Abstract

An image-reconstruction problem in x-ray fiber diffraction analysis (which is used to determine the atomic structure of biopolymers) is considered. The problem is to reconstruct an image (the electron density function) given data that are squared sums of the Fourier coefficients of the image as well as partial information (the model) on the image. A Bayesian estimation approach based on a prior for the missing part of the image is considered. Current (heuristic) approaches to this problem correspond to certain maximum a posteriori estimates. These estimates exhibit bias toward the model, and current methods to reduce the bias are based on scaling of the Fourier coefficients. A new procedure to remove bias, based on orthogonalization, is derived and shown by simulations to be superior to scaling. Bias and unbiasing are compared for the different maximum a posteriori estimates, for different amounts of missing information. These results are also compared with a new minimum mean-square-error estimate for this problem that has the form of weighted maximum a posteriori Fourier coefficients. The minimum mean-square-error estimate is free from bias and gives results superior to the unbiased maximum a posteriori estimates.

© 1999 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. Drenth, Principles of Protein X-Ray Crystallography (Springer-Verlag, Berlin, 1994).
  3. A. D. French, K. H. Gardner, eds., Fiber Diffraction Methods (American Chemical Society, Washington, D.C., 1980).
  4. R. P. Millane, “Structure determination by x-ray fiber diffraction,” in Crystallographic Computing 4: Techniques and New Technologies, N. W. Isaacs, M. R. Taylor, eds. (Oxford U. Press, Oxford, 1988), pp. 169–186.
  5. R. P. Millane, “Image reconstruction from cylindrically averaged diffraction intensities,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 137–143 (1993).
    [CrossRef]
  6. S. Baskaran, R. P. Millane, “Bayesian image reconstruction from partial image and spectral intensity data,” IEEE Trans. Image Process (to be published).
  7. S. Baskaran, R. P. Millane, “Bayesian image reconstruction from partial image and spectral amplitude data,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 227–237 (1997).
    [CrossRef]
  8. R. P. Millane, S. Baskaran, “Optimal difference Fourier synthesis in fiber diffraction,” Fiber Diffract. Rev. 6, 14–18 (1997).
  9. A. J. C. Wilson, “The probability distribution of x-ray intensities,” Acta Crystallogr. 2, 318–321 (1949).
    [CrossRef]
  10. S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
    [CrossRef]
  11. G. Stubbs, “The probability distributions of x-ray intensities in fiber diffraction: largest likely values of fiber diffraction R factors,” Acta Crystallogr. Sect. A 45, 254–258 (1989).
    [CrossRef]
  12. R. P. Millane, “Intensity distributions in fiber diffraction,” Acta Crystallogr. Sect. A 46, 552–559 (1990).
    [CrossRef]
  13. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).
  14. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 369, 665.
  15. M. M. Woolfson, “An improvement on the ‘heavy-atom’ method of solving crystal structures,” Acta Crystallogr. 9, 804–810 (1956).
    [CrossRef]
  16. G. A. Sim, “A note on the heavy-atom method,” Acta Crystallogr. 13, 511–512 (1960).
    [CrossRef]
  17. K. Namba, G. Stubbs, “Difference Fourier synthesis in fiber diffraction,” Acta Crystallogr. Sect. A 43, 533–539 (1987).
    [CrossRef]
  18. C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973).
  19. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  20. R. Henderson, J. K. Moffat, “The difference Fourier technique in protein crystallography: errors and their treatment,” Acta Crystallogr. Sect. B 27, 1414–1420 (1971).
    [CrossRef]
  21. R. J. Read, “Improved Fourier coefficients for maps using phases from partial structures with errors,” Acta Crystallogr. Sect. A 42, 140–149 (1986).
    [CrossRef]

1997 (1)

R. P. Millane, S. Baskaran, “Optimal difference Fourier synthesis in fiber diffraction,” Fiber Diffract. Rev. 6, 14–18 (1997).

1990 (2)

R. P. Millane, “Intensity distributions in fiber diffraction,” Acta Crystallogr. Sect. A 46, 552–559 (1990).
[CrossRef]

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

1989 (1)

G. Stubbs, “The probability distributions of x-ray intensities in fiber diffraction: largest likely values of fiber diffraction R factors,” Acta Crystallogr. Sect. A 45, 254–258 (1989).
[CrossRef]

1987 (1)

K. Namba, G. Stubbs, “Difference Fourier synthesis in fiber diffraction,” Acta Crystallogr. Sect. A 43, 533–539 (1987).
[CrossRef]

1986 (1)

R. J. Read, “Improved Fourier coefficients for maps using phases from partial structures with errors,” Acta Crystallogr. Sect. A 42, 140–149 (1986).
[CrossRef]

1971 (1)

R. Henderson, J. K. Moffat, “The difference Fourier technique in protein crystallography: errors and their treatment,” Acta Crystallogr. Sect. B 27, 1414–1420 (1971).
[CrossRef]

1960 (1)

G. A. Sim, “A note on the heavy-atom method,” Acta Crystallogr. 13, 511–512 (1960).
[CrossRef]

1956 (1)

M. M. Woolfson, “An improvement on the ‘heavy-atom’ method of solving crystal structures,” Acta Crystallogr. 9, 804–810 (1956).
[CrossRef]

1949 (1)

A. J. C. Wilson, “The probability distribution of x-ray intensities,” Acta Crystallogr. 2, 318–321 (1949).
[CrossRef]

1943 (1)

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
[CrossRef]

Baskaran, S.

R. P. Millane, S. Baskaran, “Optimal difference Fourier synthesis in fiber diffraction,” Fiber Diffract. Rev. 6, 14–18 (1997).

S. Baskaran, R. P. Millane, “Bayesian image reconstruction from partial image and spectral intensity data,” IEEE Trans. Image Process (to be published).

S. Baskaran, R. P. Millane, “Bayesian image reconstruction from partial image and spectral amplitude data,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 227–237 (1997).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
[CrossRef]

Drenth, J.

J. Drenth, Principles of Protein X-Ray Crystallography (Springer-Verlag, Berlin, 1994).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 369, 665.

Henderson, R.

R. Henderson, J. K. Moffat, “The difference Fourier technique in protein crystallography: errors and their treatment,” Acta Crystallogr. Sect. B 27, 1414–1420 (1971).
[CrossRef]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Millane, R. P.

R. P. Millane, S. Baskaran, “Optimal difference Fourier synthesis in fiber diffraction,” Fiber Diffract. Rev. 6, 14–18 (1997).

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

R. P. Millane, “Intensity distributions in fiber diffraction,” Acta Crystallogr. Sect. A 46, 552–559 (1990).
[CrossRef]

R. P. Millane, “Image reconstruction from cylindrically averaged diffraction intensities,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 137–143 (1993).
[CrossRef]

R. P. Millane, “Structure determination by x-ray fiber diffraction,” in Crystallographic Computing 4: Techniques and New Technologies, N. W. Isaacs, M. R. Taylor, eds. (Oxford U. Press, Oxford, 1988), pp. 169–186.

S. Baskaran, R. P. Millane, “Bayesian image reconstruction from partial image and spectral amplitude data,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 227–237 (1997).
[CrossRef]

S. Baskaran, R. P. Millane, “Bayesian image reconstruction from partial image and spectral intensity data,” IEEE Trans. Image Process (to be published).

Moffat, J. K.

R. Henderson, J. K. Moffat, “The difference Fourier technique in protein crystallography: errors and their treatment,” Acta Crystallogr. Sect. B 27, 1414–1420 (1971).
[CrossRef]

Namba, K.

K. Namba, G. Stubbs, “Difference Fourier synthesis in fiber diffraction,” Acta Crystallogr. Sect. A 43, 533–539 (1987).
[CrossRef]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

Rao, C. R.

C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973).

Read, R. J.

R. J. Read, “Improved Fourier coefficients for maps using phases from partial structures with errors,” Acta Crystallogr. Sect. A 42, 140–149 (1986).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 369, 665.

Sim, G. A.

G. A. Sim, “A note on the heavy-atom method,” Acta Crystallogr. 13, 511–512 (1960).
[CrossRef]

Stubbs, G.

G. Stubbs, “The probability distributions of x-ray intensities in fiber diffraction: largest likely values of fiber diffraction R factors,” Acta Crystallogr. Sect. A 45, 254–258 (1989).
[CrossRef]

K. Namba, G. Stubbs, “Difference Fourier synthesis in fiber diffraction,” Acta Crystallogr. Sect. A 43, 533–539 (1987).
[CrossRef]

Wilson, A. J. C.

A. J. C. Wilson, “The probability distribution of x-ray intensities,” Acta Crystallogr. 2, 318–321 (1949).
[CrossRef]

Woolfson, M. M.

M. M. Woolfson, “An improvement on the ‘heavy-atom’ method of solving crystal structures,” Acta Crystallogr. 9, 804–810 (1956).
[CrossRef]

Acta Crystallogr. (3)

M. M. Woolfson, “An improvement on the ‘heavy-atom’ method of solving crystal structures,” Acta Crystallogr. 9, 804–810 (1956).
[CrossRef]

G. A. Sim, “A note on the heavy-atom method,” Acta Crystallogr. 13, 511–512 (1960).
[CrossRef]

A. J. C. Wilson, “The probability distribution of x-ray intensities,” Acta Crystallogr. 2, 318–321 (1949).
[CrossRef]

Acta Crystallogr. Sect. A (4)

K. Namba, G. Stubbs, “Difference Fourier synthesis in fiber diffraction,” Acta Crystallogr. Sect. A 43, 533–539 (1987).
[CrossRef]

G. Stubbs, “The probability distributions of x-ray intensities in fiber diffraction: largest likely values of fiber diffraction R factors,” Acta Crystallogr. Sect. A 45, 254–258 (1989).
[CrossRef]

R. P. Millane, “Intensity distributions in fiber diffraction,” Acta Crystallogr. Sect. A 46, 552–559 (1990).
[CrossRef]

R. J. Read, “Improved Fourier coefficients for maps using phases from partial structures with errors,” Acta Crystallogr. Sect. A 42, 140–149 (1986).
[CrossRef]

Acta Crystallogr. Sect. B (1)

R. Henderson, J. K. Moffat, “The difference Fourier technique in protein crystallography: errors and their treatment,” Acta Crystallogr. Sect. B 27, 1414–1420 (1971).
[CrossRef]

Fiber Diffract. Rev. (1)

R. P. Millane, S. Baskaran, “Optimal difference Fourier synthesis in fiber diffraction,” Fiber Diffract. Rev. 6, 14–18 (1997).

J. Opt. Soc. Am. A (1)

Rev. Mod. Phys. (1)

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
[CrossRef]

Other (10)

C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, New York, 1973).

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 369, 665.

J. Drenth, Principles of Protein X-Ray Crystallography (Springer-Verlag, Berlin, 1994).

A. D. French, K. H. Gardner, eds., Fiber Diffraction Methods (American Chemical Society, Washington, D.C., 1980).

R. P. Millane, “Structure determination by x-ray fiber diffraction,” in Crystallographic Computing 4: Techniques and New Technologies, N. W. Isaacs, M. R. Taylor, eds. (Oxford U. Press, Oxford, 1988), pp. 169–186.

R. P. Millane, “Image reconstruction from cylindrically averaged diffraction intensities,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 137–143 (1993).
[CrossRef]

S. Baskaran, R. P. Millane, “Bayesian image reconstruction from partial image and spectral intensity data,” IEEE Trans. Image Process (to be published).

S. Baskaran, R. P. Millane, “Bayesian image reconstruction from partial image and spectral amplitude data,” in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE3170, 227–237 (1997).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Weighting function wn(χ) as a function of χ for six values of n.

Fig. 2
Fig. 2

True image ϱ(r), known part ϱP(r), and missing part ϱQ(r) for ΔQ=0.2, 0.4, and 0.6.

Fig. 3
Fig. 3

Correlation coefficient (C) and bias (B) as a function of ΣˆQ for the MMSE estimate. The true value of ΣQ is 40.

Fig. 4
Fig. 4

Correlation coefficient (C), bias (B), number of correct peaks (NQ) and number of model peaks (NP) before (plain curves) and after (curves with circles) orthogonalization as a function of Smax for the MMSE (solid curves), MAP1 (dashed curves), and MAP2 (dotted–dashed curves) estimates, for (a) ΔQ=0.2, (b) ΔQ=0.4, and (c) ΔQ=0.6.

Fig. 5
Fig. 5

MMSE, MAP1, and MAP2 reconstructions of the missing part, ϱQ(r), for ΔQ=0.2 and Smax=5 (top row); thresholded to show the top 20 peaks (second row); orthogonalized with respect to ϱP(r) (third row); and orthogonalized and thresholded (bottom row).

Fig. 6
Fig. 6

MMSE, MAP1, and MAP2 reconstructions of the missing part, ϱQ(r), for ΔQ=0.4 and Smax=3 (top row); thresholded to show the top 40 peaks (second row); orthogonalized with respect to ϱP(r) (third row); and orthogonalized and thresholded (bottom row).

Fig. 7
Fig. 7

MAP1 reconstructions, MAP1 estimate unbiased by scaling, and MAP1 estimate unbiased by orthogonalization (top row) and thresholded to show the top 40 peaks (bottom row), for ΔQ=0.4 and (a) Smax=1, (b) Smax=3.

Fig. 8
Fig. 8

MAP1 reconstructions, MAP1 estimate unbiased by scaling, and MAP1 estimate unbiased by orthogonalization (top row) and thresholded to show the top 60 peaks (bottom row), for ΔQ=0.6 and (a) Smax=1, (b) Smax=3.

Tables (1)

Tables Icon

Table 1 Correlation Coefficients (C) and Number of Correctly Located Peaks (NQ) for the MMSE and MAP1 Estimates Compared with Those for the MAP1 Estimates Unbiased by Scaling and Orthogonalization

Equations (44)

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Fh=|Fh|exp(iϕh)=Uϱ(r)exp(2πir·h)dr,
ϱ(r)=1VhFh exp(-2πir·h),
ϱ(r)=jNϱj(r-rj),
Fh=jNfj(h)exp(2πirj·h),
fj(h)=ϱj(r)exp(2πir·h)dr.
Ij=hSj|Fh|2,
ϱ(r)=ϱP(r)+ϱQ(r).
Fh=FhP+FhQ,
PAhQ(x)=PBhQ(x)=(πΣQ)-1/2 exp(-x2/ΣQ),
ΣQ˙=jQfj2.
Yj=Ah1Bh1Bhn/2,Xj=Ah1QBh1QBhn/2Q,Θj=Ah1PBh1PBhn/2P.
Cj=1|Sj|cos ϕh1Psin ϕh1Psin ϕhn/2P,Φj=ϕh1Pϕh2Pϕhn/2P.
PX(x)=(πΣQ)-n/2 exp(-xx/ΣQ),
PX|Θ,Ij(x)=PIj|Θ,X(x)PX(x)PIj|Θ,X(x)PX(x)dx.
PIj|Θ,X(l)=(2πσI2)-1/2 exp(-(l-X+Θ2)2/(2σI2)),
PIj|Θ,X(l)δ(l-X+Θ2),
PX|Θ,Ij(x)=Z-1 exp(-x2/ΣQ)δ(Ij-x+Θ2),
PY|Θ,Ij(y)=Z-1 exp(-y-Θ2/ΣQ)δ(Ij-y2).
y-Θ2=y2+Θ2-2|yΘcos φ,
PY|Θ,Ij(φ)exp(-(Ij+IjP-2IjIjP cos φ)/ΣQ),
YˆMMSE=yPY|Θ,Ij(y)dy.
FˆhMMSE=wn(χ)(Ij/IjP)1/2FhP,
wn(χ)=In/2(χ)In/2-1(χ),
YˆMAP1=arg maxy[PY|Θ,Ij(y)],
FˆhMAP1=(Ij/IjP)1/2FhP.
YˆMAP2=arg maxy[PY|Φ,Ij(y)].
FˆhMAP2=(Ij/|Sj|)1/2 exp(iϕP).
ΣˆQ(k+1)=Ij+IjP-2[Inj/2(χ(ΣˆQ(k)))/Inj/2-1(χ(ΣˆQ(k)))]IjIjPj
β=[Gˆ(D)-g]PG|D(g)dg=Gˆ(D)PG|D(g)dg-gP,
B=ϱˆQ(r)ϱP(r)dr[ϱP(r)]2dr[ϱˆQ(r)]2dr1/2,
β=(β(r), β(r))=β2(r)dr,
β=c1(β(r), ϱP(r)),
β=c2(ϱˆQ(r), ϱP(r))=c3B.
ϱˆQ(r)=KϱP(r)+ϱˆU(r),
K=[ϱˆQ(r)]2dr[ϱP(r)]2dr1/2B
Δ1=Z-1[(Ij/IjP)1/2Θ-y]exp(-y-Θ2/ΣQ)×δ(Ij-y2)dy,=(Ij/IjP)1/2Θ-[In/2(χ)/In/2-1(χ)](Ij/IjP)1/2Θ,
βMAP1(r)=ϱˆMAP1(r)-ϱˆMMSE(r).
Δ2=Z-1(IjC-y)exp(-y-Θ2/ΣQ)×δ(Ij-y2)dy,=IjC-[In/2(χ)/In/2-1(χ)](Ij/IjP)1/2Θ,
βMAP2(r)=ϱˆMAP2(r)-ϱˆMMSE(r).
cos2 ϑ= cos2 ϑ δ(y2-1)dyδ(y2-1)dy,
cos2 ϑ= cos2 ϑ δ(y2-1)yn-1dysinn-2 ϑ dϑdΩy-δ(y2-1)yn-1dysinn-2 ϑ dϑdΩy-.
cos2 ϑ=0π cos2 ϑ(sin ϑ)n-2dϑ0π(sin ϑ)n-2dϑ=1n,
ϱˆU(r)=ϱˆQ(r)-KϱP(r).
C=ϱˆQ(r)ϱQ(r)dr[ϱQ(r)]2dr[ϱˆQ(r)]2dr1/2.

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