M. R. Sawaya, H. Heinz, L. D. Marks, P. Ercius, and J. Miao, “Three-dimensional coordinates of individual atoms in materials revealed by electron tomography,” Nat. Mater. 14, 1099–1103 (2015).
    [Crossref] [PubMed]
  54. M. van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” J. Struct. Biol. 151, 250–262 (2005).
    [Crossref] [PubMed]
  55. M. van Heel, “Similarity measures between images,” Ultramicroscopy 21, 95–100 (1987).
    [Crossref]

2015 (7)

S. Maretzke, “A uniqueness result for propagation-based phase contrast imaging from a single measurement,” Inverse Probl. 31, 065003 (2015).
[Crossref]

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

A. Robisch, K. Kröger, A. Rack, and T. Salditt, “Near-field ptychography using lateral and longitudinal shifts,” New J. Phys. 17, 073033 (2015).
[Crossref]

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22, 867–878 (2015).
[Crossref] [PubMed]

M. Bartels, M. Krenkel, J. Haber, R. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref] [PubMed]

C. Homann, T. Hohage, J. Hagemann, A.-L. Robisch, and T. Salditt, “Validity of the empty-beam correction in near-field imaging,” Phys. Rev. A 91, 013821 (2015).
[Crossref]

R. Xu, C.-C. Chen, L. Wu, M. C. Scott, W. Theis, C. Ophus, M. Bartels, Y. Yang, H. Ramezani-Dakhel, M. R. Sawaya, H. Heinz, L. D. Marks, P. Ercius, and J. Miao, “Three-dimensional coordinates of individual atoms in materials revealed by electron tomography,” Nat. Mater. 14, 1099–1103 (2015).
[Crossref] [PubMed]

2014 (2)

M. Krenkel, M. Töpperwien, M. Bartels, P. Lingor, D. Schild, and T. Salditt, “X-ray phase contrast tomography from whole organ down to single cells,” Proc. SPIE 9212, 92120R (2014).
[Crossref]

A. Ruhlandt, M. Krenkel, M. Bartels, and T. Salditt, “Three-dimensional phase retrieval in propagation-based phase-contrast imaging,” Phys. Rev. A 89, 033847 (2014).
[Crossref]

2013 (3)

T. Hohage and F. Werner, “Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data,” Numer. Math. 123, 745–779 (2013).
[Crossref]

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 3, 1927 (2013).
[Crossref] [PubMed]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

2011 (5)

K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

S. Kalbfleisch, H. Neubauer, S. Krüger, M. Bartels, M. Osterhoff, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIP Conf. Proc. 1365, 96–99 (2011).
[Crossref]

S. Kindermann, “Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems,” Electron. Trans. Numer. Anal 38, 233–257 (2011).

A. Burvall, U. Lundström, P. A. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in X-ray phase-contrast imaging suitable for tomography,” Opt. Express 19, 10359–10376 (2011).
[Crossref] [PubMed]

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Non-linear iterative phase retrieval based on Fréchet derivative,” Opt. Express 19, 22809–22819 (2011).
[Crossref] [PubMed]

2010 (2)

K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59, 1–99 (2010).
[Crossref]

H. Quiney, “Coherent diffractive imaging using short wavelength light sources,” J. Mod. Opt. 57, 1109–1149 (2010).
[Crossref]

2009 (1)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

2008 (2)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

A. Barty, S. Marchesini, H. Chapman, C. Cui, M. Howells, D. Shapiro, A. Minor, J. Spence, U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege, A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, “Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: Determination of structural deformation mechanisms,” Phys. Rev. Lett. 101, 055501 (2008).
[Crossref] [PubMed]

2007 (3)

J. Rodenburg, A. Hurst, A. Cullis, B. 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]

S. Marchesini, “Invited article: A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[Crossref]

K. A. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24, 536–547 (2007).
[Crossref]

2006 (3)

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional X-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

M. V. Klibanov, “On the recovery of a 2-d function from the modulus of its Fourier transform,” J. Math. Anal. Appl. 323, 818–843 (2006).
[Crossref]

M. Burger and B. Kaltenbacher, “Regularizing Newton–Kaczmarz methods for nonlinear ill-posed problems,” SIAM J. Numer. Anal. 44, 153–182 (2006).
[Crossref]

2005 (1)

M. van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” J. Struct. Biol. 151, 250–262 (2005).
[Crossref] [PubMed]

2004 (3)

L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12, 2960–2965 (2004).
[Crossref] [PubMed]

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

P. Jonas and A. Louis, “Phase contrast tomography using holographic measurements,” Inverse Probl. 20, 75 (2004).
[Crossref]

2003 (1)

2002 (3)

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[Crossref]

M. Hintermüller, K. Ito, and K. Kunisch, “The primal-dual active set strategy as a semismooth Newton method,” SIAM J. Optimiz. 13, 865–888 (2002).
[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]

1999 (2)

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X-rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

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 400, 342–344 (1999).
[Crossref]

1997 (2)

B. Blaschke, A. Neubauer, and O. Scherzer, “On convergence rates for the iteratively regularized Gauss-Newton method,” IMA J. Numer. Anal. 17, 421–436 (1997).
[Crossref]

A. Pogany, D. Gao, and S. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

1993 (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E=50-30 000 eV, Z=1-92,” At. Data. Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

1992 (2)

R. Wade, “A brief look at imaging and contrast transfer,” Ultramicroscopy 46, 145–156 (1992).
[Crossref]

A. B. Bakushinskii, “The problem of the convergence of the iteratively regularized Gauss-Newton method,” Zh. Vychisl. Mat. Mat. Fiz. 32, 1503–1509 (1992).

1987 (1)

M. van Heel, “Similarity measures between images,” Ultramicroscopy 21, 95–100 (1987).
[Crossref]

1977 (1)

J. Guigay, “Fourier-transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

1971 (1)

H. Erickson and A. Klug, “Measurement and compensation of defocusing and aberrations by fourier processing of electron micrographs,” Phil. Trans. R. Soc. B 261, 105–118 (1971).
[Crossref]

1970 (1)

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[Crossref] [PubMed]

1966 (1)

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Soviet Math. Dokl. 7, 414–417 (1966).

1937 (1)

S. Kaczmarz, “Angen¨aherte Auflosüng von Systemen linearer Gleichungen,” B. Acad. Pol. Sci. 35, 355–357 (1937).

Artyukhin, A. B.

A. Barty, S. Marchesini, H. Chapman, C. Cui, M. Howells, D. Shapiro, A. Minor, J. Spence, U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege, A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, “Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: Determination of structural deformation mechanisms,” Phys. Rev. Lett. 101, 055501 (2008).
[Crossref] [PubMed]

Bakushinskii, A. B.

A. B. Bakushinskii, “The problem of the convergence of the iteratively regularized Gauss-Newton method,” Zh. Vychisl. Mat. Mat. Fiz. 32, 1503–1509 (1992).

Bartels, M.

M. Bartels, M. Krenkel, J. Haber, R. Wilke, and T. Salditt, “X-ray holographic imaging of hydrated biological cells in solution,” Phys. Rev. Lett. 114, 048103 (2015).
[Crossref] [PubMed]

T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22, 867–878 (2015).
[Crossref] [PubMed]

R. Xu, C.-C. Chen, L. Wu, M. C. Scott, W. Theis, C. Ophus, M. Bartels, Y. Yang, H. Ramezani-Dakhel, M. R. Sawaya, H. Heinz, L. D. Marks, P. Ercius, and J. Miao, “Three-dimensional coordinates of individual atoms in materials revealed by electron tomography,” Nat. Mater. 14, 1099–1103 (2015).
[Crossref] [PubMed]

A. Ruhlandt, M. Krenkel, M. Bartels, and T. Salditt, “Three-dimensional phase retrieval in propagation-based phase-contrast imaging,” Phys. Rev. A 89, 033847 (2014).
[Crossref]

M. Krenkel, M. Töpperwien, M. Bartels, P. Lingor, D. Schild, and T. Salditt, “X-ray phase contrast tomography from whole organ down to single cells,” Proc. SPIE 9212, 92120R (2014).
[Crossref]

K. Giewekemeyer, S. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A 83, 023804 (2011).
[Crossref]

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T. Salditt, M. Osterhoff, M. Krenkel, R. N. Wilke, M. Priebe, M. Bartels, S. Kalbfleisch, and M. Sprung, “Compound focusing mirror and x-ray waveguide optics for coherent imaging and nano-diffraction,” J. Synchrotron Radiat. 22, 867–878 (2015).
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A. Barty, S. Marchesini, H. Chapman, C. Cui, M. Howells, D. Shapiro, A. Minor, J. Spence, U. Weierstall, J. Ilavsky, A. Noy, S. P. Hau-Riege, A. B. Artyukhin, T. Baumann, T. Willey, J. Stolken, T. van Buuren, and J. H. Kinney, “Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: Determination of structural deformation mechanisms,” Phys. Rev. Lett. 101, 055501 (2008).
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Supplementary Material (1)

NameDescription
» Data File 1: CSV (19 KB)      The computed site coordinates of the imaged colloidal crystal underlying to figure 5.

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Figures (6)

Fig. 1
Fig. 1

Schematic setup for propagation-based phase contrast imaging with hard x-rays: GINIX at P10 beamline, DESY [45, 48]. Quasi-monocromatic x-rays are focused onto a waveguide, illuminating a downstream object by a cone beam emanating from this coherent quasi-point source. Rotation of the sample allows for tomographic measurements.

Fig. 2
Fig. 2

X-ray phase contrast imaging of a nano-structured object using the IRGNM reconstruction algorithm. The test pattern (institute logo) was defined in a thin gold film by focused ion beam milling. (a) Diffraction pattern recorded at GINIX endstation, P10 beam-line, DESY [45] (flat-field corrected, maximum photon flux per pixel ≈3400). (b) Reconstructed phase image of the entire field of view, assuming a fixed ratio of 0.21 between absorption µ and phase shifts ϕ. (c) Magnification of (b) in the framed region around the logo. For comparison, the dashed inset shows the corresponding part of the phase map reconstructed by direct inversion of the CTF (16) via the methods of [48]. Scale bars: 2 µm in the effective geometry. Fringe artifacts on the upside of the logo are due to diffraction fringes leaving the field of view in (a). The phase shifts ϕ ≈ 0.20 induced by the object correspond to a thickness of the gold layer of about 100nm.

Fig. 3
Fig. 3

Simultaneous IRGNM reconstruction of phase shifts ϕ and absorption µ from the data in Fig. 2(a) without assuming a fixed ratio µ/ϕ. Negative values of µ and ϕ in-dicate missing material in the gold film. The circular support visible in the phase- and absorption maps and negativity of µ and ϕ has been imposed as a constraint. All other parameters are retained as in the computation of Fig. 2(b). The dashed inset in the ab-sorption image shows the reconstruction without negativity constraint in the IRGNM and demonstrates that simultaneous recovery tends to introduce low-frequency artifacts. These are effectively suppressed by exploiting physical a priori knowledge on the sign of phase shifts and absorption. Scale bar: 2 µm.

Fig. 4
Fig. 4

Supplementary IRGNM reconstruction of a non-homogeneous object from a simulated near-field hologram with 2% Gaussian white noise. Constraints, reconstruction- and setup parameters are chosen as for the experimental data in Fig. 3. (a) and (b): Exact object composed of an absorbing logo structure with ϕ = 0.2, µ = 0.04 (gold) embedded into a non-absorbing disc that induces the same phase shifts ϕ = 0.2. (c) and (d): Recovered phase and absorption images. Except for a slight halo, features in ϕ and µ are correctly identified, revealing the hidden logo structure by the enabled absorption contrast.

Fig. 5
Fig. 5

X-ray phase contrast tomography of a colloidal micro-crystal composed of 415nm-size polystyrene beads, reconstructed by a regularized Newton-Kaczmarz method. The maximum photon flux is 770 per pixel corresponding to an absorbed dose 110kGy.(a) Slice visualization of the tomographic data given by 1024 × 1024 flat-field corrected holograms measured under 249 incident angles between 0 and 173° at GINIX endstation, P10 beamline, DESY [45]. (b) Central slice of the reconstructed 2563 voxel volume, plotting the increment δ of the refractive index n = 1−δ +iβ which is proportional to electron density. A priori constraints were positivity δ ≥ 0 and vanishing absorption β = 0. The scale bar is 1 µm. (c) Fourier shell correlation (FSC) computed for reconstructions from complementary data sets of 125 and 124 incident angles (green curve). For comparison, the blue curve shows the result without positivity constraint and the red dashed one plots the 1/2-bit threshold curve [54]. The intersection between the green- and the red curve indicates an achieved resolution of 95nm. (d) Binary representation of the slice in (b), determined by deconvolving the reconstructed δ with the form factor of a 415nm-sized homogeneous sphere after Gaussian filtering.

Fig. 6
Fig. 6

3d-rendering of a colloidal crystal of polystyrene nano-beads reconstructed by Newton-based phase contrast tomography. The underlying binary object, a slice of which is shown in Fig. 5(d), has been determined by deconvolving the smoothened reconstruction of the refraction parameter δ with the form factor of a homogeneous sphere of diameter 415nm. The coordinates of the colloid sites are provided in Data File 1. Scale bar: 1 µm.

Equations (20)

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I obs = F ( f ) + ε .
f k + 1 = f k + F [ f k ] 1 ( I obs F ( f k ) ) .
f k + 1 = arg min f X F ( f k ) + F [ f k ] ( f f k ) I obs Y 2 + α k f f 0 X 2
f k + 1 = f k + ( F [ f k ] * F [ f k ] + α k ) 1 ( F [ f k ] * ( I obs F ( f k ) ) + α k ( f 0 f k ) ) .
( F [ f K ] * F [ f K ] + α k ) 1 1 α k .
I I obs Y : = | I I obs | 2 d x .
I I obs Y : = I I obs max ( I 0 , I obs ) 1 2 L 2 ,
f X : = ( 1 + ξ 2 ) s 2 ( f ) ( ξ ) L 2
γ min ( 0 , f k ) min ( 0 , sign ( f k ) ) ( f f k ) L 2 2 ,
f X dis 2 = f T G X f , I Y dis 2 = I T G Y I
F dis [ f ] * = G X 1 F d i s [ f ] T G Y .
( I 1 obs , , I p obs ) = ( F 1 , , F p ) ( f ) + ε .
f k + 1 = arg min f X F j k ( f k ) + F j k [ f k ] ( f f k ) I j k obs Y j k 2 + α k ( β k f f 0 X 2 + ( 1 β k ) f f k X 2 )
I = F ( f ) : = | D ( exp ( i f ) ) | 2 ,
D ( ψ ) : = 1 ( exp ( i π ξ 2 N F ) ( ψ ) ( ξ ) )
( F ( f ) 1 ) 2 sin ( π ξ 2 N F ) ( ϕ ) ( ξ ) cos ( π ξ 2 N F ) ( μ ) ( ξ )
F ( f + h ) = F ( f ) + 2 ( i D ( exp ( i f ) ) ¯ D ( exp ( i f ) h ) ) + O ( h X 2 ) .
F [ f ] h = 2 J ( D ( exp ( i f ) ) ¯ D ( exp ( i f ) h ) ) .
ϕ θ i μ θ / 2 = k ( δ θ i β θ ) d z = k ( δ i β ) ( θ )
I tomo = F tomo ( f tomo ) : = | D ( exp ( i k ( f tomo ) ) ) | 2

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