IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998 1131
Irregular Sampling Theorems for Wavelet Subspaces
Wen Chen, Shuichi Itoh, Member, IEEE, and Junji Shiki
Abstract—From the Paley–Wiener 1/4-theorem, the finite en-
ergy signal
f
(
t
)
can be reconstructed from its irregularly sampled
values
f
(
k
+
k
)
if
f
(
t
)
is band-limited and
sup
k
j
k
j
<
1
=
4
.We
consider the signals in wavelet subspaces and wish to recover the
signals from its irregular samples by using scaling functions. Then
the way to estimate the upper bound of
sup
k
j
k
j
such that the
irregularly sampled signals can be recovered is very important.
Following the work done by Liu and Walter, we present an
algorithm which can estimate a proper upper bound of
sup
k
j
k
j
.
Compared to Paley–Wiener 1/4-theorem, this theorem can relax
the upper bound for sampling in some wavelet subspaces.
Index Terms—Biorthogonality, MRA, orthogonality, sampling,
scaling function, wavelet, Zak-transform.
I. INTRODUCTION
F
OR a finite-energy -band continuous signal
, i.e., and , the
classical Shannon Sampling Theorem gives the following
reconstruction formula:
(1)
where
and is the Fourier transform of
defined by
Unfortunately it is not appropriate for nonbandlimited
signals. However, if we let
, the
problem can be viewed as that of sampling in a wavelet
subspace, with
playing the role of
scaling function of Multi-Resolution Analysis (MRA)
. Realizing these properties,
Walter [16] extended (1) such that it holds for a class of
scaling functions. Let
be a continuous orthonormal
scaling function of MRA
such that
for some . Walter [16] showed that there is a sequence
in such that and
(2)
holds for any
.
However, in many cases the sampling is not always at the
same step. How should irregular sampling cases be dealt with?
Manuscript received July 11, 1996; revised December 15, 1997. The
material in this paper was presented in part at the IEEE International
Symposium on Information Theory, Ulm, Germany, June 29–July 4, 1997.
The authors are with the Department of Information Network Sci-
ences, Graduate School of Information Systems, University of Electro-
Communications, Chofugaoka 1-5-1, Chofu, Tokyo 182, Japan.
Publisher Item Identifier S 0018-9448(98)02351-7.
Paley–Wiener’s 1/4-Theorem (see [18, p. 151]) states that,
if
and then
(3)
holds for any
(Paley–Wiener Space), where
But it cannot deal well with nonbandlimited signals, and
sampling with the symmetricity constraint
is also
restrictive. Following Walter’s [16] work, Liu and Walter [11]
tried to extend Paley–Wiener’s 1/4-Theorem to hold for the
sampling in a class of orthonormal wavelet subspaces without
the symmetric sampling constraint
. But they could
not claim that there is a sequence
such that a
similar to (3) reconstruction formula holds when
. Then Liu [10] turned to deal with the special case, spline
wavelets, by applying the Feichtinger–Grochenig Iterative
Algorithm (see [7]). Chen, Itoh, and Shiki [2] obtained a
recovering formula for sampling in general wavelet subspaces,
but they were led to an
-bound on . In fact, they cannot
yet estimate the aforementioned
.
In this paper, we can estimate an
-bound for ,
which enables a reconstruction formula similar to (3) to hold
when
. Our theorem does not only require the
symmetric sampling constraint
, but also relaxes
the bound
for the sampling in some wavelet subspaces. In
summary, we can estimate some
, such that for
any
with , there is an
such that
(4)
holds for any
. Our idea is to let
Then we show that is a Riesz basis of .By
applying the Paley–Wiener Theorem (see [18, p. 38]), we
manage to find some
so that is another
Riesz basis of
equivalent to . Then there is a
basis sequence
biorthogonal to ,
such that
(5)
holds for any
. Since
holds, we obtain (4).
0018–9448/98$10.00 1998 IEEE
1132 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998
Furthermore, the theorems are modified to be appropriate
for the shift sampling case by using Zak-transform (see [9]).
Later, we also calculate some examples and indicate that
can be bigger than for the sampling in B-spline of order
wavelet subspace.
Let us now introduce MRA (Multi-Resolution Analysis)
which was mentioned above. (For more details see [13], [14],
or any book on wavelets such as [1], [15], and [17].)
A subspace sequence
of is said to be an
MRA if
1)
, , ,
2) Function
if and only if ,
3) There is a function (called scaling function)
such that form a Riesz basis in .
The terms Multi-Resolution Approximation and Multi-
Resolution Decomposition are also used sometimes. If
is an orthogonal (respectively, orthonormal)
Riesz basis, MRA
and scaling function are considered
orthogonal (respectively, orthonormal).
MRA pair
are considered biorthogonal if
(6)
where
and are scaling functions of
MRA
and , respectively. In that case, scaling
function pair
is also considered biorthogonal.
Finally, let us introduce some notations used in this paper.
For a measurable set
, denotes the measure of .
For the measurable functions
and , a real
number
, and an interval , we write
for
II. IRREGULAR SAMPLING THEOREM AND
ALGORITHM FOR ORTHOGONAL WAVELET
SUBSPACES
Firstly, we shall show the existence of the
described
in Section I for the sampling in orthogonal wavelet sub-
spaces. Then we shall provide an algorithm to estimate
the
.
Theorem 1: Let
be a continuous orthogonal scaling
function of MRA
with
1)
for some ,
2)
.
Then there exists a
, such that for any
there is a sequence biorthogonal to
in such that (4) holds for any
.
In order to demonstrate the theorem, we need a lemma
which can be found in Liu and Walter [11].
Lemma 1: Let
be an orthonormal continuous scaling
function of MRA
with for some
. Then
(7)
holds for any
.
Proof of Theorem: Let
. Then, is
an orthonormal continuous scaling function with
for some and .
Define
Then, Walter [16] tells us that is a Riesz basis in
, i.e., for any ,
(8)
holds for some
. If we can find a , such that
for any
, there is a such that
(9)
holds for any
, then is a Riesz
basis in
due to Paley–Wiener Theorem (see [18, p. 38]).
CHEN et al.: IRREGULAR SAMPLING THEOREMS FOR WAVELET SUBSPACES 1133
Hence, there is a sequence in biorthogonal to
, such that
(10)
holds for any
, i.e., (4) holds for any due
to Lemma 1.
In order to show (9), let
(11)
where (11) is due to the orthonormality of . Denote
then and
(12)
But
(13)
where (13) is due to the index transform
and
.
In the meantime, the assumption
for
some
and the continuity of imply that the series
converges uniformly with respect to on , and
is uniformly bounded with respect to . Hence,
(14)
and
(15)
The above argument implies that there really exists some
, such that for any
(16)
That is,
(17)
From (12), (13), and (17), we derive
Then, due to (8), we conclude that (9) holds.
The theorem tells us that there really exist some
,
such that an irregularly sampled signal with deviation
within some interval can be reconstructed. However,
we also need to know how big the
can be, so that we
can design a sampling satisfying the criterion for a concrete
signal. Therefore, we should find some algorithm to estimate
the
from the scaling function . We need at first to
introduce a function class
, ,
and to give some simple propositions of
that function class, then present the algorithm.
1134 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 3, MAY 1998
Definition 1:
if there is a constant such that for any
(18)
We also write
Proposition 1:
1)
and
if .
2)
3) If is differentiable on each interval and
then .
Proof:
1) Obviously
and
hold. Therefore, holds.
On the other hand, for any
This means that holds.
The second part is easy to see.
2) If we let
, it is easy to see
On the other hand, we have
This implies that the first inclusion holds.
3) Due to the Lagrange mean value formula, there exists a
number sequence
such that
We now show a theorem which can lead to an algorithm
to estimate the
.
Theorem 2: Suppose
is an orthogonal continuous scal-
ing function of MRA
with
1)
for some ,
2)
,
3)
.
Then for any
, there exists a se-
quence
biorthogonal to in
such that (4) holds if
(19)
Proof: Let . Then, is an orthonor-
mal continuous scaling function which satisfies the above three
assumptions in Theorem 2. Following the proof of Theorem 1,
we only need to show that for any
and any
(20)
Due to
, we have
(21)
(22)
By the way, we also have
(23)
(24)
(25)
(26)
CHEN et al.: IRREGULAR SAMPLING THEOREMS FOR WAVELET SUBSPACES 1135
(27)
(28)
where (23) and (27) are due to Parseval identity, and (28) is
due to the orthonormality of
. Following (20), (22), and
(28), we only need to show
(29)
However, (19) exactly implies (29).
Remark 1:
1)
in (19) can be . In that case, the right-
hand side of (19) is
. Then, Theorem 2 holds for any
.
2) Referring to Section IV, we will find
.
Hence
and (19) is, in fact,
3) For cardinal orthonormal scaling function (see [1], [17])
holds.
4) From Proposition 1, we know that
is big
enough. In fact, we can verify that spline, Daubechies
scaling function, and Meyer scaling function are all
included in it. And in a practical case we will often
find
, ,or .
For a sequence
, we wish to know if we can
reconstruct the original signal
from the sampled values
. Of course, we can verify the conditions in
Theorem 2, but the following method is simpler and more
convenient, because it has less restrictive constraints than
Theorem 2.
Corollary 1: In Theorem 2, if 3) is replaced by
“For a sequence
, there is a constant , such that
for any mapping
( is the integer set)
for some ,”
then there is a sequence
biorthogonal to
in such that (4) holds if
(30)
Proof: Let
. Then (22) becomes
(31)
Referring to the proof of Theorem 2 and 2) of Remark 1, we
only need
It seems that we can establish an algorithm for general
wavelet subspaces by orthonormalizing the scaling function
(refer to [1], [15], and [17]), i.e., taking
such as
. But obtaining and calculating
from involves convolution or the fast Fourier transform
(FFT) to be undertaken twice. This is inconvenient, so we
should find other proper ways to estimate
directly. First,
we extend the theorem and algorithm to those for the sampling
in biorthogonal wavelet subspaces, then we deduce the results
for the sampling in general of wavelet subspaces.
III. I
RREGULAR SAMPLING THEOREM AND ALGORITHM
FOR
BIORTHOGONAL WAVELET
SUBSPACES
Theorem 3: Suppose
is a biorthogonal contin-
uous scaling function pair of the MRA pair
(with
), which satisfies
1)
, and for some
,
2)
.
Then there exists a
, such that for any
, there is a sequence biorthogonal to
in such that (4) holds.
We need two lemmas for the proof of the theorem.
Lemma 2: Under the same assumption as Theorem 3,
is a Riesz basis of .
Proof: It is easy to see that
is well-defined
and
. Let be the linear operator on
that takes
into
for any . Since
(32)
(33)
(34)
we obtain
(35)