A primal-dual method for large-scale image reconstruction in emission tomography

SIAM Journal on Optimization

Volumn 11, Issue 3, 2000, Pages 691-715

  Johnson, Calvin A ^a   Sofer, Ariela ^b  

^a NATIONAL INSTITUTES OF HEALTH   (United States)

^b George Mason University   (United States)

Author keywords

Applications of nonlinear programming; Estimation; Large scale problems; Parallel computation; Primal dual methods; Tomography

Indexed keywords

EID: 0035238730     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623497330148     Document Type: Article

References (61)

1. H.H. BARRETT, D.W. WILSON, AND B.M.W. TSUI, Noise properties of the EM algorithm I: Theory, Phys. Med. Biol., 39 (1994), pp. 833-846.
2. R.H. CHAN AND M.K. NG, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), pp. 427-482.
3. G. CHINN AND S.-C. HUANG, A general class of preconditioners for statistical iterative reconstruction of emission computed tomography, IEEE Trans. Med. Imag., 16 (1997), pp. 1-10.
4. N.H. CLINTHORNE, T.-S. PAN, P-C. CHIAO, W.L. ROGERS, AND J.A. STAMOS, Preconditioning methods for improved convergence rates in iterative reconstructions, IEEE Trans. Med. Imag., 12 (1993), pp. 78-83.
5. A.R. CONN, N. GOULD, AND PH.L. TOINT, A primal-dual algorithm for minimizing a non-convex function subject to bound and linear equality constraints, in Nonlinear Optimization and Related Topics, Appl. Optim. 36, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, pp. 15-49.
6. R.S. DEMBO AND T. STEIHAUG, Truncated-Newton algorithms for large-scale unconstrained optimization, Math. Programming, 26 (1983), pp. 190-212.
7. G. DEMOMENT, Image reconstruction and restoration: Overview of common estimation structures and problems, IEEE Trans. Acoust. Speech Signal Process., 37 (1989), pp. 2024-2036.
8. A.P. DEMPSTER, N.M. LAIRD, AND D.B. RUBIN, Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. Ser. B, 39 (1977), pp. 1-38.
9. A.R. DEPIERRO, A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography, IEEE Trans. Med. Imag., 14 (1995), pp. 132-137.
10. A.R. DEPIERRO, On the convergence of an EM-type algorithm for penalized likelihood estimation in emission tomography, IEEE Trans. Med. Imag., 14 (1995), pp. 762-765.
11. J.-P. DUSSAULT, Numerical stability and efficiency of penalty algorithms, SIAM J. Numer, Anal., 32 (1995), pp. 296-317.
12. A.V. FIACCO AND G.P. MCCORMICK, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, New York, 1968.
13. A. FORSCHEN AND P.E. GILL, Primal-dual interior methods for nonconvex nonlinear programming, SIAM J. Optim., 8 (1998), pp. 1132-1152.
14. A. FORSGREN, P.E. GILL, AND J.R. SHINNERL, Stability of symmetric ill-conditioned systems arising in interior methods for constrained optimization, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 187-211.
15. G. GOLUB AND C. VAN LOAN, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.
16. P.J. GREEN, On use of the EM algorithm for penalized likelihood estimation, J. Roy. Statist. Soc. Ser. B., 52 (1990), pp. 443-452.
17. P.J. GREEN, Bayesian reconstructions from emission tomography data using a modified EM algorithm, IEEE. Trans. Med. Imag., 9 (1990), pp. 84-93.
18. C.C. GONZAGA, Path-following methods for linear programming, SIAM Rev., 34 (1992), pp. 167-224.
19. T. HEBERT AND R. LEAHY, A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors, IEEE Trans. Med. Imag., 8 (1989), pp. 194-202.
20. G.T. HERMAN, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.
21. M.R. HESTENES AND E. STEIFEL, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49 (1952), pp. 409-436.
22. H.M. HUDSON AND R.S. LARKIN, Accelerated image reconstruction using ordered subsets of projection data, IEEE Trans. Med. Imag., 13 (1994), pp. 601-609.
23. F. JARRE AND M.A. SAUNDERS, A practical interior-point method for convex programming, SIAM J. Optim., 5 (1995), pp. 149-171.
24. C.A. JOHNSON, Nonlinear Optimization for Volume PET Reconstructions, Ph.D. dissertation, Department of Operations Research and Engineering, George Mason University, Fairfax, VA, 1997.
25. C.A. JOHNSON, Y. YAN, R.E. CARSON, R.L. MARTINO, AND M.E. DAUBE-WITHERSPOON, A system for the 3D reconstruction of retracted-septa PET data using the EM algorithm, IEEE Trans. Nucl. Sci., 42 (1995), pp. 1223-1227.
26. C.A. JOHNSON, J. SEIDEL, R.E. CARSON, W.R. GANDLER, A. SOFER, M.V. GREEN, AND M.E. DAUBE-WITHERSPOON, Evaluation of 3D reconstruction algorithms for a small animal PET camera, IEEE Trans Nucl. Sci., 44 (1997) pp. 1303-1308.
27. L. KAUFMAN, Implementing and accelerating the EM algorithm for positron emission tomography, IEEE Trans. Med. Imag., 6 (1987), pp. 37-51.
28. L. KAUFMAN, Solving emission tomography problems on vector machines, Ann. Oper. Res., 22 (1990), pp. 325-353.
29. L. KAUFMAN, Maximum likelihood, least squares and penalized least squares for PET, IEEE Trans. Med. Imag., 12 (1993), pp. 200-214.
30. L. KAUFMAN AND A. NEUMAIER, PET regularization for envelope guided conjugate gradients, IEEE Trans. Med. Imag., 15 (1996), pp. 385-389.
31. S. KAWATA AND L. NALCIOGLU, Constrained reconstruction by the conjugate gradient method, IEEE Trans. Med. Imag., 4 (1985), pp. 65-71.
32. M. KOJIMA, S. MIZUNO, AND A. YOSHISE, A primal-dual interior point algorithm for linear programming, in Progress in Mathematical Programming: Interior-Point and Related Methods, N. Megiddo, ed., Springer-Verlag, New York, 1989, pp. 29-47.
33. A. KURUC, Probability measure estimation using "weak" loss functions in positron emission tomography, in Tomography, Impedance Imaging and Integral Geometry, Lectures in Appl. Math. 30, M. Cheney, P. Kuchment, and E.T. Quinto, eds., AMS, Providence, RI, 1994, pp. 125-142.
34. D.S. LALUSH AND B.M.W. TSUI, The importance of preconditioned in fast Poisson-based iterative reconstruction algorithms for SPECT, in Proceedings of the 1995 IEEE Nuclear Science Symposium and Medical Imaging Conference, Conference Record 3, 1995, pp. 1326-1330.
35. J. LIOW AND S. STROTHER, Practical tradeoffs between noise, quantitation and number of iterations for maximum likelihood-based reconstructions, IEEE Trans. Med. Imag., 13 (1991), pp. 601-609.
36. K. LANGE AND R. CARSON, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomogr., 8 (1984), pp. 306-316.
37. K. LANGE, M. BAHN, AND R. LITTLE, A theoretical study of some maximum likelihood algorithms for emission and transmission tomography, IEEE Trans. Med. Imag., 6 (1987), pp. 106-114.
38. K. LANGE, Convergence of EM image reconstruction algorithms with Gibbs smoothing, IEEE Trans. Med. Imag., 9 (1990), pp. 439-446.
39. E. LEVITAN AND G.T. HERMAN, A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography, IEEE Trans. Med. Imag., 6 (1987), pp. 185-192.
40. I. LUSTIG., R.E. MARSTEN, AND D.F. SHANNO, Interior point methods for linear programming: Computational state of the art, ORSA J. Comput., 6 (1994), pp. 1-14.
41. G.P. MCCORMICK, The Superlinear Convergence of a Nonlinear Primal-Dual Algorithm, Report T-550/91, Department of Operations Research, George Washington University, Washington, DC, 1991.
42. E.U. MUMCUOGLU AND R. LEAHY, Gradient projection conjugate gradient algorithm in Bayesian PET reconstruction, in Proceedings of the 1994 IEEE Nuclear Science Symposium and Medical Imaging Conference, Conference Record 3, 1995, pp. 1212-1216.
43. E.U. MUMCUOGLU, R.M. LEAHY, S.R. CHERRY, AND Z. ZHOU, Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images, IEEE Trans. Med. Imag., 13 (1994), pp. 687-701.
44. E.U. MUMCUOGLU, R.M. LEAHY, AND S.R. CHERRY, Bayesian reconstruction of PET images: Methodology and performance analysis, Phys. Med. Biol., 41 (1996), pp. 1777-1807.
45. W. MURRAY, Analytic expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions, J. Optim. Theory Appl., 7 (1971), pp. 189-196.
46. S.G. NASH, Preconditioning of truncated-Newton methods, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 599-616.
47. S.G. NASH AND A. SOFER, Block truncated-Newton methods for parallel optimization, Math. Programming, 45 (1989), pp. 529-546.
48. S.G. NASH AND A. SOFER, Assessing a search direction within a truncated-Newton method, Oper. Res. Lett., 9 (1990), pp. 219-221.
49. S.G. NASH AND A. SOFER, A general-purpose parallel algorithm for unconstrained optimization, SIAM J. Optim., 1 (1991), pp. 530-547.
50. S.G. NASH AND A. SOFER, A barrier method for large-scale constrained optimizaton, ORSA J. Comput., 5 (1993), pp. 40-53.
51. S.G. NASH AND A. SOFER, Linear and Nonlinear Programming, McGraw-Hill, New York, 1996.
52. D.B. PONCELEON, Barrier Methods for Large-Scale Quadratic Programming, Report SOL 91-2, Ph.D. thesis, Department of Operations Research, Stanford University, Stanford, CA, 1991.
53. L.A. SHEPP AND Y. VARDI, Maximum likelihood reconstruction for emission tomography, IEEE Trans Med. Imag., 1 (1982), pp. 113-122.
54. E.M. SIMANTIRAKI AND D.F. SHANNO, An infeasible-interior-point method for linear complementarity problems, SIAM J. Optim., 7 (1997), pp. 620-640.
55. D.L. SNYDER, M.I. MILLER, L.J. THOMAS, JR., AND D.G. POLITTE, Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography, IEEE Trans. Med. Imag., 6 (1987), pp. 228-238.
56. Y. VARDI, L. A. SHEPP, AND L. KAUFMAN, A statistical model for positron emission tomography, J. Amer. Statist. Assoc., 80, (1985), pp. 8-37.
57. D.W. WILSON AND B.M.W. TSUI, Noise properties of filtered-backprojection and ML-EM reconstructed emission tomography images, IEEE Trans. Nucl. Sci., 40 (1993), pp. 1198-1203.
58. M.H. WRIGHT, Interior methods for constrained optimization, in Acta Numer., Cambridge University Press, Cambridge, UK, 1991, pp. 341-407.
59. M.H. WRIGHT, Some properties of the Hessian of the logarithmic barrier function, Math. Programming, 67 (1994), pp. 265-295.
60. M.H. WRIGHT, III-conditioning and computational error in interior methods for nonlinear programming, SIAM J. Optim., 9 (1999), pp. 84-111.
61. S.J. WRIGHT, Stability of linear equations solvers in interior-point methods, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 1287-1307.