Abstract and Applied Analysis

Volume 2010 (2010), Article ID 214213, 9 pages

http://dx.doi.org/10.1155/2010/214213

## Error Estimates from Noise Samples for Iterative Algorithm in Shift-Invariant Signal Spaces

Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

Received 1 September 2010; Accepted 4 October 2010

Academic Editor: Douglas Robert Anderson

Copyright © 2010 Jun Xian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 is a vector of noise samples.(2)Add time-jitter error in the above error model where are jitter. Due to the sampling time, points are not met correctly but might differ from the exact ones by not more than a given , that is, .(3)Numerical error in the th iterative step in the iterative algorithm as shown in Theorem 2.1 is , that is,

For the first kind of error analysis, it is widely studied, see [1, 3, 7, 8, 10]. However, the second and third kinds of error analysis have not been given as much attention. The second type of error analysis is presented in band-limited signal spaces [13]. The time-jitter errors without additive noise are studied in [6, 9]. Some results of the third type of error analysis are shown in reproducing kernel spaces [14]. In this paper, we study the three types of error for the iterative algorithm in shift-invariant spaces.

The paper is organized as follows. In Section 2, we introduce some concepts such as bounded partition of unity, shift-invariant spaces, and the iterative algorithm in shift-invariant spaces. In Section 3, we give error estimates of the iterative algorithm reconstruction algorithm from noisy samples.

#### 2. Notations and Preliminaries

The shift-invariant spaces under consideration are of the form
where is the so-called * generator*. The generator belongs to a subspace of continuous functions of Wiener amalgam spaces . A measurable function belongs to Wiener amalgam spaces if it satisfies
For any , we have the norm equivalences in [3]; that is, there exist constants and such that

A set satisfying is called * separated*, where is a countable index set. For the sampling set , we can give the following definition of bounded partition of unity (BPU). *A bounded partition of unity (BPU)* associated with is a set of functions that satisfy(1), where ,(2),(3).

The operator defined by is a quasi-interpolant of the sampled values .

Aldroubi and Gröchenig presented the following iterative algorithm in [3].

Theorem 2.1. *Let be sampling set with for some and be a BPU associated with . Let be in and be a bounded projection from onto . Then, there exists density such that if is separated, then any can be recovered from its samples on sampling set by the iterative algorithm
*

#### 3. Error Estimates of Iterative Reconstruction Algorithm

First, for given data of the form , we give the estimation of in Theorem 3.3, where is recovered via the iterative approximation projection reconstruction algorithm in Theorem 2.1 from corrupted samples . If are noise with zero mean and variance, then the error estimates of and are presented in Theorem 3.1. Second, we discuss the second type of error. The estimate of will be shown in Theorem 3.4, where is recovered via the iterative approximation projection reconstruction algorithm from corrupted samples . Third, if there exists the numerical error in th iterative step in the iterative algorithm, then we will consider the numerical stability of the iterative algorithm in Theorem 3.5.

Theorem 3.1. *Suppose that the initial samples in the iterative approximation projection reconstruction algorithm as shown by Theorem 2.1 are corrupted, that is, for noise . Assume that , are noises with zero mean and variance, that is,
**
If there exists a constant such that
**
then for any **
where is recovered via the iterative approximation projection reconstruction algorithm from corrupted samples and , and are defined in Theorem 2.1.*

*Proof. *Applying (2.4) iteratively leads to
which together with the convergence of (2.4) implies that
where , .

Combining (3.5) with leads to

By , (3.2), and (3.6), we obtain

*Remark 3.2. *If we add some restricted conditions presented in [14] for generator , that is,
where , and is the dual of , then it is easy to check that (3.2) holds in shift-invariant space. Song et al. obtained the restricted conditions such that (3.2) holds for bandlimited space in [15]. From Lemma 2.1 of [16], (3.2) holds for special shift-invariant, that is spline subspace.

Theorem 3.3. *If the initial samples in the iterative approximation projection reconstruction algorithm as shown by Theorem 2.1 are corrupted, that is, for noise , then is bounded by the norm of the noise. More precisely,
**
where is recovered via the iterative approximation projection reconstruction algorithm from corrupted samples and .*

*Proof. *Let
where .

From the iterative (3.10),

Using Lemma 8.3 of [3], we may choose so small that

By (3.4),(3.11), and (3.12), we have
This implies that .

Theorem 3.4. *Assume that the initial samples in the iterative approximation projection reconstruction algorithm as shown by Theorem 2.1 are corrupted, that is, for noise with and . If there exist constants such that
**
for the generator and sampling set , then
**
where is recovered via the iterative approximation projection reconstruction algorithm from corrupted samples as defined in Theorem 3.1 and as defined in (2.3).*

*Proof. *From (3.2) and (3.5), we have

Next, we will estimate .

By the Hölder inequality and (3.14),

The last inequality follows from (2.3).

So, we have

Lastly, we will consider the numerical stability of the iterative algorithm.

Theorem 3.5. *Assume that the numerical error in th iterative step in the iterative algorithm as shown in Theorem 2.1 is , that is,
**
Then, we have the following estimation
**
where .*

*Proof. *By induction, we obtain , where and for .

Using Lemma 8.3 of [3], we may choose so small such that (3.12) holds.

Therefore,

The iterative algorithm converges exponentially too, precisely

In fact, by the iterative algorithm
From (3.12), we have .

Combining (3.21) with (3.22), we have

#### Acknowledgments

This project is partially supported by the National Natural Science Foundation of China (10801136, 10871213), the Natural Science Foundation of Guangdong Province (07300434), and the Fundamental Research Funds for the Central Universities (10lgpy27). This work is partially done when the author is visiting the Department of Mathematics, University of Central Florida. The author thanks the department for the hospitality. The authors would like to thank Professor Qiyu Sun and Professor Wenchang Sun for their discussion and suggestions.

#### References

- C. E. Shannon, “Communication in the presence of noise,”
*Proceedings of the IEEE*, vol. 37, pp. 10–21, 1949. View at Google Scholar - 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. View at Publisher · View at Google Scholar - A. Aldroubi and K. Gröchenig, “Nonuniform sampling and reconstruction in shift-invariant spaces,”
*SIAM Review*, vol. 43, no. 4, pp. 585–620, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar - B. Liu and W. Sun, “Determining averaging functions in average sampling,”
*IEEE Signal Processing Letters*, vol. 14, no. 4, pp. 244–246, 2007. View at Publisher · View at Google Scholar - S. Luo, “Error estimation for non-uniform sampling in shift invariant space,”
*Applicable Analysis*, vol. 86, no. 4, pp. 483–496, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Unser, “Sampling—50 years after Shannon,”
*Proceedings of the IEEE*, vol. 88, no. 4, pp. 569–587, 2000. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar - M. Z. Nashed and Q. Sun, “Sampling and reconstruction of signals in a reproducing kernel subspace of ${L}^{p}({R}^{d})$,”
*Journal of Functional Analysis*, vol. 258, no. 7, pp. 2422–2452, 2010. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar - W. Sun and X. Zhou, “Average sampling in spline subspaces,”
*Applied Mathematics Letters*, vol. 15, no. 2, pp. 233–237, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH