Reconstruction with noisy data: An approach via eigenvalue optimization

SIAM Journal on Optimization

Volumn 8, Issue 1, 1998, Pages 82-104

  Noll, Dominikus ^a  

^a UNIVERSITÉ DE TOULOUSE   (France)

Author keywords

Eigenvalue optimization; Finite element discretization; Fisher information; Image restoration; Inverse problems; Large scale problems; Power spectrum estimation; Semidefinite programming

Indexed keywords

EID: 0032350054     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S105262349629856X     Document Type: Article

References (40)

1. R. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. D. L. ANGWIN AND H. KAUFMAN, Nonhomogeneous image identification and restoration procedures, Springer Ser. Inform. Sci., 23 (1991), pp. 177-208.
3. J. M. BORWEIN AND A. S. LEWIS, Duality relationships for entropy-like minimization problems, SIAM J. Control Optim., 29 (1991), pp. 325-338.
4. J. M. BORWEIN AND A. S. LEWIS, Partially finite convex programming I, II, Math. Programming, 57 (1992), pp. 15-48 and 49-84.
5. J. M. BORWEIN, A. S. LEWIS, AND D. NOLL, Maximum entropy reconstruction using derivative information I: Fisher information and convex duality, Math. Oper. Res., 21 (1996), pp. 442-468.
6. J. M. BORWEIN, A. S. LEWIS, M. N. LIMBER, AND D. NOLL, Maximum entropy reconstruction using derivative information II: Computational results, Numer. Math., 69 (1995), pp. 243-256.
7. J. M. BORWEIN, M. N. LIMBER, AND D. NOLL, Fast heuristic methods for function reconstruction using derivative information, Appl. Anal., 58 (1995), pp. 241-261.
8. P. J. BROCKWELL AND R. A. DAVIS, Time Series: Theory and Methods, 2nd ed., Springer-Verlag, New York, 1991.
9. K. J. BROWN AND S. S. LIN, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight functions, J. Math. Anal. Appl., 75 (1980), pp. 112-120.
10. S. F. BURCH, S. F. GULL, AND J. K. SKILLING, Image restoration by a powerful maximum entropy method, Computer Vision, Graphics, and Image Processing, 23 (1983), pp. 113-128.
11. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
12. R. COURANT, Methoden der mathematischen Physik I, Springer Verlag, Berlin, 1931.
13. S. J. COX AND M. L. OVERTON, Optimal design of a column against buckling, SIAM J. Math. Anal., 23 (1992), pp. 287-325.
14. B. DACOROGNA, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, 1989.
15. J. L. DOOB, Probability and statistics, Trans. Amer. Math. Soc., 36 (1934), pp. 759-775.
16. G. J. ERICKSON AND C. R. SMITH, EDS., Maximum-Entropy and Bayesian Methods in Science and Engineering, Kluwer, Dordrecht, The Netherlands, 1988.
17. R. A. FISHER, Theory of statistical estimation, Philos. Trans. Roy. Soc. London, 222 (1930), pp. 309-368.
18. B. R. FRIEDEN, Restoring with maximum likelihood and maximum entropy, J. Opt. Soc. Amer., G2 (1972), pp. 511-518.
19. S. GEMAN AND D. GEMAN, Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images, IEEE Trans. Pattern Analysis and Machine Intelligence, 6 (1984), pp. 721-741.
20. B. GINSBURG AND A. IOFFE, The maximum principle in optimal control of systems governed by semilinear equations, in Proc. IMA Workshop on Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, B. Mordukhovich and B. Sussman, eds., Springer-Verlag, 1995.
21. W. HACKBUSCH, Elliptic Differential Equations: Theory and Numerical Treatment, Springer-Verlag, New York, 1992.
22. R. M. HARALICK, E. ØSTEVOLD, AND X. ZHUANG, A differential equation approach to maximum entropy image reconstruction, IEEE Trans. Acoust. Speech Signal Process., 35 (1987), pp. 208-218.
23. B. R. HUNT, Deconvolution of linear systems by constrained regression and its relationship to the Wiener theory, IEEE Trans. Automat. Control, 23 (1972), pp. 703-705.
24. R. W. JOHNSON AND J. E. SHORE, Which is the better entropy expression for speech processing: -s log s or log s?, IEEE Trans. Acoust., Speech Signal Proc., 32 (1984), pp. 129-137.
25. T. KATO, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, New York, 1984.
26. A. K. KATSAGGELOS AND K.-T. LAY, Maximum likelihood identification and restoration of images using the expectation-maximization algorithm, Springer Series in Information Sciences, 23 (1991), pp. 143-176.
27. A. S. LEWIS, The convergence of entropy-based approximations for moment problems, Optimization, 28 (1994), pp. 383-395.
28. A. S. LEWIS AND M. L. OVERTON, Eigenvalue optimisation, Acta Numerica, (1996).
29. M. K. V. MURTHY AND G. STAMPACCHIA, A variational inequality with mixed boundary conditions, Israel J. Math., 13 (1972), pp. 188-224.
30. D. NOLL, Rates of convergence for best entropy estimates, Stat. Decisions, 13 (1995), pp. 141-165.
31. D. NOLL, Restoration of degraded images with maximum entropy, J. Global Optim., 10 (1997), pp. 91-103.
32. D. NOLL, Consistency of a nonlinear deconvolution method with applications in image restoration, Adv. Math. Sci. Appl., 7 (1997), pp. 789-808.
33. M. L. OVERTON, Large scale optimisation of eigenvalues, SIAM J. Optim., 2 (1992), pp. 88-120.
34. M. L. OVERTON AND R. S. WOMERSLEY, Second derivatives for optimizing eigenvalues of symmetric matrices, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 697-718.
35. R. T. ROCKAFELLAR, Integrals which are convex functionals I, II, Pacific J. Math., 24 and 39 (1968), pp. 525-539 and 439-469.
36. R. N. SILVER, Quantum Statistical Inference, IEEE Report, Kluwer, Dordrecht, The Netherlands, 1992.
37. H. J. TRUSSELL, The relationship between image restoration by the maximum a posteriori method and a maximum entropy method, IEEE Trans. Acous. Speech Signal Proc., 28 (1980), pp. 114-117.
38. L. VANDENBERGHE AND S. BOYD, Semidefinite programming, SIAM Rev., 38 (1996), pp. 49-95.
39. W. P. ZIEMER, Weakly Differentiable Functions, Springer-Verlag, New York, 1989.
40. V. ZIVOJNOVIC, Higher-order statistics and Huber's robustness, in IEEE Signal Processing Workshop on Higher-Order Statistics, Spain, 1995.