Acta Mathematica Sinica, English Series Apr., 2009, Vol. 25, No. 4, pp. 545–552 Published online: March 25, 2009 DOI: 10.1007/s10114-009-7412-4 Http://www.ActaMath.com
Acta Mathematica Sinica, English Series The Editorial Office of AMS & Springer-Verlag 2009
General A-P Iterative Algorithm in Shift-invariant Spaces Jun XIAN Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou 510275, P. R. China E-mail :
[email protected]
Song LI1) Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China E-mail :
[email protected] Abstract A general A-P iterative algorithm in a shift-invariant space is presented. We use the algorithm to show reconstruction of signals from weighted samples and also show that the general improved algorithm has better convergence rate than the existing one. An explicit estimate for a guaranteed rate of convergence is given. Keywords irregular sampling, iterative algorithm, lattice-invariant space, shift-invariant space, spline subspace MR(2000) Subject Classification 42C15, 41A15, 46A35, 94A12
1
Introduction
In the classical sampling problem, the reconstruction of f on Rd from its samples {f (xj ) : j ∈ J}, where J is a countable indexing set, is one of the main tasks in many applications in signal or image processing. However, this problem can’t be solved without extra information on the function under consideration, and becomes meaningful only when the function f is assumed to be bandlimited, or to belong to a shift-invariant space [1–24]. Bandlimited signal of finite energy is completely characterized by its regular samples if they are taken at a sufficiently high rate (Nyquist criterion), and described by the famous classical Shannon sampling theorem. Obviously, the following shift-invariant space is not a space of bandlimited function unless the generator is bandlimited. p p c(k)ϕ(· − k) : c ∈ . V (ϕ) = k∈Zd
In real applications, sampling points are not always regular. For example, the sampling steps need to be fluctuated according to the signals so as to reduce the number of samples and there are not sufficiently many samples available. There may be also undesirable jitters that exist in sampling instants, and communication systems may suffer from random delay due to the Received August 17, 2007, Accepted June 3, 2008 This work is supported in part by the National Natural Science Foundation of China (10771190, 10801136), the Mathematical Tianyuan Foundation of China NSF (10526036), China Postdoctoral Science Foundation (20060391063), Natural Science Foundation of Guangdong Province (07300434) 1) Corresponding Author
Xian J. and Li S.
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channel traffic congestion and encoding delay and so on. If a weighted sampling is considered, we will make use of the fact that the use of weights on the samples makes the reconstruction more stable and efficient. Non-uniform sampling and reconstruction problem in general shiftinvariant spaces is a relatively recent and active research field [1–2, 5–7, 20–22]. It is well-known that spline subspaces yield many advantages in their generation and numerical treatment so that there are many practical applications for signal or image processing. Therefore, the recent research of spline subspaces has received much attention (see [4, 20–21, 24]). Clearly, we hope signal spaces to be sufficiently large to accommodate a large number of possible models. Aldroubi and Feichtinger introduced lattice-invariant space of the form Vmp (ϕ) = { k∈Zd c(k)ϕ(· − Lk) : c ∈ pm } in [3], where ϕ is a suitable generator, L is a d × d non-singular matrix, 1 ≤ p ≤ ∞ and m is a weight function. The matrix L transforms the lattice Zd to the lattice Λ. When L is the identity matrix, we can obtain the standard shiftinvariant spaces. Aldroubi and Feichtinger pointed out that a combination of generator ϕ (e.g. radial symmetric ones) with suitable lattice Λ (related to sphere packing) is a good alternative for the usual voxel representation of volume data and has been observed also in this context (cf. [25]). The family of lattice-invariant spaces is sufficiently large and flexible. For practical application and computation of the reconstruction, Goh et al., showed practical reconstruction algorithm of bandlimited signals from irregular samples in [13], Aldroubi et al., presented a A-P iterative algorithm in [1–3, 6]. We will generalize and improve the A-P iterative algorithm and also show that the algorithm has better convergence rate than the existing one. 2
A General Improved A–P Iterative Algorithm
Aldroubi presented A-P iterative algorithm in [1–3, 5–6]. In this section, we will improve the algorithm and also show that the generalized and improved algorithm has faster convergence rate than the existing one. As mentioned above, a weight function is just a non-negative function. We will use other two kinds of weight functions. Firstly, the weight functions denoted by w are always assumed to be continuous, symmetric, i.e., w(x) = w(−x), positive and submultiplicative: 0 < w(x + y) ≤ w(x)w(y), ∀ x, y ∈ Rd . Secondly, a weight function v is called moderate with respect to the weight w, or simply wmoderate, if it is continuous, symmetric and positive, and satisfies v(x + y) ≤ cw(x)v(y) for all x, y ∈ Rd , where c is constant. If w is submultiplicative and v is w-moderate, we know that Lpv is a translation-invariant space, where the Lpv is defined by the norm 1/p p |f (x)v(x)| dx . f Lpv = Rd
. Let W (Lpv ) = {f ∈ Lpv : f W (Lpv ) = ( 1 ≤ p < ∞, and
k∈Zd
esssupx∈[0,1]d |f (x + k)|p v(k)p )(1/p) < ∞} if
∞ p W (L∞ v ) = {f ∈ Lv : f W (Lv ) = sup esssupx∈[0,1]d |f (x + k)|v(k) < ∞} k∈Zd
if p = ∞.
General A-P Iterative Algorithm in Shift-invariant Spaces
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Endowed with the norm · W (Lpv ) , W (Lpv ) is a Banach space [5]. Moreover it is translationinvariant, i.e., if f ∈ W (Lpv ), then f (· − y) ∈ W (Lpv ) and f (· − y)W (Lpv ) ≤ Cw(y)f W (Lpv ) . The subspace of continuous functions W0 (Lpv ) = W (C, Lpv ) ⊂ W (Lpv ) is a closed subspace of Lpv and is also a Banach space [5]. For a generator ϕ, the shift-invariant space Vvp (ϕ) is defined by Vvp (ϕ) = c(k)ϕ(· − k) : c ∈ pv , k∈Zd
where the
pv
is defined by the norm cpv =
1/p |c(α)v(α)|
p
.
α∈Zd
For δ > 0, let Bδ (x) = {y : |y − x| ≤ δ}. Definition 2.1 Let δ be a positive number. A general bounded partition of unity (GBP U ) is a set of functions {βj1 , βj2 } that satisfy : (1) 0 ≤ βj1 , βj2 ≤ 1(∀j1 ≡ j1 (j), j2 ≡ j2 (j) ∈ J), where J be countable separated index set, (2) suppβj1 ⊂ B δ (xj1 ), suppβj2 ⊂ B δ (xj2 ), 2 2 (βj1 + βj2 ) = 1. (3) j∈J
In fact, in the case of βj1 = βj2 = βj , the above GBPU’s definition is the ordinary BPU’s definition which was used in [1, 3, 6]. Example 2.1 For the explicit expressions of βj1 and βj2 , we can choose βj1 and βj2 to be some characteristic functions. However, their smoothness are poor and this will influence the convergence in reconstruction, so we need to construct smooth βj1 and βj2 . In the following, we will construct smooth βj1 and βj2 with the bell function βn ∈ C ∞ over [xn−1 , xn ], where xn−1 < xn . The bell functions have been investigated in [26]. We begin by choosing non-negative functions f (x) and g(x) as x 0, x ≤ 0, f (1 + t)f (1 − t)dt. g(x) = f (x) = 2 e−1/x , x > 0, −∞
It follows that g(x) ∈ C ∞ and g(x) = 0 when x < −1 and g(x) > 0 when x > −1. Note g(x)2 that g(x) = g(1) for x ≥ 1. If we set θ(x) = g(−x) 2 +g(x)2 , then θ(x) + θ(−x) = 1 for x ∈ R, 2 π x ∞ and θ ∈ C . Now we put S (x) = sin ( 2 θ( )) and C (x) = cos2 ( π2 θ( x )), where is small. Obviously, we can obtain the relations S (x) + C (x) = 1 and C (x) = S (−x). The bell functions βn over [xn−1 , xn ] are defined by βn (x) = C (x − xn )S (x − xn−1 ) for all x ∈ R. It follows from [26] that bell functions βn satisfy the following basic properties: (1) suppβn = [xn−1 − , xn + ], (2) βn (x) = S (x − xn−1 ), when x ∈ [xn−1 − , xn−1 + ], (3) βn (x) = 1, when x ∈ [xn−1 + , xn − ], (4) βn (x) = C (x − xn ), when x ∈ [xn − , xn + ], (5) n βn (x) = 1 for all x ∈ R.
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Taking two bell functions βj1 and βj2 defined over [xj , tively, then {βj1 , βj2 } belongs to C ∞ and is a GBPU.
xj +xj+1 ] 2
and [
xj +xj+1 , xj+1 ], 2
respec-
For the remainder of this article, we will assume that the weight functions {ϕxj : xj ∈ X} satisfy the following properties: (i) suppϕxj ⊂ Ba (xj ), for some a ∈ Rd ,
(ii) There exists M > 0 such that Rd |ϕxj |dx ≤ M ,
(iii) Rd ϕxj dx = 1. Lemma 2.1 [6] Assume that ϕ ∈ W0 (L1w ), v is w-moderate, and that ϕ satisfies that there exist two positive constants mp and Mp such that for any c ∈ pv , mp cppv ≤ c(k)ϕ(· − k) p ≤ Mp cppv k
Lv
holds true. If f ∈ Vvp (ϕ), then we have the norm equivalences f Lpv ≈ cpv ≈ f W (Lpv ) . Lemma 2.2 [6] If ϕ ∈ W (L1w ) and c ∈ pv , then the function f = k∈Zd ck ϕ(· − k) belongs to W (Lpv ) and f W (Lpv ) ≤ cpv ϕW (L1w ) . The following Theorem 2.1 gives the reconstruction algorithm that shows faster convergence rate than the existing algorithm. Theorem 2.1 Let X = {xj : j ∈ J} be sampling set with Rd = j∈J Bδ (xj ) for some δ > 0, and {βj1 , βj2 } be a GBPU associated with X. Assume ϕ to be in W0 (L1w ), and that there exist two positive constants mp and Mp such that for any c ∈ pv , p mp cppv ≤ ck ϕ(· − k) p ≤ Mp cppv k
Lv
holds true. Let P be a bounded projection from Lpv (Rd ) onto Vvp (ϕ). If we choose the proper δ and a such that 1 M osc δ (ϕ)W (L1w ) + osca (ϕ)W (L1w ) P op 2 mp mp M 1 1 + 2 osca (ϕ)W (Lw ) osc δ (ϕ)W (Lw ) < 1, 2 mp where oscδ (f )(x) = sup|y|≤δ |f (x + y) − f (x)|, then any f ∈ Vvp (ϕ) can be recovered from its samples { f, ϕxj : xj ∈ X} by the iterative algorithm : f1 = P Af, fn+1 = P A(f − fn ) + fn , where the operator A is defined by Af = j∈J f, ϕxj1 βj1 + f, ϕxj2 βj2 . The convergence is geometric, that is,
1 M osc δ (ϕ)W (L1w ) + osca (ϕ)W (L1w ) fn+1 − f ≤ P op 2 mp mp n M + 2 osca (ϕ)W (L1w ) osc δ (ϕ)W (L1w ) f − f1 Lpv . 2 mp Proof Let Qf (x) = j∈J f (xj1 )βj1 (x) + j∈J f (xj2 )βj2 (x). Lp v
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Since f ∈ Vvp (ϕ), we have f − P Af Lpv = f − P Qf + P Qf − P Af Lpv ≤ f − P Qf Lpv + P Qf − P Af Lpv ≤ P op f − Qf Lpv + P op Qf − Af Lpv = P op (f − Qf Lpv + Qf − Af Lpv ). Note that
|f (x) − (Qf )(x)| = f (x) − f (xj1 )βj1 (x) − f (xj2 )βj2 (x) j∈J
j∈J
= (f (x) − f (xj1 ))βj1 (x) + (f (x) − f (xj2 ))βj2 (x) j∈J
≤
j∈J
|f (x) − f (xj1 )|βj1 (x) +
j∈J
≤
osc δ (f )(x)βj1 (x) +
2
j∈J
|f (x) − f (xj2 )|βj2 (x)
j∈J
osc δ (f )(x)βj2 (x) 2
j∈J
= osc δ (f )(x). Let f =
2
α∈Zd c(α)ϕ(· − α). It follows from Lemmas 2.1 and 2.2 that
f − Qf Lpv ≤ osc δ (f )Lpv 2
≤ osc δ (f )W (Lpv ) 2
≤ cpv osc δ (ϕ)W (L1w ) 2
1 ≤ f Lpv osc δ (ϕ)W (L1w ) . 2 mp To estimate Qf − Af Lpv , we have
|(Qf − Af )(x)| = (f (xj1 ) − f, ϕxj1 )βj1 (x) + (f (xj2 ) − f, ϕxj2 )βj2 (x) j∈J
=
Rd
+
j∈J
(f (xj1 ) − f (ξ) ϕxj1 (ξ)βj1 (x)
j∈J
(f (xj2 ) − f (ξ))ϕxj2 (ξ)βj2 (x))dξ
j∈J
≤M
osc δ (f )(xj1 )βj1 (x) + 2
j∈J
=M
j∈J
+
2
j∈J
|ck |osca (ϕ)(xj1 − k) βj1 (x)
k∈Zd
j∈J
k∈Zd
|ck |osca (ϕ)(xj2 − k) βj2 (x)
= MQ |ck |osca (ϕ)(x − k) . Let g(x) =
k∈Zd
k∈Zd
|ck |osca (ϕ)(x − k).
osc δ (f )(xj2 )βj2 (x)
Xian J. and Li S.
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Then Qf − Af Lpv ≤ M QgLpv ≤ M (gLpv + g − QgLpv ) 1 p p ≤ M gLv + gLv osc δ (ϕ)W (L1w ) 2 mp 1 p osc δ (ϕ)W (L1w ) ≤ M gLv 1 + 2 mp 1 M p f Lv osca (ϕ)W (L1w ) 1 + osc δ (ϕ)W (L1w ) . ≤ 2 mp mp Combining the above discussions, we can obtain 1 f Lpv osc δ (ϕ)W (L1w ) 2 mp 1 M + f Lpv osca (ϕ)W (L1w ) 1 + osc δ (ϕ)W (L1w ) 2 mp mp M 1 osc δ (ϕ)W (L1w ) + osca (ϕ)W (L1w ) = 2 mp mp M + 2 osca (ϕ)W (L1w ) osc δ (ϕ)W (L1w ) f Lpv , 2 mp
f − Qf Lpv + Qf − Af Lpv ≤
that is,
I − P Aop ≤ P op
1 M osc δ (ϕ)W (L1w ) + osca (ϕ)W (L1w ) 2 mp mp M + 2 osca (ϕ)W (L1w ) osc δ (ϕ)W (L1w ) . 2 mp
Therefore, we have fn+1 − f Lpv = fn + P A(f − fn ) − f Lpv = P A(f − fn ) − (f − fn )Lpv ≤ I − P Af − fn Lpv ≤ · · · ≤ I − P An f − f1 Lpv . Combining with the estimate of I − P A, we can obtain 1 M osc δ (ϕ)W (L1w ) + osca (ϕ)W (L1w ) fn+1 − f Lpv ≤ P nop 2 mp mp n M + 2 osca (ϕ)W (L1w ) osc δ (ϕ)W (L1w ) f1 − f Lpv . 2 mp Taking assumption P op
1 M osc δ (ϕ)W (L1w ) + osca (ϕ)W (L1w ) 2 mp mp M + 2 osca (ϕ)W (L1w ) osc δ (ϕ)W (L1w ) < 1, 2 mp
we know that the algorithm is convergent. Furthermore, the convergence is geometric. Remark 2.1
It is obvious that
1 M M osc δ (ϕ)W (L1w ) + osca (ϕ)W (L1w ) + 2 osca (ϕ)W (L1w ) osc δ (ϕ)W (L1w ) 2 2 mp mp mp 1 M M ≤ oscδ (ϕ)W (L1w ) + osca (ϕ)W (L1w ) + 2 osca (ϕ)W (L1w ) oscδ (ϕ)W (L1w ) . mp mp mp
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It follows from [1, 6] that the convergence rate of the old algorithm is 1 M M oscδ (ϕ)W (L1w ) + osca (ϕ)W (L1w ) + 2 osca (ϕ)W (L1w ) oscδ (ϕ)W (L1w ) . P op mp mp mp Hence this algorithm improves the convergence rate of the existing algorithm. In case of βj1 = βj2 = βj , Theorem 2.1 is the main result of [1, 6]. The reconstruction algorithm requires the existence of bounded projection operator from Lpv onto Vvp (ϕ). For this purpose, the following Theorem 2.2 will provide a construction method for the bounded projection from Lpv onto Vvp (ϕ). We can prove Theorem 2.2 by a method similar to the one in the proof of Theorem 6.2 in [6]. Therefore, we omit it here. In fact, Lei, Jia and Cheney also presented a similar construction of the bounded projection operator P from Lpv onto Vvp (ϕ) when v = 1 [17]. ˜ − k) ϕ(· − k) is a bounded projection Theorem 2.2 If ϕ ∈ W0 (L1w ), then P f = k∈Zd f, ϕ(· p p operator from Lv onto Vv (ϕ), where ϕ˜ is the dual of ϕ. −1 T ˆ ) j)|2 ≤ B into the assumptions Remark 2.2 If we pose the condition A ≤ j∈Zd |ϕ(ξ+(L −1 T of Theorem 2.1, where (L ) is the transpose of the matrix inverse of L, then we can easily generalize the results of Theorem 2.1 to the case of weighted lattice-invariant space. The above condition guarantees that lattice-invariant space Vvp is a closed subspace of W (Lpv ) ⊂ Lpv for all w-moderate weights v and for 1 ≤ p ≤ ∞ [3]. When p = 2, d = 1 and generator ϕ = χ[0,1] ∗ · · · ∗ χ[0,1] (N convolutions), N ≥ 1, the above results are the main results of [24]. We can find the improved algorithm with its implementation in spline subspaces in [24] too. 3
Conclusion
In this paper, we discuss in some detail the problem of the weighted sampling and reconstruction and provide a reconstruction formula in general shift-invariant spaces, which is the generalized and improved form of the results of [1, 4, 24] in shift-invariant spaces. We give the general A–P iterative algorithm in general shift-invariant spaces, and use the improved algorithm to show the reconstruction of signals from weighted samples. The algorithm shows better convergence than the old one. Acknowledgements The author wishes to thank Prof. Dr. Hans Georg Feichtinger for his careful reading of the manuscript and his fruitful suggestions. This work was partially done while the first author was visiting the Chern Institute of Mathematics at the Faculty of Mathematics, Nankai University in 2008. The visit was supported by the Chern Institute of Mathematics. References [1] Aldroubi, A.: Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces. Appl. Comput. Harmon. Anal., 13, 156–161 (2002) [2] Aldroubi, A., Feichtinger, H.: Exact iterative reconstruction algorithm for multivate irregular sampled functions in spline-like spaces: The Lp theory. Proc. Amer. Math. Soc., 126(9), 2677–2686 (1998) [3] Aldroubi, A., Feichtinger, H. G.: Non-uniform sampling: exact reconstruction from non-uniformly distributed weighted-averages. Wavelet analysis (Hong Kong, 2001), 1–8, Ser. Anal., 1, World Sci. Publishing, River Edge, NJ, 2002
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