Abstract

Several strategies in phase retrieval are unified by an iterative “difference map” constructed from a pair of elementary projections and three real parameters. For the standard application in optics, where the two projections implement Fourier modulus and object support constraints, respectively, the difference map reproduces the “hybrid” form of Fienup’s input–output map when a particular choice is made for two of the parameters. The geometric construction of the difference map illuminates the distinction between its fixed points and the recovered object, as well as the mechanism whereby the form of stagnation encountered by alternating projection schemes is avoided. When support constraints are replaced by object histogram or atomicity constraints, the difference map lends itself to crystallographic phase retrieval. Numerical experiments with synthetic data suggest that structures with hundreds of atoms can be solved.

© 2003 Optical Society of America

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References

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  1. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  2. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7, pp. 231–275.
  3. C. Giacovazzo, Direct Phasing in Crystallography (Oxford U. Press, Oxford, UK, 1998).
  4. H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).
  5. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  6. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  7. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  8. H. Takajo, T. Takahashi, R. Ueda, M. Taninaka, “Study on the convergence property of the hybrid input output algorithm used for phase retrieval,” J. Opt. Soc. Am. A 15, 2849–2861 (1998).
    [CrossRef]
  9. H. Takajo, T. Takahashi, T. Shizuma, “Further study on the convergence property of the hybrid input output algorithm used for phase retrieval,” J. Opt. Soc. Am. A 16, 2163–2168 (1999).
    [CrossRef]
  10. The “difference map” considered here should not be confused with the “map” of the electron density studied by crystallographers in “difference Fourier synthesis.” The latter, in our notation, corresponds to πmod(ρ)-ρ, whereas the difference that we consider involves a projection on a prioriconstraints as well.
  11. The term “stagnation” is interpreted to be the vanishing of the change between iterates.
  12. The term “subspace” denotes a general subset of EN and in almost all cases of interest is a smooth submanifold, possibly with boundary. When encountered in the discussion, linear or affine subspaces will be explicitly identified as such.
  13. K. Y. J. Zhang, P. Main, “Histogram matching as a new density modification technique for phase refinement and extension of protein molecules,” Acta Crystallogr. Sect. A 46, 41–46 (1990).
    [CrossRef]
  14. V. Elser, “Linear time heuristic for the bipartite Euclidean matching problem,” (manuscript available from the author: ve10@cornell.edu).
  15. In the case of support constraint with positivity, the constraint subspace is a smooth space with boundary, and the relevant dimensionality is that of the space without boundary.
  16. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  17. For a smooth subspace C with tangent space X at a∈C, the affine approximation to C at a is the space C′=X+a. By taking the set of all differences of elements in C′, one recovers the linear space: X=C′-C′.
  18. V. Elser, “Random projections and the optimization of an algorithm for phase retrieval,” J. Phys. A. Math. Gen. 35, 1–13 (2002).
  19. D. Sayre, “The squaring method: a new method for phase determination,” Acta Crystallogr. 5, 60–65 (1952).
    [CrossRef]
  20. T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXVI. Developments of the Sayre-equation tangent formula,” Acta Crystallogr., Sect. A 44, 353–357 (1988).
    [CrossRef]
  21. D. Sayre, “On least-squares refinement of the phases of crystallographic structure factors,” Acta Crystallogr., Sect. A 28, 210–212 (1972).
    [CrossRef]
  22. H. A. David, Order Statistics, 2nd ed. (Wiley, New York, 1981).
  23. C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995).
    [CrossRef]
  24. R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993).
    [CrossRef] [PubMed]
  25. H. H. Bauschke, P. L. Combettes, D. R. Luke, “Phase retrieval, Gerchberg–Saxton algorithm, and Fienup vari ants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
    [CrossRef]

2002

1999

1998

1995

C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995).
[CrossRef]

1993

R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993).
[CrossRef] [PubMed]

1990

K. Y. J. Zhang, P. Main, “Histogram matching as a new density modification technique for phase refinement and extension of protein molecules,” Acta Crystallogr. Sect. A 46, 41–46 (1990).
[CrossRef]

R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

1988

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXVI. Developments of the Sayre-equation tangent formula,” Acta Crystallogr., Sect. A 44, 353–357 (1988).
[CrossRef]

1984

1982

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

D. Sayre, “On least-squares refinement of the phases of crystallographic structure factors,” Acta Crystallogr., Sect. A 28, 210–212 (1972).
[CrossRef]

1963

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

1952

D. Sayre, “The squaring method: a new method for phase determination,” Acta Crystallogr. 5, 60–65 (1952).
[CrossRef]

Bauschke, H. H.

Blessing, R. H.

C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995).
[CrossRef]

Combettes, P. L.

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7, pp. 231–275.

David, H. A.

H. A. David, Order Statistics, 2nd ed. (Wiley, New York, 1981).

Debaerdemaeker, T.

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXVI. Developments of the Sayre-equation tangent formula,” Acta Crystallogr., Sect. A 44, 353–357 (1988).
[CrossRef]

DeTitta, G. T.

R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993).
[CrossRef] [PubMed]

Elser, V.

V. Elser, “Random projections and the optimization of an algorithm for phase retrieval,” J. Phys. A. Math. Gen. 35, 1–13 (2002).

Fienup, J. R.

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

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7, pp. 231–275.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Giacovazzo, C.

C. Giacovazzo, Direct Phasing in Crystallography (Oxford U. Press, Oxford, UK, 1998).

Hauptman, H. A.

C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995).
[CrossRef]

R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993).
[CrossRef] [PubMed]

Jones, R.

R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993).
[CrossRef] [PubMed]

Langs, D. A.

R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993).
[CrossRef] [PubMed]

Levi, A.

Luke, D. R.

Main, P.

K. Y. J. Zhang, P. Main, “Histogram matching as a new density modification technique for phase refinement and extension of protein molecules,” Acta Crystallogr. Sect. A 46, 41–46 (1990).
[CrossRef]

Millane, R. P.

Miller, R.

C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995).
[CrossRef]

R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993).
[CrossRef] [PubMed]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Sayre, D.

D. Sayre, “On least-squares refinement of the phases of crystallographic structure factors,” Acta Crystallogr., Sect. A 28, 210–212 (1972).
[CrossRef]

D. Sayre, “The squaring method: a new method for phase determination,” Acta Crystallogr. 5, 60–65 (1952).
[CrossRef]

Shizuma, T.

Smith, G. D.

C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995).
[CrossRef]

Stark, H.

Takahashi, T.

Takajo, H.

Taninaka, M.

Tate, C.

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXVI. Developments of the Sayre-equation tangent formula,” Acta Crystallogr., Sect. A 44, 353–357 (1988).
[CrossRef]

Teeter, M. M.

C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995).
[CrossRef]

Ueda, R.

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Weeks, C. M.

C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995).
[CrossRef]

R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993).
[CrossRef] [PubMed]

Woolfson, M. M.

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXVI. Developments of the Sayre-equation tangent formula,” Acta Crystallogr., Sect. A 44, 353–357 (1988).
[CrossRef]

Yang, Y.

H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).

Zhang, K. Y. J.

K. Y. J. Zhang, P. Main, “Histogram matching as a new density modification technique for phase refinement and extension of protein molecules,” Acta Crystallogr. Sect. A 46, 41–46 (1990).
[CrossRef]

Acta Crystallogr.

D. Sayre, “The squaring method: a new method for phase determination,” Acta Crystallogr. 5, 60–65 (1952).
[CrossRef]

Acta Crystallogr. Sect. A

K. Y. J. Zhang, P. Main, “Histogram matching as a new density modification technique for phase refinement and extension of protein molecules,” Acta Crystallogr. Sect. A 46, 41–46 (1990).
[CrossRef]

Acta Crystallogr., Sect. A

T. Debaerdemaeker, C. Tate, M. M. Woolfson, “On the application of phase relationships to complex structures. XXVI. Developments of the Sayre-equation tangent formula,” Acta Crystallogr., Sect. A 44, 353–357 (1988).
[CrossRef]

D. Sayre, “On least-squares refinement of the phases of crystallographic structure factors,” Acta Crystallogr., Sect. A 28, 210–212 (1972).
[CrossRef]

Acta Crystallogr., Sect. D

C. M. Weeks, H. A. Hauptman, G. D. Smith, R. H. Blessing, M. M. Teeter, R. Miller, “Crambin: a direct solution for a 400 atom structure,” Acta Crystallogr., Sect. D 51, 33–38 (1995).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

J. Phys. A. Math. Gen.

V. Elser, “Random projections and the optimization of an algorithm for phase retrieval,” J. Phys. A. Math. Gen. 35, 1–13 (2002).

Opt. Acta

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Optik (Stuttgart)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Science

R. Miller, G. T. DeTitta, R. Jones, D. A. Langs, C. M. Weeks, H. A. Hauptman, “On the application of the minimal principle to solve unknown structures,” Science 259, 1430–1433 (1993).
[CrossRef] [PubMed]

Other

For a smooth subspace C with tangent space X at a∈C, the affine approximation to C at a is the space C′=X+a. By taking the set of all differences of elements in C′, one recovers the linear space: X=C′-C′.

V. Elser, “Linear time heuristic for the bipartite Euclidean matching problem,” (manuscript available from the author: ve10@cornell.edu).

In the case of support constraint with positivity, the constraint subspace is a smooth space with boundary, and the relevant dimensionality is that of the space without boundary.

H. A. David, Order Statistics, 2nd ed. (Wiley, New York, 1981).

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), Chap. 7, pp. 231–275.

C. Giacovazzo, Direct Phasing in Crystallography (Oxford U. Press, Oxford, UK, 1998).

H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).

The “difference map” considered here should not be confused with the “map” of the electron density studied by crystallographers in “difference Fourier synthesis.” The latter, in our notation, corresponds to πmod(ρ)-ρ, whereas the difference that we consider involves a projection on a prioriconstraints as well.

The term “stagnation” is interpreted to be the vanishing of the change between iterates.

The term “subspace” denotes a general subset of EN and in almost all cases of interest is a smooth submanifold, possibly with boundary. When encountered in the discussion, linear or affine subspaces will be explicitly identified as such.

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Figures (14)

Fig. 1
Fig. 1

Examples of elementary projections: (a) object comprising five Gaussian “atoms,” (b) Fourier modulus of (a), (c) random nonnegative density, (d) Fourier modulus projection of (c) using (b), (e) histogram projection of (d), (f) atomicity projection of (d).

Fig. 2
Fig. 2

Constraint subspaces (perpendicular rods) in the neighborhood of (a) a solution and (b) a near solution. The two circular disks in (a) represent points that project to the intersection of the constraint subspaces, ρ12, under action of π1 ° f2 and π2 ° f1, respectively; their intersection is the set of fixed points of the difference map. When the constraint subspaces do not intersect, as in (b), the action of the difference map is to move the iterates along the axis of minimum separation.

Fig. 3
Fig. 3

Plot of the map fα [expression (38)] composed with itself three times for α=0.37 and c=8/3. Points of intersection with the straight line give the three fixed points λ = 0, c-1, 1; the divergent behavior near λ=-1 and λ=2 corresponds to the “uranium” instability.

Fig. 4
Fig. 4

(a) Object and (b) corresponding Fourier modulus used in subsequent phase retrieval experiments.

Fig. 5
Fig. 5

Phase retrieval using the difference map with support and nonnegativity constraints. (a) Final iterate ρ* (fixed point of the map), (b) behavior of the error ei with iteration i. The random gray contrast in (a) is removed, and the original object [Fig. 4(a)] is almost perfectly recovered, when either (c) π1 ° f2=(πpos ° πsupp) ° (2πmod-1) or (d) π2 ° f1=πmod is applied to ρ*.

Fig. 6
Fig. 6

Same as Figs. 5(a) and 5(b) but with histogram constraints. Far fewer iterations are required to recover the same object.

Fig. 7
Fig. 7

Plots of sorted pixel values, or “histograms”: (a) Cornell seal [Fig. 4(a)], (b) random disk [Fig. 8(a) below].

Fig. 8
Fig. 8

(a) Phase retrieval applied to a random disk using histogram constraints, (b) plot of the error. The recovered object (not shown) not only had the correct circular outline but reproduced in detail the pixel values within the disk.

Fig. 9
Fig. 9

(a) Object comprising 60 equal “atoms”; error plots for three different implementations of a priori information: (b) histogram constraint, (c) 3×3-pixel atoms, (d) atomicity implemented by Sayre’s equation.

Fig. 10
Fig. 10

Comparison of phase retrieval successes after 100 iterations of the difference map for histogram projection, 3×3-pixel atomicity projection, and atomicity projection through the use of Sayre’s equation. The horizontal axis is the difference-map parameter β. See the text for more details.

Fig. 11
Fig. 11

Phase retrieval becomes easier when the atoms are clustered, as in example (a) comprising 200 atoms; (b) error plot for the case of 3×3-pixel atomicity projection.

Fig. 12
Fig. 12

Phase retrieval for an object comprising 60 atoms in three dimensions: (a) first 16 layers, arranged lexicographically, of the 32×32×32-voxel array, (b) error plot for the difference map with histogram projection.

Fig. 13
Fig. 13

Phase retrieval for an object comprising 60 “atoms” in one dimension: (a) plot of object values, (b) fixed point of difference map with histogram projection, (c) error plot. A single application of Fourier modulus projection to (b) perfectly reproduces (a), up to inversion and translation (not shown).

Fig. 14
Fig. 14

Evolution of the difference-map attractor with decreasing β: (a) 0.7, (b) 0.6, (c) 0.5.

Tables (1)

Tables Icon

Table 1 Widths σ and Errors δave of Finitely Sampled Gaussians in Dimension d for Supports within a Distance R of the Origin.

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

πmod=F-1 π˜mod F,
πsupp:ρnρn=ρnifnS0ifnS.
πhist:ρo(n)hn.
D=1+βΔ,
Δ=π1 ° f2-π2 ° f1
(π1 ° f2)(ρ*)=ρ12=(π2 ° f1)(ρ*),
(π1 ° f2)-1(ρ12)(π2 ° f1)-1(ρ12).
ei=Δ(ρ(i)),
e=2 sinθ122.
fi(ρ)=(1+γi)πi(ρ)-γiρ
C1=X1+a2+b1,C2=a1+X2+b2,
π1(x1+x2+y)=x1+a2+b1,π2(x1+x2+y)=a1+x2+b2.
G(x1+x2+y)=a1+a2+b1,
D(x1+x2+y)=a1+a2+y+(1-βγ2)(x1-a1)+(1+βγ1)(x2-a2)+β(b1-b2).
Dn(x1+x2+y)=a1+a2+y+nβ(b1-b2).
D(x˜1+)=x˜1+β{πpos[(1+γ2)a˜1-γ2x˜1]-a˜1}+.
D(x2+y)=a2+y+(1+βγ1)(x2-a2)+β(b1-b2).
D: ρρ+β{π1[(1+β-1)π2(ρ)-β-1ρ]-π2[(1-β-1)π1(ρ)+β-1ρ]}.
D|γ2=1/βγ1=-1=Fhybrid:ρnρn=πmod(ρ)nifnSρn-βπmod(ρ)nifnS.
Finout:ρnρn=ρnifnSρn-βπmod(ρ)nifnS,
Foutout:ρnρn=πmod(ρ)nifnS(1-β)πmod(ρ)nifnS.
ρ12=(π2 ° f1)(ρ*)=πmod(ρ*).
RS:ρnρn=ρnifnS-ρnifnS.
D|β=-1:ρnρn=(πmod ° RS)(ρ)nifnSρn+(πmod ° RS)(ρ)nifnS.
ρ12=(π1 ° f2)(ρ*)=πsupp(ρ*).
ρ=g*(ρ×ρ),
gr=8πσd/4 exp-2|r|2σ
ρr(1)=2πσd/4 exp-|r-r0|2σ.
ρ2=1Mr|ρr|2=1,
T:arg ρ˜arg[g˜×(ρ˜*ρ˜)].
|ρ˜||g˜×(ρ˜*ρ˜)|.
V=12ρ-g*(ρ×ρ)2.
V:ρρ-g*(ρ×ρ)-2(g¯*ρ)×ρ+2[g¯*g*(ρ×ρ)]×ρ,
Smod=πmod ° (1-αV).
Smod|α=1:arg ρ˜arg{g˜×(ρ˜*ρ˜)+2(g¯˜×ρ˜)*ρ˜-2[|g˜|2×(ρ˜*ρ˜)]*ρ˜},
Snorm=πnorm ° (1-αV),
πatomπSayre=Snormk
fα:λλ-α[λ-λ2+c(λ3-λ2)]
log 2<log{[fα(λ*)]k}=k log[1+α(1-c-1)]kα(1-c-1).
0=fα(λ0)-1λ0-1=1-α(cλ02-λ0).
α<1cλ02-λ0.
ρatom=πatom(ρrand).
|ρ˜rand|q2=q02|q|2+q02,
Ψ(r, r0)=2πσd/4 exp-|r-r0|2σ,
 |Ψ|2 ddr=1.
sS|Ψ˜s(t)|2=1
δ(t)=sS|Ψ˜s(t)-Ψ(s, p0+t)|2.
Ψ˜s(t)=Ψ(s, p0+t)[ΣsS|Ψ(s, p0+t)|2]1/2,
δ(t)=sS|Ψ(s, p0+t)|21/2-12.
δave=Tδ(t)ddt.
ρ˜q=σ2πd/4A(q)exp-σ4|q|2,
A(q)=m=1M exp iq·rm
σ-1=1dΣq|q|2|ρ˜q|2Σq|ρ˜q|2.
pm-pnS-S(mn).

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