We consider error estimates of iterative algorithm in shift-invariant signal spaces. For the classical sampling and reconstruction algorithm, error estimate from its samples corrupted by white noises are widely studied, but the error analysis of noise with time jitter and iterative noise has not been given as much attention. In this paper, three types of error estimates are studied. In detail, we obtain the error estimate for reconstructing a signal from its noise samples, noise samples with time jitter, and iterative noise. 1. Introduction The famous Shannon sampling theorem [1] successful resolves the reconstruction of a function on from its samples , where is a countable index set. This is a common task in many applications in signal or image processing. The Shannon sampling theorem says that if is the bandlimited signal of finite energy, then it is completely characterized by its samples. In many engineering applications, such as MRI imaging, signals and images are not band limited. One such example is the shift-invariant spaces. Nonuniform sampling and reconstruction problems in shift-invariant spaces are a relatively recent and active research field [2–12]. The shift-invariant space model developed in the 1990s is successful for many engineering problems, where the signal to be reconstructed is assumed to live in a shift-invariant space. It has been shown to be suitable and realistic, especially for taking into account of realistic environment, for modeling signals with smooth spectrum, or for numerical implementation [8, 10, 12]. In this paper, we will assume that the functions or signals all belong to shift-invariant space of the form [2–12] Many reconstruction algorithms are studied in shift-invariant spaces. For example, the iterative algorithm is obtained in shift-invariant spaces [3]. However, the reconstructing a function from data corrupted by noise has not been given as much attention. Smale and Zhou reconstructed signals from noisy data in [7] and gave error estimates for the reconstructed signal [8]. Aldroubi et al. discussed error analysis of frame reconstruction from noisy samples in [2]. Chen et al. gave the estimate of aliasing error for reconstruction algorithm in shift-invariant spaces [4]. We will study error analysis of the iterative reconstruction algorithm from noisy samples in shift-invariant spaces. In this paper, the following three types of errors are considered for the iterative algorithm in shift-invariant spaces.(1)Signal samples are affected by some additive noise and are therefore given by where is sampling point and
References
[1]
C. E. Shannon, “Communication in the presence of noise,” Proceedings of the IEEE, vol. 37, pp. 10–21, 1949.
[2]
A. Aldroubi, C. Leonetti, and Q. Sun, “Error analysis of frame reconstruction from noisy samples,” IEEE Transactions on Signal Processing, vol. 56, no. 6, pp. 2311–2325, 2008.
[3]
A. Aldroubi and K. Gr？chenig, “Nonuniform sampling and reconstruction in shift-invariant spaces,” SIAM Review, vol. 43, no. 4, pp. 585–620, 2001.
[4]
W. Chen, B. Han, and R.-Q. Jia, “Estimate of aliasing error for non-smooth signals prefiltered by quasi-projections into shift-invariant spaces,” IEEE Transactions on Signal Processing, vol. 53, no. 5, pp. 1927–1933, 2005.
[5]
B. Liu and W. Sun, “Determining averaging functions in average sampling,” IEEE Signal Processing Letters, vol. 14, no. 4, pp. 244–246, 2007.
[6]
S. Luo, “Error estimation for non-uniform sampling in shift invariant space,” Applicable Analysis, vol. 86, no. 4, pp. 483–496, 2007.
[7]
S. Smale and D.-X. Zhou, “Shannon sampling and function reconstruction from point values,” American Mathematical Society. Bulletin. New Series, vol. 41, no. 3, pp. 279–305, 2004.
[8]
S. Smale and D.-X. Zhou, “Shannon sampling. II. Connections to learning theory,” Applied and Computational Harmonic Analysis, vol. 19, no. 3, pp. 285–302, 2005.
[9]
W. Sun and X. Zhou, “Sampling theorem for wavelet subspaces: error estimate and irregular sampling,” IEEE Transactions on Signal Processing, vol. 48, no. 1, pp. 223–226, 2000.
[10]
M. Unser, “Sampling—50 years after Shannon,” Proceedings of the IEEE, vol. 88, no. 4, pp. 569–587, 2000.
[11]
J. Xian and S. Li, “Sampling set conditions in weighted multiply generated shift-invariant spaces and their applications,” Applied and Computational Harmonic Analysis, vol. 23, no. 2, pp. 171–180, 2007.
[12]
J. Xian, S.-P. Luo, and W. Lin, “Weighted sampling and signal reconstruction in spline subspaces,” Signal Processing, vol. 86, no. 2, pp. 331–340, 2006.
[13]
A. Nordio, C.-F. Chiasserini, and E. Viterbo, “Signal reconstruction errors in jittered sampling,” IEEE Transactions on Signal Processing, vol. 57, no. 12, pp. 4711–4718, 2009.
[14]
M. Z. Nashed and Q. Sun, “Sampling and reconstruction of signals in a reproducing kernel subspace of ,” Journal of Functional Analysis, vol. 258, no. 7, pp. 2422–2452, 2010.
[15]
Z. Song, W. Sun, X. Zhou, and Z. Hou, “An average sampling theorem for bandlimited stochastic processes,” IEEE Transactions on Information Theory, vol. 53, no. 12, pp. 4798–4800, 2007.
[16]
W. Sun and X. Zhou, “Average sampling in spline subspaces,” Applied Mathematics Letters, vol. 15, no. 2, pp. 233–237, 2002.
