Abstract

A multicriterion cross-entropy minimization approach to positron emission tomographic (PET) imaging is described. An unexplored multicriterion cross-entropy optimization algorithm based on weighted-sum scalarization is used to solve this problem. The efficacy of the algorithm is compared with that of the single-criterion optimization algorithm and the convolution backprojection method for image reconstruction from computer-generated projection data and Siemens PET scanner data. The algorithms described have been implemented on a PIII/686 microcomputer.

© 2001 Optical Society of America

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References

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  1. R. L. Kashyap, M. C. Mittal, “Picture reconstruction from projections,” IEEE Trans. Comput. C-24, 915–923 (1975).
    [CrossRef]
  2. G. T. Herman, A. Lent, “Quadratic optimization for image reconstruction. I,” Comput. Vis. Graph. Image Process. 5, 319–332 (1976).
    [CrossRef]
  3. G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
    [CrossRef] [PubMed]
  4. Y. Censor, G. T. Herman, “On some optimization techniques in image reconstruction from projections,” Appl. Numer. Math. 3, 365–391 (1987).
    [CrossRef]
  5. Y. Censor, “Finite series-expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
    [CrossRef]
  6. Y. Wang, W. Lu, “Multicriterion image reconstruction and implementations,” Comput. Vis. Graph. Image Process. 46, 131–135 (1989).
    [CrossRef]
  7. Y. Wang, W. Lu, “Multicriterion maximum entropy image reconstruction from projections,” IEEE Trans. Med. Imaging 11, 70–75 (1992).
    [CrossRef] [PubMed]
  8. Y. Wang, W. Lu, “Multiobjective decision-making approach to image reconstruction from projections,” J. Opt. Soc. Am. A 8, 1649–1656 (1991).
    [CrossRef]
  9. V. Chankong, Y. Y. Haimes, Multiobjective Decision Making: Theory and Methodology (North-Holland, Amsterdam, 1993).
  10. L. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
    [CrossRef] [PubMed]
  11. Y. Vardi, L. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
    [CrossRef]
  12. L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).
    [CrossRef]
  13. S. Kullback, Information Theory and Statistics (Wiley, New York, 1959).
  14. J. L. Saaty, “A scaling method for priorities in hierarchical structures,” J. Math. Psychol. 15, 234–281 (1977).
    [CrossRef]
  15. W. I. Zangwill, Nonlinear Programming: A Unified Approach (Prentice-Hall, Englewood Cliffs, N.J., 1969).
  16. F. B. Hildegrand, Introduction to Numerical Analysis, 2nd ed. (McGraw-Hill, New York, 1974).

1992

Y. Wang, W. Lu, “Multicriterion maximum entropy image reconstruction from projections,” IEEE Trans. Med. Imaging 11, 70–75 (1992).
[CrossRef] [PubMed]

1991

1989

Y. Wang, W. Lu, “Multicriterion image reconstruction and implementations,” Comput. Vis. Graph. Image Process. 46, 131–135 (1989).
[CrossRef]

1987

Y. Censor, G. T. Herman, “On some optimization techniques in image reconstruction from projections,” Appl. Numer. Math. 3, 365–391 (1987).
[CrossRef]

1985

Y. Vardi, L. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

1983

Y. Censor, “Finite series-expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

1982

L. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef] [PubMed]

1977

J. L. Saaty, “A scaling method for priorities in hierarchical structures,” J. Math. Psychol. 15, 234–281 (1977).
[CrossRef]

1976

G. T. Herman, A. Lent, “Quadratic optimization for image reconstruction. I,” Comput. Vis. Graph. Image Process. 5, 319–332 (1976).
[CrossRef]

G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef] [PubMed]

1975

R. L. Kashyap, M. C. Mittal, “Picture reconstruction from projections,” IEEE Trans. Comput. C-24, 915–923 (1975).
[CrossRef]

1974

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).
[CrossRef]

Censor, Y.

Y. Censor, G. T. Herman, “On some optimization techniques in image reconstruction from projections,” Appl. Numer. Math. 3, 365–391 (1987).
[CrossRef]

Y. Censor, “Finite series-expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

Chankong, V.

V. Chankong, Y. Y. Haimes, Multiobjective Decision Making: Theory and Methodology (North-Holland, Amsterdam, 1993).

Haimes, Y. Y.

V. Chankong, Y. Y. Haimes, Multiobjective Decision Making: Theory and Methodology (North-Holland, Amsterdam, 1993).

Herman, G. T.

Y. Censor, G. T. Herman, “On some optimization techniques in image reconstruction from projections,” Appl. Numer. Math. 3, 365–391 (1987).
[CrossRef]

G. T. Herman, A. Lent, “Quadratic optimization for image reconstruction. I,” Comput. Vis. Graph. Image Process. 5, 319–332 (1976).
[CrossRef]

G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef] [PubMed]

Hildegrand, F. B.

F. B. Hildegrand, Introduction to Numerical Analysis, 2nd ed. (McGraw-Hill, New York, 1974).

Kashyap, R. L.

R. L. Kashyap, M. C. Mittal, “Picture reconstruction from projections,” IEEE Trans. Comput. C-24, 915–923 (1975).
[CrossRef]

Kaufman, L.

Y. Vardi, L. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

Kullback, S.

S. Kullback, Information Theory and Statistics (Wiley, New York, 1959).

Lent, A.

G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef] [PubMed]

G. T. Herman, A. Lent, “Quadratic optimization for image reconstruction. I,” Comput. Vis. Graph. Image Process. 5, 319–332 (1976).
[CrossRef]

Logan, B. F.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).
[CrossRef]

Lu, W.

Y. Wang, W. Lu, “Multicriterion maximum entropy image reconstruction from projections,” IEEE Trans. Med. Imaging 11, 70–75 (1992).
[CrossRef] [PubMed]

Y. Wang, W. Lu, “Multiobjective decision-making approach to image reconstruction from projections,” J. Opt. Soc. Am. A 8, 1649–1656 (1991).
[CrossRef]

Y. Wang, W. Lu, “Multicriterion image reconstruction and implementations,” Comput. Vis. Graph. Image Process. 46, 131–135 (1989).
[CrossRef]

Mittal, M. C.

R. L. Kashyap, M. C. Mittal, “Picture reconstruction from projections,” IEEE Trans. Comput. C-24, 915–923 (1975).
[CrossRef]

Saaty, J. L.

J. L. Saaty, “A scaling method for priorities in hierarchical structures,” J. Math. Psychol. 15, 234–281 (1977).
[CrossRef]

Shepp, L.

Y. Vardi, L. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

L. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef] [PubMed]

Shepp, L. A.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).
[CrossRef]

Vardi, Y.

Y. Vardi, L. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

L. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef] [PubMed]

Wang, Y.

Y. Wang, W. Lu, “Multicriterion maximum entropy image reconstruction from projections,” IEEE Trans. Med. Imaging 11, 70–75 (1992).
[CrossRef] [PubMed]

Y. Wang, W. Lu, “Multiobjective decision-making approach to image reconstruction from projections,” J. Opt. Soc. Am. A 8, 1649–1656 (1991).
[CrossRef]

Y. Wang, W. Lu, “Multicriterion image reconstruction and implementations,” Comput. Vis. Graph. Image Process. 46, 131–135 (1989).
[CrossRef]

Zangwill, W. I.

W. I. Zangwill, Nonlinear Programming: A Unified Approach (Prentice-Hall, Englewood Cliffs, N.J., 1969).

Appl. Numer. Math.

Y. Censor, G. T. Herman, “On some optimization techniques in image reconstruction from projections,” Appl. Numer. Math. 3, 365–391 (1987).
[CrossRef]

Comput. Biol. Med.

G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comput. Biol. Med. 6, 273–294 (1976).
[CrossRef] [PubMed]

Comput. Vis. Graph. Image Process.

G. T. Herman, A. Lent, “Quadratic optimization for image reconstruction. I,” Comput. Vis. Graph. Image Process. 5, 319–332 (1976).
[CrossRef]

Y. Wang, W. Lu, “Multicriterion image reconstruction and implementations,” Comput. Vis. Graph. Image Process. 46, 131–135 (1989).
[CrossRef]

IEEE Trans. Comput.

R. L. Kashyap, M. C. Mittal, “Picture reconstruction from projections,” IEEE Trans. Comput. C-24, 915–923 (1975).
[CrossRef]

IEEE Trans. Med. Imaging

Y. Wang, W. Lu, “Multicriterion maximum entropy image reconstruction from projections,” IEEE Trans. Med. Imaging 11, 70–75 (1992).
[CrossRef] [PubMed]

L. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21, 21–43 (1974).
[CrossRef]

J. Am. Stat. Assoc.

Y. Vardi, L. Shepp, L. Kaufman, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

J. Math. Psychol.

J. L. Saaty, “A scaling method for priorities in hierarchical structures,” J. Math. Psychol. 15, 234–281 (1977).
[CrossRef]

J. Opt. Soc. Am. A

Proc. IEEE

Y. Censor, “Finite series-expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

Other

V. Chankong, Y. Y. Haimes, Multiobjective Decision Making: Theory and Methodology (North-Holland, Amsterdam, 1993).

W. I. Zangwill, Nonlinear Programming: A Unified Approach (Prentice-Hall, Englewood Cliffs, N.J., 1969).

F. B. Hildegrand, Introduction to Numerical Analysis, 2nd ed. (McGraw-Hill, New York, 1974).

S. Kullback, Information Theory and Statistics (Wiley, New York, 1959).

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Figures (3)

Fig. 1
Fig. 1

Illustration of a gray-level image of the phantom.

Fig. 2
Fig. 2

Images reconstructed by (a) the MCEOT, (b) the MLE algorithm, and (c) the FBP method.

Fig. 3
Fig. 3

Images reconstructed by (a) the MCEOT, (b) the MLE algorithm, and (c) the FBP method.

Tables (1)

Tables Icon

Table 1 Numerical Results of Phantom Reconstruction

Equations (44)

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

k=1makl=1.
y=Ax,
f1(x)=c1j=1nxj lnxjzj,
f2(x)=12c2(xTx+xTSx),
12xTSx=12 jNxj-18 kNjxk2.
f3(x)=c3i=1mj=1naijxjln j=1naijxjyi,
min f(x)=(f1(x), f2(x), f3(x))suchthatAx=y,x0,
min v(f )=q=13wqfq(x)suchthatAx=y,x0,
v(f1, f2, f3)=b,
qij=vfj/vfi=wjwi,i, j=1, 2, 3.
qij=limΔi0 ΔiΔj,i, j=1, 2, 3.
Qw=λmaxw,
γ(μ, x)=v(x)+μh(x),
v(x)=q=13wqfq(x),h(x)=12(Ax-y)T(Ax-y),
Minimizeγ(μ(k), x).
limkK v(x(k))=v(x0).
limkK γ(μ(k), x(k))=γ*v*.
limkK μ(k)h(x(k))=γ*-v(x0).
limkK h(x(k))=0.
v(x0)=limkK v(x(k))v*.
min γ(μ, x)=q=13wqfq(x)+12μ(Ax-y)T(Ax-y),
γ(x)=γ(x)x1, , γ(x)xn.
q=13wq fq(x)xi+μi=1majir=1nairxr-yi=0,j=1, , n.
w1c1[ln(xj/zj)+1]+w2c2r=1nsjrxr+xj+w3c3i=1maijlnr=1nairxr/yi+yi+μi=1majir=1nairxr-yi=0,i=1, , n.
xj=zj exp-w1-1c1-1w2c2r=1nsjrxr+xj+w3c3i=1maijlnr=1nairxr/yi+yi+μi=1majir=1nairxr-yi,j=1, , n,
ϕj(x, μ)=exp-w1-1c1-1w2c2r=1nsjrxr+xj+w3c3i=1maijlnr=1nairxr/yi+yi+μi=1majir=1nairxr-yi,j=1, , n;
xj=zjϕj(x, μ),j=1,, n.
xj=zjϕj(x, μ)j=1nzjϕj(x, μ),j=1, 2, , n.
xj(k+1)=Ψj(x(k), μ(k)),j=1, , n,
Ψj(x, μ)=zjϕj(x, μ)j=1nzjϕj(x, μ),j=1, , n,
Ψ(x, μ)=(Ψ1(x, μ), , Ψn(x, μ))T.
xj(1)=zj exp(ATy)j,j=1, , n,
c1=j=1nxj(1) ln[xj(1)/zj]-1,
c2=12[x(1)TSx(1)+x(1)Tx(1)]-1,
c3=i=1mj=1naijxj(1)lnj=1naijxj(1)yi-1.
w(1)=13,13,13.
a.μ(k)=10k-1,
ϕj(x(k), μ(k))=exp-w2(k)c2r=1nsjrxr(k)+xj(k)+w3(k)c3i=1maijlnr=1nairxr(k)yi+yi+μ(k)i=1majir=1nairxr(k)-yiw1(k)c1,j=1, , n,
Ψj(x(k), μ(k))=zjϕj(x(k), μ(k))j=1nzjϕj(x(k), μ(k)),j=1, , n,
d.xj(k+1)=Ψj(x(k), μ(k)),j=1, , n.
Δq=|fˆq(x(k+1))-fˆq(x(k))|,q=1, 2, 3.
qij=Δj/Δi,i=1, 2, 3, j=1, 2, 3.
Qw(k)=λmaxw(k).
d=i=1nxi(k)-xi02i=1nxi0-x¯021/2,

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