Abstract

In this paper, we study the success rate of the reconstruction of objects of finite extent given the magnitude of its Fourier transform and its geometrical shape. We demonstrate that the commonly used combination of the hybrid input output and error reduction algorithm is significantly outperformed by an extension of this algorithm based on randomized overrelaxation. In most cases, this extension tremendously enhances the success rate of reconstructions for a fixed number of iterations as compared to reconstructions solely based on the traditional algorithm. The good scaling properties in terms of computational time and memory requirements of the original algorithm are not influenced by this extension.

© 2012 OSA

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    [CrossRef] [PubMed]
  8. A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]

2011

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

2009

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nat Mater8, 291–298 (2009).
[CrossRef] [PubMed]

A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
[CrossRef] [PubMed]

2008

V. Elser and R. P. Millane, “Reconstruction of an object from its symmetry-averaged diffraction pattern,” Acta Crystallogr., Sect. A: Found. Crystallogr.64, 273–279 (2008).
[CrossRef]

A. A. Minkevich, T. Baumbach, M. Gailhanou, and O. Thomas, “Applicability of an iterative inversion algorithm to the diffraction patterns from inhomogeneously strained crystals,” Phys. Rev. B78, 174110 (2008).
[CrossRef]

2007

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

A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B76, 104106 (2007).
[CrossRef]

S. Marchesini, “Phase retrieval and saddle-point optimization,” J. Opt. Soc. Am. A24, 3289–3296 (2007).
[CrossRef]

2006

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature442, 63–66 (2006).
[CrossRef] [PubMed]

2004

F. v. d. Veen and F. Pfeiffer, “Coherent x-ray scattering,” Phys J..: Condens. Matter16, 5003–5030 (2004).
[CrossRef]

2003

2002

1999

H. Takajo, T. Takahashi, and T. Shizuma, “Further study on the convergence property of the hybrid input–output algorithm used for phase retrieval,” J. Opt. Soc. Am. A16, 2163–2168 (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,” Nature400, 342–344 (1999).
[CrossRef]

1998

1996

1990

1989

L. Auslander and F. A. Grunbaum, “The Fourier transform and the discrete Fourier transform,” Inverse Prob.5, 149 (1989).
[CrossRef]

1986

1984

1982

D. C. Youla and H. Webb, “Image restoration by the method of convex projections: part 1 – theory,” IEEE Trans. Med. Imaging1, 81 –94 (1982).
[CrossRef] [PubMed]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik (Stuttgart)61, 247–262 (1982).

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

Afanasiev, G. N.

G. N. Afanasiev, Vavilov-Cherenkov and Synchrotron Radiation: Foundations and Applications (Springer, Netherlands, 2004).

Auslander, L.

L. Auslander and F. A. Grunbaum, “The Fourier transform and the discrete Fourier transform,” Inverse Prob.5, 149 (1989).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik (Stuttgart)61, 247–262 (1982).

Baumbach, T.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

A. A. Minkevich, T. Baumbach, M. Gailhanou, and O. Thomas, “Applicability of an iterative inversion algorithm to the diffraction patterns from inhomogeneously strained crystals,” Phys. Rev. B78, 174110 (2008).
[CrossRef]

U. Pietsch, V. Holy, and T. Baumbach, High-Resolution X-ray Scattering From Thin Films to Lateral Nanostructures (Springer, New York, 2004).

Bauschke, H. H.

Biermanns, A.

A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
[CrossRef] [PubMed]

Chamard, V.

A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B76, 104106 (2007).
[CrossRef]

Chapman, H. N.

Charalambous, P.

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,” Nature400, 342–344 (1999).
[CrossRef]

Charlet, B.

A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B76, 104106 (2007).
[CrossRef]

Combettes, P. L.

Davydok, A.

A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
[CrossRef] [PubMed]

Diaz, A.

A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
[CrossRef] [PubMed]

Elser, V.

V. Elser and R. P. Millane, “Reconstruction of an object from its symmetry-averaged diffraction pattern,” Acta Crystallogr., Sect. A: Found. Crystallogr.64, 273–279 (2008).
[CrossRef]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A20, 40–55 (2003).
[CrossRef]

Fienup, J.

Fienup, J. R.

Fohtung, E.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

Gailhanou, M.

A. A. Minkevich, T. Baumbach, M. Gailhanou, and O. Thomas, “Applicability of an iterative inversion algorithm to the diffraction patterns from inhomogeneously strained crystals,” Phys. Rev. B78, 174110 (2008).
[CrossRef]

A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B76, 104106 (2007).
[CrossRef]

Gottschalch, V.

A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
[CrossRef] [PubMed]

Grigoriev, D.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

Grunbaum, F. A.

L. Auslander and F. A. Grunbaum, “The Fourier transform and the discrete Fourier transform,” Inverse Prob.5, 149 (1989).
[CrossRef]

Harder, R.

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nat Mater8, 291–298 (2009).
[CrossRef] [PubMed]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature442, 63–66 (2006).
[CrossRef] [PubMed]

Holy, V.

U. Pietsch, V. Holy, and T. Baumbach, High-Resolution X-ray Scattering From Thin Films to Lateral Nanostructures (Springer, New York, 2004).

Holý, V.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

Irvine, A. C.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

Kirz, J.

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,” Nature400, 342–344 (1999).
[CrossRef]

Levi, A.

Luke, D. R.

Marchesini, S.

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

S. Marchesini, “Phase retrieval and saddle-point optimization,” J. Opt. Soc. Am. A24, 3289–3296 (2007).
[CrossRef]

Metzger, T.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

Metzger, T. H.

A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
[CrossRef] [PubMed]

Miao, J.

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,” Nature400, 342–344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15, 1662–1669 (1998).
[CrossRef]

Micha, J.-S.

A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B76, 104106 (2007).
[CrossRef]

Millane, R.

Millane, R. P.

V. Elser and R. P. Millane, “Reconstruction of an object from its symmetry-averaged diffraction pattern,” Acta Crystallogr., Sect. A: Found. Crystallogr.64, 273–279 (2008).
[CrossRef]

R. P. Millane, “Multidimensional phase problems,” J. Opt. Soc. Am. A13, 725–734 (1996).
[CrossRef]

Minkevich, A. A.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

A. A. Minkevich, T. Baumbach, M. Gailhanou, and O. Thomas, “Applicability of an iterative inversion algorithm to the diffraction patterns from inhomogeneously strained crystals,” Phys. Rev. B78, 174110 (2008).
[CrossRef]

A. A. Minkevich, M. Gailhanou, J.-S. Micha, B. Charlet, V. Chamard, and O. Thomas, “Inversion of the diffraction pattern from an inhomogeneously strained crystal using an iterative algorithm,” Phys. Rev. B76, 104106 (2007).
[CrossRef]

Novák, V.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

Paetzelt, H.

A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
[CrossRef] [PubMed]

Pfeifer, M. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature442, 63–66 (2006).
[CrossRef] [PubMed]

Pfeiffer, F.

F. v. d. Veen and F. Pfeiffer, “Coherent x-ray scattering,” Phys J..: Condens. Matter16, 5003–5030 (2004).
[CrossRef]

Pietsch, U.

A. Biermanns, A. Davydok, H. Paetzelt, A. Diaz, V. Gottschalch, T. H. Metzger, and U. Pietsch, “Individual GaAs nanorods imaged by coherent x-ray diffraction,” J. Synchrotron Radiat.16, 796–802 (2009).
[CrossRef] [PubMed]

U. Pietsch, V. Holy, and T. Baumbach, High-Resolution X-ray Scattering From Thin Films to Lateral Nanostructures (Springer, New York, 2004).

Riotte, M.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, T. Metzger, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Strain field in (Ga,Mn)As/GaAs periodic wires revealed by coherent x-ray diffraction,” Europhys. Lett.94, 66001 (2011).
[CrossRef]

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

Robinson, I.

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nat Mater8, 291–298 (2009).
[CrossRef] [PubMed]

Robinson, I. K.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature442, 63–66 (2006).
[CrossRef] [PubMed]

Sayre, D.

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,” Nature400, 342–344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15, 1662–1669 (1998).
[CrossRef]

Schmidbauer, M.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

Seldin, J. H.

Shizuma, T.

Slobodskyy, T.

A. A. Minkevich, E. Fohtung, T. Slobodskyy, M. Riotte, D. Grigoriev, M. Schmidbauer, A. C. Irvine, V. Novák, V. Holý, and T. Baumbach, “Selective coherent x-ray diffractive imaging of displacement fields in (Ga,Mn)As/GaAs periodic wires,” Phys. Rev. B84, 054113 (2011).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Graphical illustration of the (HIO+OR)+ER-algorithm and its building blocks.

Fig. 2
Fig. 2

Pure phase objects used for the investigation of the convergence of the (HIO+OR)+ER-algorithm. The magnitude of both objects is constant. The phase is plotted using a HSV color-bar, however, the region outside the shape �� has been set to black. The oversampling ratio is σ = 8.456.

Fig. 3
Fig. 3

Investigation of the (HIO+OR)+ER-algorithm for a purely real object fin with strong variation of its magnitude over short length scales (oversampling ratio σ = 5.06). In Fig. (b), continuous lines represent the (HIO+OR)+ER-algorithm, isolated dots the HIO+ER-algorithm. A pure HIO+OR-calculation without ER is included as black, dash-dotted curve.

Fig. 4
Fig. 4

Comparison of the success rate of reconstructions of pure phase objects (see Eq. (20), (22) and (23)) with the HIO+ER- and the (HIO+OR)+ER-algorithm. The parameter β was fixed to 0.85. Continuous lines represent results of the (HIO+OR)+ER-algorithm, isolated dots of the HIO+ER-algorithm. A pure HIO+OR-calculation without ER is included as black, dash-dotted curve.

Fig. 5
Fig. 5

Long-term stagnation of the success rate of the traditional HIO-algorithm (without overrelaxation and without randomization) for the first phase object (see Eq. 22) for different choices of the internal parameters (NHIO, NER), but fixed β = 0.85.

Fig. 6
Fig. 6

Investigation of the sensitivity of the (HIO+OR)+ER-algorithm with NHIO = 50 and NER = 20 on the choice of the parameter β and on the range of the uniform distribution determining the relaxation parameter λA for the first phase object (see Eq. 22).

Fig. 7
Fig. 7

Success rate of the HIO+ER-algorithm for fixed overrelaxation λA (no randomization). Parameters are chosen as (NHIO = 130, NER = 10) and β = 0.85.

Fig. 8
Fig. 8

Comparison of the success rate of both frameworks which provide a generalization of the traditional HIO-algorithm based on randomization. Continuous lines illustrate the behavior for randomized overrelaxation of PA (see Eq. (10)), whereas dots represent the behavior of the success rate resulting from independent randomization of the coefficients in a projection polynomial (see Eq. (13) and Eq. (14)). In both cases, the deterministic contribution is equivalent to the traditional HIO-algorithm with parameters NHIO = 140 and β = 0.85. No ER has been performed.

Equations (26)

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f in ( x ) = k 𝕄 [ j = 1 d exp ( 2 π i N j k j x j ) ] A k exp ( i Φ k ) x 𝕊 ,
2 ( N N 𝕊 ) N N 2 N 𝕊 ,
f x ( i + 1 ) = P 𝕊 P A f x ( i ) H ^ ER f x ( i ) .
P 𝕊 f x ( i ) = { f x ( i ) if x 𝕊 , 0 if x 𝕊
P A g k ( i ) = A k exp ( iarg ( g k ( i ) ) ) .
f x ( i + 1 ) = { P A f x ( i ) if x 𝕊 , f x ( i ) β P A f x ( i ) if x 𝕊 ,
f x ( i + 1 ) = [ 1 P 𝕊 β P A + ( 1 + β ) P 𝕊 P A ] f x ( i ) H ^ HIO ( β ) f x ( i ) .
Q μ ; λ μ 1 + λ μ ( P μ 1 ) ,
f x ( i + 1 ) = [ 1 + β ( P 𝕊 Q A ; λ A P A Q 𝕊 ; λ 𝕊 ) ] f x ( i ) H ^ Diff ( β , λ A , λ 𝕊 ) f x ( i ) ,
Q A ; λ A = 1 + λ A ( P A 1 ) .
f x ( i + 1 ) = [ 1 P 𝕊 β Q A ; λ A + ( 1 + β ) P 𝕊 Q A ; λ A ] f x ( i ) H ^ HIO + O R ( β , λ A ) f x ( i ) .
H ^ HIO + O R ( β , λ A ) [ 1 + β ( λ A 1 ) ] + [ β λ A β λ A ] P 𝕊 β λ A P A + [ ( 1 + β ) λ A ] P 𝕊 P A .
H ^ Proj ( b , c 𝕊 , c A ) = b 1 + n = 1 n Max even [ c 𝕊 , 2 n ( P 𝕊 P A ) n + c A , 2 n ( P A P 𝕊 ) n ] + n = 0 n Max odd [ c 𝕊 , 2 n + 1 P 𝕊 ( P A P 𝕊 ) n + c A , 2 n + 1 P A ( P 𝕊 P A ) n ] ,
b = 1 n = 1 p [ c n , 𝕊 + c n , A ] .
c ξ , n = c ξ , n ( D ) + r ξ , n c ξ , n ( R ) .
H ^ HIO R ( b , c 𝕊 , 1 , c 𝕊 , 1 , c A , 1 , c A , 2 ) b 1 + c 𝕊 , 1 P 𝕊 + c A , 1 P A + c 𝕊 , 2 P 𝕊 P A + c A , 2 P A P 𝕊
c 𝕊 , 1 ( D ) = 1 , c A , 1 ( D ) = β , c 𝕊 , 2 ( D ) = 1 + β , c A , 2 ( D ) = 0 .
c 𝕊 , 1 = β λ A β λ A , c A , 1 = β λ A , c 𝕊 , 2 = ( 1 + β ) λ A , c A , 2 = 0 ,
c 𝕊 , 1 = 1 γ A ( 1 + β ) , c A , 1 = β ( 1 + γ A ) , c 𝕊 , 2 = ( 1 + β ) ( 1 + γ A ) , c A , 2 = 0 ,
φ ( i ) = arccos ( | f ( i ) ; f in | / f ( i ) ; f ( i ) f in ; f in ) .
ε ( i ) = | g ˜ ( i ) | A ; | g ˜ ( i ) | A A ; A = 1 A 2 2 k ( | g ˜ ( i ) ( k ) | A ( k ) ) 2 ,
χ ( i ) = arccos ( | f ( i 1 ) ; f ( i ) | / f ( i 1 ) ; f ( i 1 ) f ( i ) ; f ( i ) ) .
f ( x ) = exp ( i ξ ( x ) ) Ω 𝕊 ( x ) ,
Ω 𝕊 ( x ) = { 0 if x 𝕊 , 1 if x 𝕊 .
ξ 1 ( x , y ) = ( 2 π ) 2 [ ( x b 1 ) 2 + ( y c 1 ) 2 ] ,
ξ 2 ( x , y ) = ( 2 π ) [ ( x b 2 ) 3 + ( y c 2 ) 2 + x 2 y 3 c 2 2 b 2 3 ]

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