Reconstruction of sparse multiband wavelet signals
from Fourier measurements
Yang Chen
Cheng Cheng
Qiyu Sun
Department of Mathematics
Hunan Normal University
Changsha, Hunan, China
Email: yang chenww123@163.com
Department of Mathematics
University of Central Florida
Orlando, FL 32816
Email: cheng.cheng@knights.ucf.edu
Department of Mathematics
University of Central Florida
Orlando, FL 32816
Email: qiyu.sun@ucf.edu
Abstract—In this paper, we consider the problem of reconstructing sparse multiband wavelet signals of finite levels from
their samples in Fourier domain. We show that those sparse
signals are determined by their Fourier measurements on a set
of size proportional to their sparsity.
I. I NTRODUCTION
Sparse representations of signals in redundant dictionaries
can improve pattern recognition, compression, noise reduction,
image inpainting, and source separation ([10], [16], [24], [25]).
Fourier bases and wavelet bases provide sparse representation
of many signals and images of interest. However, wavelets are
well localized and fewer coefficients are needed to represent
local transient structures and singularities ([8], [17]).
The problem of reconstructing sparse wavelet signals from
their Fourier measurements has many practical applications,
and it has received a great deal of attentions in recent years
since the emergence of compressive sensing ([4], [11], [26],
[27]). The convenient `q -minimization method in compressive
sensing cannot be immediately applied for the above sparse
wavelet signal recovery problem as wavelet bases are coherent
at different levels ([2], [5], [9], [21]), or it requires much more
Fourier samples than sparsity of the wavelet signal.
Denote by #E the cardinality of a finite set E. Let φ be
a scaling function of a multiresolution analysis with dilation
M ≥ 2, and ψm , 1 ≤ m ≤ M − 1, be wavelet functions, see
Section II for their definitions. Consider a sparse multiband
wavelet signal f with sparsity s = (s0 , · · · , sJ ),
f=
X
J M
−1 X
X
X
a0 (n)φ(·−n)+
bm,j (n)M j ψm (M j ·−n),
j=0 m=1 n∈Z
n∈Z
(I.1)
where
sj :=
max{#K0 , #K1,0 , . . . , #KM −1,0 } if j = 0
max{#K1,j , . . . , #KM −1,j } if j = 1, . . . , J
is sparsity of the signal f at level j, and where K0 and
Km,j are supports of coefficient vectors (a0 (n))n∈Z and
(bm,j (n))n∈Z , 1 ≤ m ≤ M − 1, 0 ≤ j ≤ J respectively.
Prony’s method extracts frequency information of a sinusoid
signal with multiple frequencies from its uniform samples,
978-1-4673-7353-1/15/$31.00 c 2015 IEEE
and it has been used in power system, speech processing and
signal processing ([13], [14], [18], [23]). In [27], Zhang and
Daragotti applied Prony’s method for the exact reconstruction
of sparse wavelet signals with dilation 2.
Define Fourier transform of an integrable function f on the
real line R by
Z
ˆ
f (ξ) =
f (t)e−itξ dt.
R
Under the assumption that Fourier transform of the scaling
function φ does not vanish on (−π, π),
φ̂(ξ) 6= 0, ξ ∈ (−π, π),
(I.2)
Zhang and Daragotti proved in [27] that compactly-supported
sparse signals of the form (I.1) with dilation 2 can be reconstructed from their Fourier measurements on a set of size about
twice of signal sparsity. In this paper, we extend their result
to sparse wavelet signals with arbitrary dilation M without
nonvanishing condition (I.2) on the scaling function φ and
finite-duration assumption for sparse wavelet signals.
The paper is organized as follows. In Section II, we recall
some concepts of multiband wavelets and a preliminary result
on Fourier transform of scaling functions and wavelets ([8],
[17], [20]). In Section III, we construct a sampling set Ω
explicitly for any given sparsity s = (s1 , . . . , sJ ), and show
that sparse multiband wavelet signals with sparsity s can be
reconstructed from their Fourier measurements on Ω.
II. M ULTIBAND WAVELETS
Take an integer M ≥ 2 and a family {Vj }j∈Z of closed
subspaces of L2 (R). We say that {Vj }∞
j=−∞ is a multiresolution analysis with dilation M if the following conditions are
satisfied:
(i) Vj ⊂ Vj+1 for all j ∈ Z;
(ii) S
Vj+1 = {f (M ·), f ∈ Vj } for all j ∈ Z;
(iii) Tj∈Z Vj = L2 (R);
(iv) j∈Z Vj = {0}; and
(v) there exists a function φ ∈ V0 such that {φ(·−n), n ∈ Z}
is a Riesz basis for V0 .
The function φ in the above definition is known as a scaling
function. The multiresolution analysis {Vj }j∈Z is generated
by its scaling function φ, since the scaling space Vj has
{M j/2 φ(M j · −n), n ∈ Z} as its Riesz basis for every
j ∈ Z. Due to the Riesz basis property (v), there exist positive
constants A and B such that
X
A≤
|φ̂(ξ + 2kπ)|2 ≤ B, ξ ∈ R.
(II.1)
k∈Z
For the case that A = B = 1, {φ(· − n), n ∈ Z} is an
orthonormal basis for V0 .
The scaling function φ of a multiresolution analysis with
dilation M satisfies a refinement equation,
b ξ) = H0 (ξ)φ̂(ξ), ξ ∈ R,
φ(M
(II.2)
where H0 (ξ) is a 2π-periodic function. In this paper, we
always assume that
φ̂ is continuous on R and φ̂(0) = 1.
(II.3)
From (II.1), (II.2) and (II.3), it follows that H0 (ξ) is continuous, H0 (0) = 1, and
M
−1
X
2lπ A
H0 ξ +
≤
B
M
2
l=0
B
≤ , ξ ∈ R.
A
M
−1
X
am,m0 (ξ)c(m)c(m0 ) ≤ B 0 , ξ ∈ R
(II.5)
(II.6)
PM −1
for all c = (c(1), . . . , c(M − 1))T with m=1 |c(m)|2 = 1,
where A0 and B 0 are positive constants, and
M
−1
X
l=0
2lπ 2lπ Hm ξ +
Hm 0 ξ +
, ξ∈R
M
M
0
for 0 ≤ m, m ≤ M − 1.
(II.7)
Then {M j/2 ψm (M j · −n), 1 ≤ m ≤ M − 1, j, n ∈ Z} forms
a Riesz basis for L2 (R), and for every j ∈ Z, {M j/2 ψm (M j ·
−n), 1 ≤ m ≤ M − 1, n ∈ Z} is a Riesz basis of Wj :=
Vj+1 Vj , the orthogonal complement of Vj in Vj+1 . For the
case that A = B = 1 in (II.1) and A0 = B 0 = 1 in (II.6),
{M j/2 ψm (M j · −n), 1 ≤ m ≤ M − 1, j, n ∈ Z} is an
orthonormal basis for L2 (R).
As {φ(·−n), n ∈ Z}∪{ψm (·−n), 1 ≤ m ≤ M −1, n ∈ Z}
is a Riesz basis for the scaling space V1 , the M × Z matrices
b + 2kπ)
Φ(ξ
k∈Z
have rank M for all ξ ∈ R, where
b
Φ(ξ)
= (φ̂(ξ), ψ̂1 (ξ), . . . , ψ̂M −1 (ξ))T .
M
−1
X
am (ξ)ψ̂m (ξ),
(II.11)
m=1
where am , 0 ≤ m ≤ M − 1, are trigonometric functions, we
have that
b + 2k0 π), Φ(ξ
b + 2k1 π), . . . , Φ(ξ
b + 2kM −1 π)
A(ξ) Φ(ξ
= fˆ(ξ + 2k0 π), . . . , fˆ(ξ + 2kM −1 π) ,
(II.12)
where A(ξ) = (a0 (ξ), . . . , aM −1 (ξ)). This together with (II.8)
leads to the following result on the reconstruction of signals
in V1 from their Fourier transform.
Proposition II.1. Let f be a signal in V1 , and am , 0 ≤
m ≤ M − 1, be trigonometric functions in (II.11). If integers
km , 0 ≤ m ≤ M − 1, satisfy (II.9) and (II.10), then
am (ξ), 0 ≤ m ≤ M − 1, are completely determined by
fˆ(ξ + 2km π), 0 ≤ m ≤ M − 1.
Take N ≥ 1 and define
Define wavelet functions ψm , 1 ≤ m ≤ M − 1, by
ψbm (ξ) = Hm (ξ/M )φ̂(ξ/M ), 1 ≤ m ≤ M − 1.
ξ + 2k π m
6= 0, 0 ≤ m ≤ M − 1.
(II.10)
M
The existence of such integers follows from (II.1), while (II.8)
holds because of (II.4), (II.5), (II.6), and
b + 2k0 π), Φ(ξ
b + 2k1 π), . . . , Φ(ξ
b + 2kM −1 π)
Φ(ξ
ξ + 2mπ =
Hm0
M
0≤m0 ,m≤M −1
ξ + 2k π ξ + 2k
0
M −1 π
×diag φ̂
, . . . , φ̂
.
M
M
φ̂
fˆ(ξ) = a0 (ξ)φ̂(ξ) +
m,m0 =1
am,m0 (ξ) =
and
(II.4)
for all 1 ≤ m ≤ M − 1, and
A0 ≤
b + 2k0 π), Φ(ξ
b + 2k1 π), . . . , Φ(ξ
b + 2kM −1 π) = M,
rank Φ(ξ
(II.8)
provided that
km ∈ m + M Z
(II.9)
For any signal f ∈ V1 with
Let Hm , 1 ≤ m ≤ M − 1, be continuous 2π-periodic
functions such that
a0,m (ξ) = 0, ξ ∈ R
In fact,
Λ = {2(p/N + km (p))π, 0 ≤ p ≤ N − 1, 0 ≤ m ≤ M − 1},
(II.13)
where km (p) ∈ m + M Z are chosen so that φ̂(2π(p/N +
km (p))/M ) 6= 0. Recall that any trigonometric polynomial of
degree N − 1 is completely determined by its evaluation on
{2pπ/N, 0 ≤ p ≤ N − 1}. This together with Proposition II.1
leads to the following result on reconstruction of non-sparse
signals in V1 from samples of their Fourier transforms.
Corollary II.2. Let N ≥ 1 and Λ be as in (II.13). Then any
signal f ∈ V1 of the form
f=
N
−1
X
n=0
a0 (n)φ(· − n) +
M
−1 N
−1
X
X
bm (n)ψm (· − n)
m=1 n=0
can be reconstructed from samples of fˆ on the set Λ of size
MN.
III. R ECOVERY OF SPARSE MULTIBAND WAVELET SIGNALS
Therefore by (III.1), (III.2), (III.7) and Proposition II.1,
aJ (M −J (p+1/2)hπ) and bm,J (M −J (p+1/2)hπ), 1 ≤ m ≤
M − 1, −sJ ≤ p ≤ sJ − 1, are uniquely determined from
−i
i −1
Ω0i = ∪sp=−s
{M
(p+1/2)hπ+2k
(i,
p)π,
0
≤
m
≤
M
−1},
samples of fˆJ = fˆ on M J Ω0J ⊂ Ω.
m
i
(III.1)
Recall from (III.5) that
where integers km (i, p) ∈ m + M Z, 0 ≤ m ≤ M − 1, satisfy
X
−inM −J hπ p+1/2
−J
,
b
(n)
e
b
(M
(p+1/2)hπ)
=
−i−1
−1
m,J
m,J
φ̂ M
(p + 1/2)hπ + 2M k (i, p)π 6= 0.
(III.2)
Take h > 0 and sparsity vector s = (s0 , . . . , sJ ). For 0 ≤
i ≤ J, let
m
n∈Km,J
Define
Ω = ∪Ji=0 M i Ω0i .
Set ksk∞ = sup0≤j≤J |sj | and ksk1 =
(III.3)
P
0≤j≤J
|sj |. Then
Ω ⊂ {(p + 1/2)hπ, −ksk∞ ≤ p ≤ ksk∞ − 1} + 2πZ
and
#Ω ≤
J
X
#Ω0i =
i=0
J
X
2M si = 2M ksk1 .
i=0
Theorem III.1. Let M ≥ 2, φ be a scaling function satisfying
(II.3), ψm , 1 ≤ m ≤ M − 1, be wavelet functions in (II.7),
and let Ω be the set in (III.3) with irrational h > 0. Then any
sparse multiband wavelet signal with sparsity vector s can be
reconstructed from its Fourier measurements on Ω.
Proof. Let the sparse multiband wavelet signal f have the
representation (I.1). Taking Fourier transform at both sides of
the equation (I.1) gives
fˆ(ξ) = a0 (ξ)φ̂(ξ) +
J M
−1
X
X
bm,j (M −j ξ)ψbm (M −j ξ),
where
X
a0 (n)e−inξ
(III.4)
n∈Z
and
bm,j (ξ) =
X
bm,j (n)e−inξ
(III.5)
n∈Z
for 1 ≤ m ≤ M − 1 and 0 ≤ j ≤ J.
Define fi , 0 ≤ i ≤ J, by
fˆi (ξ) = a0 (ξ)φ̂(ξ) +
i M
−1
X
X
bm,j (M −j ξ)ψbm (M −j ξ).
j=0 m=1
Then fJ = f , and
fˆi (M i ξ)
=
=
fˆi−1 (M i ξ) +
ai (ξ)φ̂(ξ) +
M
−1
X
bm,i (ξ)ψbs (ξ)
m=1
M
−1
X
bm,i (ξ)ψbs (ξ), (III.6)
m=1
where ai , 1 ≤ i ≤ J, are 2π-periodic functions.
Applying (III.6) with i = J, we see that
fˆ(M J ξ) = aJ (ξ)φ̂(ξ) +
Following similar steps, we can reconstruct functions fi−1 −
fi by induction on i = J, J − 1, . . . , 1. Finally we recover the
function f0 from samples of its Fourier transform on {(p +
1/2)hπ, −s0 ≤ p ≤ s0 − 1} ⊂ Ω. By (III.4) and (III.5),
X
p+1/2
a0 ((p + 1/2)hπ) =
a0 (n) e−inhπ
,
n∈K0
and
bm,0 ((p + 1/2)hπ) =
X
bm,0 (n) e−inhπ
p+1/2
,
n∈Km,0
j=0 m=1
a0 (ξ) =
where −sJ ≤ p ≤ sJ − 1. Applying Prony’s method ([11],
−J
[23], [27]) and observing that e−inM hπ , n ∈ Km,J , are
distinct each other as h is irrational, we can recover sparse
trigonometric polynomials bm,J (ξ), 1 ≤ m ≤ M − 1, from
their evaluations on M −J (p+1/2)hπ, −sJ ≤ p ≤ sJ −1. This
together with (III.6) provides a reconstruction of the function
fJ−1 − fJ .
M
−1
X
m=1
bm,J (ξ)ψbs (ξ).
(III.7)
where −s0 ≤ p ≤ s0 − 1. Then sparse trigonometric polynomials a0 and bm,0 , 1 ≤ m ≤ M − 1, are uniquely determined
from their samples on (p + 1/2)hπ, −s0 ≤ p ≤ s0 − 1 by
Prony’s method. This completes the proof.
Theorem III.1 can be thought as a generalization of Zhang
and Dragotti’s result in [27], where M = 2 and the scaling
function φ satisfies (I.2). The nonzero assumption (I.2) on
scaling functions is satisfied from B-splines and Daubechies’
scaling functions with dilation M ≥ 2 ([1], [8], [12], [22]).
Under the above additional assumption on the scaling function
φ, the sampling set Ω in (III.3) can be constructed as follow,
see Figure 1:
i −1
Ω = ∪Ji=0 ∪sp=−s
((p+1/2)hπ+2M i πZ)∩(−M i+1 π, M i+1 π).
i
(III.8)
Corollary III.2. Let M ≥ 2, φ be a scaling function satisfying
(I.2) and (II.3), and let ψm , 1 ≤ m ≤ M − 1, be wavelet
functions in (II.7). Then any sparse multiband wavelet signal
with sparsity vector s can be reconstructed from samples of
its Fourier transform on the set Ω in (III.8) with irrational
h > 0.
We finish this section with a remark on the irrational
requirement on h in Theorem III.1 and Corollary III.2. In
most of practical applications, the scaling function φ and the
wavelet functions ψm , 1 ≤ m ≤ M −1, have compact support.
Fig. 1. The sampling set Ω in (III.8) associated with sparse wavelet signals
with M = 3, J = 1 and s = (3, 3).
Thus sparse multiband signals of the form (I.1) will have finite
duration if there exist a < b such that
K0 ⊂ [a, b) and Km,j ⊂ [−M j a, M j b)
(III.9)
for all 1 ≤ m ≤ M − 1 and 0 ≤ j ≤ J, c.f. a = 0 and b =
1 in [27]. Under the above additional requirement on sparse
representation (I.1) of a signal, the irrational requirement on
h in Theorem III.1 and Corollary III.2 can be replaced by the
following quantitative condition,
(b − a)h ≤ 2.
The reason is that the irrational requirement on h in Theorem
III.1 is used in the proof only to guarantee that e−inhπ , n ∈
−j
K0 , are distinct each other, and also that e−inM hπ , n ∈
Km,j , are distinct each other for all 1 ≤ m ≤ M − 1 and
0 ≤ j ≤ J.
IV. C ONCLUSIONS
In this paper, we show that sparse multiband wavelet signals
can be reconstructed from their Fourier measurements on a set
Ω, whose cardinality is almost proportional to signal sparsity.
Similar result could be established for high-dimensional sparse
wavelet signals with arbitrary dilation. More challenging problem on this aspect is exact reconstruction of signals having
sparse wavelet-like (e.g. wavelet packet, curvelet, framelet and
shearlet) representation from their partial Fourier information
([3], [6], [7], [15], [19]).
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