Nonideal sampling and regularization theory

IEEE Transactions on Signal Processing

Volumn 56, Issue 3, 2008, Pages 1055-1070

  Ramani, Sathish ^a   Van De Ville, Dimitri ^a   Blu, Thierry ^a   Unser, Michael ^a  

^a EPFL   (Switzerland)

Author keywords

Mateacute; rn class; Minimax and minimum mean square error (MMSE) reconstruction; Nonideal sampling; Regularization; Shift invariant spaces; Smoothing splines; Tikhonov criterion; Wiener solution

Indexed keywords

ADDITIVE NOISE; MEAN SQUARE ERROR; OPTIMIZATION; RANDOM PROCESSES; SAMPLING; VARIATIONAL TECHNIQUES;

REGULARIZATION THEORY; SHIFT-INVARIANT SPACE; SMOOTHING SPLINE ESTIMATORS; TIKHONOV CRITERION; WIENER SOLUTION;

SIGNAL RECONSTRUCTION;

EID: 40749094435     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2007.908997     Document Type: Article

References (59)

1. C. E. Shannon, "Communication in the presence of noise," Proc. IRE vol. 37, pp. 10-21, 1949.
2. J. M. Whittaker, "The Fourier theory of the cardinal functions," Proc. Math. Soc. Edinburgh, vol. 1, pp. 169-176, 1929.
3. D. A. Linden, "A discussion of sampling theorems," Proc. IRE, vol. 47, pp. 1219-1226, 1959.
4. A. J. Jerri, "The Shannon sampling theorem - Its various extensions and applications: A tutorial review," Proc. IEEE, vol. 65, no. 11, pp. 1565-1596, 1977.
5. M. Unser, "Sampling-50 years after Shannon," Proc. IEEE, vol. 88, no. 4, pp. 569-587, 2000.
6. P. L. Butzer, "A survey of the Whittaker-Shannon sampling theorem and some of its extensions," J. Math. Res. Expo., vol. 3, pp. 185-212, 1983.
7. E. T. Whittaker, "On the functions which are represented by the expansion of interpolating theory," Proc. R. Soc. Edinburgh, vol. 35, pp. 181-194, 1915.
8. J. J. Benedetto, "Frames, sampling and seizure prediction," in Advances in Wavelets, K.-S. Lau, Ed. New York: Springer-Verlag, 1998, pp. 1-15.
9. J. R. Higgins, Sampling Theory in Fourier and Signal Analysis. Oxford, U.K.: Clarendon, 1996.
10. D. P. Petersen and D. Middleton, "Sampling and reconstruction of wave-number-limited functions in N-dimensional Euclidean spaces," Inf. Control, vol. 5, pp. 279-323, 1962.
11. J. L. Brown, Jr. and K. Sangsari, "Sampling reconstruction of N dimensional band-limited images after multilinear filtering," IEEE Trans. Circuits Syst., vol. 36, no. 7, pp. 1035-1038, 1989.
12. A. Feuer and G. C. Goodwin, "Reconstruction of multidimensional bandlimited signals from nonuniform and generalized samples," IEEE Trans. Signal Process., vol. 53, no. 11, pp. 4273-4282, 2005.
13. S. H. Izen, "Generalized sampling expansion on lattices," IEEE Trans. Signal Process., vol. 53, no. 6, pp. 1949-1963, 2005.
14. H. R. Kunsch and F. A. Hamprecht, "Optimal lattices for sampling," IEEE Trans. Inf. Theory, vol. 51, no. 2, pp. 634-647, 2005.
15. A. Papoulis, "Generalized sampling expansion," IEEE Trans. Circuits Syst., vol. 24, pp. 652-654, 1977.
16. A. Feuer, "On the necessity of Papoulis' result for multidimensional GSE," IEEE Signal Process. Lett., vol. 11, no. 4, pp. 420-422, 2004.
17. d)," Trans. Amer. Math. Soc. vol. 341, no. 2, pp. 787-806, 1994.
18. A. Aldroubi and M. Unser, "Sampling procedures in function spaces and asymptotic equivalence with Shannon's sampling theory," Numer. Funct. Anal. Optim., vol. 15, no. 1 & 2, pp. 1-21, 1994.
19. A. Aldroubi and K. Gröchenig, "Nonuniform sampling and reconstruction in shift invariant spaces," SIAM Rev., vol. 43, pp. 585-620, 2001.
20. W. Chen, S. Itoh, and J. Shiki, "On sampling in shift-invariant spaces," IEEE Trans. Inf. Theory, vol. 48, no. 10, pp. 2802-2810, 2002.
21. K. Yao, "Applications of reproducing kernel Hilbert spaces - Bandlimited signal models," Inf. Control, vol. 11, pp. 429-444, 1967.
22. M. Z. Nashed and G. G. Walter, "General sampling theorems for functions in reproducing kernel Hilbert spaces," Math. Contr., Signals, Syst. vol. 4, pp. 373-412, 1991.
23. Y. C. Eldar and T. Werther, "General framework for consistent sampling in Hilbert spaces," Int. J. Wavelets, Multiresolution Inf. Process. vol. 3, no. 4, pp. 497-509, 2005.
24. Y. C. Eldar and T. G. Dvorkind, "A minimum squared-error framework for generalized sampling," IEEE Trans. Signal Process., vol. 54, no. 6, pp. 2155-2167, 2006.
25. G. G. Walter, "A sampling theorem for wavelet subspaces," IEEE Trans. Inf. Theory, vol. 38, pp. 881-884, 1992.
26. X. G. Xia and Z. Zhang, "On sampling theorem, wavelets, and wavelet transforms," IEEE Trans. Signal Process., vol. 41, pp. 3524-3535, 1993.
27. I. Djokovic and P. P. Vaidyanathan, "Generalized sampling theorems in multiresolution subspaces," IEEE Trans. Signal Process., vol. 45, pp. 583-599, 1997.
28. I. W. Selesnick, "Interpolating multiwavelets bases and the sampling theorem," IEEE Trans. Signal Process., vol. 47, no. 6, pp. 1615-1621, 1999.
29. G. Demoment, "Image reconstruction and restoration - Overview of common estimation structures and problems," IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 12, pp. 2024-2036, 1989.
30. N. B. Karayiannis and A. N. Venetsanopoulos, "Regularization theory in image restoration - The stabilizing functional approach," IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 7, pp. 1155-1179, 1990.
31. M. R. Banham and A. K. Katsaggelos, "Digital image restoration," IEEE Signal Process. Mag., vol. 14, no. 2, pp. 24-41, 1997.
32. M. Unser and A. Aldroubi, "A general sampling theory for nonideal acquisition devices," IEEE Trans. Signal Process., vol. 42, no. 11, pp. 2915-2925, 1994.
33. Y. C. Eldar and M. Unser, "Non-ideal sampling and interpolation from noisy observations in shift-invariant spaces," IEEE Trans. Signal Process., vol. 54, no. 7, pp. 2636-2651, 2006.
34. C. Rabut, "Elementary m-harmonic cardinal B-splines," Numer. Algorithms, vol. 2, pp. 39-62, 1992.
35. S. Tirosh, D. Van De Ville, and M. Unser, "Polyharmonic smoothing splines and the multi-dimensional Wiener filtering of fractal-like signals," IEEE Trans. Image Process., vol. 15, no. 9, pp. 2616-2630, 2006.
36. M. Unser and T. Blu, "Fractional splines and wavelets," SIAM Rev. vol. 42, no. 1, pp. 43-67, 2000.
37. M. Unser and T. Blu, "Generalized smoothing splines and the optimal discretization of the Wiener filter," IEEE Trans. Signal Process. vol. 53, no. 6, pp. 2146-2159, 2005.
38. S. Saitoh, V. K. Tuan, and M. Yamamotom, "Convolution inequalities and applications," J. Inequal. Pure Appl. Math., vol. 42, no. 3, 2003.
39. 1 norms for image restoration," SIAM J. Scientif. Comput., vol. 27, no. 6, pp. 1881-1902, 2006. RAMANI et al.: NONIDEAL SAMPLING AND REGULARIZATION THEORY 1069
40. M. Nikolova, "Minimizers of cost-functions involving nonsmooth data-fidelity terms: Application to the processing of outliers," SIAM J. Numer. Anal., vol. 40, no. 3, pp. 965-994, 2002.
41. S. Osher, L. Rudin, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D, vol. 60, pp. 259-268, 1992.
42. 1 function approximation," SIAM J. Appl. Math., vol. 65, no. 5, pp. 1817-1837, 2005.
43. Y. Katznelson, An Introduction to Harmonic Analysis. New York: Dover, 1976.
44. L. Schwartz, Théorie des distributions. Paris: Hermann, 1966.
45. J. Duchon, "Splines minimizing rotation-invariant semi-norms in Sobolev spaces," in Multivariate Approximation Theory, W. Schempp and K. Zeller, Eds. Basel: Birkhäuser-Verlag, 1979, pp. 85-100.
46. T. Blu and M. Unser, "Approximation error for quasi-interpolators and (multi-) wavelet expansions," Appl. Comp. Harm. Anal., vol. 6, no. 2, pp. 219-251, Mar. 1999.
47. D. Pierre, Optimization Theory With Applications. New York: Dover, 1986.
48. I. Schoenberg, "Contribution to the problem of approximation of equidistant data by analytic functions," Quart. Appl. Math., vol. 4, pp. 45-99, 1946, and 112-141.
49. I. Schoenberg, "Cardinal interpolation and spline functions," J. Approx. Theory, vol. 2, pp. 167-206, 1969.
50. I. Schoenberg, "Cardinal spline interpolation," CBMS-NSF Regional Conf. Ser. Appl. Math., vol. 12, 1973.
51. W. R. Madych and S. A. Nelson, "Polyharmonic cardinal spline," J. Approx. Theory, vol. 60, pp. 141-156, 1990.
52. P. Thévenaz and M. Unser, "Precision isosurface rendering of 3-D image data," IEEE Trans. Image Process., vol. 12, 2003.
53. D. Van De Ville, T. Blu, and M. Unser, "Isotropic polyharmonic B-splines: Scaling functions and wavelets," IEEE Trans. Image Process., vol. 14, no. 11, pp. 1798-1813, 2005.
54. M. L. Stein, Interpolation of Spatial Data-Some Theory for Kriging New York: Springer-Verlag, 1999.
55. B. Whitcher, "Wavelet-based bootstrapping of spatial patterns on a finite lattice," Computat. Statist. Data Anal., vol. 50, no. 9, pp. 2399-2421, 2006.
56. B. Pesquet-Popescu and J. L. Véhel, "Stochastic fractal models for image processing," IEEE Signal Process. Mag., vol. 19, no. 5, pp. 48-62, 2002.
57. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 9th ed. New York: Dover, 1965.
58. S. M. Kay, Statistical Signal Processing Vol. I: Estimation Theory Englewood Cliffs, NJ: Prentice-Hall, 1995.
59. C. L. Fales, F. O. Huck, J. A. McCormick, and S. K. Park, "Wiener restoration of sampled image data: End-to-end analysis," J. Opt. Soc. Amer. A, vol. 5, no. 3, pp. 300-314, 1988.