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[9] F. O’Sullivan, “A study of least squares and maximum likelihood for image reconstruction in positron emission tomography,” Ann. Statist., vol. 23, pp. 1267–1300, 1995. [10] D. G. Politte and D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron emission tomography,” IEEE Trans. Med. Imag., vol. 10, pp. 82–89, Mar. 1991. [11] B. W. Silverman, M. C. Jones, D. W. Nychka, and J. D. Wilson, “A smoothed EM algorithm to indirect estimation problems, with particular reference to stereology and emission tomography,” J. Roy. Statist. Soc. Ser. B, vol. 52, pp. 271–324, 1990. [12] D. L. Snyder and I. M. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci., vol. NS-32, pp. 3864–3872, Oct. 1985. [13] Y. Vardi, L. A. Shepp, and L. Kaufman, “A statistical model for positron emission tomography,” J. Amer. Statist. Assoc., vol. 80, pp. 8–37, 1985. [14] J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imag., vol. 13, pp. 290–300, June 1994. [15] P. Moulin and J. Liu, “Statistical imaging and complexity regularization,” IEEE Trans. Inform. Theory, vol. 46, pp. 1762–1777, Aug. 2000. [16] L. Cavalier, “Asymptotically efficient estimation of a density in tomography,” Ann. Statist., vol. 28, pp. 330–347, 2000. , “On the problem of local adaptive estimation in tomography,” [17] Bernoulli, vol. 7, pp. 63–78, 2001. [18] L. Cavalier and A. B. Tsybakov, “Sharp adaptation for inverse problems with random noise,” Probab. Theory Related Fields, to be published. [19] D. L. Donoho and M. G. Low, “Renormalization exponents and optimal pointwise rates of convergence,” Ann. Statist., vol. 20, pp. 944–970, 1992. [20] S. Geman and D. E. McClure, “Statistical methods for tomographic image reconstruction,” Proc. 46 Sect. ISI, Bull. ISI, vol. 52, pp. 5–21, 1987. [21] I. M. Johnstone, “Wavelet shrinkage for correlated data and inverse problems: Adaptivity results,” Statist. Sinica, vol. 9, pp. 51–83, 1999. [22] M. C. Jones and B. W. Silverman, “An orthogonal series density estimation approach to reconstructing positron emission tomography images,” J. Appl. Statist., vol. 16, pp. 177–191, 1989. [23] I. M. Johnstone and B. W. Silverman, “Speed of estimation in positron emission tomography and related inverse problems,” Ann. Statist., vol. 18, pp. 251–280, 1990. [24] J.-Y. Koo, “Convergence rates for logspline tomography,” J. Multiv. Anal., vol. 67, pp. 367–384, 1998. [25] J.-Y. Koo and H.-Y. Chung, “Log-density estimation in linear inverse problems,” Ann. Statist., vol. 26, pp. 335–362, 1998. [26] A. P. Korostelev and A. B. Tsybakov, “Optimal rates of convergence of estimators in a probabilistic setup of tomography problem,” Probl. Inform. Transm., vol. 27, pp. 73–81, 1991. [27] S. R. Deans, The Radon Transform and Some of Its Applications. New York: Wiley, 1983. [28] F. Natterer, The Mathematics of Computerized Tomography. Chichester, U.K.: Wiley, 1986. [29] E. J. Hoffman, S. C. Huang, M. E. Phelps, and D. E. Kuhl, “Quantitation in positron emission computed tomography: 4. Effect of accidental coincidences,” J. Comput. Assist. Tomog., vol. 5, pp. 391–400, 1981. [30] D. L. Snyder, J. A. O’Sullivan, B. R. Whiting, R. J. Murphy, J. Benac, J. A. Cataldo, D. G. Politte, and J. F. Williamson, “Deblurring subject to nonnegativity constraints when known functions are present with application to object-constrained computerized tomography,” IEEE Trans. Med. Imag., vol. 20, pp. 1009–1017, Oct. 2001. [31] D. L. Snyder, T. J. Schulz, and J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Processing, vol. 40, pp. 1143–1150, May 1992. [32] Y. Choi, J.-Y. Koo, and N.-Y. Lee, “Image reconstruction using the wavelet transform for positron emission tomography,” IEEE Trans. Med. Imag., vol. 20, pp. 1188–1193, Nov. 2001. [33] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992. [34] W. Härdle, G. Kerkyacharian, D. Picard, and A. B. Tsybakov, Wavelets, Approximation and Statistical Applications. New York: Springer-Verlag, 1998. [35] Y. Meyer, Wavelets and Operators. Cambridge, U.K.: Cambridge Univ. Press, 1992.
[36] V. Solo, “Limits to estimation in stochastic ill-conditioned inverse problems,” IEEE Trans. Inform. Theory, vol. 46, pp. 1872–1880, Aug. 2000. [37] J.-Y. Koo, “Optimal rates of convergence for nonparametric statistical inverse problems,” Ann. Statist., vol. 21, pp. 590–599, 1993. [38] L. Birgé, “Approximation dans les espaces métriques et théorie de l’estimation,” Z. Wahrsch. Verw. Geiete, vol. 65, pp. 181–237, 1983. [39] J. F. C. Kingman, Poisson Processes. London, U.K.: Clarendon, 1993. [40] R. D. Reiss, A Course on Point Processes. New York: Springer-Verlag, 1993. [41] W. Feller, An Introduction to Probability Theory and Its Applications. New York: Wiley, 1971, vol. 2. [42] A. I. Sakhanenko, “Berry–Esseen type estimates for large deviation probabilities,” Siberian Math. J., vol. 32, pp. 647–656, 1991.
On Sampling in Shift Invariant Spaces Wen Chen, Shuichi Itoh, Member, IEEE, and Junji Shiki
Abstract—In this correspondence, a necessary and sufficient condition for sampling in the general framework of shift-invariant spaces is derived. Then this result is applied, respectively, to the regular sampling and the perturbation of regular sampling in shift-invariant spaces. A simple necessary and sufficient condition for regular sampling in shift-invariant spaces is attained. Furthermore, an improved estimate for the perturbation is derived for the perturbation of regular sampling in shift-invariant spaces. The derived estimate is easy to calculate, and shown to be optimal in some shift-invariant spaces. The algorithm to calculate the reconstruction frame is also presented in this correspondence. Index Terms—Frame, generator, irregular sampling, sampling, shift-invariant space, Zak transform.
I. INTRODUCTION AND PRELIMINARIES In digital signal and image processing and digital communications, a continuous signal is usually represented and processed by using its discrete samples. Then a fundamental problem is how to represent a continuous signal in terms of a discrete sequence. For a band-limited signal of finite energy, it is completely characterized by its samples, by the famous classical Shannon sampling theorem. Observing that the Shannon function sinc = sin(1()1) is, in fact, a scaling function of a multiresolution analysis (MRA) [11], Walter [19] first extended Shannon sampling theorem to the setting of wavelet subspaces. Following Walter’s work [19], Janssen [13] studied the so-called shift sampling in wavelet subspaces by using the Zak transform, which can cover some extra cases of Walter’s result. Walter [20], Xia [22], and the authors of [7], [8] also studied the oversampling procedure in wavelet subspaces. On the other hand, Aldroubi and Unser [1]–[3], [18] studied the sampling procedure in shift-invariant spaces. Chen and Itoh [9] improved the works
Manuscript received August 16, 1999; revised June 9, 2002. This work was supported by the Japan Society for the Promotion of Sciences (JSPS). W. Chen is with the Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada (e-mail: wenchen@ math.ualberta.ca). S. Itoh and J. Shiki are with the Department of Information Network Sciences, Graduate School of Information Systems, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan (e-mail:
[email protected]). Communicated by J. A. O’Sullivan, Associate Editor for Detection and Estimation. Publisher Item Identifier 10.1109/TIT.2002.802646.
0018-9448/02$17.00 © 2002 IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 10, OCTOBER 2002
of Walter [19], and Aldroubi and Unser [3], and found a necessary and sufficient condition for sampling in shift-invariant spaces. However, in many real applications, sampling points are not always measured regularly. Sometimes the sampling steps need to be fluctuated according to the signals so as to reduce the number of samples and computational complexity. There are also many cases where undesirable jitter exists in sampling instants. Some communication systems may suffer from random delay due to channel traffic congestion and encoding delay. In such cases, the system will become more efficient when a perturbation factor is considered. For the band-limited signals of finite energy, a generalization of the Shannon sampling theorem, known as Kadec’s theorem [24], can be used. Following the work on sampling in wavelet subspaces, Liu and Walter [16], Liu [15], and the authors [4] tried to extend Kadec’s theorem to a class of wavelet subspaces. But they actually did not get a real extension of Kadec’s theorem. Then the authors [6] introduced a function class L [a; b] ( > 0, 2 [0; 1), and 0 2 [a; b] [01; 1]), a norm k1 kL [a; b] of L [a; b], and found an algorithm to treat the perturbation of regular sampling in wavelet subspaces as follows. Some notations used in the theorem will be defined in the next sections. Theorem 1 [6]: Suppose that ' is a continuous scaling function of an MRA fVm gm in L [a; b]. If ' = O((1 + j 1 j)0s ) for some s > 1, and the Zak transform Z' (; 1) 6= 0, then there is an r; ' 2 (0; 1] such that for any scalar sequence frk gk [0r; ' ; r; ' ] \ [a; b], there is a Riesz basis1 fS; k gk of V0 such that
f
=
2V
holds in L2 (R) for any f
r; '
1=2. For any f 2 V ('), there is a fck gk 2 l2 such that f = k ck '(1 0 k) in L2 (R). Since k
ck '(1 0 k)
2
jck j
2
j'(1 0 k)j
2
k k the series k ck '( k) point-wise converges to a continuous function in V ('). Without loss of generality, we can take any f V (') as a continuous function. When we try to find an algorithm to reconstruct a continuous signal f V (') by using its discrete samples f (tk ) k , obviously the samples cannot be arbitrary; that is, some constraints should be imposed on the sampling points tk k or the samples f (tk ) k . The weaker the constraints are the better the reconstruction method is evaluated. Our objective in this section is to find a necessary and sufficient condition on the samples f (tk ) k such that a reconstruction formula like (1) holds. The result is formulated in the following theorem.
10
2
2
g
f
f g
f
f
g
g
Theorem 2: Suppose that a continuous stable generator ' is such that ' = O((1 + j 1 j)0s ) for some s > 1=2. Then there is a frame fSn gn of V (') such that ff (tn )gn is a moment sequence to the frame fSn gn and
f
=
n
f (tn )Sn
(6)
2 V (') if and only if there is a constant
holds in L2 (R) for any f C 1 such that
C 01 kf k2
jf (tn)j C kf k 2
n
2
01 ^ = 'G ^ ' :
k
k
(tn
=
p1
=
p1
2
2
0 k)j k
k
2
l
1 G0 ' (! )
2
3
gl '(tn 0 k 0 l)eik!
3
k
1 G0 ' (! ) 1
k
2l
2
'(tn 0 k 0 l)ei(k+l)!
k
'(tn 0 k)eik!
1 G0 ' (! ) 1
Therefore, f (tn 0 k)gk s > 1=2. Let
k
3
3
'(tn 0 k)eik!
j'(tn 0 k)j
1=2
3
:
2
due to ' = O((1 + j 1 j)0s ) for some
q' (tn ; 1) =
0
10
(tn k)'( k): (8) k Then q' (tn ; ) is continuous in V ('). For any function f V ('), k) there is a scalar sequence ck k l2 such that f = k ck '( holds in L2 (R). Following the Parseval identity, we derive
1
2
f g 2
10
hf; q'(tn ; 1)i 1
^ q^' (tn ; 1) f;
1
'^
=
2
=
2
=
2
=
2
j'^j
2
2
0
k
k
k
0
k
ck e0ik1 ;
1 G0 ' (! )
2
2
=
k
'^ (tn 0 k)e0ik1
k
ck e0ik1 ;
ck e0ik!
1
=
k
2
1
2
=
2
1
=
ck e0ik1 ;
k
j'^j
1
l
l
gl '(tn 0 k 0 l)e0ik1
gl e0ik1
k
j'^(! + 2k)j
k
'(tn 0 k)e0ik1
2
'(tn 0 k)eik! d!
k
ck e0ik!
k
'(tn 0 k)eik! d!
ck '(tn 0 k)
f (tn ):
jhf; q' (tn ; 1)ij C kf k : 2
2
n It implies that fq' (tn ; 1)gn is a frame of V ('). For any n 2 Z , take Sn = Tq01 (q' (tn ; 1)), where Tq is the frame transform of the frame fq'(tn ; 1)gn . Then fSngn is a dual frame of the frame fq' (tn ; 1)gn in V (') and such that
f
hf; q' (tn ; 1)i =
Sn (t) n holds in L2 (R) for any f any n Z , the fact
2
0 k)eik!
l
C 01 kf k2
gk '(1 0 k) 2 V (')
(tn
p1
gl e0il!
From (7), we derive
in L2 (R). The fact ' = O((1 + j 1 j)0s ) for some s > 1=2 shows that can be chosen to be continuous. Moreover, the formula = g ' ( 1 0 k ) also pointwise holds. Given t 2 R , we have n k k 1=2
j
=
2
=
Then we have =
p1
p21
(7)
holds for any f 2 V ('). Proof: Sufficiency. 1 2 Since G0 ' 2 L [0; 2] is a 2 -periodic function, we can assume 1 G0 gk e0ik! ' (! ) = k in L2 (R) with scalar sequence fgk g 2 l2 . Define in L2 (R) by
=
=
2
f (tn )Sn n V ('). Meanwhile, for any f
f (tn ) = hf; Tq (Sn )i = hTq (f ); Sn i
2 V (') and
shows that ff (tn )g is a moment sequence to the frame fSn gn . Necessity. On the contrary, if there is a frame fSn gn of V (') such that ff (tn )gn is a moment sequence to the frame fSn gn and (6) holds in
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 10, OCTOBER 2002
2 V ('), then f (tn ) = hTs01 (f ); Sni, where Ts is the frame transform of the frame fSn gn (see Section I). Therefore,
L2 (R) for any f
n
and n
jf (tn)j2 =
jhTs01(f ); Sn ij2
n
Corollary 1: In Theorem 2, the reconstruction frame fSn gn is derived by
Sn = 2
C 01 kTs01(f )k2 C 02 kf k2
hold for some constant C
where (dmn ) (Amn ), and
Amn =
1.
The frame fSn gn of the shift-invariant space V (') such that (6) holds in L2 (R) for any f 2 V (') is called a reconstruction frame. In real-world application, we need to know the expression of the reconstruction frame fSn gn . Since Tq is the frame transform of the frame fq' (tn ; 1)gn , we have
=
hq'(tn ; 1); q' (tm ; 1)i q'(tm ; 1)
m 1
2
m
(tn
k
k =
1
2
m
j'^j2
k
k =
1
2
m
j'^j2
l
1
2
2
m k
q' (tm ; 1)
k
2 0
0
1 G0 ' (! )
'(tn 0 k)e0ik!
k
Anm =
0
'
k
(a.e.) and S^0 = '= ^ ' ^ (a.e.). Proof: For any f 2 V ('), there is a scalar sequence fck gk 2 l2 2 such that f = k ck '(1 0 k) holds both in L (R) and pointwise. Since
kf k2 = 21
=
k
'(tm 0
k)eik! d!:
q' (tn ; 1) =
1
2
m
Anm Sm :
1
2 1
2
f^ '^
2
ck e0ik1
k
G1'=2
k
2
ck e0ik1
and k
Then we have
'(k) exp(ik1):
k'^3 k1 < 1. 2) There is a frame fSn gn of V (') such that ff (n)gn is a moment sequence to the frame fSn gn and f = n f (n)Sn for any f 2 V ('). 3) There is a Riesz basis fSn gn of V (') such that f = n f (n)Sn for any f 2 V ('). In these cases, the fSn gn is defined by the formulas Sn = S0 (1 0 n) 3
=
'(tn 0 k)e0ik!
'(tm 0 k): (10)
III. REGULAR SAMPLING IN SHIFT-INVARIANT SPACES
'(tm 0 k)eik! d!:
G01 (!)
k
3 1) 0 < k' ^ k0
'(tn 0 k)e0ik1 ;
1 G0 ' (! )
'(tn 0 k)e0ik!
In the following sections, we will explore the relation between the matrix (Amn ) and the reconstruction frame fSn gn in more detail.
Let 2
k
are equivalent.
'(tm 0 k)e0ik1 q' (tm ; 1)
q' (tm ; 1)
2
Theorem 3: Let ' be a continuous stable generator such that ' = O((1+ j1j)0s ) for some s > 1=2. Then the following three statements
gl '(tm 0 k 0 l)e0ik1 q' (tm ; 1) 2
(9)
k 3 2 Obviously, ' ^ is a 2 -periodic function in L [0; 2].
gl '(tn 0 k 0 l)e0ik1 ;
gl eik1
dmn q' (tm ; 1)
is the inverse matrix of the infinite-dimensional matrix
'^3 =
0ik1 0 k)'e ^
l
l
k =
0ik1 ; 0 k)'e ^
(tm
n
An important case of sampling is the so-called regular sampling, i.e., tn = n for all n 2 Z . As we mentioned in the Introduction, many people have contributed to this area of research. The following theorem contributes in a different direction. Since it is not so difficult to understand, we just give a swift proof here. We will also need the 3 notation ' ^ defined by
Tq (q' (tn ; 1)) =
dmn q' (tn ; 1)
n
for any m 2 Z . This is formulated in the following corollary.
jhTs01(f ); Sn ij2
n
Since Tq is a frame transform, the infinite-dimensional matrix A = 01 = (dmn ) the inverse matrix of A. Then we derive
(Anm ) is invertible. Denote by A
Sm = 2
C kTs01(f )k2 C 2kf k2 jf (tn)j2 =
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jf (k)j2 = 21 =
1
2
k
'^3
l
k
2
3
;
cl '(k 0 l)e0ik1 ck e0ik1
2
3
2
3
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 10, OCTOBER 2002
we can conclude that the fact C
for any f
01 kf k2
jf (k)j C kf k 2
k
2
2 V (') is equivalent to the fact 0 = 3 = C kG' k j' ^ j C kG' k1 : 1
1 2
1 2
0
By Theorem 2, these arguments show the equivalence of statement 1) and statement 2). Consider the q' defined by(8). For any scalar sequence fck g 2 l2 , we have k =
=
=
2
1
ck q' (k; ) 1
' ^
2 1
k
ik1
1=2
G'
2
0 k)ck e0ik1
(n
l
gl e
01=2 '^3
1
0ik1
G'
2
k
'(n
ck e
k
k'^3 k kG' k 2 0
jck j 2
0
k
k
0 k)ck e0ik1
f =
2
3
1
2
ck q' (k; )
k'^3 k1 kG' k1
j j
2
ck k for any scalar sequence ck k l2 . It implies that the function sequence q' (n; ) n is a Riesz basis of V ('). Then there is a unique dual Riesz basis Sn n of the Riesz basis q' (n; ) n biorthogonal to q' (n; ) n such that statement 3) holds. Since statement 3) implies statement 2), thereafter, statement 1) automatically, it shows the equivalence of statement 1) and statement 3). 3 ^0 = '= Finally, the formula Sn = S0 ( n) and S ^ ' ^ can follow from [9] and [19]. But we would like to use Corollary 1 to derive it so as to show the potential of Corollary 1. From (10), we have 2 3 2 0i(n0m)! ' ^ Anm = e d!: G' 0
f
f g 2
1g f g
1g
f
1g
10
j j
Suppose that fck gk are the Fourier coefficients of the 2 -periodic function j'^G j in L2 [0; 2]. Then, we derive Anm = cn0m for any m; n 2 Z , and m
1
^m = q^' (n; ) = cn0m S
j'^3 j2 G'
2 '^'^3 e0in1 :
Suppose that the scalar sequence fdn gn is the Fourier coefficients of G the 2 -periodic function j'^ j in L2 [0; 2 ]. Make an infinite-dimensional matrix D = (dm0n ). Then the matrix D is the inverse matrix of the matrix A, because the formula n ck0n dn0j = k0j is equivalent to the formula ik!
k
ck e
n
in! eij! = eij!
dn e
(11)
and (11) automatically holds. The k0j used here is the Dirac function, which takes value 1 whenever k = j and takes value 0 otherwise. 3 2 Define the function S0 by the formula S^0 = '= ^ ' ^ in L (R). From (11), we have
3 ^ j ^m = j' S
2
G'
^0 S
k
0in1 = S^0 e0im1 : dm0n e
It shows Sm = S0 (1 0 m) for any m 2 Z .
1 0 k) 1 0 k)
sin (
(
IV. PERTURBATION OF REGULAR SAMPLING IN SHIFT-INVARIANT SPACES
:
2
f
f (k)
k automatically follows from Theorem 3.
2
3
Then statement 1) shows
2
This theorem implies that for regular sampling in shift-invariant spaces, the reconstruction frame fSn gn is an exact frame, i.e., a Riesz basis. This fact is also determined by the samples f'(k)gk of the 3 generator ', i.e., C 01 j' ^ j C for some constant C 1. 3 When ' is a stable continuous cardinal generator, that is, ' ^ = ^0 = ' ^, i.e., S0 = '. Therefore, the reconstruction 1 (a.e.). Then S formula becomes f = k f (k)'(1 0 k) for any f 2 V ('). By the way, the fact that statement 1) implies statement 3) is also found by Walter [19] when ' = O((1 + j 1 j)0s ) for some s > 1. The formula Sm = S0 (1 0 m) for any m 2 Z has also been found by Walter [19], Aldroubi and Unser [3], and Chen and Itoh [9] under their assumed conditions on the generator '. Take ' = sinc. Then ' is a stable continuous cardinal generator. The shift-invariant space V (') is the collection of all -band signals of finite energy. Then, the classical Shannon sampling formula
Another important case of sampling is the perturbation of regular sampling, tn = n + rn (rn 2 (01; 1)) for any n 2 Z . A fundamental question in this case is to estimate the range of the perturbation frk gk . Following Kadec’s theorem for the band-limited signals of finite energy, we have found an estimate for perturbation in the setting of wavelet subspaces by using Riesz basis theory. In the following, we will derive a sharp estimate by using frame theory. In order to establish the algorithm for perturbation of regular sampling in shift-invariant spaces, we need to introduce the function class L [a; b] ( > 0, 2 [0; 1), 0 2 [a; b] [01; 1]) defined and used in our previous work [6]. We have reasoned that this class is an appropriate collection by giving some propositions in that paper. Here we will recall the definition and give an additional proposition, which are useful to the piecewise differentiable stable generators in many real applications. Definition 1: L [a; b] ( > 0, 2 [0; 1) and 0 2 [a; b] [01; 1]) consists of all the measurable functions f , for which the norm
kf kL
[a; b] =
sup
fr g [a;b]
j
0 j j
j 1:
k f (k + + rk ) f (k + ) < supk rk
The following proposition is very useful to the piecewise differentiable generators in many real applications. Proposition 1: Let function f on R be differentialable on the intervals k + + [a; b] ( 2 [0; 1), 0 2 [a; b] [01; 1]) for any k 2 Z . If the series k sup[a;b] jf 0 (1 + k + )j < 1 then f 2 L1 [a; b], where f 0 is the derivative of f . Proof: Since r 0 jf (k + + rk ) 0 f (k + )j = f (k + + t) dt 0 k k sup jr j
k
sup jrk j it implies f
2 L [a; b].
k
jf 0 (k + + t)jdt
0
k
j0
1j
sup f (k + + ) [a;b]
1
Now we show the theorem for perturbation of regular sampling in shift-invariant spaces. We will also discuss some special cases after the proof.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 10, OCTOBER 2002
Theorem 4: Suppose that ' is a continuous stable generator such that ' O j 1 j 0s for some s > = , and
2807
12 = ((1 + ) ) 3 3 1) 0 < k' ^ k0 k'^ k1 < 1. 2) ' 2 L0 [a; b]. Then for any frk gk [0r' ; r' ] \ [a; b], there is a frame fSk gk of V (') such that the scalar sequence ff (k +rk )gk is a moment sequence to the frame fSk gk and f = k f (k + rk )Sk holds in L (R) for any f 2 V (') if = k'^3 (!)k r' < : (12)
Hence
= + rk . By Theorem 2, we only need to show 1 such that 0 C kf k jf (tk )j C kf k k holds for any f 2 V ('). If we can show that there is a positive number < 1 such that jf (tk ) 0 f (k)j jf (k)j (13) k k holds for any f 2 V ('), then (1 0 ) jf (k)j jf (tk )j (1 + ) jf (k)j : k k k By Theorem 3, statement 1) implies that there is a constant C 1 such
that is, k'kL [a; b] (12).
1
k'^3 k
2 0
1
Proof: Let tk k that there is a constant C 1
2
2
2
2
2
2
2
2
k
kk 2
jf (k)j C kf k 2
k
Anm
2 V ('). These arguments show that C 0 (1 0 ) kf k jf (tk )j C (1 + ) kf k 1
2
2
2
2
2
k
()
for any f 2 V ' . By Theorem 2, Theorem 4 is shown. All that remains to be shown is (13). For an f 2 V ' , there is a scalar sequence fck gk 2 l2 such that f k ck ' 1 0 k holds both in L2 R and pointwise. Let
=
( )
1=
jf (tk ) 0 f (k)j
k
=
k
=
l
n
k; l
() )
(
2
0
n
2 ('(tn 0 l) 0 '(n 0 l)): Take ak; l
=
' tn k ' n k ' tn l ' n l : n al; k holds for any k; l Z . Following the argument in
=
1=
2
k; l
akl ck cl
k
ck2
sup k
jakl j
l
and
sup k
l
jakl j k'kL
[a; b]
sup jr j
2
:
2 0
k
ck2
0
( + rm )e0il! ei m0n ! d!: (
(+ )
l
Then we have the formulas An(n+k)
2
=
=
(+
l
' l
l
0
)
' l rn+k e0il! eik! d!:
( + rn )eil!
1
Anm eimt
l
' l rn eil!
( ) = G0' (!)
Bnk !
( + rn )eil!
' l
' l
2
1 G0 '
k
1 G0 '
l
= k. Then
( + rn 0 k)e0ik!
' n
( + rm 0 k)eik! d!
0
)
( + rn k )e0il! :
' l
l
()
+
()
Bnk ! eik! ! d!
An(n+k) ei(n+k)t
k
= eint
2
eikt
k
0
()
Bnk ! eik! d!:
Let Cn
=
k
eikt
2 0
()
Bnk ! eik! d!:
Then fCn gn are the eigenvalues of the matrix A. Since the matrix A is invertible with the inverse matrix A01 Cn01 rn0m , from Corol0 1 lary 1, we have Sn Cn q' n rn ; 1 for any n 2 Z . Simple calculation shows that the formula
^ =2
( ( 0 ) 0 ( 0 ))( ( 0 ) 0 ( 0 ))
Then ak; l [6], we have
=
k
2 ('(tn 0 l) 0 '(n 0 l)) = ck cl ('(tn 0 k) 0 '(n 0 k)) k; l
Let
cl ' tk
( ( 0 k) 0 '(n 0 k))
2
' m
k
2 An(n+k)
1 G0 '
2
=
2
ck cl ' tn
sup jr j k'^3 k
[a; b]
2
=
and
( ( 0 l) 0 '(k 0 l))
k
0
2
Let m 0 n
for any f
2
Once again, we are interested in the expression of the reconstruction frame fSn gn . By Corollary 1
2
2
jf (k)j :
sup jr j k'^3 k . This is exactly implied by
that
C 01 f
k
2
2
k'kL
ck2
2
2
ck2
sup jr j ) :
Therefore, we only need to show
0
[a;b]
[a; b]
Recall the proof of Theorem 3. We know
2
k'kL
(k'kL
ck2
k
^( +
=( )
^ ( + rn ; 1) = e0in1 G'0 '^ 1
q' n
l
)
( + rn )eil1
' l
holds for any n 2 Z . Finally, we get the formula to calculate the reconstruction frame fSn gn as
^ = 2
Sn
ik1 k ' k rn e 'e0in1 : Cn G'
( + )
^
In a regular sampling case, rn for any n 2 Z , and Cn G' =j'3 j2 . 0 in! 3 Then we have Sn e '=' and S0 '='3 , that is, exactly the result of Theorem 3.
^ =
=0 ^^
^ =^ ^
=
^
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 10, OCTOBER 2002
The estimate (12) is also derived in our previous work [6] when the continuous stable generator ' is orthonormal and satisfies ' = O((1 + j 1 j)0s ) for some s > 1. But in Theorem 4, the estimate (12)
also holds for a nonorthonormal continuous stable generator whenever it satisfies ' = O((1 + j 1 j)0s ) for some s > 1=2. The authors [6] also derived an estimate for perturbation of regular sampling in shift-invariant spaces with a nonorthonormal continuous stable generator when ' satisfies ' = O((1 + j 1 j)0s ) for some s > 1. But that estimate of [6] is somewhat awkward and rough. The examples in Section VI will make it more clear. It is also worth noting that the reconstruction frame may turn out to be a Riesz basis under the assumption of Theorem 4. It is also possible to derive the result of Theorem 4 directly by Riesz basis theory. But we do not touch Riesz basis in the preceding argument, and derive the result swiftly from Theorem 2. We leave the Riesz basis discussion to future work.
2 V (') if kZ' (; 1)k r; ' < k'kL a;b
holds in L2 (R) for any f
[
Z' (; 1) =
'( + n) exp(in1):
n
sup
Since the proofs of the shift sampling theorems are very similar to those of Theorems 3 and 4, respectively, we here only to give the statements of the shift sampling theorems by omitting the proofs. Theorem 5: Suppose ' is a continuous stable generator such that 0s = O ((1 + j 1 j) ) for some s > 1=2. Then, for a 2 [0; 1), the following three items are equivalent.
'
1) 0 < kZ' (; 1)k0 kZ' (; 1)k1 < 1. 2) There is a frame fS; n gn of V (') such that ff (n + )gn is a moment sequence to the frame fS; n gn and
f=
f (n + )S; n
n
holds in L2 (R) for any f 2 V (').
3) There is a Riesz basis fS; n gn of V (') such that
f
=
holds in L (R) for any f 2
n
f (n)Sn
The following is the shift-sampling version of the algorithm for perturbation of regular sampling in shift-invariant spaces. Theorem 6: Suppose ' is a continuous stable generator such that ' = O((1 + j 1 j)0s ) for some s > 1=2, and
1) 0 < kZ' (; 1)k0 kZ' (; 1)k1 < 1. 2) ' 2 L [a; b]. Then, for any frk gk [0r; ' ; r; ' ] \ [a; b], there is a frame fS; k gk of V (') such that ff (k + + rk )gk is a moment sequence to the frame fS; k gk and
f
=
k
f (k + + rk )S; k
kZ' (; !)k k'kL a; b
0
[
]
is optimal for a general continuous stable generator; and a question remains of what is the optimal estimate for a general continuous stable generator. Example 1 [10]: Take the B-spline of degree 1
N1 (t) = t [0; 1) + (2 0 t) [1; 2) :
^ 3 = ei! . Since kN1 k Then N 1 L [01; 1] = 3, we derive the estimate rN < 1=3. Suppose rk 0 (or rk 0) for all k 2 Z . Then
kN kL 0 ; 1
[
1 0]
=
kN kL 1
[0; 1]
= 2:
Therefore, the estimate is rN < 1=2. Liu and Walter [16] have shown that the estimate rN < 1=2 is optimal when rk p0 (or rk 0) for all k 2 Z . Compared to the estimate rN < 1=(2 3) derived in our previous work, the present result does improve our previous one [6]. Again, we want to know the structure of the reconstruction frame. For regular sampling in V (N1 ), from Theorem 3 we have S^n = N^1 e0in! =ei! . Therefore, Sn = N1 (1 0 n 0 1) for any n 2 Z . For perturbation of regular sampling in V (N1 ), from Corollary 1, we have N1 (k + rn )eik! =(Cn GN ): S^n = 2N^1 e0in! k Suppose rn 0 for all n 2 Z . Then 2 Cn = eikt 0 k
2 V (').
In these cases, the reconstruction frame fS; n gn is derived by the formulas S; n = S; 0 (1 0 n) and S^; 0 = '=Z ^ ' (; 1).
(15)
Since the Daubechies mother wavelets and the Meyer mother wavelet are all the orthonormal continuous stable generators [11], [17], [21], the estimates by theorems in this correspondence are the same to those derived by our previous works (see the discussion after Theorem 4). We here calculate the B-spline of degree 1 (denoted by N1 ). We derive the estimate rNp < 1=2, which is better than our previous estimate rN < 1=(2 3), and which is shown by Liu and Walter to be optimal [16]. Unfortunately, we are not yet sure that the estimate 2(01;1)
(14)
:
]
VI. EXAMPLES TO SHOW THE ALGORITHM
V. SHIFT SAMPLING IN SHIFT-INVARIANT SPACES Unfortunately, there are some important continuous stable genera3 tors '’s with k' ^ k0 = 0. An obvious example is the B-spline of degree 2, which has been calculated in our previous works [4]–[6]. As done by Janssen [13] for Walter’s sampling theorem [19], and the authors [6] for irregular sampling theorem, we also treat it by shift sampling. Then the shift-sampling versions of sampling theorems can be attained by using the Zak transform Z' (; 1) ( 2 [0; 1)) defined by
1=
0
2 =
k
l
N1 (l + rn )eil!
l
N1 (l + rn+k )eil!
GN
eikt
eik! d!
2 0
i! )(r
i! n+k + (1 + rn+k )e ) ik! e d!: GN 2 01 eik! d! . Then Let unk = 0 (rn + (1 + rn )ei! )GN Cn (!) = eik! unk rn+k + eik! un(k+1) (1 + rn+k ) k k i! rn + (1 + rn )e 0i! e = 2 GN
2 (rn + (1 + rn )e
+
k
eik! unk (rn+k + rn+k01 e0i! ):
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 10, OCTOBER 2002
p p Then we have e02 = =e K 3 = . From Theorem 3, for regular sampling in V K , the reconstruction frame fSn gn is given by p 0! =2 0in! e e = e0(!+2k) =2 : Sn k For perturbation of regular sampling in V K , the fact K 0 p t =2 1 implies K 2 L0 0 ; and = te p p kK kL [01; 1] = e01=4 e01=2 :
^ 2 +2 ( )
Let
Bn =
k
(rn k + rn
+k
+
01 e0i! )unk eik! :
= j sin(1=2)j
finally we have
S^n =
4
j 1 =2 + kj0
4
k
1 2
2(rn + (1 + rn )ei! )N^ e0in! : 2(rn e0i! + (1 + rn )) + GN Bn
2 6 t 0 2t 0 3 (3 0 t) N (t) = ; + 2 2 ; + 2 ; : ^ 3 (!) = ei! (ei! + 1)=2 = 0 at the point ! = . So we have Then N 2
2
2
[0 1)
[1 2)
[2 3)
2
to use the shift sampling theorem. The Zak transform of N2 is ZN = ; ! = ei! = e2i! = :
(1 2 ) = 1 8 + 3
The facts
4+
8
jZN (1=2; !)j 3=4 0 1=8 0 1=8 = 1=2
and
ZN
(1=2; ) = 1=2
0
N 0 (t) = t [0; 1) + (3 0 4t) [1; 2) + (3 0 t) [2; 3) 2
( ) ^ =8^
(1+6 + ( )
)
The Daubechies mother wavelets, the Meyer mother wavelet, and the B-splines are all scaling functions of some MRAs. But the next continuous stable generator ceases to be a scaling function of an MRA.
p Example 3: Take the Gaussian kernel K = e0t =2 . Then K e0! =2 . Therefore, K is not a scaling function of any MRA since
^= ^ (2k) 6= 0 for any k 6= 0. However, K GK
=
e0(!+2k)
k
= e0! + 2
2 [0; 2], we derive 1 0x GK e0 + 2 e
For any !
4
2
and
GK
1+2
0
=1 2 1
k=1
dx = e04
e0(!+2k)
p
+ 2=e
2
1 0x p e dx = 1 + 2:
( )
It shows that K is a stable generator of the shift-invariant space V K . By Poisson summation formula, we know that p = e0(!+2k) =2 : K3
^ =1 2
k
p
)
+2
1 4
expression of the reconstruction frame fSn gn . But a computer-based solution of fSn gn can be derived from Corollary 1.
The next example is calculated to compare to the traditional Kadec’s theorem. sin (1) Example 4: Take the Shannon sampling function (1) . 3 Then . This implies [0; ] , , and Gsinc that is a continuous cardinal orthonormal stable generator of the shift-invariant space V . For regular sampling in V , the reconstruction frame fSn gn is given by Sn 10 n = 10 n . For perturbation of regular sampling in V , Theorem 4 cannot rk = k rk j is generally dibe used, since the series k j vergent. To find a theorem like Theorem 4 that covers the Shannon sampling function is also a very interesting open problem. At present , however, we can only use Theorem 2. For any f 2 V , a simple calculation shows
sinc = =1 (sinc) = sin( ( )) ( ( (sinc) sin( ) (( + ) )
^ = sinc sinc
^ =1 sinc (sinc)
1=
=2
by Proposition 1, we have kN2 kL [01=4; 1=4] . Finally, we derive the estimate r1=2; N < = . For regular sampling in V N2 , by Theorem 3, the reconstruction frame fSn gn is determined by Sn ei! e2i! for N2 e0in! = any n 2 Z . For perturbation of regular sampling in V N2 , since the calculation to find an explicit expression of the reconstruction frame fSn gn is very complicated, we leave it to future work. Anyway, the computer-based solution can follow from Corollary 1. Shifting by different , the estimate derived by shift sampling theorem is, in general, different. To find a shift to derive an optimal estimate is a useful and interesting future work.
14
+4
))
(sinc)
( ; 1)k = 1=2. Since the derivative
imply kZN
1 2
=
2 ) ( 2e = + 4) 0:35: The right-hand side is larger than 1=4. It is difficult to give an explicit (
Example 2 [10]: Take the B-spline of degree
( )
[ 1 1] 1 2 ( 2
Therefore, we derive an estimate p rK < e1=202 e01=2 =
1
The next example shows the importance of shift sampling theorem.
t2
1 2 +2
^ = 2
Since
GN
2809
jf (k + rk ) 0 f (k)j
2
k
(1 0 cos r + sin r)
2
jf (k)j
2
k where r rsinc . Since 0 r r < when r < = , it deduces the estimate rsinc < = . Kadec [24] has shown that the estimate rsinc < = is optimal. The reconstruction frame can be derived from Corollary 1.
=
14
1 cos + sin 14
1
14
ACKNOWLEDGMENT The authors thank Prof. R.-Q. Jia and Prof. B. Han for finding some flaws in our manuscript. They also thank the referees for their very helpful suggestions. REFERENCES [1] A. Aldroubi and M. Unser, “Families of wavelet transforms in connection with Shannon sampling theory and the Gabor transform,” in Wavelets: A Tutorial in Theory and Applications, C. K. Chui, Ed. New York: Academic, 1992, pp. 509–528. [2] , “Families of multi-resolution and wavelet spaces with optimal properties,” Numer. Funct. Anal. Optimiz., vol. 14, pp. 417–446, 1993. [3] , “Sampling procedures in function spaces and asymptotic equivalence with Shannon sampling theorem,” Numer. Funct. Anal. Optimiz., vol. 15, pp. 1–21, 1994. [4] W. Chen, S. Itoh, and J. Shiki, “Sampling theorem by wavelets for irregularly sampled signals,” Denshi Joho Tsushin Gaikkai Ronbunshi, vol. J79-A, no. 12, pp. 1941–1949, Dec. 1996. [5] W. Chen and S. Itoh, “On irregular sampling in wavelet subspaces,” IEICE Trans. Fundamentals, vol. E80-A, no. 7, pp. 1299–1307, July 1997. [6] W. Chen, S. Itoh, and J. Shiki, “Irregular sampling theorem for wavelet subspaces,” IEEE Trans. Inform. Theory, vol. 44, pp. 1131–1142, May 1998. [7] W. Chen and S. Itoh, “Oversampling theorem for wavelet subspaces,” IEICE Trans. Fundamentals, vol. E81-A, no. 1, pp. 131–138, Jan. 1998.
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